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K-stability and large complex structure limits

Jacopo Stoppa

Abstract

We discuss how, under suitable assumptions, a Kähler test configuration admits a mirror Landau-Ginzburg model, giving a corresponding expression for the Donaldson-Futaki invariant as a residue pairing. We study the general behaviour of such mirror formulae under large scaling of the Kähler form. We exploit the observation that this scaling trivially preserves $K$-stability, but takes the mirror Landau-Ginzburg model to a large complex structure limit. In certain cases the mirror formulae for the Donaldson-Futaki invariant simplify in this limit. We focus on a special type of limiting behaviour, when the Donaldson-Futaki invariant concentrates at a single critical point of the Landau-Ginzburg potential, and show that this leads to new formulae for the Donaldson-Futaki invariant in terms of theta functions on the mirror. We provide a main application, which shows that such limiting behaviour actually occurs for test configurations in several nontrivial examples, both toric and non-toric, in the case of slope (in)stability for polarised surfaces.

K-stability and large complex structure limits

Abstract

We discuss how, under suitable assumptions, a Kähler test configuration admits a mirror Landau-Ginzburg model, giving a corresponding expression for the Donaldson-Futaki invariant as a residue pairing. We study the general behaviour of such mirror formulae under large scaling of the Kähler form. We exploit the observation that this scaling trivially preserves -stability, but takes the mirror Landau-Ginzburg model to a large complex structure limit. In certain cases the mirror formulae for the Donaldson-Futaki invariant simplify in this limit. We focus on a special type of limiting behaviour, when the Donaldson-Futaki invariant concentrates at a single critical point of the Landau-Ginzburg potential, and show that this leads to new formulae for the Donaldson-Futaki invariant in terms of theta functions on the mirror. We provide a main application, which shows that such limiting behaviour actually occurs for test configurations in several nontrivial examples, both toric and non-toric, in the case of slope (in)stability for polarised surfaces.
Paper Structure (24 sections, 11 theorems, 248 equations)

This paper contains 24 sections, 11 theorems, 248 equations.

Table of Contents

  1. Introduction
  2. General argument
  3. Mirror Donaldson-Futaki invariant
  4. Limiting behaviour
  5. Special test configurations
  6. Perturbations
  7. Main examples
  8. Product configurations
  9. Toric del Pezzo case
  10. Asymptotics of critical points
  11. Asymptotics of residues
  12. Donaldson-Futaki invariants
  13. Toric weak del Pezzos
  14. Non-toric case
  15. Degree $5$ del Pezzo
  16. ...and 9 more sections

Key Result

Proposition 2.3

For $k >0$ we have an expansion In particular, if $(\mathcal{X}, [\omega_{\mathcal{X}}])$ is strictly destabilising, we must have for all sufficiently large $k$.

Theorems & Definitions (27)

  • Example 1.1
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • Remark 2.8
  • Example 2.9: Degeneration to the normal cone of a point
  • ...and 17 more