[Paper Review] Twisted cscK metrics and Kähler slope stability

Jacopo Stoppa|ArXiv.org|Apr 2, 2008

Geometry and complex manifolds28 references3 citations

TL;DR

This paper introduces a cohomological obstruction to solving the twisted constant scalar curvature Kähler (cscK) equation on Kähler manifolds, extending Ross-Thomas slope stability to general Kähler classes and effective divisors. It proves that when the twist vanishes, the obstruction reduces to slope stability, yielding new examples of non-projective, slope-unstable manifolds and obstructing cscK metrics on general type threefolds in adiabatic classes.

ABSTRACT

We introduce a cohomological obstruction to solving the constant scalar curvature Kähler (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 Kähler manifold. Thus we find examples of non-projective slope unstable manifolds.

Motivation & Objective

• To extend the Ross-Thomas slope stability obstruction from polarized projective varieties to general Kähler manifolds with effective divisors.
• To establish a cohomological obstruction for the existence of twisted cscK metrics, generalizing the cscK case.
• To apply the obstruction to adiabatic limits in holomorphic submersions, producing new examples of general type threefolds without cscK metrics.
• To construct non-projective Kähler manifolds that are slope unstable and admit no cscK metrics in certain Kähler classes.

Proposed method

• Introduces the twisted cscK equation: $ S(\omega) - \Lambda_\omega \alpha = \widehat{S}_\alpha $, where $ \alpha $ is a semipositive (1,1)-form.
• Defines the Seshadri constant $ \epsilon(D, \Omega) $ for an effective divisor $ D $, measuring the positivity of the Kähler class $ \Omega $ relative to $ D $.
• Uses the lower bound on the K-energy from Chen-Tian to derive a necessary condition for the existence of solutions to the twisted cscK equation.
• Applies the obstruction to adiabatic classes $ \Omega_r = c_1(L) + r\pi^*\Omega_B $ in holomorphic submersions, linking solutions to twisted cscK metrics on the base.
• Constructs non-projective examples via projective bundles over Voisin's non-projective Kähler manifolds, using Mumford destabilization of vector bundles.
• Applies deformation theory (Demailly-Eckl-Peternell) to show that the resulting manifolds have no projective deformations.

Experimental results

Research questions

• RQ1 Can the slope stability obstruction for cscK metrics be generalized beyond polarized projective varieties to arbitrary Kähler manifolds with effective divisors?
• RQ2 What cohomological condition must be satisfied for the twisted cscK equation to admit a solution in a given Kähler class?
• RQ3 Do adiabatic limits in holomorphic submersions lead to obstructions for cscK metrics on the total space, and if so, what are the numerical conditions?
• RQ4 Can non-projective Kähler manifolds be constructed that are slope unstable and admit no cscK metrics in natural Kähler classes?
• RQ5 Are there general type threefolds that do not admit cscK metrics in adiabatic classes, and what numerical invariants obstruct this?

Key findings

• The paper establishes a cohomological obstruction to solving the twisted cscK equation, which reduces to slope stability when the twist vanishes.
• For general type threefolds in adiabatic classes $ \Omega_r = a c_1(K_{Z|S''}) + r\pi^*\Omega_s $, no cscK metrics exist for $ a > 0 $, $ s \ll 1 $, $ r \gg 0 $.
• The construction yields non-projective Kähler manifolds that are slope unstable and admit no cscK metrics in classes $ \mathcal{O}_{\mathbb{P}}(1) + r\pi^*\Omega_M $ for $ r \gg 0 $.
• The resulting manifold $ X = \mathbb{P}(\mathcal{O}_M(-E_q) \oplus \mathcal{O}_M) $ has no projective deformations, as shown by deformation theory of projective bundles.
• The obstruction arises from the Seshadri constant $ \epsilon(\mathbb{P}(F), \Omega_r) = 1 $ for large $ r $, tied to Mumford destabilization of the bundle.
• The method confirms a conjecture of Ross-Thomas by extending slope stability to general Kähler classes on Kähler manifolds.