Scalar curvature and deformations of complex structures

TL;DR

This paper introduces a novel coupled system linking scalar curvature and complex structure deformations on compact Kähler manifolds, with a variational framework, stability criteria, and verification in toric cases.

Contribution

It formulates a new coupled system of equations, develops a variational approach, defines a Futaki invariant and a generalized K-stability, and verifies the conjecture in toric manifolds.

Findings

System derived from hyperkähler reduction.

Variational characterization of the equations.

Verification of stability conjecture in toric case.

Abstract

We study a system of equations on a compact complex manifold, that couples the scalar curvature of a Kaehler metric with a spectral function of a first-order deformation of the complex structure. The system comes from an infinite-dimensional Kaehler reduction, which is a hyperkaehler reduction for a particular choice of the spectral function. The system can be formally complexified using a flat connection on the space of first-order deformations that are compatible with a Kaehler metric. We describe a variational characterization of the equations, a Futaki invariant for the system, and a generalization of K-stability that is conjectured to characterize the existence of solutions to the system. We verify a particular case of this conjecture in the context of toric manifolds.