The holonomic rank of the A-hypergeometric system M_A(β) is the degree of the toric ideal I_A for generic parameters; in general, this is only a lower bound. To the semigroup ring of A we attach the ranking arrangement and use this algebraic invariant and the exceptional arrangement of nongeneric parameters to construct a combinatorial formula for the rank jump of M_A(β). As consequences, we obtain a refinement of the stratification of the exceptional arrangement by the rank of M_A(β) and show that the Zariski closure of each of its strata is a union of translates of linear subspaces of the parameter space. These results hold for generalized A-hypergeometric systems as well, where the semigroup ring of A is replaced by a nontrivial weakly toric module M contained in \CC[\ZZ A]. We also provide a direct proof of the result of M. Saito and W. Traves regarding the isomorphism classes of M_A(β).
Multivariate hypergeometric functions associated with toric varieties were introduced by Gel'fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeometric function is a product of resultants, that is, a product of special discriminants arising from Cayley configurations. This conjecture is proved for toric hypersurfaces and for toric varieties of dim…
The holonomic rank of the A-hypergeometric system H_A(β) is shown to depend on the parameter vector βwhen the underlying toric ideal I_A is a non Cohen Macaulay codimension 2 toric ideal. The set of exceptional parameters is usually infinite.
We first retell in the K-theoretic context the heuristics of $S^1$-equivariant Floer theory on loop spaces which gives rise to $D_q$-module structures, and in the case of toric manifolds, vector bundles, or super-bundles to their explicit $q$-hypergeometric solutions. Then, using the fixed point localization technique developed in Parts II--IV, we prove that these $q$-hypergeometric solutions represent K-theoretic Gromov-Witten invariants.
We study $A$-hypergeometric functions introduced by Gelfand-Kapranov-Zelevinsky and prove a formula for the eigenvalues of their monodromy automorphisms defined by the analytic continuaions along large loops contained in complex lines parallel to the coordinate axes. A method of toric compactifications will be used to prove our main theorem.
The Euler-Koszul complex is the fundamental tool in the homological study of A-hypergeometric differential systems and functions. We compare Euler-Koszul homology with D-module direct images from the torus to the base space through orbits in the corresponding toric variety. Our approach generalizes a result by Gel'fand et al. and yields a simpler, more algebraic proof. In the process we extend the Euler-Koszul functor a category of infinite toric modules and describe multigraded localizations of Euler-Koszul homology.
We study aspects related to Kontsevich's homological mirror symmetry conjecture in the case of Calabi-Yau complete intersections in toric varieties. In a 1996 lecture at Rutgers University, Kontsevich indicated how his proposal implies that the groups of automorphisms of the two types of categories involved in the homological mirror symmetry conjecture should also be identified. Our main results provide an explicit geometric construction of the correspondence between the automorphisms of the two types of categories.
We generalize the combinatorial description of the orbifold (Chen--Ruan) cohomology and of the Grothendieck ring of a Deligne--Mumford toric stack and its associated stacky fan in a lattice $N$ in the presence of a deformation parameter $β\in N \otimes {\mathbb C}.$ As an application, we construct a topological mirror symmetry map that produces a complete system of $Γ$--series solutions to the better behaved version of the GKZ hypergeometric system for $β\in N \otimes {\mathbb C}.$
We produce a decomposition of the parameter space of the $A$-hypergeometric system associated to a projective monomial curve as a union of an arrangement of lines and its complement, in such a way that the analytic behavior of the solutions of the system is explicitly controlled within each term of the union.
We introduce a notion of balanced configurations of vectors. This is motivated by the study of rational A-hypergeometric functions in the sense of Gelfand, Kapranov and Zelevinsky. We classify balanced configurations of seven plane vectors up to GL(2,R) equivalence and deduce that the only gkz-rational toric four-folds in complex projective space P^6 are those varieties associated with an essential Cayley configuration. In this case, we study a suitable hyperplane arrangement and show that all rational A-hypergeometric functions may be described in terms of toric residues.
The holonomic rank of an A-hypergeometric system $H_A(β)$ is conjectured to be independent of the parameter vector $β$ if and only if the toric ideal $I_A$ is Cohen Macaulay. We prove this conjecture in the case that $I_A$ is generic by explicitly constructing more than $\vol(A)$ many linearly independent hypergeometric functions for parameters $β$ coming from embedded primes of certain initial ideals of $I_A$.
