Evgeny Materov - Academia.edu (original) (raw)
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Papers by Evgeny Materov
Discrete & Computational Geometry
Let V be a real vector space of dimension n and let M ⊂ V be a lattice. Let P ⊂ V be an n-dimensi... more Let V be a real vector space of dimension n and let M ⊂ V be a lattice. Let P ⊂ V be an n-dimensional polytope with vertices in M , and let ϕ : V → C be a homogeneous polynomial function of degree d. For q ∈ Z >0 and any face F of P , let D ϕ,F (q) be the sum of ϕ over the lattice points in the dilate qF. We define a generating function G ϕ (q, y) ∈ Q[q][y] packaging together the various D ϕ,F (q), and show that it satisfies a functional equation that simultaneously generalizes Ehrhart-Macdonald reciprocity and the Dehn-Sommerville relations. When P is a simple lattice polytope (i.e., each vertex meets n edges), we show how G ϕ can be computed using an analogue of Brion-Vergne's Euler-Maclaurin summation formula. (For instance, if P is a simplex, then h(P, t) = t n + t n−1 + • • • + 1.) The Dehn-Sommerville relations say that h k (P) = h n−k (P) for all k.
Using the Cayley trick, we define the notions of mixed toric residues and mixed Hessians associat... more Using the Cayley trick, we define the notions of mixed toric residues and mixed Hessians associated with r Laurent polynomials f 1 ,. .. , f r. We conjecture that the values of mixed toric residues on the mixed Hessians are determined by mixed volumes of the Newton polytopes of f 1 ,. .. , f r. Using mixed toric residues, we generalize our Toric Residue Mirror Conjecture to the case of Calabi-Yau complete intersections in Gorenstein toric Fano varieties obtained from nef-partitions of reflexive polytopes.
The purpose of this paper is to give an explicit formula which allows one to compute the dimensio... more The purpose of this paper is to give an explicit formula which allows one to compute the dimension of the cohomology groups of the sheaf OmegaPp(D)\Omega_{\P}^p(D)OmegaPp(D) of p-th differential forms of Zariski twisted by an ample invertible sheaf on a complete simplicial toric variety. The formula involves some combinatorial sums of integer points over all faces of the support polytope
We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture whi... more We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture which has close relations to toric mirror symmetry. Our conjecture, we call it Toric Residue Mirror Conjecture, claims that the generating functions of intersection numbers of divisors on a special sequence of simplicial toric varieties are power series expansions of some rational functions obtained as toric residues. We expect that this conjecture holds true for all Gorenstein toric Fano varieties associated with reflexive polytopes and give some evidences for that. The proposed conjecture suggests a simple method for computing Yukawa couplings for toric mirror Calabi-Yau hypersurfaces without solving systems of differential equations. We make several explicit computations for Calabi-Yau hypersurfaces in weighted projective spaces and in products of projective spaces.
Discrete & Computational Geometry
Let V be a real vector space of dimension n and let M ⊂ V be a lattice. Let P ⊂ V be an n-dimensi... more Let V be a real vector space of dimension n and let M ⊂ V be a lattice. Let P ⊂ V be an n-dimensional polytope with vertices in M , and let ϕ : V → C be a homogeneous polynomial function of degree d. For q ∈ Z >0 and any face F of P , let D ϕ,F (q) be the sum of ϕ over the lattice points in the dilate qF. We define a generating function G ϕ (q, y) ∈ Q[q][y] packaging together the various D ϕ,F (q), and show that it satisfies a functional equation that simultaneously generalizes Ehrhart-Macdonald reciprocity and the Dehn-Sommerville relations. When P is a simple lattice polytope (i.e., each vertex meets n edges), we show how G ϕ can be computed using an analogue of Brion-Vergne's Euler-Maclaurin summation formula. (For instance, if P is a simplex, then h(P, t) = t n + t n−1 + • • • + 1.) The Dehn-Sommerville relations say that h k (P) = h n−k (P) for all k.
Using the Cayley trick, we define the notions of mixed toric residues and mixed Hessians associat... more Using the Cayley trick, we define the notions of mixed toric residues and mixed Hessians associated with r Laurent polynomials f 1 ,. .. , f r. We conjecture that the values of mixed toric residues on the mixed Hessians are determined by mixed volumes of the Newton polytopes of f 1 ,. .. , f r. Using mixed toric residues, we generalize our Toric Residue Mirror Conjecture to the case of Calabi-Yau complete intersections in Gorenstein toric Fano varieties obtained from nef-partitions of reflexive polytopes.
The purpose of this paper is to give an explicit formula which allows one to compute the dimensio... more The purpose of this paper is to give an explicit formula which allows one to compute the dimension of the cohomology groups of the sheaf OmegaPp(D)\Omega_{\P}^p(D)OmegaPp(D) of p-th differential forms of Zariski twisted by an ample invertible sheaf on a complete simplicial toric variety. The formula involves some combinatorial sums of integer points over all faces of the support polytope
We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture whi... more We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture which has close relations to toric mirror symmetry. Our conjecture, we call it Toric Residue Mirror Conjecture, claims that the generating functions of intersection numbers of divisors on a special sequence of simplicial toric varieties are power series expansions of some rational functions obtained as toric residues. We expect that this conjecture holds true for all Gorenstein toric Fano varieties associated with reflexive polytopes and give some evidences for that. The proposed conjecture suggests a simple method for computing Yukawa couplings for toric mirror Calabi-Yau hypersurfaces without solving systems of differential equations. We make several explicit computations for Calabi-Yau hypersurfaces in weighted projective spaces and in products of projective spaces.