[Paper Review] The Dirichlet problem for degenerate complex Monge-Ampere equations

D. H. Phong, Jacob Sturm|ArXiv.org|Apr 13, 2009

Geometry and complex manifolds18 references18 citations

TL;DR

This paper establishes $C^{1,eta}$ regularity estimates for solutions to degenerate complex Monge-Ampère equations on compact Kähler manifolds with boundary, when the cohomology class is non-negative and satisfies a positivity condition relative to a divisor. The key result is the construction of $C^{1,eta}$ geodesic rays in the space of Kähler potentials for each test configuration, resolving a regularity gap in the degenerate case.

ABSTRACT

The Dirichlet problem for a Monge-Ampere equation corresponding to a nonnegative, possible degenerate cohomology class on a Kaehler manifold with boundary is studied. C^{1,α} estimates away from a divisor are obtained, by combining techniques of Blocki, Tsuji, Yau, and pluripotential theory. In particular, C^{1,α} geodesic rays in the space of Kaehler potentials are constructed for each test configuration

Motivation & Objective

• To address the regularity problem for geodesic rays in the space of Kähler potentials when the underlying cohomology class is degenerate.
• To extend prior results on the Dirichlet problem for degenerate complex Monge-Ampère equations beyond the strictly positive definite case.
• To construct $C^{1,eta}$ geodesic rays using pluripotential theory and a priori estimates, particularly in the context of test configurations.
• To prove that the solution to the Monge-Ampère equation remains regular away from a divisor, even when the cohomology class is non-negative but not strictly positive.

Proposed method

• Combines techniques from Blocki, Tsuji, Yau, and pluripotential theory to derive $C^{1,eta}$ estimates for degenerate Monge-Ampère equations.
• Imposes a key condition: $ abla_0 - eta[E] > 0$ for some $eta > 0$, where $E$ is an effective divisor disjoint from the boundary.
• Constructs a non-degenerate form $ ilde{ abla}$ on the compactified total space $ ilde{rak X}_D$ of a test configuration using a metric on an ample line bundle.
• Applies Theorem 2 (a priori estimates) to the Dirichlet problem on $ ilde{rak X}_D$ with zero boundary data, under the condition that $ abla_0 - eta[E]$ is Kähler.
• Uses rotation-invariant metrics and curvature constructions to ensure the existence of a strictly positive form $ abla_eta$ satisfying the required positivity condition.
• Relies on the Donaldson embedding theorem and meromorphic sections to define a divisor $E$ on the central fiber such that $ abla_0 - eta[E]$ becomes a Kähler class.

Experimental results

Research questions

• RQ1 Can $C^{1,eta}$ regularity be established for solutions to degenerate complex Monge-Ampère equations on Kähler manifolds with boundary when the cohomology class is non-negative?
• RQ2 Does the existence of geodesic rays in the space of Kähler potentials persist in the degenerate cohomology case, particularly for test configurations?
• RQ3 Can the method of a priori estimates and pluripotential theory be extended to handle the case where $ abla_0$ is only semi-positive?
• RQ4 Is it possible to construct a Kähler form $ abla_eta$ such that $ abla_0 - eta[E]$ is Kähler for some effective divisor $E$?
• RQ5 How can the geodesic ray equation be solved in the generalized sense when the initial data is not fully prescribed at $w=0$?

Key findings

• The paper establishes $C^{1,eta}$ regularity estimates for solutions to the degenerate complex Monge-Ampère equation away from a divisor $E$.
• A solution $ ilde{ abla}$ is constructed on the compactified total space $ ilde{rak X}_D$ such that $ ilde{ abla}_0 - eta[E]$ is a Kähler form for some $eta > 0$.
• The existence of a $C^{1,eta}$ geodesic ray in the space of Kähler potentials is proven for each test configuration, extending previous results to the degenerate case.
• The construction relies on a metric $H_0$ on $p^*{rak L}^m$ with curvature $ ilde{ abla}_0 eq 0$, which is rotation-invariant and positive on fibers.
• The key technical step is the construction of a metric $H$ such that $ ilde{ abla}_0 - eta[E] = abla_eta - eta rac{i}{2} ar{ abla} ext{log} orm{ ext{sec}}^2$, where $ ext{sec}$ is the canonical section of $O(E)$.
• The result confirms the existence of generalized geodesic rays in the space of Kähler potentials under the degenerate cohomology condition, supporting the stability conjecture of Donaldson.