[Paper Review] Orthogonal projection of a test configuration to vector fields

Tomoyuki Hisamoto|arXiv (Cornell University)|Oct 23, 2016

Geometry and complex manifolds24 references8 citations

TL;DR

This paper establishes the analytic limit of the orthogonal projection of a test configuration onto holomorphic vector fields by proving moment convergence of weight distributions. It shows that the $L^p$-norm of the projection converges to the $L^p$-norm of the tangent vector of the associated weak geodesic ray, providing a refined analytic characterization of the algebraic projection introduced by Székelyhidi.

ABSTRACT

Given a polarized complex manifold, projection of a torus-equivariant test configuration to holomorphic vector fields was introduced by G. Székelyhidi, as the limit of the associated $\mathbb{C}^*$-actions. We show that there actually holds the moment convergence of the weight distributions. Our analytic approach at the same time specifies the limit in terms of the weak geodesic ray associated with the test configuration. Related to the result, we discuss about the reduced $L^p$-norm of the test configuration in attempt to describe the uniform K-stability of the polarization relative to the automorphism group.

Motivation & Objective

• To provide an analytic description of the orthogonal projection of a test configuration onto holomorphic vector fields, previously defined algebraically by Székelyhidi.
• To establish the convergence of the $L^p$-norm of the projected weights to the $L^p$-norm of the tangent vector of the weak geodesic ray associated with the test configuration.
• To clarify the relationship between the algebraic projection $P_k(A_k)$ and the geometric limit $P(\dot{\varphi}^0)$ via moment convergence.
• To support the notion of uniform K-stability by relating the reduced $L^p$-norm of test configurations to the norm of the projected vector field.

Proposed method

• Use of the weak geodesic ray $\varphi^t$ associated with a $T$-equivariant normal test configuration $(\mathcal{X}, \mathcal{L})$ to define the tangent vector $\dot{\varphi}^0$ at $t=0$.
• Application of the $dd^c$-lemma to identify tangent vectors in $\mathcal{H}$ with smooth functions, and define the $L^2$ inner product using the Monge-Ampère measure.
• Construction of the orthogonal projection $P_k: \mathfrak{sl}(H^0(\mathcal{X}_0, \mathcal{L}_0^{ ens k})) \to (\mathfrak{t}/\mathbb{C})_{\mathbb{R}}$ via the Killing form on the representation space.
• Use of the equivariant Riemann-Roch-Hirzebruch theorem to analyze the asymptotic behavior of traces $\operatorname{Tr}A_k^p$ and $\operatorname{Tr}A_k B_{j,k}$ as $k \to \infty$, showing rationality of coefficients.
• Proof of moment convergence via the identity $2\operatorname{Tr}A_k B_{j,k} = \operatorname{Tr}(A_k + B_{j,k})^2 - \operatorname{Tr}A_k^2 - \operatorname{Tr}B_{j,k}^2$, which yields rational coefficients.
• Reduction to the case of a 1-PS action via base change $\tau \mapsto \tau^d$, followed by construction of a modified test configuration $\mathcal{X}'$ with generator $A - B$, and showing its $L^2$-norm vanishes to conclude $A_1 = B_1$.

Experimental results

Research questions

• RQ1 Does the algebraic projection $P_k(A_k)$ of the weight matrix $A_k$ onto the Lie algebra $\mathfrak{t}/\mathbb{C}$ converge to an analytic limit as $k \to \infty$?
• RQ2 Can the limit of the $L^p$-norm of the projected weights be expressed as an integral of the $p$-th power of the Hamilton function associated with the weak geodesic ray?
• RQ3 Is the tangent vector $\dot{\varphi}^0$ of the weak geodesic ray a rational linear combination of Hamilton functions of 1-PS generators of the torus $T$?
• RQ4 Does the vanishing of the $L^2$-norm of a modified test configuration imply triviality of the configuration, thereby recovering the original generator?
• RQ5 How does the moment convergence result support the definition of uniform K-stability via the reduced $L^p$-norm of test configurations?

Key findings

• The moment convergence holds: $\lim_{k\to\infty} \frac{\operatorname{Tr}P_k(A_k)^p}{k^p N_k} = \frac{1}{V} \int_X P(\dot{\varphi}^0)^p \omega^n$ for all $p \geq 1$, establishing a precise analytic limit of the algebraic projection.
• The Hamilton function $\dot{\varphi}^0$ associated with the initial tangent vector of the weak geodesic ray is a rational linear combination of Hamilton functions of 1-PS generators of the torus $T$, proving rationality of the limit projection.
• The generator $A_k$ of the test configuration coincides with the generator $B_k$ of the 1-PS action when the configuration is base-changed to a multiple root, implying $A_1 = B_1$.
• The $L^2$-norm of the modified test configuration $\mathcal{X}'$ with generator $A - B$ vanishes, so $\| (\mathcal{X}', \mathcal{L}') \|_2 = 0$, which implies triviality by [BHJ15], hence $A_1 = B_1$.
• The result provides a new analytic proof of the convergence result in [His12] for the $\mathbb{C}^*$-case and generalizes it to arbitrary torus actions.
• The convergence result supports the definition of uniform K-stability via the reduced $L^p$-norm, as the limit norm of the projection matches the $L^p$-norm of the tangent vector of the geodesic ray.