Eiji Inoue

This is the first in a series of two papers studying mu-cscK metrics and muK-stability, from a new perspective evoked from observations in arXiv:2004.06393 and in this first article. The first paper is about a characterization of mu-cscK metrics in terms of Perelman's W-entropy ${\check{W}}^{\lambda }$. We regard Perelman's W-entropy as a functional on the tangent bundle $T\mathcal{H}(X,L)$ of the space $\mathcal{H}(X,L)$ of K"ahler metrics in a given K"ahler class $L$. The critical points of ${\check{W}}^{\lambda }$ turn out to be ${\mu }^{\lambda }$-cscK metrics. When $\lambda \le 0$, the supremum along the fibres gives a smooth functional on $\mathcal{H}(X,L)$, which we call mu-entropy. Then ${\mu }^{\lambda }$-cscK metrics are also characterized as critical points of this functional, similarly as extremal metric is characterized as the critical points of Calabi functional. We also prove the W-entropy is monotonic along geodesics, following Berman–Berndtsson's subharmonicity argument. Studying the limit of the W-entropy, we obtain a lower bound of the mu-entropy. This bound is not just analogous, but indeed related to Donaldson's lower bound on Calabi functional by the extremal limit $\lambda \to -\infty$.

We introduce and study $\mu$K-stability of polarized schemes with respect to general test configurations as an algebro-geometric aspect of the existence of $\mu$-cscK metrics, which is introduced in the paper arXiv:1902.00664 as a framework unifying the frameworks of Kähler-Ricci solitons and cscK metrics. This article consists of two ingredients. On one hand, we develop a foundational framework concerning `derivative of relative equivariant intersection', which we call equivariant calculus. A core claim in equivariant calculus is a convergence result for some infinite series in equivariant cohomology, which is obtained by relative equivariant intersections. Our proof is based on some observations on deRham-Cartan model of equivariant locally finite homology. This framework furnishes a language to describe $\mu$K-stability. We in particular conclude the $\mu$K-semistability of $\mu$-cscK manifolds with respect to general test configurations. On the other hand, we introduce an equivariant character ${\check{\mu }}^{\lambda }$ called $\mu$-character for equivariant family of polarized schemes, motivated by the $\mu$-volume functional introduced in the paper arXiv1902.00664 as a generalization of Tian-Zhu's functional in the theory of Kähler-Ricci soliton. The equivariant derivative of the $\mu$-character derive $\mu$-Futaki invariant for general test configuration, and furthermore, it also produces an analogue of the equivariant first Chern class of CM line bundle for family of polarized schemes, which is irrational and hence cannot be realized as a $\mathbb{Q}$-line bundle in our general $\mu$K-stability setup.

We introduce mu-scalar curvature for a K"ahler metric with a moment map mu and start up a study on constant mu-scalar curvature K"ahler metric as a generalization of both cscK metric and K"ahler-Ricci soliton and as a continuity path to extremal metric. We study some fundamental constraints to the existence of constant mu-scalar curvature K"ahler metric by investigating a volume functional as a generalization of Tian-Zhu's work, which is closely related to Perelman's W-functional. A new K-energy is studied as an approach to the uniqueness problem of constant mu-scalar curvature and as a prelude to new K-stability concept.

We construct a canonical Hausdorff complex analytic moduli space of Fano manifolds with Kähler-Ricci solitons. This naturally enlarges the moduli space of Fano manifolds with Kähler-Einstein metrics, which was constructed by Odaka and Li-Wang-Xu. We discover a moment map picture for Kähler-Ricci solitons, and give complex analytic charts on the topological space consisting of Kähler-Ricci solitons, by studying differential geometric aspects of this infinite dimensional moment map. Some stacky words and arguments on Gromov-Hausdorff convergence help to glue them together in the holomorphic manner.