Affine Geometry
and the Discrete Legendre Transfrom
Andrey Novoseltsev
Department of Mathematics
University of Washington
April 25, 2008
Andrey Novoseltsev (UW)
Affine Geometry
April 25, 2008
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Outline
1
Combinatorial Constructions in Mirror Symmetry
2
Integral Tropical Manifolds
3
Discriminant Locus
4
Local Monodromy
5
Functions on Tropical Manifolds
6
Discrete Legendre Transform
Andrey Novoseltsev (UW)
Affine Geometry
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Batyrev
Hypersurfaces in toric varieties coming from reflexive polytopes.
Duality: ∆ ↔ ∆◦ .
These polygons are polar (and we will see them again):
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Affine Geometry
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Batyrev–Borisov
Complete intersections in toric varieties coming from
NEF-partitions of reflexive polytopes.
Duality: {∆1 , . . . , ∆r } ↔ {∇1 , . . . , ∇r }, such that
∆ = Conv {∆1 , . . . , ∆r } ,
∆◦ = ∇1 + · · · + ∇r ,
∇ = Conv {∇1 , . . . , ∇r } ,
∇◦ = ∆1 + · · · + ∆r .
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Affine Geometry
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Degenerations
Calabi-Yau manifolds in Grassmannians and (partial) flags.
Duality is obtained using degeneration to toric varieties.
Gross and Siebert propose another degeneration approach.
Duality: discrete Legendre transform of a polarized positive
integral tropical manifold (B, P, ϕ) ↔ (B̌, P̌, ϕ̌).
Let’s make sense of the last sentence!
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Affine Geometry
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Polyhedral Complex
Definition
An integral (or lattice) rational convex polyhedron σ is the (possibly
unbounded) intersection of finitely many closed affine half-spaces in
Rn with at least one vertex, s.t. functions defining these half-spaces
can be taken with rational coefficients and all vertices are integral. Let
LPoly be the category of integral convex polyhedra with integral affine
isomorphisms onto faces as morphisms.
Definition
An integral polyhedral complex is a category P and a functor
F : P → LPoly s.t. if σ ∈ F (P) and τ is a face of σ, then τ ∈ F (P).
To avoid self-intersections: there is at most one morphism between
any two objects of P. (This requirement is not essential.)
Andrey Novoseltsev (UW)
Affine Geometry
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Topological Manifold and Tangent Spaces
Definition
Let B be the topological space associated to P: the quotient of
`
σ∈P F (σ) by the equivalence relation of face inclusion.
From now on we will denote F (σ) just by σ and call them cells.
Definition
Λσ ' Zdim σ is the free abelian group of integral vector fields along σ.
For any y ∈ Int σ there is a canonical injection Λσ → Tσ,y , inducing the
isomorphism Tσ,y ' Λσ,R = Λσ ⊗Z R.
Using the exponential map, we can identify σ with a polytope σ
e ⊂ Tσ,y .
(Different choices of y will correspond to translations of σ
e.)
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Affine Geometry
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Fan Structure
Definition
Let Nτ be a lattice of rank k and Nτ,R = Nτ ⊗Z R. A fan structure along
τ ∈ P is a continuous map Sτ : Uτ → Nτ,R , where Uτ is the open star
of τ , s.t.
Sτ−1 (0) = Int τ
for each e : τ → σ the restriction Sτ |Int σ is induced by an
epimorphism Λσ → W ∩ Nτ for some vector subspace W ⊂ Nτ,R
cones Ke = R>0 Sτ (σ ∩ Uτ ), e : τ → σ, form a finite fan Στ in Nτ,R
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Affine Geometry
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Tropical Manifold
Definition
If τ ⊂ σ, the fan structure along σ induced by Sτ is the composition
S
τ
Uσ → Uτ −→
Nτ,R → Nτ,R /Lσ = Nσ,R ,
where Lσ ⊂ NR is the linear span of Sτ (Int σ).
