We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset.
We introduce elliptic hypertoric varieties, which is an elliptic analogue of hypertoric varieties and multiplicative hypertoric varieties. We also prove an elliptic version of Hikita conjecture, which relates elliptic (resp. additive and mulitplicative) hypertoric varieties to the equivariant elliptic cohomology (resp. ordinary cohomology and K-theory) of their 3d mirror hypertoric varieties.
We give a complete description of the equivariant quantum cohomology ring of any smooth hypertoric variety, and find a mirror formula for the quantum differential equation.
A hypertoric variety is a quaternionic analogue of a toric variety. Just as the topology of toric varieties is closely related to the combinatorics of polytopes, the topology of hypertoric varieties interacts richly with the combinatorics of hyperplane arrangements and matroids. Using finite field methods, we obtain combinatorial descriptions of the Betti numbers of hypertoric varieties, both for ordinary cohomology in the smooth case and intersection cohomology in the singular case. We also introduce a conjectural ring structure on the intersection cohomology of a hypertoric variety.
We establish a connection between the orbifold cohomology of hypertoric varieties and the Ehrhart theory of Lawrence polytopes. More specifically, we show that the dimensions of the orbifold cohomology groups of a hypertoric variety are equal to the coefficients of the Ehrhart $δ$-polynomial of the associated Lawrence polytope. As a consequence, we deduce a formula for the Ehrhart $δ$-polynomial of a Lawrence polytope and use the injective part of the Hard Lefschetz Theorem for hypertoric varieties to deduce some inequalities between the coefficients of the $δ$-polynomial.
For any conical symplectic resolution, we give a conjecture relating the intersection cohomology of the singular cone to the quantum cohomology of its resolution. We prove this conjecture for hypertoric varieties, recovering the ring structure on hypertoric intersection cohomology that was originally constructed by Braden and the second author.
We show that the equivariant Chow cohomology ring of a toric variety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan. This gives a large class of singular spaces for which localization holds in equivariant Chow cohomology with integer coefficients. We also compute the equivariant Chow cohomology of toric prevarieties and general complex hypertoric varieties in terms of piecewise polynomial functions.
We provided two explicit formulas for the intersection cohomology (as a graded vector space with pairing) of the symplectic quotient by a circle in terms of the $S^1$ equivariant cohomology of the original symplectic manifold and the fixed point data. The key idea is the construction of a small resolution of the symplectic quotient.
The purpose of this paper is to provide a way to compute the intersection cohomology of the GIT quotient of a nonsingular projective variety. We show that the middle perversity intersection cohomology of the GIT quotient $M//G$ is naturally embedded into the $G$ equivariant cohomology of $M^{ss}$ under an assumption satisfied by many interesting spaces. This embedding enables us to compute the intersection pairing in terms of the cup product structure of the equivariant cohomology ring. We also show that the image is a subspace defined by truncation. Overall, we can compute the intersection co…
We show by studying the symplectic geometry of the extended moduli space that the intersection cohomology of the representation space $Hom(π_1(Σ),G)/G$ for a simply connected compact Lie group $G$ is naturally embedded into the $G$ equivariant cohomology of $Hom(π_1(Σ),G)$ where $Σ$ is a closed Riemann surface. This enables us to compute the intersection cohomology as a graded vector space with intersection pairing, in terms of the equivariant cohomology ring. The case where $G=SU(2)$ -- the moduli space of rank 2 holomorphic vector bundles of even degree -- is discussed in detail.
