[Paper Review] A Numerical Criterion for K-Energy maps of Algebraic Manifolds

Sean Timothy Paul|arXiv (Cornell University)|Oct 2, 2012

Geometry and complex manifolds3 references15 citations

TL;DR

This paper establishes a numerical criterion for the uniform boundedness below of the Mabuchi K-energy on Bergman metrics of a smooth, linearly normal complex projective variety. It proves that the Mabuchi energy is uniformly bounded below on the space of Bergman metrics if and only if the weight polytope of the X-resultant is contained in that of the X-hyperdiscriminant for all maximal tori, reducing the problem to a polyhedral-combinatorial condition via geometric invariant theory and Hilbert-Mumford numerical invariants.

ABSTRACT

Let X be a projective manifold. We prove that the Mabuchi Energy of X is bounded below on all degenerations in B (the space of Bergman metrics) if and only if it is bounded below uniformly on B.

Motivation & Objective

• To establish a numerical criterion for the uniform boundedness below of the Mabuchi energy on the space of Bergman metrics associated to a smooth, linearly normal complex projective variety.
• To reduce the global lower bound problem to checking boundedness on each algebraic one-parameter subgroup in the space of Bergman metrics.
• To characterize the uniform boundedness of the Mabuchi energy in terms of the inclusion of weight polytopes of the X-resultant and X-hyperdiscriminant under all maximal tori actions.
• To connect the K-energy behavior to geometric invariant theory and numerical invariants via the Kempf-Ness functional and Hilbert-Mumford criterion.
• To provide a polyhedral-combinatorial characterization of K-stability via the relative position of weight polytopes of canonical invariants.

Proposed method

• The paper uses the Hilbert-Mumford numerical criterion for semistability, applied to the pair of invariants: the X-resultant and X-hyperdiscriminant.
• It defines the weight polytope of a vector in a representation as the convex hull of its weight support under a maximal torus action.
• The numerical semistability condition is expressed via the inequality $ w_\lambda(w) \leq w_\lambda(v) $ for all one-parameter subgroups $ \lambda $, where $ w_\lambda $ denotes the weight of $ \lambda $ on a vector.
• The main result is derived by applying Theorem 4.1, which relates the Mabuchi energy to the Kempf-Ness functional of the pair $ (R, \Delta) $, with an error bounded by a constant depending only on the Hilbert polynomial and metric.
• The proof reduces the uniform lower bound on the Mabuchi energy to the inclusion $ \mathcal{N}(R) \subset \mathcal{N}(\Delta) $ across all maximal tori, using the asymptotic behavior of the energy under degenerations.
• The construction relies on the $ G $-equivariance of the resultant and discriminant maps into irreducible representations of $ SL(N+1,\mathbb{C}) $, with explicit highest weight descriptions.

Experimental results

Research questions

• RQ1 Under what conditions is the Mabuchi energy uniformly bounded below on the space of Bergman metrics for a smooth, linearly normal projective variety?
• RQ2 Can the global lower bound on the Mabuchi energy be reduced to checking boundedness on each algebraic one-parameter subgroup?
• RQ3 What is the precise polyhedral-combinatorial condition that characterizes the uniform lower boundedness of the Mabuchi energy in terms of the weight polytopes of the X-resultant and X-hyperdiscriminant?
• RQ4 How do the numerical invariants of the resultant and discriminant relate to the K-energy functional via the Kempf-Ness functional?
• RQ5 Is the condition $ \mathcal{N}(R) \subset \mathcal{N}(\Delta) $ across all maximal tori both necessary and sufficient for the uniform lower bound of the Mabuchi energy?

Key findings

• The Mabuchi energy of $ (X, \omega_{FS}|_X) $ is uniformly bounded below on the space $ \mathcal{B} $ of Bergman metrics if and only if it is bounded below on every algebraic one-parameter subgroup in $ \mathcal{B} $, establishing a reduction principle.
• The uniform lower bound holds if and only if the weight polytope of the X-resultant is contained in the weight polytope of the X-hyperdiscriminant for all maximal tori $ H \leq SL(N+1,\mathbb{C}) $, as stated in Corollary 4.1.
• The Mabuchi energy is related to the Kempf-Ness functional of the pair $ (R, \Delta) $ via the inequality $ |(n+1)\nu_{\omega_{FS}|_X}(\sigma) - \frac{1}{c_n^2} p_{R(X)\Delta(X)}(\sigma)| \leq M $, where $ M $ depends only on the Hilbert polynomial and the metric.
• The resultant $ R(X) $ and discriminant $ \Delta(X) $ are nonzero sections in irreducible representations of $ SL(N+1,\mathbb{C}) $, with explicit highest weights given by $ (r,\dots,r,0,\dots,0) $, where $ r $ is the common degree of the two invariants.
• The degree $ r $ of the resultant and discriminant is given by $ r = d(n+1)(n(n+1)d - d\mu) $, where $ d $ is the degree of $ X $ and $ \mu $ is the average scalar curvature of $ \omega_{FS}|_X $.
• The $ G $-equivariance of the maps $ X \mapsto R(X) $ and $ X \mapsto \Delta(X) $ ensures that the weight polytopes transform consistently under group actions, enabling the global criterion.