Definition
An integral tropical manifold of dimension n is a (countable) integral
polyhedral complex P with a fan structure Sv : Uv → Nv ,R ' Rn at
each vertex v ∈ P s.t.
S
for any vertex v the support |Σv | = C∈Σv C is (non-strictly)
convex with nonempty interior;
if v and w are vertices of τ , then the fan structures along τ
induced from Sv and Sw are equivalent.
Andrey Novoseltsev (UW)
Affine Geometry
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Tropical Manifolds from Reflexive Polytopes
Let ∆ be a reflexive
polytope.
Let P = ∂∆.
The fan structures
are given by
projections along
the vertices.
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Affine Geometry
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Constructing Discriminant Locus
Fan structures at vertices define an affine structure on B away
from the closed discriminant locus ∆ of codimension two.
For each bounded τ ∈ P, s.t. dim τ 6= 0, n, choose aτ ∈ Int τ :
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Affine Geometry
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Constructing Discriminant Locus
For each unbounded τ , s.t. dim τ 6= n, choose a 0 6= aτ ∈ Λτ,R , s.t.
aτ + τ ⊂ τ .
For each chain τ1 ⊂ · · · ⊂ τn−1 with dim τi = i and τi bounded for
i 6 r , where r > 1, let
X
∆τ1 ...τn−1 = Conv {aτi : 1 6 i 6 r } +
R>0 · aτi ⊂ τn−1 .
i>r
Let ∆ be the union of such polyhedra.
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Affine Geometry
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Constructing Discriminant Locus
(Only “visible” half of ∆ is shown for the bounded case.)
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Affine Geometry
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Remarks on Discriminant Locus
If % ∈ P and dim % = n − 1, the connected components of % \ ∆
are in bijection with the vertices of %.
Interiors of top dimensional cells and fan structures at vertices
define an affine structure on B \ ∆ (fan structures give charts via
the exponential maps).
This defines a flat connection on TB\∆ which we may use for
parallel transport.
There is some flexibility in constructing ∆.
“Generic” discriminant locus: coordinates of aτ are “as
algebraically independent as possible.” Then ∆ contains no
rational points.
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Affine Geometry
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Local Monodromy
Choose the following data:
ω ∈ P — a bounded edge (i.e. a one-dimensional cell)
v + and v − — vertices of ω
% ∈ P, dim % = n − 1, % 6⊂ ∂B, ω ⊂ %
σ + and σ − — top dimensional cells containing %
Follow the change of affine charts given by
the fan structure at v +
the polyhedral structure of σ +
the fan structure at v −
the polyhedral structure of σ −
again the fan structure at v +
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Affine Geometry
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Local Monodromy
We obtain a transformation Tω% ∈ SL(Λv + ), where Λv + is the lattice
of integral tangent vectors to B at the point v + . (Not the integral
vector fields along the zero-dimensional face v + !)
Let dω ∈ Λω ⊂ Λv + be the primitive vector from v + to v − .
∗
+
Let ď% ∈ Λ⊥
% ⊂ Λv + be the primitive vector s.t. hď% , σ i > 0.
Then Tω% (m) = m + κω% hm, ď% idω for some constant κω% .
κω% is independent on the ordering of v ± and σ ± .
If m ∈ Λ% , then Tω% (m) = m.
For any m we have Tω% (m) − m ∈ Λ% .
κω% > 0 for “geometrically meaningful manifolds”.
If all κω% > 0, B is called positive.
The affine structure extends to a neighborhood of τ ∈ P iff
κω% = 0 for all ω and % s.t. ω ⊂ τ ⊂ %.
Andrey Novoseltsev (UW)
Affine Geometry
April 25, 2008
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Functions on Tropical Manifolds
Definition
An affine function on an open set U ⊂ B is a continuous map U → R
that is affine on U \ ∆.
Definition
A piecewise-linear (PL) function on U is a continuous map ϕ : U → R
s.t. for all fan structures Sτ : Uτ → Nτ,R along τ ∈ P we have
ϕ|U∩Uτ = λ + Sτ∗ (ϕτ ) for some affine function λ : Uτ → R and a
function ϕτ : Nτ,R → R which is PL w.r.t. the fan Στ .
