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Some contributions of Edoardo Ballico to Moduli spaces and their applications

Elizabeth Gasparim

TL;DR

Ballico's contributions span moduli spaces of vector bundles and coherent sheaves, with notable intersections with Homological Mirror Symmetry and noncommutative Calabi–Yau geometry. The paper surveys results on the number of components and the existence of singularities in moduli spaces, and presents noncommutative deformations and Poisson structures that affect moduli, including the KKP conjecture in select cases. It also includes a section on real-moduli flexibility by Kucharz and a reflective methodology piece on the Ballico Game, illustrating the broad interdisciplinary impact. Overall, the work situates Ballico's influence at the crossroads of algebraic geometry, symplectic geometry, and mathematical physics.

Abstract

This is a contribution to the Special Volume in Celebration of the 70th Birthday of Edoardo Ballico. First, I describe how some results of Ballico on moduli of vector bundles and categories coherent sheaves were useful for solving problems in a variety of areas: Homological Mirror Symmetry, symplectic geometry, Hodge theory, mathematical physics, noncommutative geometry. Second, I summarise some strong results of Ballico about the number of components of moduli scheme of sheaves and about the existence of singularities on moduli of vector bundles. Third, the text includes a section written by Wojciech Kucharz, about the work of Ballico on moduli flexibility of real manifolds.

Some contributions of Edoardo Ballico to Moduli spaces and their applications

TL;DR

Ballico's contributions span moduli spaces of vector bundles and coherent sheaves, with notable intersections with Homological Mirror Symmetry and noncommutative Calabi–Yau geometry. The paper surveys results on the number of components and the existence of singularities in moduli spaces, and presents noncommutative deformations and Poisson structures that affect moduli, including the KKP conjecture in select cases. It also includes a section on real-moduli flexibility by Kucharz and a reflective methodology piece on the Ballico Game, illustrating the broad interdisciplinary impact. Overall, the work situates Ballico's influence at the crossroads of algebraic geometry, symplectic geometry, and mathematical physics.

Abstract

This is a contribution to the Special Volume in Celebration of the 70th Birthday of Edoardo Ballico. First, I describe how some results of Ballico on moduli of vector bundles and categories coherent sheaves were useful for solving problems in a variety of areas: Homological Mirror Symmetry, symplectic geometry, Hodge theory, mathematical physics, noncommutative geometry. Second, I summarise some strong results of Ballico about the number of components of moduli scheme of sheaves and about the existence of singularities on moduli of vector bundles. Third, the text includes a section written by Wojciech Kucharz, about the work of Ballico on moduli flexibility of real manifolds.

Paper Structure

This paper contains 12 sections, 12 theorems, 14 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A Landau--Ginzburg model without projective mirrors
  3. Vector bundles, Lagrangian skeleta and duality
  4. Hodge theoretic invariants of Landau--Ginzburg models
  5. Existence of instantons and elementary transformations
  6. Moduli of bundles on noncommutative Calabi--Yau varieties
  7. The number of components of moduli schemes of sheaves
  8. Existence of singularities on moduli of vector bundles
  9. Moduli flexibility of real manifolds (written by Kucharz)
  10. Methodology and how to play the Ballico Game
  11. Acknowledgements
  12. Conflict of Interest

Key Result

Lemma 1

$Fuk(\mathrm{LG}(2))$ is generated by two Lagrangians $L_0$ and $L_1$ with morphisms: where we think of $\mathbb Z$ as a complex concentrated in degree $0$ and $\mathbb Z [-1]$ as its shift, concentrated in degree $1$, and the products $m_k$ all vanish except for $m_2(\,\cdot\,, \hbox{id})$ and $m_2({\hbox{id},}\,\cdot\,)$.

Figures (2)

  • Figure : Duality between Lagrangians and vector bundles.
  • Figure :

Theorems & Definitions (14)

  • Lemma
  • Remark
  • Theorem
  • Example 1
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • ...and 4 more