Fano manifolds whose elementary contractions are smooth ¶1

Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds Fano manifolds whose elementary contractions are smooth P1 -fibrations RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Gianluca Occhetta with R. Muñoz, L.E. Solá Conde, K. Watanabe and J. Wi´sniewski Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Madrid, January 2014 Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Motivation Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Fano bundles Fano manifolds and P1 -fibrations Gianluca Occhetta Fano bundles Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture A vector bundle E on a smooth complex projective variety X is a Fano bundle iff PX (E) is a Fano manifold. Fano manifolds and P1 -fibrations Gianluca Occhetta Fano bundles Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture A vector bundle E on a smooth complex projective variety X is a Fano bundle iff PX (E) is a Fano manifold. If E is a Fano bundle on X then X is a Fano manifold. Fano manifolds and P1 -fibrations Gianluca Occhetta Fano bundles Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds A vector bundle E on a smooth complex projective variety X is a Fano bundle iff PX (E) is a Fano manifold. If E is a Fano bundle on X then X is a Fano manifold. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Fano bundles of rank 2 on projective spaces and quadrics have been classified in the ’90s (Ancona, Peternell, Sols, Szurek, Wi´sniewski). Fano manifolds and P1 -fibrations Gianluca Occhetta Fano bundles Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds A vector bundle E on a smooth complex projective variety X is a Fano bundle iff PX (E) is a Fano manifold. If E is a Fano bundle on X then X is a Fano manifold. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Fano bundles of rank 2 on projective spaces and quadrics have been classified in the ’90s (Ancona, Peternell, Sols, Szurek, Wi´sniewski). Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Generalization: Classification of Fano bundles of rank 2 on (Fano) manifolds with b2 = b4 = 1 (Muñoz, , Solá Conde, 2012). Fano manifolds and P1 -fibrations Gianluca Occhetta Fano bundles Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds A vector bundle E on a smooth complex projective variety X is a Fano bundle iff PX (E) is a Fano manifold. If E is a Fano bundle on X then X is a Fano manifold. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Fano bundles of rank 2 on projective spaces and quadrics have been classified in the ’90s (Ancona, Peternell, Sols, Szurek, Wi´sniewski). Dynkin diagram Bott-Samelson varieties Construction Generalization: Classification of Fano bundles of rank 2 on (Fano) manifolds with b2 = b4 = 1 (Muñoz, , Solá Conde, 2012). Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Later the assumption b4 = 1 was removed by Watanabe. Fano manifolds and P1 -fibrations Gianluca Occhetta Varieties with two P1 -bundle structures Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture As a special case we have the classification of Fano manifolds of Picard number two with two P1 -bundle structures. Fano manifolds and P1 -fibrations Gianluca Occhetta Varieties with two P1 -bundle structures Motivation Fano bundles The problem Lie Algebras Cartan decomposition As a special case we have the classification of Fano manifolds of Picard number two with two P1 -bundle structures. Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture X P1 P2 P3 Q3 Q5 K(G2 ) Y P1 P2 Q3 P3 K(G2 ) Q5 Bundle O⊕O TP2 N S C Q Fano manifolds and P1 -fibrations Gianluca Occhetta Varieties with two P1 -bundle structures Motivation Fano bundles The problem Lie Algebras Cartan decomposition As a special case we have the classification of Fano manifolds of Picard number two with two P1 -bundle structures. Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture X P1 P2 P3 Q3 Q5 K(G2 ) • N Null-correlation bundle; Y P1 P2 Q3 P3 K(G2 ) Q5 Bundle O⊕O TP2 N S C Q Fano manifolds and P1 -fibrations Gianluca Occhetta Varieties with two P1 -bundle structures Motivation Fano bundles The problem Lie Algebras Cartan decomposition As a special case we have the classification of Fano manifolds of Picard number two with two P1 -bundle structures. Cartan matrix X P1 P2 P3 Q3 Q5 K(G2 ) Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • N Null-correlation bundle; • S spinor bundle; Y P1 P2 Q3 P3 K(G2 ) Q5 Bundle O⊕O TP2 N S C Q Fano manifolds and P1 -fibrations Gianluca Occhetta Varieties with two P1 -bundle structures Motivation Fano bundles The problem Lie Algebras Cartan decomposition As a special case we have the classification of Fano manifolds of Picard number two with two P1 -bundle structures. Cartan matrix X P1 P2 P3 Q3 Q5 K(G2 ) Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • N Null-correlation bundle; • S spinor bundle; • C Cayley bundle; Y P1 P2 Q3 P3 K(G2 ) Q5 Bundle O⊕O TP2 N S C Q Fano manifolds and P1 -fibrations Gianluca Occhetta Varieties with two P1 -bundle structures Motivation Fano bundles The problem Lie Algebras Cartan decomposition As a special case we have the classification of Fano manifolds of Picard number two with two P1 -bundle structures. Cartan matrix X P1 P2 P3 Q3 Q5 K(G2 ) Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Y P1 P2 Q3 P3 K(G2 ) Q5 • N Null-correlation bundle; • S spinor bundle; • C Cayley bundle; • Q universal quotient bundle. Bundle O⊕O TP2 N S C Q Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Varieties with two P1 -bundle structures A different perspective Fano manifolds and P1 -fibrations Varieties with two P1 -bundle structures Gianluca Occhetta A different perspective Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Theorem Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture A Fano manifold with Picard number 2 and two P1 -bundle structures is isomorphic to a complete flag manifold of type A1 × A1 , A2 , B2 or G2 . Fano manifolds and P1 -fibrations Generalization Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • Classify Fano manifolds whose elementary contractions are P1 -bundles (o P1 -fibrations). Fano manifolds and P1 -fibrations Generalization Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix • Classify Fano manifolds whose elementary contractions are P1 -bundles (o P1 -fibrations). Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • Understand the relation between P1 -fibrations and homogeneity (if any). Fano manifolds and P1 -fibrations Generalization Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix • Classify Fano manifolds whose elementary contractions are P1 -bundles (o P1 -fibrations). Dynkin diagrams Rational homogeneous manifolds RH manifolds • Understand the relation between P1 -fibrations and homogeneity (if any). Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Theorem X Fano manifold whose elementary contractions are smooth P1 fibrations such that X is not a product. If dim X 6= 24 then X is a complete flag manifold. Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Lie algebras and complete flags Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Lie Algebras Cartan decomposition Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture G semisimple Lie group, g associated Lie algebra, Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Lie Algebras Cartan decomposition Fano bundles The problem G semisimple Lie group, g associated Lie algebra, Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture h ⊂ g Cartan subalgebra (maximal abelian subalgebra). Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Lie Algebras Cartan decomposition Fano bundles The problem G semisimple Lie group, g associated Lie algebra, Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture h ⊂ g Cartan subalgebra (maximal abelian subalgebra). The action of h on g defines an eigenspace decomposition, called Cartan decomposition of g: M g=h⊕ gα . α∈h∨ \{0} Fano manifolds and P1 -fibrations Lie Algebras Gianluca Occhetta Cartan decomposition Motivation Fano bundles The problem G semisimple Lie group, g associated Lie algebra, Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections h ⊂ g Cartan subalgebra (maximal abelian subalgebra). The action of h on g defines an eigenspace decomposition, called Cartan decomposition of g: M g=h⊕ gα . α∈h∨ \{0} The spaces gα are defined by Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture gα = {g ∈ g | [h, g] = α(h)g, for every h ∈ h} , and the elements α ∈ h∨ \ {0} such that gα 6= 0 are called roots of g. The (finite) set Φ of such elements is called root system of g. Fano manifolds and P1 -fibrations Lie Algebras Gianluca Occhetta Cartan decomposition Motivation Fano bundles The problem G semisimple Lie group, g associated Lie algebra, Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections h ⊂ g Cartan subalgebra (maximal abelian subalgebra). The action of h on g defines an eigenspace decomposition, called Cartan decomposition of g: M g=h⊕ gα . α∈h∨ \{0} The spaces gα are defined by Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture gα = {g ∈ g | [h, g] = α(h)g, for every h ∈ h} , and the elements α ∈ h∨ \ {0} such that gα 6= 0 are called roots of g. The (finite) set Φ of such elements is called root system of g. A basis of h∨ formed by elements of Φ such that the coordinates of every element of Φ are integers, all ≥ 0 or all ≤ 0 is a system of simple roots of g. Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Lie Algebras Weyl group Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture On the space h∨ there is a symmetric bilinear form κ coming from the Killing form of g; this form, restricted to the real vector space E generated by the roots is positive definite. Fano manifolds and P1 -fibrations Lie Algebras Gianluca Occhetta Weyl group Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models On the space h∨ there is a symmetric bilinear form κ coming from the Killing form of g; this form, restricted to the real vector space E generated by the roots is positive definite. The root system Φ is invariant with respect to the isometries of (E, κ) defined by: Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture σα (x) = x − hx, αiα, where hx, αi := 2 κ(x, α) . κ(α, α) Fano manifolds and P1 -fibrations Lie Algebras Gianluca Occhetta Weyl group Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models On the space h∨ there is a symmetric bilinear form κ coming from the Killing form of g; this form, restricted to the real vector space E generated by the roots is positive definite. The root system Φ is invariant with respect to the isometries of (E, κ) defined by: Fibrations and reflections Dynkin diagram σα (x) = x − hx, αiα, where hx, αi := 2 Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture which are the reflections with respect to the roots. κ(x, α) . κ(α, α) Fano manifolds and P1 -fibrations Lie Algebras Gianluca Occhetta Weyl group Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models On the space h∨ there is a symmetric bilinear form κ coming from the Killing form of g; this form, restricted to the real vector space E generated by the roots is positive definite. The root system Φ is invariant with respect to the isometries of (E, κ) defined by: Fibrations and reflections Dynkin diagram σα (x) = x − hx, αiα, where hx, αi := 2 Bott-Samelson varieties Construction Properties κ(x, α) . κ(α, α) which are the reflections with respect to the roots. Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture W ⊂ Gl(E) generated by {σα , α ∈ Φ} is the Weyl group of g. Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Lie Algebras Cartan matrix Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Given a set of simple roots {α1 , . . . , αn } of g, the Cartan matrix di g is the matrix whose entries are the integers hαi , αj i. Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Lie Algebras Cartan matrix Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Given a set of simple roots {α1 , . . . , αn } of g, the Cartan matrix di g is the matrix whose entries are the integers hαi , αj i. A and all its principal minors are positive definite and moreover Fano manifolds and P1 -fibrations Lie Algebras Gianluca Occhetta Cartan matrix Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Given a set of simple roots {α1 , . . . , αn } of g, the Cartan matrix di g is the matrix whose entries are the integers hαi , αj i. A and all its principal minors are positive definite and moreover • aii = 2 for every i, Fano manifolds and P1 -fibrations Lie Algebras Gianluca Occhetta Cartan matrix Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Given a set of simple roots {α1 , . . . , αn } of g, the Cartan matrix di g is the matrix whose entries are the integers hαi , αj i. A and all its principal minors are positive definite and moreover Flag manifolds • aii = 2 for every i, Homogeneous models • aij = 0 iff aji = 0, Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Fano manifolds and P1 -fibrations Lie Algebras Gianluca Occhetta Cartan matrix Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Given a set of simple roots {α1 , . . . , αn } of g, the Cartan matrix di g is the matrix whose entries are the integers hαi , αj i. A and all its principal minors are positive definite and moreover Flag manifolds • aii = 2 for every i, Homogeneous models • aij = 0 iff aji = 0, Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • if aij 6= 0, i 6= j, then aij , aji ∈ Z− and aij aji = 1, 2 or 3. Fano manifolds and P1 -fibrations Lie Algebras Gianluca Occhetta Cartan matrix Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Given a set of simple roots {α1 , . . . , αn } of g, the Cartan matrix di g is the matrix whose entries are the integers hαi , αj i. A and all its principal minors are positive definite and moreover Flag manifolds • aii = 2 for every i, Homogeneous models • aij = 0 iff aji = 0, Fibrations and reflections • if aij 6= 0, i 6= j, then aij , aji ∈ Z− and aij aji = 1, 2 or 3. Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture In particular the possible 2 × 2 principal minors are (up to transposition) 2 0 2 −1 2 −1 2 −1 0 2 −1 2 −2 2 −3 2 Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Lie Algebras Dynkin diagrams Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Set D = {1, . . . , n}, where n is the number of simple roots of g. With the matrix A is associated a finite Dynkin diagram D, in the following way Fano manifolds and P1 -fibrations Lie Algebras Gianluca Occhetta Dynkin diagrams Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Set D = {1, . . . , n}, where n is the number of simple roots of g. With the matrix A is associated a finite Dynkin diagram D, in the following way • D is a graph whose set of nodes is D, Fano manifolds and P1 -fibrations Lie Algebras Gianluca Occhetta Dynkin diagrams Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds Set D = {1, . . . , n}, where n is the number of simple roots of g. With the matrix A is associated a finite Dynkin diagram D, in the following way • D is a graph whose set of nodes is D, RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • the nodes i and j are joined by aij aji edges, Fano manifolds and P1 -fibrations Lie Algebras Gianluca Occhetta Dynkin diagrams Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds Set D = {1, . . . , n}, where n is the number of simple roots of g. With the matrix A is associated a finite Dynkin diagram D, in the following way • D is a graph whose set of nodes is D, RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • the nodes i and j are joined by aij aji edges, • if |aij | > |aji | the edge is directed towards the node i. Fano manifolds and P1 -fibrations Lie Algebras Gianluca Occhetta Dynkin diagrams Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds Set D = {1, . . . , n}, where n is the number of simple roots of g. With the matrix A is associated a finite Dynkin diagram D, in the following way • D is a graph whose set of nodes is D, RH manifolds Flag manifolds Homogeneous models • the nodes i and j are joined by aij aji edges, • if |aij | > |aji | the edge is directed towards the node i. Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture 2 0 0 2 2 −1 −1 2 2 −2 −1 2 2 −3 −1 2 Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Dynkin diagrams of (semi)simple Lie algebras An SLn+1 Bn SO2n+1 Cn Sp2n Dn SO2n Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds CLASSICAL Flag manifolds Homogeneous models E6 Fibrations and reflections Dynkin diagram Bott-Samelson varieties E7 Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture E8 F4 G2 EXCEPTIONAL Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Rational homogeneous manifolds Subgroups P ⊂ G s.t. G/P is a projective variety are called parabolic. Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Rational homogeneous manifolds Subgroups P ⊂ G s.t. G/P is a projective variety are called parabolic. Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture A parabolic subgroup is determined by the choice of a set of simple roots, i.e. by a subset I ⊂ D, and the corresponding variety is denoted by marking the nodes of I. Fano manifolds and P1 -fibrations Rational homogeneous manifolds Gianluca Occhetta Motivation Subgroups P ⊂ G s.t. G/P is a projective variety are called parabolic. Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture A parabolic subgroup is determined by the choice of a set of simple roots, i.e. by a subset I ⊂ D, and the corresponding variety is denoted by marking the nodes of I. Example Set G = SL(4) Fano manifolds and P1 -fibrations Rational homogeneous manifolds Gianluca Occhetta Motivation Subgroups P ⊂ G s.t. G/P is a projective variety are called parabolic. Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds A parabolic subgroup is determined by the choice of a set of simple roots, i.e. by a subset I ⊂ D, and the corresponding variety is denoted by marking the nodes of I. Example Set G = SL(4) Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture P3 G(1, 3) (P3 )∗ Fano manifolds and P1 -fibrations Rational homogeneous manifolds Gianluca Occhetta Motivation Subgroups P ⊂ G s.t. G/P is a projective variety are called parabolic. Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds A parabolic subgroup is determined by the choice of a set of simple roots, i.e. by a subset I ⊂ D, and the corresponding variety is denoted by marking the nodes of I. Example Set G = SL(4) Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram P3 G(1, 3) (P3 )∗ Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture F(0, 1) F(1, 2) F(0, 2) Fano manifolds and P1 -fibrations Rational homogeneous manifolds Gianluca Occhetta Motivation Subgroups P ⊂ G s.t. G/P is a projective variety are called parabolic. Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds A parabolic subgroup is determined by the choice of a set of simple roots, i.e. by a subset I ⊂ D, and the corresponding variety is denoted by marking the nodes of I. Example Set G = SL(4) Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram P3 G(1, 3) (P3 )∗ Bott-Samelson varieties Construction Properties Uniqueness F(0, 1) F(1, 2) CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture F(0, 1, 2) F(0, 2) Fano manifolds and P1 -fibrations Gianluca Occhetta Complete flag manifolds Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture A complete flag manifold G/B is a rational homogeneous manifolds s.t. in its Dynkin diagram all the nodes are marked. B, called Borel subgroup, is the smallest parabolic subgroup. Fano manifolds and P1 -fibrations Gianluca Occhetta Complete flag manifolds Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture A complete flag manifold G/B is a rational homogeneous manifolds s.t. in its Dynkin diagram all the nodes are marked. B, called Borel subgroup, is the smallest parabolic subgroup. If D = An , then G/B is the manifold parametrizing complete flags of linear subspaces in Pn . Fano manifolds and P1 -fibrations Gianluca Occhetta Complete flag manifolds Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture A complete flag manifold G/B is a rational homogeneous manifolds s.t. in its Dynkin diagram all the nodes are marked. B, called Borel subgroup, is the smallest parabolic subgroup. If D = An , then G/B is the manifold parametrizing complete flags of linear subspaces in Pn . • Elementary contractions of G/B are P1 -bundles. Fano manifolds and P1 -fibrations Complete flag manifolds Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture A complete flag manifold G/B is a rational homogeneous manifolds s.t. in its Dynkin diagram all the nodes are marked. B, called Borel subgroup, is the smallest parabolic subgroup. If D = An , then G/B is the manifold parametrizing complete flags of linear subspaces in Pn . • Elementary contractions of G/B are P1 -bundles. • Every rational homogenous manifold is dominated by a complete flag manifold. Fano manifolds and P1 -fibrations Complete flag manifolds Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties A complete flag manifold G/B is a rational homogeneous manifolds s.t. in its Dynkin diagram all the nodes are marked. B, called Borel subgroup, is the smallest parabolic subgroup. If D = An , then G/B is the manifold parametrizing complete flags of linear subspaces in Pn . • Elementary contractions of G/B are P1 -bundles. • Every rational homogenous manifold is dominated by a complete flag manifold. Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • If f : Z → G/B is a surjective morphism from a rational homogenous manifold, then Z = G/B × Z 0 , and f is the projection. Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Cartan matrix Geometric interpretation Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Cartan matrix Geometric interpretation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture G/B complete flag manifold, with Dynkin diagram D. Fano manifolds and P1 -fibrations Cartan matrix Gianluca Occhetta Geometric interpretation Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams G/B complete flag manifold, with Dynkin diagram D. Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • pi : G/B → G/Pi elementary contraction corresponding to the unmarking of node i; Fano manifolds and P1 -fibrations Cartan matrix Gianluca Occhetta Geometric interpretation Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams G/B complete flag manifold, with Dynkin diagram D. Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • pi : G/B → G/Pi elementary contraction corresponding to the unmarking of node i; • Γi fiber of pi ; Fano manifolds and P1 -fibrations Cartan matrix Gianluca Occhetta Geometric interpretation Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams G/B complete flag manifold, with Dynkin diagram D. Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • pi : G/B → G/Pi elementary contraction corresponding to the unmarking of node i; • Γi fiber of pi ; • −Ki = −KG/B + p∗i KG/Pi relative anticanonical. Fano manifolds and P1 -fibrations Cartan matrix Gianluca Occhetta Geometric interpretation Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams G/B complete flag manifold, with Dynkin diagram D. Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram • pi : G/B → G/Pi elementary contraction corresponding to the unmarking of node i; • Γi fiber of pi ; • −Ki = −KG/B + p∗i KG/Pi relative anticanonical. Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture The Cartan matrix of D is the intersection matrix −Ki · Cj . Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Fano manifolds whose elementary contractions are smooth P1 -fibrations Fano manifolds and P1 -fibrations Notation Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture X Fano manifold with Picard number n. Fano manifolds and P1 -fibrations Notation Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture X Fano manifold with Picard number n. πi : X → X i elementary contration. Fano manifolds and P1 -fibrations Notation Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds X Fano manifold with Picard number n. πi : X → X i elementary contration. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Ki relative canonical. Fano manifolds and P1 -fibrations Notation Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds X Fano manifold with Picard number n. πi : X → X i elementary contration. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Ki relative canonical. Γi fiber of πi . Fano manifolds and P1 -fibrations Notation Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds X Fano manifold with Picard number n. πi : X → X i elementary contration. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Ki relative canonical. Γi fiber of πi . D = {1, . . . , n}. Fano manifolds and P1 -fibrations Reflection group Gianluca Occhetta Relative duality Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds Lemma π : M → Y smooth P1 -fibration (M, Y smooth). Γ fiber, K relative canonical, D divisor on M, l := D · Γ. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections H i (M, D) ∼ = H i+sgn(l+1) (M, D + (l + 1)K), ∀i ∈ Z, se l 6= −1 i H (M, D) ∼ = {0} for every i ∈ Z, if l = −1. Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Corollary χ(M, D) = −χ(M, D + (l + 1)K) Fano manifolds and P1 -fibrations Gianluca Occhetta Reflection group Motivation Fano bundles The problem Lie Algebras Cartan decomposition For every contraction πi : X → X let us consider the affine involution ri0 : N 1 (X) → N 1 (X) Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture ri0 (D) := D + (D · Γi + 1)Ki . Fano manifolds and P1 -fibrations Gianluca Occhetta Reflection group Motivation Fano bundles The problem Lie Algebras Cartan decomposition For every contraction πi : X → X let us consider the affine involution ri0 : N 1 (X) → N 1 (X) Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture ri0 (D) := D + (D · Γi + 1)Ki . Setting T(D) := D + KX /2 the maps ri := T −1 ◦ ri0 ◦ T are liner involutions of N 1 (X) given by ri (D) = D + (D · Γi )Ki , Fano manifolds and P1 -fibrations Gianluca Occhetta Reflection group Motivation Fano bundles The problem Lie Algebras Cartan decomposition For every contraction πi : X → X let us consider the affine involution ri0 : N 1 (X) → N 1 (X) Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction ri0 (D) := D + (D · Γi + 1)Ki . Setting T(D) := D + KX /2 the maps ri := T −1 ◦ ri0 ◦ T are liner involutions of N 1 (X) given by ri (D) = D + (D · Γi )Ki , We have ri (Ki ) = −Ki and ri fixes pointwise the hyperplane Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Mi := {D|D · Γi = 0} ⊂ N 1 (X). Fano manifolds and P1 -fibrations Gianluca Occhetta Reflection group Motivation Fano bundles The problem Lie Algebras Cartan decomposition For every contraction πi : X → X let us consider the affine involution ri0 : N 1 (X) → N 1 (X) Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction ri0 (D) := D + (D · Γi + 1)Ki . Setting T(D) := D + KX /2 the maps ri := T −1 ◦ ri0 ◦ T are liner involutions of N 1 (X) given by ri (D) = D + (D · Γi )Ki , We have ri (Ki ) = −Ki and ri fixes pointwise the hyperplane Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Mi := {D|D · Γi = 0} ⊂ N 1 (X). Let W ⊂ Gl(N 1 (X)) be the group generated by the ri ’s. Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Reflection group Finiteness Fano manifolds and P1 -fibrations Reflection group Gianluca Occhetta Finiteness Motivation Fano bundles The problem Let χX : N 1 (X) → R be the polynomial (of degree ≤ dim X) such that Lie Algebras Cartan decomposition Cartan matrix χX (m1 , . . . , mn ) = χ(X, m1 K1 + · · · + mn Kn ) Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture and let χTX := χX ◦ T. Fano manifolds and P1 -fibrations Reflection group Gianluca Occhetta Finiteness Motivation Fano bundles The problem Let χX : N 1 (X) → R be the polynomial (of degree ≤ dim X) such that Lie Algebras Cartan decomposition χX (m1 , . . . , mn ) = χ(X, m1 K1 + · · · + mn Kn ) Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture and let χTX := χX ◦ T. Lemma For every R-divisor D and every ri χTX (D) = −χTX (ri (D)). Fano manifolds and P1 -fibrations Reflection group Gianluca Occhetta Finiteness Motivation Fano bundles The problem Let χX : N 1 (X) → R be the polynomial (of degree ≤ dim X) such that Lie Algebras Cartan decomposition χX (m1 , . . . , mn ) = χ(X, m1 K1 + · · · + mn Kn ) Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models and let χTX := χX ◦ T. Lemma For every R-divisor D and every ri Fibrations and reflections χTX (D) = −χTX (ri (D)). Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Corollary For every R-divisor D and every w ∈ W χTX (D) = ±χTX (w(D)). Fano manifolds and P1 -fibrations Reflection group Gianluca Occhetta Finiteness Motivation Fano bundles The problem Let χX : N 1 (X) → R be the polynomial (of degree ≤ dim X) such that Lie Algebras Cartan decomposition χX (m1 , . . . , mn ) = χ(X, m1 K1 + · · · + mn Kn ) Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models and let χTX := χX ◦ T. Lemma For every R-divisor D and every ri Fibrations and reflections χTX (D) = −χTX (ri (D)). Dynkin diagram Bott-Samelson varieties Construction Properties Corollary For every R-divisor D and every w ∈ W Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Theorem W is a finite group. χTX (D) = ±χTX (w(D)). Fano manifolds and P1 -fibrations Gianluca Occhetta Cartan matrix Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture there is a scalar product h , i on N 1 (X), which is W-invariant. In particular the ri ’s are euclidean reflections. Fano manifolds and P1 -fibrations Cartan matrix Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix there is a scalar product h , i on N 1 (X), which is W-invariant. In particular the ri ’s are euclidean reflections. Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Moreover −D · Γi = 2 hD, Ki i , for every i. hKi , Ki i Fano manifolds and P1 -fibrations Cartan matrix Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix there is a scalar product h , i on N 1 (X), which is W-invariant. In particular the ri ’s are euclidean reflections. Dynkin diagrams Rational homogeneous manifolds Moreover −D · Γi = 2 RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture hD, Ki i , for every i. hKi , Ki i The set Φ := {w(−Ki ) |w ∈ W, i = 1, . . . , n } ⊂ N 1 (X), is a root system, whose Weyl group is W. Fano manifolds and P1 -fibrations Cartan matrix Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix there is a scalar product h , i on N 1 (X), which is W-invariant. In particular the ri ’s are euclidean reflections. Dynkin diagrams Rational homogeneous manifolds Moreover −D · Γi = 2 RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture hD, Ki i , for every i. hKi , Ki i The set Φ := {w(−Ki ) |w ∈ W, i = 1, . . . , n } ⊂ N 1 (X), is a root system, whose Weyl group is W. The Cartan matrix A of this root system is the n × n matrix with entries aij := −Ki · Γj ∈ Z. Fano manifolds and P1 -fibrations Cartan matrix Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture By the properties of root systems Fano manifolds and P1 -fibrations Cartan matrix Gianluca Occhetta Motivation Fano bundles By the properties of root systems The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • aii = 2 for every i, Fano manifolds and P1 -fibrations Cartan matrix Gianluca Occhetta Motivation Fano bundles By the properties of root systems The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • aii = 2 for every i, • aij = 0 iff aji = 0, Fano manifolds and P1 -fibrations Cartan matrix Gianluca Occhetta Motivation Fano bundles By the properties of root systems The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • aii = 2 for every i, • aij = 0 iff aji = 0, • se aij 6= 0, i 6= j, then aij , aji ∈ Z and aij aji = 1, 2 o 3. Fano manifolds and P1 -fibrations Cartan matrix Gianluca Occhetta Motivation Fano bundles By the properties of root systems The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • aii = 2 for every i, • aij = 0 iff aji = 0, • se aij 6= 0, i 6= j, then aij , aji ∈ Z and aij aji = 1, 2 o 3. Using geometric properties of X one can show that aij with i 6= j are nonpositive. /X ? HS sj Γj πi πi |Γj / Xi Fano manifolds and P1 -fibrations Cartan matrix Gianluca Occhetta Motivation Fano bundles By the properties of root systems The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties • aii = 2 for every i, • aij = 0 iff aji = 0, • se aij 6= 0, i 6= j, then aij , aji ∈ Z and aij aji = 1, 2 o 3. Using geometric properties of X one can show that aij with i 6= j are nonpositive. /X ? HS sj Γj πi πi |Γj / Xi As a consequence we can show Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Theorem Every connected component of D is one of the following: An , Bn , Cn , Dn , E6 , E7 , E8 , F4 , or G2 . Fano manifolds and P1 -fibrations Gianluca Occhetta Homogeneous models Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture We associate with X a semi simple Lie group G, determined by D. Fano manifolds and P1 -fibrations Gianluca Occhetta Homogeneous models Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture We associate with X a semi simple Lie group G, determined by D. Given a Borel subgroup B we consider the morphism ψ : N 1 (X) → N 1 (G/B), defined by ψ(Ki ) = K i . Fano manifolds and P1 -fibrations Homogeneous models Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds We associate with X a semi simple Lie group G, determined by D. Given a Borel subgroup B we consider the morphism ψ : N 1 (X) → N 1 (G/B), defined by ψ(Ki ) = K i . Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Proposition Λ ⊂ Pic(X) generated by the Ki ’s. • dim X = dim G/B; • hi (X, D) = hi (G/B, ψ(D)) for every D ∈ Λ, i ∈ Z. Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture x ∈ X, ` = (l1 , . . . , lr ), li ∈ D, and `[s] := (l1 , . . . , lr−s ). We introduce manifolds Z`[s] , with morphisms f`[s] : Z`[s] → X, called Bott-Samelson varieties associated with the subsequences `[s], in the following way: Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture x ∈ X, ` = (l1 , . . . , lr ), li ∈ D, and `[s] := (l1 , . . . , lr−s ). We introduce manifolds Z`[s] , with morphisms f`[s] : Z`[s] → X, called Bott-Samelson varieties associated with the subsequences `[s], in the following way: If s = r we set Z`[r] := {x} and f`[r] : {x} → X is the inclusion. Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds x ∈ X, ` = (l1 , . . . , lr ), li ∈ D, and `[s] := (l1 , . . . , lr−s ). We introduce manifolds Z`[s] , with morphisms f`[s] : Z`[s] → X, called Bott-Samelson varieties associated with the subsequences `[s], in the following way: If s = r we set Z`[r] := {x} and f`[r] : {x} → X is the inclusion. Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture If s < r we build Z`[s] on Z`[s+1] in the following way: Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds x ∈ X, ` = (l1 , . . . , lr ), li ∈ D, and `[s] := (l1 , . . . , lr−s ). We introduce manifolds Z`[s] , with morphisms f`[s] : Z`[s] → X, called Bott-Samelson varieties associated with the subsequences `[s], in the following way: If s = r we set Z`[r] := {x} and f`[r] : {x} → X is the inclusion. Flag manifolds Homogeneous models If s < r we build Z`[s] on Z`[s+1] in the following way: Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction f`[s] Z`[s] p`[s+1] /X = f`[s+1] πlr−s Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Z`[s+1] g`[s+1] / Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds x ∈ X, ` = (l1 , . . . , lr ), li ∈ D, and `[s] := (l1 , . . . , lr−s ). We introduce manifolds Z`[s] , with morphisms f`[s] : Z`[s] → X, called Bott-Samelson varieties associated with the subsequences `[s], in the following way: If s = r we set Z`[r] := {x} and f`[r] : {x} → X is the inclusion. Flag manifolds Homogeneous models If s < r we build Z`[s] on Z`[s+1] in the following way: Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction f`[s] Z`[s] p`[s+1] f`[s+1] Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Z`[s+1] g`[s+1] /X < πlr−s / Xlr−s Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds x ∈ X, ` = (l1 , . . . , lr ), li ∈ D, and `[s] := (l1 , . . . , lr−s ). We introduce manifolds Z`[s] , with morphisms f`[s] : Z`[s] → X, called Bott-Samelson varieties associated with the subsequences `[s], in the following way: If s = r we set Z`[r] := {x} and f`[r] : {x} → X is the inclusion. Flag manifolds Homogeneous models If s < r we build Z`[s] on Z`[s+1] in the following way: Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction f`[s] Z`[s] p`[s+1] f`[s+1] Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Z`[s+1] g`[s+1] /X < πlr−s / Xlr−s Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds x ∈ X, ` = (l1 , . . . , lr ), li ∈ D, and `[s] := (l1 , . . . , lr−s ). We introduce manifolds Z`[s] , with morphisms f`[s] : Z`[s] → X, called Bott-Samelson varieties associated with the subsequences `[s], in the following way: If s = r we set Z`[r] := {x} and f`[r] : {x} → X is the inclusion. Flag manifolds Homogeneous models If s < r we build Z`[s] on Z`[s+1] in the following way: Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction f`[s] Z`[s] p`[s+1] f`[s+1] Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Z`[s+1] g`[s+1] /X < πlr−s / Xlr−s Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Motivation Fano bundles The problem f`[s] Z`[s] J Lie Algebras Cartan decomposition /X < Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture σ`[s+1] p`[s+1] Z`[s+1] f`[s+1] g`[s+1] πlr−s / Xl r−s Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Motivation Fano bundles The problem f`[s] Z`[s] J Lie Algebras Cartan decomposition /X < Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds σ`[s+1] p`[s+1] Z`[s+1] f`[s+1] g`[s+1] πlr−s / Xl r−s Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture the map f`[s+1] factors via Z`[s] , and gives a section σ`[s+1] di p`[s+1] . Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Motivation Fano bundles The problem f`[s] Z`[s] J Lie Algebras Cartan decomposition /X < Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds σ`[s+1] p`[s+1] Z`[s+1] f`[s+1] g`[s+1] πlr−s / Xl r−s Homogeneous models Fibrations and reflections the map f`[s+1] factors via Z`[s] , and gives a section σ`[s+1] di p`[s+1] . Dynkin diagram Bott-Samelson varieties Construction Properties In particular p`[s+1] is a P1 -bundle, given by the projectivization of an ∗ extension F`[s] of OZ`[s+1] with OZ`[s+1] (f`[s+1] (Klr−s )): Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture ∗ 0 → OZ`[s+1] (f`[s+1] (Klr−s )) −→ F`[s] −→ OZ`[s+1] → 0, ∗ determined by ζ`[s] ∈ H 1 (Z`[s+1] , f`[s+1] (Klr−s )). Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Bott-Samelson varieties Geometric interpretation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture The image of Z` in X is the set of points belonging to chains of rational curves Γl1 , Γl2 . . . Γlr starting from x. Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Bott-Samelson varieties Geometric interpretation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture The image of Z` in X is the set of points belonging to chains of rational curves Γl1 , Γl2 . . . Γlr starting from x. In the homogeneous case such loci are the Schubert varieties. Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Bott-Samelson varieties Geometric interpretation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds The image of Z` in X is the set of points belonging to chains of rational curves Γl1 , Γl2 . . . Γlr starting from x. In the homogeneous case such loci are the Schubert varieties. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Given w ∈ W, lits length λ(w) is the minimum t such that w = ri1 ◦ · · · ◦ rit ; such an expression for w is called reduced. In W there exists a unique w0 of length dim X, and all the other elements are shorter. Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Bott-Samelson varieties Geometric interpretation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds The image of Z` in X is the set of points belonging to chains of rational curves Γl1 , Γl2 . . . Γlr starting from x. In the homogeneous case such loci are the Schubert varieties. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Given w ∈ W, lits length λ(w) is the minimum t such that w = ri1 ◦ · · · ◦ rit ; such an expression for w is called reduced. In W there exists a unique w0 of length dim X, and all the other elements are shorter. Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture In the homogeneous case dim f` (Z` ) = λ(w(`)); moreover, if w(`) is reduced then f` : Z` → f (Z` ) is birational. Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Bott-Samelson varieties Geometric interpretation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds The image of Z` in X is the set of points belonging to chains of rational curves Γl1 , Γl2 . . . Γlr starting from x. In the homogeneous case such loci are the Schubert varieties. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Given w ∈ W, lits length λ(w) is the minimum t such that w = ri1 ◦ · · · ◦ rit ; such an expression for w is called reduced. In W there exists a unique w0 of length dim X, and all the other elements are shorter. Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture In the homogeneous case dim f` (Z` ) = λ(w(`)); moreover, if w(`) is reduced then f` : Z` → f (Z` ) is birational. We show that the same properties hold in general. Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Bott-Samelson varieties Uniqueness Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture X, with connected diagram D, G/B homogeneus model, ` sequence. Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Uniqueness Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds X, with connected diagram D, G/B homogeneus model, ` sequence. Z` , Z ` Bott-Samelson varieties of X and G/B RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Let `0 be a list corresponding to the longest element in W: w(`0 ) = w0 . If Z`0 ' Z `0 , then X ' G/B. Fano manifolds and P1 -fibrations Bott-Samelson varieties Gianluca Occhetta Uniqueness Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds X, with connected diagram D, G/B homogeneus model, ` sequence. Z` , Z ` Bott-Samelson varieties of X and G/B RH manifolds Flag manifolds Homogeneous models Let `0 be a list corresponding to the longest element in W: w(`0 ) = w0 . If Z`0 ' Z `0 , then X ' G/B. Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Proposizione If D = 6 F4 , G2 there exists ` = (l1 , . . . , lm ) with w(`) = w0 such that Z`[s] ' Z `[s] for every s = 0, . . . , m − 1. Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Uniqueness Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Z` is defined by the extension Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture ∗ (Klr )) −→ F` −→ OZ`[1] → 0. 0 → OZ`[1] (f`[1] Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Uniqueness Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Z` is defined by the extension Dynkin diagrams Rational homogeneous manifolds ∗ (Klr )) −→ F` −→ OZ`[1] → 0. 0 → OZ`[1] (f`[1] RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture One shows easily that the following are equivalent Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Uniqueness Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Z` is defined by the extension Dynkin diagrams Rational homogeneous manifolds ∗ (Klr )) −→ F` −→ OZ`[1] → 0. 0 → OZ`[1] (f`[1] RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture One shows easily that the following are equivalent ∗ • F` ' OZ`[1] (f`[1] (Klr )) ⊕ OZ`[1] ; Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Uniqueness Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Z` is defined by the extension Dynkin diagrams Rational homogeneous manifolds ∗ (Klr )) −→ F` −→ OZ`[1] → 0. 0 → OZ`[1] (f`[1] RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture One shows easily that the following are equivalent ∗ • F` ' OZ`[1] (f`[1] (Klr )) ⊕ OZ`[1] ; • h1 (Z`[1] , f`∗ (Kj )) = 0; Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Uniqueness Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Z` is defined by the extension Dynkin diagrams Rational homogeneous manifolds ∗ (Klr )) −→ F` −→ OZ`[1] → 0. 0 → OZ`[1] (f`[1] RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture One shows easily that the following are equivalent ∗ • F` ' OZ`[1] (f`[1] (Klr )) ⊕ OZ`[1] ; • h1 (Z`[1] , f`∗ (Kj )) = 0; • the index j does not appear in `[1]. Fano manifolds and P1 -fibrations Gianluca Occhetta Bott-Samelson varieties Uniqueness Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Z` is defined by the extension Dynkin diagrams Rational homogeneous manifolds ∗ (Klr )) −→ F` −→ OZ`[1] → 0. 0 → OZ`[1] (f`[1] RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties One shows easily that the following are equivalent ∗ • F` ' OZ`[1] (f`[1] (Klr )) ⊕ OZ`[1] ; • h1 (Z`[1] , f`∗ (Kj )) = 0; • the index j does not appear in `[1]. Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture So we have to show that if the index j appears in `[1] then h1 (Z`[1] , f`∗ (Kj )) ≤ 1. Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Special cases Fano manifolds and P1 -fibrations Special cases Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • G2 : no expression (there are 2) of w0 works. An (easy) ad hoc argument is possible. Fano manifolds and P1 -fibrations Special cases Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds • G2 : no expression (there are 2) of w0 works. An (easy) ad hoc argument is possible. Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture • F4 : no expression (there are 2144892) of w0 works. We are working on an ad hoc argument. Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Campana-Peternell Conjecture Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Positivity of the tangent bundle X smooth complex projective variety. Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles Positivity of the tangent bundle X smooth complex projective variety. The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Theorem (Mori (1979)) TX ample ⇔ X = Pm . Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles Positivity of the tangent bundle X smooth complex projective variety. The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Theorem (Mori (1979)) TX ample ⇔ X = Pm . • TX nef ⇒ ?? Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles Positivity of the tangent bundle X smooth complex projective variety. The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Theorem (Mori (1979)) TX ample ⇔ X = Pm . • TX nef ⇒ ?? 2 Abelian • Examples: Homogeneous manifolds , Razional, Semisimple G = P Parabolic Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Fano bundles Positivity of the tangent bundle X smooth complex projective variety. The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Theorem (Mori (1979)) TX ample ⇔ X = Pm . • TX nef ⇒ ?? 2 Abelian • Examples: Homogeneous manifolds , Razional, Semisimple G = P Parabolic Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Theorem (Demailly, Peternell and Schneider (1994)) ( e´ tale F X ←− X 0 −→ A TX nef ⇒ A Abelian, F Fano, TF nef Fano manifolds and P1 -fibrations Gianluca Occhetta Campana-Peternell Conjecture Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Conjecture (Campana-Peternell (1991)) Every Fano manifold with nef tangent bundle is homogeneous. Fano manifolds and P1 -fibrations Gianluca Occhetta Campana-Peternell Conjecture Motivation Fano bundles The problem Lie Algebras Cartan decomposition Conjecture (Campana-Peternell (1991)) Every Fano manifold with nef tangent bundle is homogeneous. Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Definition A CP-manifold is a Fano manifold with nef tangent bundle. Fano manifolds and P1 -fibrations Campana-Peternell Conjecture Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Conjecture (Campana-Peternell (1991)) Every Fano manifold with nef tangent bundle is homogeneous. Cartan matrix Dynkin diagrams Rational homogeneous manifolds Definition A CP-manifold is a Fano manifold with nef tangent bundle. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Results: 2 dim X = 3 [Campana Peternell(1991)] Fano manifolds and P1 -fibrations Campana-Peternell Conjecture Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Conjecture (Campana-Peternell (1991)) Every Fano manifold with nef tangent bundle is homogeneous. Cartan matrix Dynkin diagrams Rational homogeneous manifolds Definition A CP-manifold is a Fano manifold with nef tangent bundle. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Results: 2 dim X = 3 [Campana Peternell(1991)] 2 dim X = 4 [CP (1993), Mok (2002), Hwang (2006)] Fano manifolds and P1 -fibrations Campana-Peternell Conjecture Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Conjecture (Campana-Peternell (1991)) Every Fano manifold with nef tangent bundle is homogeneous. Cartan matrix Dynkin diagrams Rational homogeneous manifolds Definition A CP-manifold is a Fano manifold with nef tangent bundle. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Results: 2 dim X = 3 [Campana Peternell(1991)] 2 dim X = 4 [CP (1993), Mok (2002), Hwang (2006)] 2 dim X = 5 e ρX > 1 [Watanabe (2012)] Fano manifolds and P1 -fibrations Campana-Peternell Conjecture Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Conjecture (Campana-Peternell (1991)) Every Fano manifold with nef tangent bundle is homogeneous. Cartan matrix Dynkin diagrams Rational homogeneous manifolds Definition A CP-manifold is a Fano manifold with nef tangent bundle. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Results: 2 2 2 2 dim X = 3 [Campana Peternell(1991)] dim X = 4 [CP (1993), Mok (2002), Hwang (2006)] dim X = 5 e ρX > 1 [Watanabe (2012)] TX big e 1-ample [Solá-Conde Wi´sniewski (2004)] Fano manifolds and P1 -fibrations Campana-Peternell Conjecture Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Conjecture (Campana-Peternell (1991)) Every Fano manifold with nef tangent bundle is homogeneous. Cartan matrix Dynkin diagrams Rational homogeneous manifolds Definition A CP-manifold is a Fano manifold with nef tangent bundle. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Results: 2 2 2 2 dim X = 3 [Campana Peternell(1991)] dim X = 4 [CP (1993), Mok (2002), Hwang (2006)] dim X = 5 e ρX > 1 [Watanabe (2012)] TX big e 1-ample [Solá-Conde Wi´sniewski (2004)] • The above results are based on detailed classifications of the manifolds satisfying the required properties; Fano manifolds and P1 -fibrations Campana-Peternell Conjecture Gianluca Occhetta Motivation Fano bundles The problem Lie Algebras Cartan decomposition Conjecture (Campana-Peternell (1991)) Every Fano manifold with nef tangent bundle is homogeneous. Cartan matrix Dynkin diagrams Rational homogeneous manifolds Definition A CP-manifold is a Fano manifold with nef tangent bundle. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture Results: 2 2 2 2 dim X = 3 [Campana Peternell(1991)] dim X = 4 [CP (1993), Mok (2002), Hwang (2006)] dim X = 5 e ρX > 1 [Watanabe (2012)] TX big e 1-ample [Solá-Conde Wi´sniewski (2004)] • The above results are based on detailed classifications of the manifolds satisfying the required properties; • homogeneity is checked a posteriori. Fano manifolds and P1 -fibrations Gianluca Occhetta Motivation Campana-Peternell Conjecture A possible strategy Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture In the paper in which the conjecture is introduced, Campana and Peternell proposed the following approach: Fano manifolds and P1 -fibrations Campana-Peternell Conjecture Gianluca Occhetta A possible strategy Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix In the paper in which the conjecture is introduced, Campana and Peternell proposed the following approach: Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture 1 Prove the conjecture for CP-manifolds with Picard number one. Fano manifolds and P1 -fibrations Campana-Peternell Conjecture Gianluca Occhetta A possible strategy Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix In the paper in which the conjecture is introduced, Campana and Peternell proposed the following approach: Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture 1 Prove the conjecture for CP-manifolds with Picard number one. 2 Show that, given a CP-manifold X and a contraction f : X → Y, from the homogeneity of Y and of the fibers of f one can reconstruct the homogeneity of X. Fano manifolds and P1 -fibrations Campana-Peternell Conjecture Gianluca Occhetta A possible strategy Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix In the paper in which the conjecture is introduced, Campana and Peternell proposed the following approach: Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture 1 Prove the conjecture for CP-manifolds with Picard number one. 2 Show that, given a CP-manifold X and a contraction f : X → Y, from the homogeneity of Y and of the fibers of f one can reconstruct the homogeneity of X. The Picard number one case turned out to be very difficult. Fano manifolds and P1 -fibrations Campana-Peternell Conjecture Gianluca Occhetta A possible strategy Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix In the paper in which the conjecture is introduced, Campana and Peternell proposed the following approach: Dynkin diagrams Rational homogeneous manifolds RH manifolds Flag manifolds Homogeneous models Fibrations and reflections 1 Prove the conjecture for CP-manifolds with Picard number one. 2 Show that, given a CP-manifold X and a contraction f : X → Y, from the homogeneity of Y and of the fibers of f one can reconstruct the homogeneity of X. The Picard number one case turned out to be very difficult. Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture A possible alternative strategy is: Fano manifolds and P1 -fibrations Campana-Peternell Conjecture Gianluca Occhetta A possible strategy Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix In the paper in which the conjecture is introduced, Campana and Peternell proposed the following approach: Dynkin diagrams Rational homogeneous manifolds 1 Prove the conjecture for CP-manifolds with Picard number one. 2 Show that, given a CP-manifold X and a contraction f : X → Y, from the homogeneity of Y and of the fibers of f one can reconstruct the homogeneity of X. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections The Picard number one case turned out to be very difficult. Dynkin diagram Bott-Samelson varieties Construction Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture A possible alternative strategy is: 1 Prove the conjecture for CP-manifolds with “maximal” Picard number. Fano manifolds and P1 -fibrations Campana-Peternell Conjecture Gianluca Occhetta A possible strategy Motivation Fano bundles The problem Lie Algebras Cartan decomposition Cartan matrix In the paper in which the conjecture is introduced, Campana and Peternell proposed the following approach: Dynkin diagrams Rational homogeneous manifolds 1 Prove the conjecture for CP-manifolds with Picard number one. 2 Show that, given a CP-manifold X and a contraction f : X → Y, from the homogeneity of Y and of the fibers of f one can reconstruct the homogeneity of X. RH manifolds Flag manifolds Homogeneous models Fibrations and reflections The Picard number one case turned out to be very difficult. Dynkin diagram Bott-Samelson varieties Construction A possible alternative strategy is: 1 Prove the conjecture for CP-manifolds with “maximal” Picard number. 2 Show that any CP-manifold is dominated by a CP-manifold with “maximal” Picard number. Properties Uniqueness CampanaPeternell Conjecture Positivity of the tangent bundle Campana-Peternell Conjecture

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