Twisted cscK metrics and K\"ahler slope stability

TL;DR

This paper introduces a new cohomological obstruction to the existence of twisted cscK metrics, impacting the understanding of stability and existence of such metrics on complex manifolds, especially in the context of general type threefolds.

Contribution

It develops a novel cohomological obstruction for twisted cscK metrics, extending slope stability concepts and providing new examples of manifolds without cscK representatives.

Findings

Obstruction prevents certain manifolds from admitting twisted cscK metrics.

Many general type threefolds are shown to lack cscK classes due to this obstruction.

Extension of slope stability to effective divisors on K"ahler manifolds.

Abstract

We introduce a cohomological obstruction to solving the constant scalar curvature K\"ahler (cscK) equation twisted by a semipositive form, appearing in works of Fine and Song-Tian. Geometrically this gives an obstruction for a manifold to be the base of a holomorphic submersion carrying a cscK metric in certain ``adiabatic'' classes. In turn this produces many new examples of general type threefolds with classes which do not admit a cscK representative. When the twist vanishes our obstruction extends the slope stability of Ross-Thomas to effective divisors on a K\"ahler manifold. Thus we find examples of non-projective slope unstable manifolds.