Intersection homology
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Singular Hodge theory for combinatorial geometries Open
We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy con…
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Combinatorics and topology of proper toric maps Open
We study the topology of toric maps. We show that if f : X → Y {f\colon X\to Y} is a proper toric morphism, with X simplicial, then the cohomology of every fiber of f is pure and of Hodge–Tate type. When the map is a fibration, we gi…
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Donaldson–Thomas invariants versus intersection cohomology of quiver moduli Open
The main result of this paper is the statement that the Hodge theoretic Donaldson–Thomas invariant for a quiver with zero potential and a generic stability condition agrees with the compactly supported intersection cohomology of the closur…
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Blown-up intersection cohomology Open
In previous works, we have introduced the blown-up intersection cohomology\nand used it to extend Sullivan's minimal models theory to the framework of\npseudomanifolds, and to give a positive answer to a conjecture of M. Goresky\nand W. Pa…
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Polyadic approximations, fibrations and intersection types Open
Starting from an exact correspondence between linear approximations and non-idempotent intersection types, we develop a general framework for building systems of intersection types characterizing normalization properties. We show how this …
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On Calabi–Yau generalized complete intersections from Hirzebruch varieties and novel $K3$-fibrations Open
We consider the construction of Calabi-Yau varieties recently generalized to\nwhere the defining equations may have negative degrees over some projective\nspace factors in the embedding space. Within such "generalized complete\nintersectio…
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Hodge theory for intersection space cohomology Open
Given a perversity function in the sense of intersection homology theory, the method of intersection spaces assigns to certain oriented stratified spaces cell complexes whose ordinary reduced homology with real coefficients satisfies Poinc…
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Bounds on spectral norms and barcodes Open
We investigate the relations between algebraic structures, spectral invariants, and persistence modules, in the context of monotone Lagrangian Floer homology with Hamiltonian term. Firstly, we use the newly introduced method of filtered co…
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Computing cohomology intersection numbers of GKZ hypergeometric systems Open
In this review article, we report on some recent advances on the computational aspects of cohomology intersection numbers of GKZ systems. We also discuss the relation between intersection theory and evaluation of an integral of a product o…
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Intersection Homology. General perversities and topological invariance Open
Topological invariance of the intersection homology of a pseudomanifold without codimension one strata, proven by Goresky and MacPherson, is one of the main features of this homology. This property is true for codimension-dependent pervers…
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AN ALGORITHM OF COMPUTING COHOMOLOGY INTERSECTION NUMBER OF HYPERGEOMETRIC INTEGRALS Open
We show that the cohomology intersection number of a twisted Gauss–Manin connection with regularization condition is a rational function. As an application, we obtain a new quadratic relation associated to period integrals of a certain fam…
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On some incomplete theta integrals Open
In this paper we construct indefinite theta series for lattices of arbitrary signature $(p,q)$ as ‘incomplete’ theta integrals, that is, by integrating the theta forms constructed by the second author with J. Millson over certain singular …
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INTERSECTION COHOMOLOGY OF RANK 2 CHARACTER VARIETIES OF SURFACE GROUPS Open
For $G = \mathrm {GL}_2, \mathrm {SL}_2, \mathrm {PGL}_2$ we compute the intersection E-polynomials and the intersection Poincaré polynomials of the G -character variety of a compact Riemann surface C and of the moduli space of G -Higgs bu…
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A resolution of singularities for Drinfeld’s compactification by stable maps Open
Drinfeld’s relative compactification plays a basic role in the theory of automorphic sheaves, and its singularities encode representation-theoretic information in the form of intersection cohomology. We introduce a resolution of singularit…
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POINCARÉ DUALITY, CAP PRODUCT AND BOREL–MOORE INTERSECTION HOMOLOGY Open
Using a cap product, we construct an explicit Poincaré duality isomorphism between the blown-up intersection cohomology and the Borel–Moore intersection homology, for any commutative ring of coefficients and second-countable, oriented pseu…
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Homology and cohomology intersection numbers of GKZ systems Open
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 inter…
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Quantum singularity theory via cosection localization Open
We generalize the cosection localized Gysin map to intersection homology and Borel-Moore homology, which provides us with a purely topological construction of the Fan-Jarvis-Ruan-Witten invariants and some GLSM invariants.
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Relative twisted homology and cohomology groups associated with Lauricella's $F_D$ Open
We introduce relative twisted homology and cohomology groups associated with Euler type integrals of solutions to Lauricella's system $F_D(a,b,c)$ of hypergeometric differential equations. We define an intersection form between relative tw…
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Intersection Space Constructible Complexes Open
We present an obstruction theoretic inductive construction of intersection space pairs, which generalizes Banagl's construction of intersection spaces for arbitraty depth stratifications. We construct intersection space pairs for pseudoman…
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On intersection cohomology and Lagrangian fibrations of irreducible symplectic varieties Open
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…
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Local Euler Obstruction and Chern-Mather classes of Determinantal Varieties Open
For $m\geq n$, Let $K$ be an algebraic closed base field, and define $τ_{m,n,k}$ to be the set of $m\times n$ matrices over $K$ with kernel dimension $\geq k$. This is a projective subvariety of $\mathbb{P}^{mn-1}$, and is usually called d…
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Intersection homology duality and pairings: singular, PL and sheaf-theoretic Open
We compare the sheaf-theoretic and singular chain versions of Poincare\nduality for intersection homology, showing that they are isomorphic via\nnaturally defined maps. Similarly, we demonstrate the existence of canonical\nisomorphisms bet…
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An approach to intersection theory on singular varieties using motivic complexes Open
We introduce techniques of Suslin, Voevodsky, and others into the study of singular varieties. Our approach is modeled after Goresky–MacPherson intersection homology. We provide a formulation of perversity cycle spaces leading to perversit…
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A desingularization of Kontsevich's compactification of twisted cubics\n in $V_5$ Open
By definition, the del Pezzo $3$-fold $V_5$ is the intersection of\n$\\mathrm{Gr}(2,5)$ with three hyperplanes in $\\mathbb{P}^9$ under the Pl\\"ucker\nembedding. Rational curves in $V_5$ have been studied in various contents of\nFano geom…
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Intersection cohomology of rank two character varieties of surface groups Open
For $G = \mathrm{GL}_2, \mathrm{SL}_2, \mathrm{PGL}_2$ we compute the intersection E-polynomials and the intersection Poincaré polynomials of the $G$-character variety of a compact Riemann surface $C$ and of the moduli space of $G$-Higgs b…
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Coloured Jones and Alexander polynomials as topological intersections of cycles in configuration spaces Open
Coloured Jones and Alexander polynomials are sequences of quantum invariants recovering the Jones and Alexander polynomials at the first terms. We show that they can be seen conceptually in the same manner, using topological tools, as inte…
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A new cohomology class on the moduli space of curves Open
We define a collection $Θ_{g,n}\in H^{4g-4+2n}(\overline{\cal M}_{g,n},\mathbb{Q})$ for $2g-2+n>0$ of cohomology classes that restrict naturally to boundary divisors. We prove that the intersection numbers $\int_{\overline{\cal M}_{g,n}}Θ_…
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Localization formulas of cohomology intersection numbers Open
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 lo…
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Mixed Hodge structures on the intersection homotopy type of complex varieties with isolated singularities Open
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 interse…
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Intersection Cohomology and Perverse Eigenspaces of the Monodromy Open
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.