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Pluripotential Chern-Ricci Flows

Quang-Tuan Dang

TL;DR

The paper generalizes the parabolic pluripotential approach to degenerate complex Monge–Ampère equations from the Kähler setting to compact Hermitian manifolds, defining pluripotential solutions and proving existence, uniqueness, and stability under natural data assumptions. It establishes sharp a priori estimates, including oscillation bounds, Lipschitz and semi-concavity in time, and, under extra regularity, partial smoothness of the pluripotential flow away from singular loci. By regularization and comparison principles, it obtains a robust weak Chern–Ricci flow on complex varieties with log terminal singularities, showing that the flow exists, is unique, and smooths the initial current on nonsingular parts for t>0. The results extend the parabolic pluripotential framework beyond the Kähler case, enabling analysis of Chern–Ricci flows on mildly singular Hermitian varieties with controlled densities and data, and providing a solid foundation for geometric applications in complex geometry.

Abstract

Extending a recent theory developed on compact Kähler manifolds by Guedj-Lu-Zeriahi and the author, we define and study pluripotential solutions to degenerate parabolic complex Monge-Ampère equations on compact Hermitian manifolds. Under natural assumptions on the Cauchy boundary data, we show that the pluripotential solution is semi-concave in time and continuous in space and that such a solution is unique. We also establish a partial regularity of such solutions under some extra assumptions of the densities and apply it to prove the existence and uniqueness of the weak Chern-Ricci flow on complex compact varieties with log terminal singularities.

Pluripotential Chern-Ricci Flows

TL;DR

The paper generalizes the parabolic pluripotential approach to degenerate complex Monge–Ampère equations from the Kähler setting to compact Hermitian manifolds, defining pluripotential solutions and proving existence, uniqueness, and stability under natural data assumptions. It establishes sharp a priori estimates, including oscillation bounds, Lipschitz and semi-concavity in time, and, under extra regularity, partial smoothness of the pluripotential flow away from singular loci. By regularization and comparison principles, it obtains a robust weak Chern–Ricci flow on complex varieties with log terminal singularities, showing that the flow exists, is unique, and smooths the initial current on nonsingular parts for t>0. The results extend the parabolic pluripotential framework beyond the Kähler case, enabling analysis of Chern–Ricci flows on mildly singular Hermitian varieties with controlled densities and data, and providing a solid foundation for geometric applications in complex geometry.

Abstract

Extending a recent theory developed on compact Kähler manifolds by Guedj-Lu-Zeriahi and the author, we define and study pluripotential solutions to degenerate parabolic complex Monge-Ampère equations on compact Hermitian manifolds. Under natural assumptions on the Cauchy boundary data, we show that the pluripotential solution is semi-concave in time and continuous in space and that such a solution is unique. We also establish a partial regularity of such solutions under some extra assumptions of the densities and apply it to prove the existence and uniqueness of the weak Chern-Ricci flow on complex compact varieties with log terminal singularities.
Paper Structure (31 sections, 33 theorems, 177 equations)

This paper contains 31 sections, 33 theorems, 177 equations.

Table of Contents

  1. Preliminaries
  2. Recap on elliptic pluripotential theory
  3. Big forms
  4. Monge-Ampère operators.
  5. Minimum principle
  6. Stability
  7. Parabolic potentials
  8. Families of quasi-plurisubharmonic functions.
  9. Parabolic complex Monge-Ampère operators
  10. Assumptions and Notations
  11. Assumptions on the forms
  12. Assumptions on the densities
  13. A priori estimates
  14. Bounding the oscillation of $\varphi_t$
  15. Lipschitz control in time
  16. ...and 16 more sections

Key Result

Theorem A

Let $\varphi_0$ be a bounded $\omega_0$-psh function. Then there exists a parabolic potential $\varphi\in\mathcal{P}(X_T,\omega)$ to cmaf such that

Theorems & Definitions (70)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3
  • proof
  • Proposition 1.4
  • ...and 60 more