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A Hilbert--Mumford criterion for generalised Monge--Ampère equations

Rémi Reboulet

Abstract

We give a new numerical criterion in the spirit of GIT for existence of solutions to inverse Hessian equations, including in particular the J-equation. Our criterion is formulated in terms of stability of pairs in the sense of Paul. To that end, we build on previous work of the author with Dervan, and generalise a result of Zhang, proving isometry between generalised Chow line bundles and mixed Deligne pairings.

A Hilbert--Mumford criterion for generalised Monge--Ampère equations

Abstract

We give a new numerical criterion in the spirit of GIT for existence of solutions to inverse Hessian equations, including in particular the J-equation. Our criterion is formulated in terms of stability of pairs in the sense of Paul. To that end, we build on previous work of the author with Dervan, and generalise a result of Zhang, proving isometry between generalised Chow line bundles and mixed Deligne pairings.

Paper Structure

This paper contains 16 sections, 15 theorems, 47 equations.

Table of Contents

  1. Organisation of the paper.
  2. Acknowledgements.
  3. Preliminaries.
  4. Deligne pairings.
  5. Deligne metrics.
  6. A general Deligne pairing isometry.
  7. The mixed Chow line.
  8. Isomorphism.
  9. Isometry.
  10. Application to mixed Monge--Ampère equations.
  11. Mixed Monge--Ampère equations and Deligne pairings.
  12. Stability of pairs.
  13. Models and non-Archimedean functionals.
  14. $J$-functionals as log-norm functionals.
  15. Proof of the main result.
  16. ...and 1 more sections

Key Result

Theorem 1.2

By the equality symbol below, we mean "there exists an isomorphism of line bundles over $S$". Furthermore, all isomorphisms commute with each other. In particular, if $L_i=\mathcal{O}_X$ for some $i$, then $\langle L_0,\dots,L_n\rangle_{X/S}=\mathcal{O}_X$.

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3: Projection formula, elkik:1
  • Theorem 1.4
  • Lemma 1.5
  • Remark 1.6: Polarisation of a hypersurface via Deligne pairings
  • Remark 2.1: Action of the $SL(V_i)$
  • Remark 2.2: Interpretation of the incidence variety
  • Definition 2.3
  • Theorem 2.4
  • ...and 19 more