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Cohomological $χ$-dependence of ring structure for the moduli of one-dimensional sheaves on $\mathbb{P}^2$

Woonam Lim, Miguel Moreira, Weite Pi

Abstract

We prove that the cohomology rings of the moduli space $M_{d,χ}$ of one-dimensional sheaves on the projective plane are not isomorphic for general different choices of the Euler characteristics. This stands in contrast to the $χ$-independence of the Betti numbers of these moduli spaces. As a corollary, we deduce that $M_{d,χ}$ are topologically different unless they are related by obvious symmetries, strengthening a previous result of Woolf distinguishing them as algebraic varieties.

Cohomological $χ$-dependence of ring structure for the moduli of one-dimensional sheaves on $\mathbb{P}^2$

Abstract

We prove that the cohomology rings of the moduli space of one-dimensional sheaves on the projective plane are not isomorphic for general different choices of the Euler characteristics. This stands in contrast to the -independence of the Betti numbers of these moduli spaces. As a corollary, we deduce that are topologically different unless they are related by obvious symmetries, strengthening a previous result of Woolf distinguishing them as algebraic varieties.
Paper Structure (14 sections, 12 theorems, 78 equations)

This paper contains 14 sections, 12 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Moduli of sheaves.
  3. BPS invariants and $\chi$-independence.
  4. Structure of the proof of Theorem \ref{['main']}.
  5. Relations to other works.
  6. Acknowledgments.
  7. Tautological classes and relations in AM
  8. Tautological classes.
  9. Tautological relations
  10. Three relations in $A^{d}(M_{d,\chi})$.
  11. Truncated relations and proof of the main result
  12. Type I
  13. Type II
  14. Type II -- extended matrices

Key Result

Theorem 1.1

For $d\geq 3$, there is an isomorphism of algebraic varieties if and only if $\chi \equiv \pm \chi' \mod d$.

Theorems & Definitions (19)

  • Theorem 1.1: Woolf
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4: MS_GT
  • Proposition 2.1
  • Theorem 2.2: PSYY5
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • ...and 9 more