Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Positivity of Segre-MacPherson classes

Paolo Aluffi, Leonardo C. Mihalcea, Jörg Schürmann, Changjian Su

TL;DR

The paper proves that when $X$ is a nonsingular complex variety with globally generated $TX$, the signed Segre-MacPherson class ${\check{s}}_*(\varphi,X)$ of a constructible function with effective characteristic cycle is effective in $A_*(Z)$. This fundamental link between characteristic cycles and Segre-SM classes unifies existing positivity results and yields new ones across settings including abelian and semi-abelian varieties, Schubert cells, hyperplane complements, and Donaldson-Thomas type invariants. The approach hinges on relating CC$(\varphi)$ to Segre classes of conormal cycles and exploiting the global generation to preserve positivity through Chern operations and pushforwards. The work also extends to intersection cohomology, Milnor classes, and semi-small morphisms, providing a broad, cohesive framework for positivity phenomena in algebraic geometry with potential applications to moduli spaces and invariants in algebraic topology and string theory.

Abstract

Let $X$ be a complex nonsingular variety with globally generated tangent bundle. We prove that the signed Segre-MacPherson (SM) class of a constructible function on $X$ with effective characteristic cycle is effective. This observation has a surprising number of applications to positivity questions in classical situations, unifying previous results in the literature and yielding several new results. We survey a selection of such results in this paper. For example, we prove general effectivity results for SM classes of subvarieties which admit proper (semi-)small resolutions and for regular or affine embeddings. Among these, we mention the effectivity of (signed) Segre-Milnor classes of complete intersections if $X$ is projective and an alternation property for SM classes of Schubert cells in flag manifolds; the latter result proves and generalizes a variant of a conjecture of Fehér and Rimányi. Among other applications we prove the positivity of Behrend's Donaldson-Thomas invariant for a closed subvariety of an abelian variety and the signed-effectivity of the intersection homology Chern class of the theta divisor of a non-hyperelliptic curve; and we extend the (known) non-negativity of the Euler characteristic of perverse sheaves on a semi-abelian variety to more general varieties dominating an abelian variety.

Positivity of Segre-MacPherson classes

TL;DR

The paper proves that when is a nonsingular complex variety with globally generated , the signed Segre-MacPherson class of a constructible function with effective characteristic cycle is effective in . This fundamental link between characteristic cycles and Segre-SM classes unifies existing positivity results and yields new ones across settings including abelian and semi-abelian varieties, Schubert cells, hyperplane complements, and Donaldson-Thomas type invariants. The approach hinges on relating CC to Segre classes of conormal cycles and exploiting the global generation to preserve positivity through Chern operations and pushforwards. The work also extends to intersection cohomology, Milnor classes, and semi-small morphisms, providing a broad, cohesive framework for positivity phenomena in algebraic geometry with potential applications to moduli spaces and invariants in algebraic topology and string theory.

Abstract

Let be a complex nonsingular variety with globally generated tangent bundle. We prove that the signed Segre-MacPherson (SM) class of a constructible function on with effective characteristic cycle is effective. This observation has a surprising number of applications to positivity questions in classical situations, unifying previous results in the literature and yielding several new results. We survey a selection of such results in this paper. For example, we prove general effectivity results for SM classes of subvarieties which admit proper (semi-)small resolutions and for regular or affine embeddings. Among these, we mention the effectivity of (signed) Segre-Milnor classes of complete intersections if is projective and an alternation property for SM classes of Schubert cells in flag manifolds; the latter result proves and generalizes a variant of a conjecture of Fehér and Rimányi. Among other applications we prove the positivity of Behrend's Donaldson-Thomas invariant for a closed subvariety of an abelian variety and the signed-effectivity of the intersection homology Chern class of the theta divisor of a non-hyperelliptic curve; and we extend the (known) non-negativity of the Euler characteristic of perverse sheaves on a semi-abelian variety to more general varieties dominating an abelian variety.

Paper Structure

This paper contains 16 sections, 16 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Characteristic classes via characteristic cycles; proof of the main theorem
  3. Characteristic cycles
  4. Proof of the main theorem
  5. Effective characteristic cycles (I)
  6. Abelian varieties
  7. Affine embeddings
  8. Schubert cells in flag manifolds and a conjecture of Fehér and Rimányi
  9. Complements of hyperplane arrangements
  10. Donaldson-Thomas type invariants
  11. Characteristic classes from intersection cohomology
  12. Effective charactersitic cycles (II)
  13. Further applications
  14. Semi-small maps
  15. Regular embeddings and Milnor classes
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $X$ be a complex nonsingular variety, and assume that the tangent bundle $TX$ is globally generated. Let $Z\subseteq X$ be a closed subvariety of $X$. Then the following hold: (a) The class $(-1)^{\dim Z}{{\check{s}}_{\text{Ma}}}(Z,X)\in A_*(Z)$ is effective. (b) Assume that the inclusion $U \ho

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:mather']}
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Corollary 3.1
  • Corollary 4.1
  • ...and 17 more