Holomorphic exterior differential systems on
algebraic manifolds
Ben McKay
Monday July 15, 2013
Algebraic manifolds
An algebraic manifold means a smooth complex projective variety.
Parabolic geometries
• Suppose that G is a complex semisimple Lie group acting holomorphically,
faithfully and transitively on a compact complex manifold X = G/P .
• Roughly: a holomorphic (G, X)-geometry is a holomorphic geometric
structure on a complex manifold M with dim M = dim X which makes
M look infinitesimally like X up to G-action.
• See talks by Čap, Eastwood, Landsberg for the definition.
Examples
• M = X itself only the obvious holomorphic (G, X)-geometry iff c1 (M ) > 0
• M any complex torus many holomorphic (G, X)-geometries all are translation invariant iff c1 (M ) = 0
• The classification of holomorphic (G, X)-geometries on compact Riemann
surfaces is well known (Schwarzian derivative).
Examples II
• M locally Hermitian symmetric Riemannian manifolds, dimC M ≥ 2 unique
normal holomorphic (G, X)-geometry c1 (M ) < 0 (what others have c1 (M ) <
0?)
• More generally, pick Γ ⊂ G discrete subgroup, U ⊂ X Γ-invariant open
subset, and let M = Γ\U . Other than locally Hermitian symmetric manifolds, I only know of a countable set of examples like this, all in complex
dimension 3: Jahnke–Radloff.
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Classifying
• The classification of holomorphic (G, X)-geometries on algebraic manifolds in dimension at most 3 is complete.
• It is also complete if G = G2 or if X is an adjoint variety, and for many
other choices of (G, X).
Curves
• An entire curve is a nonconstant holomorphic C → M .
• A rational curve is a nonconstant holomorphic P1 → M .
• Terng: DEs for curves in homogeneous spaces; similar ones are defined
on any M with (G, X)-geometry.
• Green–Griffiths: all rational or entire curves satisfy all holomorphic DEs.
Old trick: rational curves
• If an algebraic manifold M has a holomorphic (G, X)-geometry and contains a rational curve then the geometry “drops”, is induced by a lower
dimensional geometry.
• So assume from now on no rational curves in M .
New trick: EDSs
• G-invariant holomorphic EDSs on X = G/P are common.
• Each one induces an EDS on any manifold M with (G, X)-geometry.
New trick: EDSs 2
• Take a holomorphic line bundle L → M . Write L & 0 if L admits a
Hermitian metric (maybe not continuous but L1 ) with curvature ≥ 0 as a
current.
• Tensor products of & 0 are & 0.
New tricks: EDSs 3
Campana, Peternell: If I a holomorphic EDS on an algebraic manifold M ,
and M has no rational curves, then det (Ω∗ /I) & 0.
New trick: EDSs 4
Demailly: If I ⊂ Ω1 a holomorphic linear Pfaffian system on an algebraic
manifold M with no rational curves and det I & 0 then I is Frobenius.
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New trick: EDSs 5
• Combining above tricks: if an algebraic manifold M has no rational curves,
and a (G, X)-geometry, then it probably has many induced Frobenius
linear Pfaffian systems.
• The corresponding systems on X = G/P are probably not Frobenius.
• M has curvature 6= 0.
• Curvature helps reduce structure group.
• If the structure group reduces to {1}, then M is a torus, and the geometry
is translation invariant.
Summary
• So far, no classification of holomorphic (G, X)-geometries on algebraic
manifolds.
• Not even a conjectured classification.
• Many obstructions.
The Jahnke–Radloff examples
• Pick α, β > 0 in Q.
• Take
√
α
0
B=
0
,
− α
1
0
√
−β
0
Q-subalgebra of 2 × 2-matrices.
• Take O ⊂ B any subring which is a Z-lattice.
• Take Γ ⊂ O× a finite index torsion-free group with det > 0.
The Jahnke–Radloff examples II
• Let
Γ0 =
I2×2
02×2
λ
γ
λ ∈ O, γ ∈ Γ ⊂ PSL(4, C) .
• Let U = C2 × (upper half plane) ⊂ P3 .
• Γ0 is a group of projective transformations preserving U .
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The Jahnke–Radloff examples III
• Let M = Γ0 \U .
• Flat holomorphic (G, X) = PSL(4, C) , P3 -geometry on M .
• Action preserves fibration (z, τ ) ∈ M → τ ∈ C to a compact Riemann
surface of high genus.
• Fibers are complex tori, complex dimension 2.
• (S. Dumitrescu): This (G, X)-geometry admits nonflat deformations.
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