We formulate general conjectures about the relationship between the A-model connection on the cohomology of a $d$-dimensional Calabi-Yau complete intersection $V$ of $r$ hypersurfaces $V_1, \ldots, V_r$ in a toric variety ${\bf P}_Σ$ and the system of differential operators annihilating the special hypergeometric function $Φ_0$ depending on the fan $Σ$. In this context, the Mirror Symmetry phenomenon can be interpreted as the following twofold characterization of the series $Φ_0$. First, $Φ_0$ is defined by intersection numbers of rational curves in ${\bf P}_Σ$ with the hypersurfaces $V_i$ and…
We investigate the space of solutions to certain $A$-hypergeometric $\mathscr{D}$-modules, which were defined and studied by Gelfand, Kapranov, and Zelevinsky. We show that the solution space can be identified with certain relative cohomology group of the toric variety determined by $A$, which generalizes the results of Huang, Lian, Yau, and Zhu. As a corollary, we also prove the existence of rank one points for Calabi--Yau complete intersections in toric varieties.
We present here the K-theoretic version of mirror models of toric manifold. First, we recall the construction of cohomological mirrors for toric manifolds, i.e. representations of the toric hypergeometric functions from quantum cohomology theory by complex oscillating integrals. Then we repeat the construction in the K-theoretic situation, and obtain complex oscillating integrals representing q-hypergeometric functions from quantum K-theory of toric manifolds, bundles, and super-bundles. Finally, we examine the Lagrangian varieties parameterized by critical points of the phase functions of the…
We present a detailed study of the generalized hypergeometric system introduced by Gel'fand, Kapranov and Zelevinski (GKZ-hypergeometric system) in the context of toric geometry. GKZ systems arise naturally in the moduli theory of Calabi-Yau toric varieties, and play an important role in applications of the mirror symmetry. We find that the Gröbner basis for the so-called toric ideal determines a finite set of differential operators for the local solutions of the GKZ system. At the special point called the large radius limit, we find a close relationship between the principal parts of the oper…
We investigate the solution space of hypergeometric systems of differential equations in the sense of Gelfand, Graev, Kapranov and Zelevinsky. For any integer $d \geq 2$ we construct a matrix $A_d \in \N^{d \times 2d}$ and a parameter vector $β_d$ such that the holonomic rank of the $A$-hypergeometric system $H_{A_d}(β_d)$ exceeds the simplicial volume $\vol(A_d)$ by at least $d-1$. The largest previously known gap between rank and volume was two. Our argument is elementary in that it uses only linear algebra, and our construction gives evidence to the general observation that rank-jumps seem …
We prove that the Cox ring of the projectivization P(E) of a rank two toric vector bundle E, over a toric variety X, is a finitely generated k-algebra. As a consequence, P(E) is a Mori dream space if the toric variety X is projective and simplicial.
Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities leads to a pair of conjectures on certain hypergeometric systems of PDEs. We explain these conjectures and verify them in some cases.
The toric Hilbert scheme of a lattice L in Z^n is the multigraded Hilbert scheme parameterizing all ideals in k[x_1,...,x_n] with Hilbert function value one for every degree in the grading monoid N^n/L. In this paper we show that if L is two-dimensional, then the toric Hilbert scheme of L is smooth and irreducible. This result is false for lattices of dimension three and higher as the toric Hilbert scheme of a rank three lattice can be reducible.
The dimension of the space of holomorphic solutions at nonsingular points (also called the holonomic rank) of a $A$--hypergeometric system $M_A (β)$ is known to be bounded above by $ 2^{2d}\operatorname{vol}(A)$, where $d$ is the rank of the matrix $A$ and $\vol (A)$ is its normalized volume. This bound was thought to be very vast because it is exponential on $d$. Indeed, all the examples we have found in the literature verify that $\operatorname{rank}(M_A (β))<2 \vol (A)$. We construct here, in a very elementary way, some families of matrices $A_{(d)}\in \ZZ^{d \times n}$ and parameter vector…
The global Okounkov body of a projective variety is a closed convex cone that encodes asymptotic information about every big line bundle on the variety. In the case of a rank two toric vector bundle E on a smooth projective toric variety, we use its Klyachko filtrations to give an explicit description of the global Okounkov body of P(E). In particular, we show that this is a rational polyhedral cone and that P(E) is a Mori dream space.
