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 1 / 24
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 April 25, 2008 2 / 24
Batyrev Hypersurfaces in toric varieties coming from reflexive polytopes. Duality:. These polygons are polar (and we will see them again): Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 3 / 24
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. Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 4 / 24
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) (ˇB, ˇP, ˇϕ). Let s make sense of the last sentence! Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 5 / 24
Polyhedral Complex Definition An integral (or lattice) rational convex polyhedron σ is the (possibly unbounded) intersection of finitely many closed affine half-spaces in R n 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 April 25, 2008 6 / 24
Topological Manifold and Tangent Spaces Definition Let B be the topological space associated to P: the quotient of F(σ) by the equivalence relation of face inclusion. σ P From now on we will denote F(σ) just by σ and call them cells. Definition Λ σ Z dim σ is the free abelian group of integral vector fields along σ. For any y Int σ there is a canonical injection Λ σ T σ,y, inducing the isomorphism T σ,y Λ σ,r = Λ σ Z R. Using the exponential map, we can identify σ with a polytope σ T σ,y. (Different choices of y will correspond to translations of σ.) Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 7 / 24
Fan Structure Definition Let N τ be a lattice of rank k and N τ,r = N τ Z R. A fan structure along τ P is a continuous map S τ : U τ N τ,r, where U τ is the open star of τ, s.t. Sτ 1 (0) = Int τ for each e : τ σ the restriction S τ Int σ is induced by an epimorphism Λ σ W N τ for some vector subspace W N τ,r cones K e = R 0 S τ (σ U τ ), e : τ σ, form a finite fan Σ τ in N τ,r Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 8 / 24
Tropical Manifold Definition If τ σ, the fan structure along σ induced by S τ is the composition U σ U τ S τ Nτ,R N τ,r /L σ = N σ,r, where L σ N R is the linear span of S τ (Int σ). Definition An integral tropical manifold of dimension n is a (countable) integral polyhedral complex P with a fan structure S v : U v N v,r R n at each vertex v P s.t. for any vertex v the support Σ v = C Σ v C is (non-strictly) convex with nonempty interior; if v and w are vertices of τ, then the fan structures along τ induced from S v and S w are equivalent. Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 9 / 24
Tropical Manifolds from Reflexive Polytopes Let be a reflexive polytope. Let P =. The fan structures are given by projections along the vertices. Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 10 / 24
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 0, n, choose a τ Int τ: Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 11 / 24
Constructing Discriminant Locus For each unbounded τ, s.t. dim τ n, choose a 0 a τ Λ τ,r, s.t. a τ + τ τ. For each chain τ 1 τ n 1 with dim τ i = i and τ i bounded for i r, where r 1, let τ1...τ n 1 = Conv {a τi : 1 i r} + i>r R 0 a τi τ n 1. Let be the union of such polyhedra. Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 12 / 24
Constructing Discriminant Locus (Only visible half of is shown for the bounded case.) Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 13 / 24
Remarks on Discriminant Locus If ϱ P and dim ϱ = n 1, the connected components of ϱ \ are in bijection with the vertices of ϱ. 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 T B\ which we may use for parallel transport. There is some flexibility in constructing. Generic discriminant locus: coordinates of a τ are as algebraically independent as possible. Then contains no rational points. Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 14 / 24
Local Monodromy Choose the following data: ω P a bounded edge (i.e. a one-dimensional cell) v + and v vertices of ω ϱ P, dim ϱ = n 1, ϱ B, ω ϱ σ + and σ top dimensional cells containing ϱ 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 + Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 15 / 24
Local Monodromy We obtain a transformation T ωϱ SL(Λ v +), where Λ v + 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 d ω Λ ω Λ v + be the primitive vector from v + to v. Let ďϱ Λ ϱ Λ v + be the primitive vector s.t. ďϱ, σ + 0. Then T ωϱ (m) = m + κ ωϱ m, ďϱ d ω for some constant κ ωϱ. κ ωϱ is independent on the ordering of v ± and σ ±. If m Λ ϱ, then T ωϱ (m) = m. For any m we have T ωϱ (m) m Λ ϱ. κ ωϱ 0 for geometrically meaningful manifolds. If all κ ωϱ 0, B is called positive. The affine structure extends to a neighborhood of τ P iff κ ωϱ = 0 for all ω and ϱ s.t. ω τ ϱ. Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 16 / 24
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 S τ : U τ N τ,r along τ P we have ϕ U Uτ = λ + S τ (ϕ τ ) for some affine function λ : U τ R and a function ϕ τ : N τ,r R which is PL w.r.t. the fan Σ τ. (This definition ensures that ϕ is good enough near.) Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 17 / 24
Multivalued PL Functions and Polarizations Definition A multivalued piecewise-linear (MPL) function ϕ on U is a collection of PL functions {ϕ i } on an open cover {U i } of U s.t. ϕ i s differ by affine functions on overlaps. MPL functions can be given by specifying maps ϕ τ : N τ,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, we also require that Σ τ is convex for every τ P.) Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 18 / 24
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 ψ v s.t. ψ v (v) = 1 = ψ(v) and let ϕ v = ψ ψ v on U v. Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 19 / 24
Discrete Legendre Transform The discrete Legendre transform is a duality transformation of the set of polarized (positive) integral tropical manifolds (B, P, ϕ) (ˇB, ˇP, ˇϕ). As a category, ˇP is the opposite of P. The functor ˇF : ˇP LPoly is given by ˇF(ˇτ) = Newton(ϕ τ ), where ˇτ = τ as objects and Newton(ϕ τ ) is the Newton polyhedron of ϕ τ : Newton(ϕ τ ) = { x N τ,r : ϕ τ + x 0 }. For the fans from our example we get: Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 20 / 24
Discrete Legendre Transform These data are enough to construct ˇB 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 ˇB, but is not a polyhedral complex. In our example we get the boundary of the polar reflexive polygon: Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 21 / 24
Discrete Legendre Transform We still need to give fan structures and a polarization. Let σ P, dim σ = n. Then ˇσ is a vertex. Let Σˇσ be the normal fan of σ in Λ σ,r. Using the parallel transport, we can identify Λ σ,r with Λ v,r = T B,v for a vertex v of σ. Let σ T B,v be the inverse image of σ under the exponential map. Let ˇϕˇσ : Σˇσ R be given by ˇϕˇσ (m) = inf(m( σ)). If Σˇσ is incomplete, extend ˇϕˇσ to Λ τ,r by infinity. A different choice of v corresponds to the translation of σ by an integral vector and the change of ˇϕˇσ by an integral affine function. In our example we get the following (same as if we repeated construction with vectors!): Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 22 / 24
Cone Identification Consider the cones { m Λ v,r : m( σ) 0 } Σˇσ, { m N v,r : m(ds v ( σ)) 0 } Σ v. ds v is an integral affine identification of cones corresponding to cells containing v ( tangent wedges ) in T B,v Λ v,r and N v,r. Cones above can be identified via ds v : N v,r Λ v,r. These maps induce the fan structure at ˇσ. This gives the polarized integral tropical manifold (ˇB, ˇP, ˇϕ). Can take ˇ = (using homeomorphism B ˇB). Discrete Legendre transform of a positive manifolds is positive. Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 23 / 24
The End Thank you! Andrey Novoseltsev (UW) Affine Geometry April 25, 2008 24 / 24