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Moduli spaces and the algebra of conformal blocks

Yanglong Zhang, Mingshuo Zhou

Abstract

For a classical simple and simply connected group $G$, let $\mathcal{M}_{G,ω}$ be the moduli space of $ω$-semistable parabolic $G$-bundles on a complex smooth projective curve of genus $g$. We prove two results in this article: (1) $\mathcal{M}_{G,ω}$ is of Fano type when $g\geq 3$; (2) the algebra of conformal blocks on any $n$-pointed stable curve for a classical simple Lie algebra is finitely generated.

Moduli spaces and the algebra of conformal blocks

Abstract

For a classical simple and simply connected group , let be the moduli space of -semistable parabolic -bundles on a complex smooth projective curve of genus . We prove two results in this article: (1) is of Fano type when ; (2) the algebra of conformal blocks on any -pointed stable curve for a classical simple Lie algebra is finitely generated.
Paper Structure (8 sections, 26 theorems, 125 equations)

This paper contains 8 sections, 26 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Moduli spaces of parabolic $G$-bundles
  4. Codimension estimates
  5. Algebra of conformal blocks
  6. Anti-canonical bundles of moduli spaces
  7. Fano type of moduli spaces
  8. Finite generation of the algebra of conformal blocks

Key Result

Theorem 1.1

When $G$ is a classical simple and $\text{simply}$ connected group, let $\mathcal{M}_{G,\omega}$ be the moduli space of $\omega$-semistable parabolic $G$-bundles on a complex smooth projective curve of genus $g$. Then $\mathcal{M}_{G,\omega}$ is of Fano type when $g\geq 3$.

Theorems & Definitions (50)

  • Theorem 1.1: =Theorem \ref{['thm Fano type of moduli space']}
  • Theorem 1.2: =Theorem \ref{['f.g. for singular']}
  • Corollary 1.3: =Corollary \ref{['corollary compactify']} $\&$ Remark \ref{['rmk 5.7']}
  • Definition 2.1
  • Proposition 2.2: see Proposition 5.1.1 of Heinloth
  • Lemma 2.3
  • Definition 2.4
  • Example 2.5
  • Theorem 2.6: see Kumar
  • Theorem 2.7: see Theorem 6.1.15 and Theorem 6.1.17 of Kumar
  • ...and 40 more