1. Scalar curvature and deformations of complex structures
- Author
-
Carlo Scarpa
- Subjects
Mathematics - Differential Geometry ,Mathematics - Algebraic Geometry ,Differential Geometry (math.DG) ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,32Q15 (Primary) 32G05, 32Q26 (Secondary) ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,Algebraic Geometry (math.AG) - 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., Comment: v3: 34 pages. Corrected the statement of Lemma 3.3, improved the exposition in Section 2. Comments are welcome!
- Published
- 2023