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About Wess-Zumino-Witten equation and Harder-Narasimhan potentials

Siarhei Finski

TL;DR

The paper studies the Wess-Zumino-Witten equation on boundaryless fibrations and identifies algebraic obstructions to the existence of exact solutions via Harder–Narasimhan filtrations. It introduces an auxiliary Monge-Ampère-type equation weighted by the Harder-Narasimhan potential and proves that approximate solutions exist for any polarized family, which closely approximate true WZW solutions when they do not exist and minimize a Yang–Mills-type functional. A detailed framework links submultiplicative filtrations, Toeplitz/quantization methods, and geodesic rays to construct δ-approximate critical Hermitian structures, and then dequantizes these to obtain ω_ε that saturate the WZW lower bound. The work also yields a Mehta–Ramanathan-type formula for WZW, connects asymptotic cohomology with the absolute Monge-Ampère functional, and ties the results to the Kobayashi–Hitchin correspondence and Hessian quotient equations in the fibration setting. Overall, it provides a unified approach to approximate geometric structures in the absence of exact WZW solutions and reveals deep links between algebraic stability data and differential-geometric obstructions.

Abstract

For a polarized family of complex projective manifolds, we identify the algebraic obstructions that govern the existence of approximate solutions to the Wess-Zumino-Witten equation. When this is specialized to the fibration associated with a projectivization of a vector bundle, we recover a version of Kobayashi-Hitchin correspondence. More broadly, we demonstrate that a certain auxiliary Monge-Ampère type equation, generalizing the Wess-Zumino-Witten equation by taking into account the weighted Bergman kernel associated with the Harder-Narasimhan filtrations of direct image sheaves, admits approximate solutions over any polarized family. These approximate solutions are shown to be the closest counterparts to true solutions of the Wess-Zumino-Witten equation whenever the latter do not exist, as they minimize the associated Yang-Mills functional. As an application, in a fibered setting, we prove an asymptotic converse to the Andreotti-Grauert theorem conjectured by Demailly.

About Wess-Zumino-Witten equation and Harder-Narasimhan potentials

TL;DR

The paper studies the Wess-Zumino-Witten equation on boundaryless fibrations and identifies algebraic obstructions to the existence of exact solutions via Harder–Narasimhan filtrations. It introduces an auxiliary Monge-Ampère-type equation weighted by the Harder-Narasimhan potential and proves that approximate solutions exist for any polarized family, which closely approximate true WZW solutions when they do not exist and minimize a Yang–Mills-type functional. A detailed framework links submultiplicative filtrations, Toeplitz/quantization methods, and geodesic rays to construct δ-approximate critical Hermitian structures, and then dequantizes these to obtain ω_ε that saturate the WZW lower bound. The work also yields a Mehta–Ramanathan-type formula for WZW, connects asymptotic cohomology with the absolute Monge-Ampère functional, and ties the results to the Kobayashi–Hitchin correspondence and Hessian quotient equations in the fibration setting. Overall, it provides a unified approach to approximate geometric structures in the absence of exact WZW solutions and reveals deep links between algebraic stability data and differential-geometric obstructions.

Abstract

For a polarized family of complex projective manifolds, we identify the algebraic obstructions that govern the existence of approximate solutions to the Wess-Zumino-Witten equation. When this is specialized to the fibration associated with a projectivization of a vector bundle, we recover a version of Kobayashi-Hitchin correspondence. More broadly, we demonstrate that a certain auxiliary Monge-Ampère type equation, generalizing the Wess-Zumino-Witten equation by taking into account the weighted Bergman kernel associated with the Harder-Narasimhan filtrations of direct image sheaves, admits approximate solutions over any polarized family. These approximate solutions are shown to be the closest counterparts to true solutions of the Wess-Zumino-Witten equation whenever the latter do not exist, as they minimize the associated Yang-Mills functional. As an application, in a fibered setting, we prove an asymptotic converse to the Andreotti-Grauert theorem conjectured by Demailly.
Paper Structure (14 sections, 54 theorems, 204 equations, 1 figure)

This paper contains 14 sections, 54 theorems, 204 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Harder-Narasimhan potentials and submultiplicativity
  3. A minimizing sequence for the Wess-Zumino-Witten functional
  4. Dequantization of approximate critical Hermitian structures
  5. Curvature of direct images and semiclassical extension theorem
  6. Weight operators of submultiplicative filtrations
  7. Inductive construction of approximate critical Hermitian structures
  8. Restriction of the weight operator over the geodesic ray
  9. Symmetric powers of geodesic rays and holomorphic extension theorem
  10. Geodesic rays of symmetric powers and fibration degenerations
  11. Approximate critical Hermitian structures and geodesic rays
  12. Mehta-Ramanathan type formula for the Wess-Zumino-Witten functional
  13. Asymptotic cohomology and the absolute Monge-Ampère functional
  14. Kobayashi-Hitchin correspondence and Hessian quotient equations

Key Result

Theorem 1.1

For any $t \in \mathbb{R}$, we have

Figures (1)

  • Figure 1: Sharp and trivial lower bounds on the Wess-Zumino-Witten functional.

Theorems & Definitions (102)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 2.1: FinSubmToepl
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • ...and 92 more