In this thesis we study toric rank functions for chip firing games and prove special cases of a conjectural Riemann-Roch. The original motivation for an investigation into this area of study came for the adaptation (due to Matt Baker) of Riemann-Roch into a graph theoretic analogue through the use of chip-firing games. Here, we collect known results and present new observations that indicate Riemann--Roch holds for trees and polygons. We also prove an asymptotic case of Riemann--Roch (i.e.~Riemann--Roch for divisors of large degree). Finally, we also provide magma code and computational eviden…
We show that any normal toric variety over a rank one valuation ring admits an equivariant open embedding in a normal toric variety which is proper over the valuation ring, after a base-change by a finite extension of valuation rings. If the value group $Γ$ is discrete or divisible then no base-change is needed. We give explicit examples which show that existing methods do not produce such normal equivariant completions. Our approach is combinatorial and proceeds by showing that $Γ$-admissible fans admit $Γ$-admissible completions. In order to show this we prove a combinatorial analog of noeth…
We give a dimension formula for the space of logarithm-free series solutions to an A-hypergeometric (or a GKZ hypergeometric) system. In the case where the convex hull spanned by A is a simplex, we give a rank formula for the system, characterize the exceptional set, and prove the equivalence of the Cohen-Macaulayness of the toric variety defined by A with the emptiness of the exceptional set. Furthermore we classify A-hypergeometric systems as analytic D-modules.
We study a hypergeometric function in two variables and a system of hypergeometric differential equations associated with this function. This is a regular holonomic system of rank $9$. We give a fundamental system of solutions to this system in terms of this hypergeometric series. We give circuit matrices along generators of the fundamental group of the complement of its singular locus with respect to our fundamental system.
In $1980$ White conjectured that every element of the toric ideal of a matroid is generated by quadratic binomials corresponding to symmetric exchanges. We prove White's conjecture for high degrees with respect to the rank. This extends our result arXiv:1302.5236 confirming White's conjecture `up to saturation'. Furthermore, we study degrees of Gröbner bases and Betti tables of the toric ideals of matroids of a fixed rank.
We provide a sharp upper bound on the quotient of the rank of an A-hypergeometric system with a three-dimensional torus action by the normalized volume of A; in this case, the upper bound is two.
We prove, using invariant Zariski-Riemann spaces, that every normal toric variety over a valuation ring of rank one can be embedded as an open dense subset into a proper toric variety equivariantly. This extends a well known theorem of Sumihiro for toric varieties over a field to this more general setting.
We give a combinatorial criterion for the tangent bundle on a smooth toric variety to be stable with respect to a given polarisation in terms of the corresponding lattice polytope. Furthermore, we show that for a smooth toric surface and a smooth toric variety of Picard rank 2, there exists an ample line bundle with respect to which the tangent bundle is stable if and only if it is an iterated blow-up of projective space.
The semi-simplicity of the Hodge group is proved for a simple Abelian variety with a stable reduction of odd toric (reductive) rank. If, besides, the dimension of the Abelian variety is an odd integer, then the Hodge conjecture on algebraic cycles holds for it.
We give a complete classification of equivariant vector bundles of rank two over smooth complete toric surfaces and construct moduli spaces of such bundles. This note is a direct continuation of an earlier note where we developed a general description of equivariant sheaves on toric varieties. Here we give a first application of that description.
We study $A$-hypergeometric systems $H_A(β)$ in the sense of Gelfand, Kapranov and Zelevinsky under two aspects: the structure of their holonomically dual system, and reducibility of their rank module. We prove first that rank-jumping parameters always correspond to reducible systems, and we show that the property of being reducible is ``invariant modulo the lattice''. In the second part we study a conjecture of Nobuki Takayama which states that the holonomic dual of $H_A(β)$ is of the form $H_A(β')$ for suitable $β'$. We prove the conjecture for all matrices $A$ and generic parameter $β$, exh…
In the mid 70's, Hartshorne conjectured that, for all n > 7, any rank 2 vector bundles on P^n is a direct sum of line bundles. This conjecture remains still open. In this paper, we construct indecomposable rank two vector bundles on a large class of Fano toric varieties. Unfortunately, this class does not contain P^n
The rank of an $A$-hypergeometric $D$-module $M_A(β)$, associated with a full rank $(d\times n)$-matrix $A$ and a vector of parameters $β\in \mathbb{C}^d$, is known to be the normalized volume of $A$, denoted $\mathrm{vol}(A)$, when $β$ lies outside the exceptional arrangement $\mathcal{E}(A)$, an affine subspace arrangement of codimension at least two. If $β\in \mathcal{E}(A)$ is simple, we prove that $d-1$ is a tight upper bound for the ratio $\mathrm{rank}(M_A(β))/\mathrm{vol}(A)$ for any $d\geq 3$. We also prove that the set of parameters $β$ such that this ratio is at least $2$ is an affi…
We undertake the study of bivariate Horn systems for generic parameters. We prove that these hypergeometric systems are holonomic, and we provide an explicit formula for their holonomic rank as well as bases of their spaces of complex holomorphic solutions. We also obtain analogous results for the generalized hypergeometric systems arising from lattices of any rank.