Let $M$ be a proper Hamiltonian $K$-space with proper moment map $μ$. The symplectic quotient $X=μ^{-1}(0)/K$ is in general a singular stratified space. In this paper we first generalize the Kirwan map to this symplectic setting which maps the $K$ equivariant cohomology of $μ^{-1}(0)$ to the middle perversity intersection cohomology $IH^*(X)$. Next, we show that there is a natural right inverse of the Kirwan map, under an assumption coming from the cosupport axiom for the intersection homology sheaf. Furthermore, we can identify the subspace of $H^*_K(μ^{-1}(0))$ which corresponds to $IH^*(X)$…
The method of intersection spaces associates cell-complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that Poincaré duality holds between the ordinary rational cohomology groups of the cell-complexes associated to complementary perversities. The cohomology of these intersection spaces defines a cohomology theory HI for singular spaces, which is not isomorphic to intersection cohomology IH. Mirror symmetry tends to interchange IH and HI. The theory IH can be tied to type IIA string theory, while HI can be tied to IIB theory. For pseudomanifolds with …
Given a symplectic cohomology class of degree 1, we define the notion of an equivariant Lagrangian submanifold. The Floer cohomology of equivariant Lagrangian submanifolds has a natural endomorphism, which induces a grading by generalized eigenspaces. Taking Euler characteristics with respect to the induced grading yields a deformation of the intersection number. Dehn twists act naturally on equivariant Lagrangians. Cotangent bundles and Lefschetz fibrations give fully computable examples. A key step in computations is to impose the "dilation" condition stipulating that the BV operator applied…
We continue the approach toward a purely combinatorial "virtual" intersection cohomology for possibly non-rational fans, based on our investigation of equivariant intersection cohomology for toric varieties (see math.AG/9904159). Fundamental objects of study are "minimal extension sheaves" on "fan spaces". These are flabby sheaves of graded modules over a sheaf of polynomial rings, satisfying three relatively simple axioms that characterize the properties of the equivariant intersection cohomology sheaf on a toric variety, endowed with the finite topology given by open invariant subsets. These…
We say that a complex analytic space, $X$, is an intersection cohomology manifold if and only if the shifted constant sheaf on $X$ is isomorphic to intersection cohomology; this is quickly seen to be equivalent to $X$ being a homology manifold. Given an analytic function $f$ on an intersection cohomology manifold, we describe a simple relation between $V(f)$ being an intersection cohomology manifold and the vanishing cycle Milnor monodromy of $f$. We then describe how the Sebastiani-Thom isomorphism allows us to easily produce intersection cohomology manifolds with arbitrary singular sets. Fin…
A homotopical treatment of intersection cohomology recently developed by Chataur-Saralegui-Tanré associates a "perverse algebraic model" to every topological pseudomanifold, extending Sullivan's presentation of rational homotopy to intersection cohomology. In this context, there is a notion of "intersection-formality", measuring the vanishing of Massey products in intersection cohomology. In the present paper, we study the perverse algebraic model of complex projective varieties with isolated singularities. We endow such invariant with natural mixed Hodge structures. This allows us to prove so…
Let $X$ be any subanalytic compact pseudomanifold. We show a De Rham theorem for $L^\infty$ forms. We prove that the cohomology of $L^\infty$ forms is isomorphic to intersection cohomology in the maximal perversity.
We describe the relationship between intersection cohomology with twisted coefficients and the perverse sheaves which play the role of the eigenspaces for the Milnor monodromy of an affine hypersurface.
We study intersection cohomology of character varieties for punctured Riemann surfaces with prescribed monodromies around the punctures. Relying on previous result from Mellit for semisimple monodromies we compute the intersection cohomology of character varieties with monodromies of any Jordan type. This proves the Poincaré polynomial specialization of a conjecture from Letellier.
In this paper we shall give formulas for the pairings of intersection cohomology classes of complementary dimensions in the intersection cohomology of geometric invariant theoretic quotients for which semistability is not necessarily the same as stability (although we make some weaker assumptions on the action). We also give formulas for intersection pairings on resolutions of singularities (or more precisely partial resolutions, since orbifold singularities are allowed) of the quotients.
We prove that the prime torsion in the local integral intersection cohomology of Schubert varieties in the flag variety of the general linear group grows exponentially in the rank. The idea of the proof is to find a highly singular point in a Schubert variety and calculate the Euler class of the normal bundle to the (miraculously smooth) fibre in a particular Bott-Samelson resolution. The result is a geometric version of an earlier result established using Soergel bimodule techniques.
We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori, in terms of structure of the zero- and one-dimensional orbits. The class of varieties to which our formula applies includes Schubert varieties in flag varieties and affine flag varieties. We also prove a monotonicity result on local intersection cohomology stalks.
We introduce the semi-infinite category of sheaves on the affine Grassmannian, and construct a particular object in it, which we call the the semi-infinite intersection cohomology sheaf. We relate it to several other entities naturally appearing in the geometric Langlands theory.
We construct Landau-Ginzburg models for numerically effective complete intersections in toric manifolds as partial compactifications of families of Laurent polynomials. We show a mirror statement saying that the quantum D-module of the ambient part of the cohomology of the submanifold is isomorphic to an intersection cohomology D-module defined from this partial compactification and we deduce Hodge properties of these differential systems.
