[Paper Review] The K-energy on small deformations of constant scalar curvature Kahler manifolds

Valentino Tosatti|arXiv (Cornell University)|Oct 9, 2010

Geometry and complex manifolds18 references3 citations

TL;DR

This paper provides a simplified proof that the K-energy is bounded below on small deformations of polarized constant scalar curvature Kähler (cscK) manifolds, leveraging a smooth test configuration with a cscK central fiber. By analyzing the asymptotic behavior of Kähler potentials under a one-parameter group action, it shows the K-energy infimum equals the limit of the K-energy along a specific path, establishing uniform lower bounds without relying on weak geodesics.

ABSTRACT

In this note we give a simplified proof of a recent result of X.X. Chen, which together with work of G. Szekelyhidi implies that on a sufficiently small deformation of a polarized constant scalar curvature Kahler manifold the K-energy has a lower bound.

Motivation & Objective

• To establish a uniform lower bound for the Mabuchi K-energy on small deformations of polarized cscK manifolds.
• To simplify X.X. Chen's proof of the K-energy lower bound in the context of smooth test configurations.
• To show that the infimum of the K-energy on the generic fiber equals the limit of the K-energy along a specific path converging to a cscK metric on the central fiber.
• To avoid reliance on weak geodesics in the space of Kähler potentials, using instead a key inequality (3.4) valid for all paths.

Proposed method

• Utilizes a smooth test configuration with a central fiber admitting a cscK metric and generic fibers diffeomorphic but not necessarily biholomorphic to it.
• Constructs a one-parameter family of Kähler metrics ω_t = ρ_{e^{-t}}^*Ω on the generic fiber, evolving via the C*-action on the test configuration.
• Applies a key inequality (3.3) relating the difference in K-energy to the Calabi energy and L² norm of the time derivative of the potential.
• Uses exponential decay of the derivative of the K-energy and the Calabi energy along the path to bound the K-energy variation.
• Splits the integral in the K-energy inequality into two parts: a fixed initial segment and a tail, with the latter bounded uniformly using uniform control on the L² norm of the time derivative.
• Takes the limit as t → ∞ to show the K-energy is bounded below by a uniform constant, with the infimum equal to the limit of the K-energy along the path.

Experimental results

Research questions

• RQ1 Does the K-energy remain bounded below on small deformations of a polarized cscK manifold?
• RQ2 Can the lower bound of the K-energy on the generic fiber of a smooth test configuration be computed as the limit of the K-energy along a specific path?
• RQ3 Is it possible to prove the K-energy lower bound without using weak geodesics in the space of Kähler potentials?
• RQ4 Does the infimum of the K-energy depend on the choice of cscK metric on the central fiber?
• RQ5 Can the K-energy infimum be computed uniformly across different paths converging to a cscK metric on the central fiber?

Key findings

• The K-energy on the generic fiber of a smooth test configuration with a cscK central fiber is uniformly bounded below.
• The infimum of the K-energy on the generic fiber equals the limit of the K-energy along a specific path ω_t as t → ∞.
• The derivative of the K-energy decays exponentially fast along the constructed path, ensuring convergence of the K-energy to a finite limit.
• The lower bound is uniform and independent of the initial Kähler potential, relying only on the geometry of the test configuration.
• The proof avoids the use of weak geodesics by relying on a general inequality (3.4) valid for all paths, simplifying Chen’s original argument.
• The infimum value is independent of the choice of cscK metric on the central fiber, provided the path converges to such a metric modulo diffeomorphisms.