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Stringy Kähler moduli, mutation and monodromy

Will Donovan, Michael Wemyss

Abstract

This paper gives the first description of derived monodromy on the stringy Kähler moduli space (SKMS) for a general irreducible flopping curve C in a 3-fold X with mild singularities. We do this by constructing two new infinite helices: the first consists of sheaves supported on C, and the second comprises vector bundles in a tubular neighbourhood. We prove that these helices determine the simples and projectives in iterated tilts of the category of perverse sheaves, and that all objects in the first helix induce a twist autoequivalence for X. We show that these new derived symmetries, along with established ones, induce the full monodromy on the SKMS. The helices have many further applications. We (1) prove representability of noncommutative deformations of all successive thickenings of a length l flopping curve, via tilting theory, (2) control the representing objects, characterise when they are not commutative, and their central quotients, and (3) give new and sharp theoretical lower bounds on Gopakumar-Vafa invariants for a curve of length l. When X is smooth and resolves an affine base, we furthermore (4) prove that the second helix classifies all tilting reflexive sheaves on X, and thus that (5) all noncommutative crepant resolutions arise from tilting bundles on X.

Stringy Kähler moduli, mutation and monodromy

Abstract

This paper gives the first description of derived monodromy on the stringy Kähler moduli space (SKMS) for a general irreducible flopping curve C in a 3-fold X with mild singularities. We do this by constructing two new infinite helices: the first consists of sheaves supported on C, and the second comprises vector bundles in a tubular neighbourhood. We prove that these helices determine the simples and projectives in iterated tilts of the category of perverse sheaves, and that all objects in the first helix induce a twist autoequivalence for X. We show that these new derived symmetries, along with established ones, induce the full monodromy on the SKMS. The helices have many further applications. We (1) prove representability of noncommutative deformations of all successive thickenings of a length l flopping curve, via tilting theory, (2) control the representing objects, characterise when they are not commutative, and their central quotients, and (3) give new and sharp theoretical lower bounds on Gopakumar-Vafa invariants for a curve of length l. When X is smooth and resolves an affine base, we furthermore (4) prove that the second helix classifies all tilting reflexive sheaves on X, and thus that (5) all noncommutative crepant resolutions arise from tilting bundles on X.

Paper Structure

This paper contains 8 sections, 11 theorems, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. SKMS and Monodromy
  3. The Simples Helix
  4. Monodromy on the SKMS
  5. Tilts of Perverse Sheaves
  6. Further Applications
  7. Thickenings, Perversity and Dynkin Combinatorics
  8. Thickenings and Katz sequences

Key Result

Theorem 1.2

Let $Y\to Y_{\mathrm{con}}$ be a flopping contraction of quasi-projective $3$-folds, where $Y$ has at worst Gorenstein terminal singularities. For any contracted curve, consider the simples helix $\{\EuScript{S}_i\}_{i\in\mathbb{Z}}$. Then $\EuScript{E}_i$ is perfect on $Y$, and there is an autoequi

Figures (2)

  • Figure 1: Monodromy on $\mathcal{M}_{\EuScript{S}\EuScript{K}}$ for length one flops.
  • Figure 2: The simples and projectives helices for length five flops.

Theorems & Definitions (14)

  • Remark 1.1
  • Theorem 1.2: \ref{['simples global auto']}
  • Theorem 1.3: \ref{['prop geom monod']}
  • Theorem 1.4: \ref{['monodromy main']}
  • Proposition 1.5: \ref{['LemmaA']}, \ref{['V pos are bundles']}, \ref{['rank scrV eq rank V']}
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9: \ref{['all tilting bundles']}
  • Theorem 1.10: \ref{['global rep']}
  • ...and 4 more