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Moduli of polarised manifolds via canonical Kähler metrics

Ruadhaí Dervan, Philipp Naumann

TL;DR

The paper provides an analytic solution to the moduli problem for polarised manifolds admitting constant scalar curvature Kähler metrics by combining a refined deformation theory (Kuranishi-type), analytic GIT, and Chen–Sun gluing to produce a Hausdorff complex moduli space. It shows that this moduli space carries a natural Kähler structure, with a Weil–Petersson-type metric arising from fibre integrals and a CM line bundle whose curvature recovers the WP form; both descend from local data and glue globally. The approach accommodates manifolds with continuous automorphisms and yields a framework that parallels algebraic moduli in spirit, including control of degenerations via analytically K-semistable limits and a modular interpretation of families. The results connect deformation theory, stability notions, and Kähler geometry to construct and study the geometry of the moduli space, with implications for compactifications and projectivity of moduli subspaces.

Abstract

We construct a moduli space of polarised manifolds which admit a constant scalar curvature Kähler metric. We show that this space admits a natural Kähler metric.

Moduli of polarised manifolds via canonical Kähler metrics

TL;DR

The paper provides an analytic solution to the moduli problem for polarised manifolds admitting constant scalar curvature Kähler metrics by combining a refined deformation theory (Kuranishi-type), analytic GIT, and Chen–Sun gluing to produce a Hausdorff complex moduli space. It shows that this moduli space carries a natural Kähler structure, with a Weil–Petersson-type metric arising from fibre integrals and a CM line bundle whose curvature recovers the WP form; both descend from local data and glue globally. The approach accommodates manifolds with continuous automorphisms and yields a framework that parallels algebraic moduli in spirit, including control of degenerations via analytically K-semistable limits and a modular interpretation of families. The results connect deformation theory, stability notions, and Kähler geometry to construct and study the geometry of the moduli space, with implications for compactifications and projectivity of moduli subspaces.

Abstract

We construct a moduli space of polarised manifolds which admit a constant scalar curvature Kähler metric. We show that this space admits a natural Kähler metric.

Paper Structure

This paper contains 14 sections, 26 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Analytic geometric invariant theory
  4. Constant scalar curvature Kähler metrics
  5. Deformation theory
  6. Construction of the moduli space
  7. Stabiliser preservation
  8. Orbit structure
  9. Taking the quotient
  10. Gluing the local moduli spaces
  11. Degenerations of cscK manifolds
  12. Kähler geometry of the moduli space
  13. The Weil--Petersson metric
  14. The CM line bundle

Key Result

Theorem 1.1

There exists a Hausdorff complex space which is a moduli space of polarised manifolds which admit a cscK metric.

Theorems & Definitions (51)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6: Chen--Sun
  • ...and 41 more