For even dimensional smooth complete intersections, of dimension at least 4, of two quadric hypersurfaces in a projective space, we study the genus zero Gromov-Witten invariants by the monodromy group of its whole family. We compute the invariants of length 4 and show that, besides a special invariant, all genus zero Gromov-Witten invariants can be reconstructed from the invariants of length 4. In dimension 4, we compute the special invariant by solving a curve counting problem. We show that the generating function of genus zero Gromov-Witten invariants has a positive radius of convergence. We…
The eigenvalues of Frobenius acting on $\ell$-adic cohomology of a complete intersection over a finite field have the divisibility predicted by the theorem of Ax and Katz. We have corrected some unforgivable typos.
We completely determine the intersection cohomology of the Satake compactifications of the moduli space of principally polarized abelian varieties in genera 2,3,4, except for the degree 10 intersection cohomology in genus 4. We also determine all the ingredients appearing in the decomposition theorem applied to the map from a toroidal compactification to the Satake compactification in these genera. As a byproduct we obtain in addition several results about the intersection cohomology of the link bundles involved.
This paper applies the decomposition theorem in intersection cohomology to geometric invariant theory quotients, relating the intersection cohomology of the quotient to that of the semistable points for the action. Suppose a connected reductive complex algebraic group $G$ acts linearly on a complex projective variety $X$. We prove that if $1 \to N \to G \to H \to 1$ is a short exact sequence of connected reductive groups, and $X^{ss}$ the set of semistable points for the action of $N$ on $X$, then the $H$-equivariant intersection cohomology of the geometric invariant theory quotient $X^{ss}//N…
We improve the known Hodge type bound for the exotic cohomology of complete intersections. In the revised version, we included a simplification of our original argument due to Pierre Deligne. The note appears in the C. R. de l'Aca. des Sc. de Paris.
We present several approaches to equivariant intersection cohomology. We show that for a complete algebraic variety acted by a connected algebraic group $G$ it is a free module over $H^*(BG)$. The result follows from the decomposition theorem or from a weight argument.
We investigate the equivariant intersection cohomology of a toric variety. Considering the defining fan of the variety as a finite topological space with the subfans being the open sets (that corresponds to the "toric" topology given by the invariant open subsets), equivariant intersection cohomology provides a sheaf (of graded modules over a sheaf of graded rings) on that "fan space". We prove that this sheaf is a "minimal extension sheaf", i.e., that it satisfies three relatively simple axioms which are known to characterize such a sheaf up to isomorphism. In the verification of the second o…
We present an obstruction theoretic inductive construction of intersection space pairs, which generalizes Banagl's construction of intersection spaces for arbitrary depth stratifications. We construct intersection space pairs for pseudomanifolds with compatible trivial structures at the link fibrations; this includes the case of toric varieties. We define intersection space complexes in an axiomatic way, similar to Goresky-McPherson axioms for intersection cohomology. We prove that if the intersection space exists, then the pseudomanifold has an intersection space complex whose hypercohomology…
For a normal variety X defined over an algebraically closed field with an action of the multiplicative group G_m, we consider the ``hyperbolic localization'' functor from D^b(X) to D^b(X^T), which localizes using closed supports in the directions flowing into the fixed points, and compact supports in the directions flowing out. We show that the hyperbolic localization of the intersection cohomology sheaf is a direct sum of intersection cohomology sheaves.
Let $X^{2n}\subseteq \mathbb{P} ^N$ be a smooth projective variety. Consider the intersection cohomology complex of the local system $R^{2n-1}π{_*}\mathbb{Q}$, where $π$ denotes the projection from the universal hyperplane family of $X^{2n}$ to ${(\mathbb{P} ^N)}^{\vee}$. We investigate the cohomology of the intersection cohomology complex $IC(R^{2n-1}π{_*}\mathbb{Q})$ over the points of a Severi's variety, parametrizing nodal hypersurfaces, whose nodes impose independent conditions on the very ample linear system giving the embedding in $\mathbb{P} ^N$.