In this paper, we give the complete classification of full exceptional collections on smooth toric Fano threefolds and fourfolds with Picard rank two. To be precise, we give a partial answer to the conjecture in \cite{Kuz} and \cite{LYY}: we provide all the exceptional collections of maximal length and prove that they are in fact full exceptional collections.
We study a family of hypergeometric motives $H(α,β|t)$ attached to a pair of tuples $α=(1/4,1/2,3/4)$, $β=(0,0,0)$. To each such motive we can attach a system of $\ell$--adic realisations with the trace of geometric Frobenius given by the evaluation of the finite field analogue of complex hypergeometric function. Geometry of elliptic fibrations makes it possible to to realise the motive $H(α,β|t)$ as a pure Chow motive attached to a suitable K3 surface $V_{t}$.
Normal toric varieties over a field or a discrete valuation ring are classified by rational polyhedral fans. We generalize this classification to normal toric varieties over an arbitrary valuation ring of rank one. The proof is based on a generalization of Sumihiro's theorem to this non-noetherian setting. These toric varieties play an important role for tropicalizations.
We study the Picard rank of smooth toric Fano varieties possessing families of minimal rational curves of given degree. We discuss variants of a conjecture of Chen-Fu-Hwang and prove a version of their statement that recovers the original conjecture in sufficiently high dimension. We also prove new cases of the original conjecture for high degrees in all dimensions. Our main tools come from toric Mori theory and the combinatorics of Fano polytopes.
Let $I_{G} \subset K[x_{1},...,x_{m}]$ be the toric ideal associated to a finite graph $G$. In this paper we study the binomial arithmetical rank and the $G$-homogeneous arithmetical rank of $I_G$ in 2 cases: $G$ is bipartite, $I_G$ is generated by quadratic binomials. In both cases we prove that the binomial arithmetical rank and the $G$-arithmetical rank coincide with the minimal number of generators of $I_G$.
Let $I_M$ and $I_N$ be defining ideals of toric varieties such that $I_M$ is a projection of $I_N$, i.e. $I_N \subseteq I_M$. We give necessary and sufficient conditions for the equality $I_M=rad(I_N+(f_1,...,f_s))$, where $f_1,...,f_s$ belong to $I_M$. Also a method for finding toric varieties which are set-theoretic complete intersection is given. Finally we apply our method in the computation of the arithmetical rank of certain toric varieties and provide the defining equations of the above toric varieties.
Hartshorne's conjecture about vector bundles on projective space states that any rank 2 vector bundle on n-dimensional projective space splits as soon as n is at least 7. Klyachko has shown that Hartshorne's conjecture is true when the vector bundles are torus equivariant. Moreover, recent work of Ilten and Süss generalizes Klyachko's work to the case of a smaller rank torus action on projective space. In this note we give a new, direct proof that torus rank 2 bundles split that avoids a description of the category of torus equivariant vector bundles.
The Gamma-series of Gel'fand-Kapranov-Zelevinsky are adapted so that they give solutions for certain resonant systems of GKZ hypergeometric differential equations. For this some complex parameters in the Gamma-series are replaced by nilpotent elements from a ring $R_{A,T}$. The adapted Gamma-series is a function $Ψ$ with values in the finite dimensional vector space $R_{A,T}\otimes C$. Applications of these results in the context of toric Mirror Symmetry are described. Building on work of Batyrev we show that the relative cohomology module of a certain hypersurface in a torus is a GKZ hypergeo…
We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems H_A(β) arising from a d x n integer matrix A and a parameter β\in \CC^d. To do so we introduce an Euler-Koszul functor for hypergeometric families over \CC^d, whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter βis rank-jump…
We show that the moduli space of rank three toric vector bundles on smooth projective toric varieties satisfies Murphy's law, that is, every singularity type of finite type arises somewhere on this space.
