On Pseudo-Effectivity and Volumes of Adjoint Classes in K\"ahler Families with Projective Central Fiber

TL;DR

This paper investigates how pseudo-effectivity and volumes of adjoint classes behave under deformations in K"ahler families, establishing stability results and confirming Siu's invariance conjecture in three dimensions.

Contribution

It proves the deformation invariance of volumes and plurigenera for K"ahler threefolds, extending stability results to non-projective cases using the K"ahler minimal model program.

Findings

Pseudo-effectivity of canonical divisors is globally stable in certain K"ahler families.

Volumes of adjoint classes remain locally constant in smooth K"ahler families with projective central fiber.

Confirmed Siu's invariance of plurigenera conjecture for three-dimensional K"ahler manifolds.

Abstract

This paper is devoted to studying the deformation behavior of pseudo-effective canonical divisors and volumes of adjoint classes in K\"ahler families. Based on recent developments in the K\"ahler minimal model program, for flat families with fiberwise canonical singularities, we establish the global stability of the pseudo-effectivity of canonical divisors and uniruledness, assuming in addition that one fiber is projective, while the same conclusion for K\"ahler threefolds is also true without the projectivity assumption of the central fiber. For a smooth K\"ahler family 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 K\"ahler threefolds, we verify the deformation invariance of volumes of adjoint classes and plurigenera for smooth families of K\"ahler threefolds, thereby confirming Siu's invariance of plurigenera conjecture in dimension three.