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On Weighted Twisted K-Energy and Its Applications

Xia Xiao

Abstract

We establish the convexity of the weighted twisted Mabuchi K-energy functional along geodesics in the finite energy space $\mathcal{E}^{1,T}(X,ω)$, covering the case of divisors with mixed cusp and conic singularities. We then prove that coercivity (relative to the complex torus) of this functional is an open condition under cone angle perturbations. This is obtained from a general result of independent interest, which shows the stability of the coercivity under perturbations by certain twist currents. In particular, this yields the openness for the existence for cscK cone metrics and proves that coercivity at the cusp limit implies existence of cscK cone metrics for small cone angles.

On Weighted Twisted K-Energy and Its Applications

Abstract

We establish the convexity of the weighted twisted Mabuchi K-energy functional along geodesics in the finite energy space , covering the case of divisors with mixed cusp and conic singularities. We then prove that coercivity (relative to the complex torus) of this functional is an open condition under cone angle perturbations. This is obtained from a general result of independent interest, which shows the stability of the coercivity under perturbations by certain twist currents. In particular, this yields the openness for the existence for cscK cone metrics and proves that coercivity at the cusp limit implies existence of cscK cone metrics for small cone angles.
Paper Structure (25 sections, 47 theorems, 193 equations)

This paper contains 25 sections, 47 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Weighted twist formalism
  3. Moment maps
  4. Weighted Monge--Ampère Operators and Weighted Energy
  5. Weighted Monge--Ampère operators
  6. Weighted energy
  7. Weighted twisted scalar curvature
  8. Futaki invariant
  9. Weighted twisted extremal function
  10. Extension to $\mathcal{E}^{1,T}(X, \omega)$
  11. Weighted Monge--Ampère operator
  12. Weighted twisted energies
  13. Extension to finite energy space
  14. Convexity of weighted twisted K-energy
  15. Weak geodesics
  16. ...and 10 more sections

Key Result

Theorem A

Let $(X, \omega)$ be a compact Kähler manifold, $T \subset \mathop{\mathrm{Aut}}\nolimits_{red}(X,\chi)$ a compact torus, and $\chi$ a $T$-invariant positive $(1,1)$-current satisfying eq: ChiProp. For smooth positive weight functions $v, \mathrm{w}$ on the moment polytope, the weighted twisted Mabu admits a unique greatest lower semicontinuous extension $\mathcal{M}_{v,\mathrm{w}}^{\chi}: \mathca

Theorems & Definitions (106)

  • Theorem A: Convexity of weighted twisted K-energy, see Theorem \ref{['thm:main']}
  • Corollary 1.1: Weighted K-energy convexity for mixed cusp and conic singularities, see Corollary \ref{['prop:extention-K-energy-divisor']}
  • Theorem B: Openness of coercivity for mixed cusp and conic singularities, see Theorem \ref{['thm:coeropen0']}
  • Corollary 1.2: Openness of existence of conic cscK metrics
  • Corollary 1.3: From cusp coercivity to small-angle conic existence
  • Conjecture 1.4: Existence of $(1,1,2\pi[D])$-extremal Poincaré-type Kähler metrics
  • Remark 1.5
  • Example 2.1
  • Definition 3.1
  • Lemma 3.2
  • ...and 96 more