(This definition ensures that ϕ is “good enough” near ∆.)
Andrey Novoseltsev (UW)
Affine Geometry
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Multivalued PL Functions and Polarizations
Definition
A multivalued piecewise-linear (MPL) function ϕ on U is a collection of
PL functions {ϕi } on an open cover {Ui } of U s.t. ϕi ’s differ by affine
functions on overlaps.
MPL functions can be given by specifying maps ϕτ : Nτ,R → R.
All of the above definitions can be restricted to integral functions in the
obvious way.
Definition
If all local PL representatives of an integral MPL function ϕ are strictly
convex, ϕ is a polarization of (B, P) and (B, P, ϕ) is a polarized integral
tropical manifold.
(If ∂B 6= ∅, we also require that |Στ | is convex for every τ ∈ P.)
Andrey Novoseltsev (UW)
Affine Geometry
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Example of a Polarization
Suppose (B, P) is constructed from a reflexive polytope ∆.
Let ψ be a PL function on the fan generated by ∆, s.t. ψ|∂∆ ≡ 1.
For each vertex v choose an integral affine function ψv s.t.
ψv (v ) = 1 = ψ(v ) and let ϕv = ψ − ψv on Uv .
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Affine Geometry
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Discrete Legendre Transform
The discrete Legendre transform is a duality transformation of the
set of polarized (positive) integral tropical manifolds
(B, P, ϕ) ↔ (B̌, P̌, ϕ̌).
As a category, P̌ is the opposite of P.
The functor F̌ : P̌ → LPoly is given by F̌ (τ̌ ) = Newton(ϕτ ), where
τ̌ = τ as objects and Newton(ϕτ ) is the Newton polyhedron of ϕτ :
∗
Newton(ϕτ ) = x ∈ Nτ,R
: ϕτ + x > 0 .
For the fans from our example we get:
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Affine Geometry
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Discrete Legendre Transform
These data are enough to construct B̌ as a topological manifold.
It also can be obtained as the dual cell complex of (B, P) using
the barycentric subdivision, which gives a homeomorphism
between B and B̌, but is not a polyhedral complex.
In our example we get the boundary of the polar reflexive polygon:
Andrey Novoseltsev (UW)
Affine Geometry
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Discrete Legendre Transform
We still need to give fan structures and a polarization.
Let σ ∈ P, dim σ = n. Then σ̌ is a vertex.
Let Σσ̌ be the normal fan of σ in Λ∗σ,R .
∗
Using the parallel transport, we can identify Λ∗σ,R with Λ∗v ,R = TB,v
for a vertex v of σ.
Let σ
e ⊂ TB,v be the inverse image of σ under the exponential map.
Let ϕ̌σ̌ : |Σσ̌ | → R be given by ϕ̌σ̌ (m) = − inf(m(e
σ )).
∗
If Σσ̌ is incomplete, extend ϕ̌σ̌ to Λτ,R by infinity.
A different choice of v corresponds to the translation of σ
e by an
integral vector and the change of ϕ̌σ̌ by an integral affine function.
In our example we get the following (same as if we repeated
construction with vectors!):
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Affine Geometry
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Cone Identification
Consider the cones
m ∈ Λ∗v ,R : m(e
σ ) > 0 ∈ Σσ̌ , m ∈ Nv∗,R : m(dSv (e
σ )) > 0 ∈ Σ∗v .
dSv is an integral affine identification of cones corresponding to
cells containing v (“tangent wedges”) in TB,v ' Λv ,R and Nv ,R .
Cones above can be identified via dSv∗ : Nv∗,R → Λ∗v ,R .
These maps induce the fan structure at σ̌.
This gives the polarized integral tropical manifold (B̌, P̌, ϕ̌).
ˇ = ∆ (using homeomorphism B → B̌).
Can take ∆
Discrete Legendre transform of a positive manifolds is positive.
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Affine Geometry
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The End
Thank you!
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