We revisit the localization formulas of cohomology intersection numbers associated to a logarithmic connection. The main contribution of this paper is threefold: we prove the localization formula of the cohomology intersection number of logarithmic forms in terms of residue of a connection; we prove that the leading term of the Laurent expansion of the cohomology intersection number is Grothendieck residue when the connection is hypergeometric; and we prove that the leading term of stringy integral discussed by Arkani-Hamed, He and Lam is nothing but the self-cohomology intersection number of …
Under a certain condition A we give a construction to calculate the intersection cohomology of a rank one local system on the complement to a hyperplane-like divisor
We prove a general formula for the intersection form of two arbitrary monomials in boundary divisors. Furthermore we present a tree basis of the cohomology of $\overline {M}_{0,n}$. With the help of the intersection form we determine the Gram matrix for this basis and give a formula for its inverse. This enables us to calculate the tensor product of the higher order multiplications arising in quantum cohomology and formal Frobenius manifolds. In the context of quantum cohomology this establishes the explicit Künneth formula.
We describe the homology intersection form associated to regular holonomic GKZ systems in terms of the combinatorics of regular triangulations. Combining this result with the twisted period relation, we obtain a formula of cohomology intersection numbers in terms of a Laurent series. We show that the cohomology intersection number depends rationally on the parameters. We also prove a conjecture of F. Beukers and C. Verschoor on the signature of the monodromy invariant hermitian form. This is a continuation of the previous work arXiv:1904.00565.
We extend the methods developed in our earlier work to algorithmically compute the intersection cohomology Betti numbers of reductive varieties. These form a class of highly symmetric varieties that includes equivariant compactifications of reductive groups. Thereby, we extend a well-known algorithm for toric varieties.
We study intersection cohomology of moduli spaces of semistable vector bundles on a complex algebraic surface. Our main result relates intersection Poincare polynomials of the moduli spaces to Donaldson-Thomas invariants of the surface. In support of this result, we compute explicitly intersection Poincare polynomials for sheaves with rank two and three on ruled surfaces.
The purpose of this article is to investigate the intersection cohomology for algebraic varieties with torus action. Given an algebraic torus $\mathbb{T}$, one of our result determines the intersection cohomology Betti numbers of any normal projective $\mathbb{T}$-variety admitting an algebraic curve as global quotient. The calculation is expressed in terms of a combinatorial description involving a divisorial fan which is the analogous of the defining fan of a toric variety. Our main tool to obtain this computation is a description of the decomposition theorem in this context.
We study a Pfaffian of the system of differential equations annihilating Appell's hypergeometric series $F_4(a,b,c;x)$ by twisted cohomology groups associated with integrals representing solutions to this system. We simplify its connection matrix by the pull-back under a double cover of the complement of the singular locus. We express the simplified connection matrix in terms of the intersection form of the twisted cohomology groups.
We consider a hypersurface in $\mathbb{C}^n$ with an isolated singular point at the origin, and study the mixed Hodge structure of the stalk of its intersection cohomology complex at the origin. In particular we express the dimension of each graded piece of the weight filtration of this mixed Hodge structure in terms of the numbers of the Jordan blocks in the Milnor monodromy.
This paper studies intersection theory on the compactified moduli space M(n,d) of holomorphic bundles of rank n and degree d over a fixed compact Riemann surface of genus g > 1 where n and d may have common factors. Because of the presence of singularities we work with the intersection cohomology groups defined by Goresky and MacPherson and the ordinary cohomology groups of a certain partial resolution of singularities of M(n,d). Based on our earlier work, we give a precise formula for the intersection cohomology pairing and provide a method to calculate pairings on the partial resolution of s…
We determine the product structure on Hochschild cohomology of commutative algebras in low degrees, obtaining the answer in all degrees for complete intersection algebras. As applications, we consider cyclic extension algebras as well as Hochschild and ordinary cohomology of finite abelian groups.
We prove several results concerning the intersection cohomology and the perverse filtration associated with a Lagrangian fibration of an irreducible symplectic variety. We first show that the perverse numbers only depend on the deformation equivalence class of the ambient variety. Then we compute the border of the perverse diamond, which further yields a complete description of the intersection cohomology of the Lagrangian base and the invariant cohomology classes of the fibers. Lastly, we identify the perverse and Hodge numbers of intersection cohomology when the irreducible symplectic variet…
For each nonsingular hyperelliptic curve of arbitrary genus, we construct a natural injection from the Galois cohomology of 2-torsion subgroups of Jacobian varieties of the curve to the set of isomorphism classes of nonsingular complete intersections of two quadrics. This gives a generalization of the result of Flynn and Skorobogatov.
