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An asymptotic slope of the α-K-energy for the Kahler-Yang-Mills equations

Akito Futaki, Jianwei Shi

Abstract

For holomorphic vector bundles over compact Kähler manifolds, we establish a formula for the asymptotic slope of the α-K-energy associated with the Kahler-Yang-Mills equations.

An asymptotic slope of the α-K-energy for the Kahler-Yang-Mills equations

Abstract

For holomorphic vector bundles over compact Kähler manifolds, we establish a formula for the asymptotic slope of the α-K-energy associated with the Kahler-Yang-Mills equations.

Paper Structure

This paper contains 11 sections, 10 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Organisation of the paper.
  3. Coupled Kähler-Yang-Mills equations and $\alpha$-K-energy
  4. Multivariate energy functionals and its asymptotic slope
  5. Multivariate energy functionals and modified multivariate energy functionals
  6. Cohomological test configurations and $\mathcal{C}^\infty$-compatible rays
  7. Asymptotic slope of modified multivariate energy functionals
  8. Expanding the $\alpha$-K-energy
  9. Multivariate functional for $Q_1$ and $Q_2$
  10. The functional $\operatorname{M}'$
  11. An asymptotic slope of $\alpha$-K-energy

Key Result

Theorem 1

Let $(X,\omega)$ be a compact Kähler manifold with $\omega$ located in the cohomology class $\Omega\in H^{1,1}(X,\mathbb{R})$, and $\pi:E\to X$ an irreducible holomoprhic vector bundle over $X$. Then for any Hermitian metric $H$ of $E$, any $(\mathcal{X}, \mathcal{A})$ a cohomological test configura where $b_t=(\omega+dd^c\varphi_t,H)$, $\operatorname{M}^{\operatorname{NA}}(\mathcal{X},\mathcal{A}

Theorems & Definitions (21)

  • Theorem 1
  • Remark 2
  • Theorem 3: alvarez2013coupled
  • Corollary 4
  • Definition 5: sjostrom2018k
  • Proposition 6: sjostrom2018k
  • Definition 7: Dervan2023
  • Remark 8
  • Remark 9
  • Definition 10: sjostrom2018k
  • ...and 11 more