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Non-collapsing volume estimate for local Kähler metrics in big cohomology classes

Thai Duong Do, Duc-Bao Nguyen, Duc-Viet Vu

TL;DR

This work proves a uniform local non-collapsing volume bound for a broad family of Kähler metrics in big cohomology classes under a local density condition. The authors develop a robust regularization theory for complex Sobolev functions, establish a global integration-by-parts formula and a Cauchy–Schwarz inequality in the singular current setting, and derive energy estimates together with a Sobolev inequality adapted to big cohomology classes. These tools let the local density control yield a lower bound Vol_T(B_T(x,r)) \\ge C r^m V_T for small r, with constants depending only on the ambient geometry and density bounds. The approach extends prior global results to local, big-class contexts and provides a framework applicable to Gromov–Hausdorff convergence and degeneration analyses in Kähler geometry.

Abstract

We prove a uniform local non-collapsing volume estimate for a large family of Kähler metrics in the big cohomology classes. The key ingredient is a generalization of a mixed energy estimate for functions in the complex Sobolev space to the setting of big cohomology classes.

Non-collapsing volume estimate for local Kähler metrics in big cohomology classes

TL;DR

This work proves a uniform local non-collapsing volume bound for a broad family of Kähler metrics in big cohomology classes under a local density condition. The authors develop a robust regularization theory for complex Sobolev functions, establish a global integration-by-parts formula and a Cauchy–Schwarz inequality in the singular current setting, and derive energy estimates together with a Sobolev inequality adapted to big cohomology classes. These tools let the local density control yield a lower bound Vol_T(B_T(x,r)) \\ge C r^m V_T for small r, with constants depending only on the ambient geometry and density bounds. The approach extends prior global results to local, big-class contexts and provides a framework applicable to Gromov–Hausdorff convergence and degeneration analyses in Kähler geometry.

Abstract

We prove a uniform local non-collapsing volume estimate for a large family of Kähler metrics in the big cohomology classes. The key ingredient is a generalization of a mixed energy estimate for functions in the complex Sobolev space to the setting of big cohomology classes.

Paper Structure

This paper contains 8 sections, 33 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Regularization of functions in complex Sobolev spaces
  3. Integration by parts formula
  4. First step
  5. Second step
  6. Cauchy-Schwarz inequality
  7. Energy estimates
  8. Sobolev inequality for complex Sobolev functions

Key Result

Theorem 1.1

Let $A > 0,K > 0$ and $p_0>1$ be constants. Let $T \in \mathcal{M}_{\mathop{\mathrm{big}}\nolimits}(X,A)$, $x_0 \in U_T$ and $R_0 \in (0, \operatorname{d}_T(x_0,\partial U)]$. Assume that where $f:= V_T^{-1} T^n /\omega_X^n$ defined on $U_T$. Then there exist constants $m = m(n)>0, C= C(\omega_X,p_0,A,K)>0$ independent of $T,x_0$ and $R_0$ such that for every $r \in (0,R_0/2]$.

Theorems & Definitions (60)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 50 more