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arXiv · 2401.02655

CscK metrics near the canonical class

Bin Guo·Wangjian Jian·Yalong Shi·Jian Song
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Abstract

Let $X$ be a Kähler manifold with semi-ample canonical bundle $K_X$. It is proved by Jian-Shi-Song that for any Kähler class $γ$, there exists $δ>0$ such that for all $t\in (0, δ)$ there exists a unique cscK metric $g_t$ in $K_X+ t γ$. In this paper, we prove that $\{ (X, g_t) \}_{ t\in (0, δ)} $ have uniformly bounded Kähler potentials, volume forms and diameters. As a consequence, these metric spaces are pre-compact in the Gromov-Hausdorff sense.

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Bin Guo, Wangjian Jian, Yalong Shi, Jian Song. 2024-01-05. CscK metrics near the canonical class. https://arxiv.org/abs/2401.02655

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