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A new generic vanishing theorem on homogeneous varieties and the positivity conjecture for triple intersections of Schubert cells

Part of: (Co)homology theory Cycles and subschemes Projective and enumerative geometry Special varieties Singularities

Published online by Cambridge University Press:  10 March 2025

Jörg Schürmann ,
Connor Simpson  and
Botong Wang
Jörg Schürmann
Affiliation:
Mathematisches Institut, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany [email protected]
Connor Simpson
Affiliation:
Institute for Advanced Study, 1 Einstein Dr, Princeton, NJ 08540, USA [email protected]
Botong Wang
Affiliation:
Institute for Advanced Study, 1 Einstein Dr, Princeton, NJ 08540 [email protected] Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706-1388, USA
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Abstract

In this paper we prove a new generic vanishing theorem for $X$ a complete homogeneous variety with respect to an action of a connected algebraic group. Let $A, B_0\subset X$ be locally closed affine subvarieties, and assume that $B_0$ is smooth and pure dimensional. Let ${\mathcal {P}}$ be a perverse sheaf on $A$ and let $B=g B_0$ be a generic translate of $B_0$. Then our theorem implies $(-1)^{\operatorname {codim} B}\chi (A\cap B, {\mathcal {P}}|_{A\cap B})\geq 0$. As an application, we prove in full generality a positivity conjecture about the signed Euler characteristic of generic triple intersections of Schubert cells. Such Euler characteristics are known to be the structure constants for the multiplication of the Segre–Schwartz–MacPherson classes of these Schubert cells.

Keywords

vanishing theoremperverse sheaveshomogeneous varietyabelian varietyflag varietySchubert cellChern–Schwartz–MacPherson classSegre–Schwartz–MacPherson classEuler characteristictriple intersectionpositivity conjecture

MSC classification

Primary: 14C17: Intersection theory, characteristic classes, intersection multiplicities 14F17: Vanishing theorems 14M15: Grassmannians, Schubert varieties, flag manifolds 14M17: Homogeneous spaces and generalizations
Secondary: 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14F45: Topological properties 14N15: Classical problems, Schubert calculus 32S20: Global theory of singularities; cohomological properties 32S60: Stratifications; constructible sheaves; intersection cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 1 , January 2025 , pp. 1 - 12
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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