Complete Geodesic Metrics in Big Classes

TL;DR

The work extends the complete geodesic metric framework to high-energy spaces $\\mathcal{E}^{p}(X,\\theta)$ for a smooth $\\theta$ representing a big cohomology class on a compact Kähler manifold. It introduces a robust approximation scheme via model analytic singularities and desingularization, importing the established geometry from the big and nef setting on a desingularized space and transferring it back to the original manifold. The main results are that $\\mathcal{E}^{p}(X,\\theta)$ admits a complete geodesic metric $d_{p}$ with weak geodesics as metric geodesics, and that for $p>1$ this metric space is uniformly convex (CAT(0) in the $p=2$ case), enabling unique geodesics and potential gradient-flow analysis. The contraction property of the singularity projection and the consistency with known metrics in the nef/Kähler regimes provide a unified and stable framework for studying geodesics and variational flows in complex geometry beyond the Kähler case. This sets the stage for future work on Mabuchi-type convexity, gradient flows, and geodesic rays in the general big setting.

Abstract

Let $(X,ω)$ be a compact Kähler manifold and $θ$ be a smooth closed real $(1,1)$-form that represents a big cohomology class. In this paper, we show that for $p\geq 1$, the high energy space $\mathcal{E}^{p}(X,θ)$ can be endowed with a metric $d_{p}$ that makes $(\mathcal{E}^{p}(X,θ),d_{p})$ a complete geodesic metric space. The weak geodesics in $\mathcal{E}^{p}(X,θ)$ are the metric geodesic for $(\mathcal{E}^{p}(X,θ), d_{p})$. Moreover, for $p > 1$, the geodesic metric space $(\mathcal{E}^{p}(X,θ), d_{p})$ is uniformly convex.

Theorems & Definitions (121)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• proof
• Lemma 2.2: darvas2023relative
• Theorem 2.3: EleonoraTrapaniMAmeasureoncontactsets
• Theorem 2.4: EleonoraTrapaniMAmeasureoncontactsets
• Corollary 2.5
• proof
• Lemma 2.6: Darvas2019GeometricPT