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On Pseudo-Effectivity and Volumes of Adjoint Classes in Kähler Families with Projective Central Fiber

Christopher D. Hacon, Yi Li, Sheng Rao

TL;DR

This work tackles how canonical positivity and adjoint-class volumes behave under deformations in Kähler settings, proving global stability of pseudo-effectivity when a central fiber is projective and extending to Kähler threefolds without that assumption. It develops and applies an analytic MMP framework with extension across nearby fibers, together with base-point-free and Stokes-type arguments, to show invariance of volumes and plurigenera in relevant cases. A core achievement is the deformation invariance of volumes of adjoint classes and the deformation invariance of plurigenera in dimension three, yielding Siu-type results beyond the projective category. The results have broad implications for the stability of uniruledness and the deformation theory of canonical invariants in Kähler geometry, leveraging generalized pairs, nef/big/pseudo-effective theory, and the transcendental base-point-free machinery.

Abstract

This paper is devoted to studying the deformation behavior of pseudo-effective canonical divisors and volumes of adjoint classes in Kähler families. Based on recent developments in the Kähler minimal model program, for flat families with fiberwise canonical singularities, we establish the global stability of the pseudo-effectivity of canonical divisors, assuming in addition that one fiber is projective, while the same conclusion for Kähler threefolds is also true without the projectivity assumption of the central fiber. For smooth Kähler families whose central fiber is projective with a big adjoint class, we show that its volume remains locally constant. Finally, using the (relative) minimal model program for Kähler threefolds, we verify the deformation invariance of volumes of adjoint classes and plurigenera for smooth families of Kähler threefolds, thereby confirming Siu's invariance of plurigenera conjecture in dimension three.

On Pseudo-Effectivity and Volumes of Adjoint Classes in Kähler Families with Projective Central Fiber

TL;DR

This work tackles how canonical positivity and adjoint-class volumes behave under deformations in Kähler settings, proving global stability of pseudo-effectivity when a central fiber is projective and extending to Kähler threefolds without that assumption. It develops and applies an analytic MMP framework with extension across nearby fibers, together with base-point-free and Stokes-type arguments, to show invariance of volumes and plurigenera in relevant cases. A core achievement is the deformation invariance of volumes of adjoint classes and the deformation invariance of plurigenera in dimension three, yielding Siu-type results beyond the projective category. The results have broad implications for the stability of uniruledness and the deformation theory of canonical invariants in Kähler geometry, leveraging generalized pairs, nef/big/pseudo-effective theory, and the transcendental base-point-free machinery.

Abstract

This paper is devoted to studying the deformation behavior of pseudo-effective canonical divisors and volumes of adjoint classes in Kähler families. Based on recent developments in the Kähler minimal model program, for flat families with fiberwise canonical singularities, we establish the global stability of the pseudo-effectivity of canonical divisors, assuming in addition that one fiber is projective, while the same conclusion for Kähler threefolds is also true without the projectivity assumption of the central fiber. For smooth Kähler families whose central fiber is projective with a big adjoint class, we show that its volume remains locally constant. Finally, using the (relative) minimal model program for Kähler threefolds, we verify the deformation invariance of volumes of adjoint classes and plurigenera for smooth families of Kähler threefolds, thereby confirming Siu's invariance of plurigenera conjecture in dimension three.
Paper Structure (21 sections, 65 theorems, 183 equations)

This paper contains 21 sections, 65 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: generalized Kähler pairs and positivities of classes
  3. Generalized pairs
  4. Nef, big and pseudo-effective classes
  5. Lelong numbers and restricted non-Kähler locus
  6. Volume of cohomology classes
  7. Bigness and pseudo-effectivity in families
  8. Uniruled varieties
  9. Pseudo-effective threshold
  10. Singularities and positivities of line bundles in families
  11. Deformation of singularities
  12. Relative ampleness between families
  13. Relative pseudo-effectivity of line bundles in projective families
  14. Global generation and semi-ampleness of line bundles in families
  15. Extension of the analytic MMP and stability of nefness and bigness
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let X be a compact Kähler manifold. Then X is uniruled if and only if the canonical line bundle $K_X$ is not pseudo-effective.

Theorems & Definitions (159)

  • Theorem 1.1: Ou25
  • Theorem 1.2: Theorem \ref{['GlobalPseff']}
  • Corollary 1.3: Corollary \ref{['DefUniruled']}
  • Conjecture 1.4
  • Conjecture 1.5: Siu02, CP23Pluri
  • Theorem 1.6: Theorem \ref{['InvarVol']}
  • Theorem 1.7: Theorem \ref{['InvarVol2']}
  • Theorem 1.8: Theorem \ref{['DefVol3fold']}
  • Definition 2.1: Generalized pair, DHY23
  • Definition 2.2: Generalized discrepancies
  • ...and 149 more