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We construct a canonical Hausdorff complex analytic moduli space of Fano manifolds with Kähler-Ricci solitons. This enlarges the moduli space of Fano manifolds with Kähler-Einstein metrics. 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 moment map. Some stacky words and arguments on Gromov-Hausdorff convergence help to glue them together in the holomorphic manner.

We introduce µ-scalar curvature for a Kähler metric with a moment map µ and start up a study on constant µ-scalar curvature Kähler metric as a generalization of both cscK metric and Kähler-Ricci soliton and as a continuity path to extremal metric. We study some fundamental constraints to the existence of constant µ-scalar curvature Kähler 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 µ-scalar curvature and as a prelude to new K-stability concept. Contents 18 4. µK-energy and µK-stability 25 4.1. µK-energy functional 25 4.2. A prelude to µK-stability 29 5. Openness of Kähler class and λ admitting µ-cscK 30 5.1. Regularity 30 5.2. Perturbation 30 References 32

This is the first in a series of two papers (cf. [Ino4]) studying µ-cscK metrics and µK-stability, from a new perspective evoked from observations in [Ino3] and in this first paper.The first paper is about a characterization of µ-cscK metrics in terms of Perelman's W -entropy W λ . We regard Perelman's W -entropy as a functional on the tangent bundle T H(X, L) of the space H(X, L) of Kähler metrics in a given Kähler class L. The critical points of W λ turn out to be µ λ -cscK metrics. When λ ≤ 0, the supremum along the fibres gives a smooth functional on H(X, L), which we call µ-entropy. Then µ λ -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 µ-entropy. This bound is not just analogous, but indeed related to Donaldson's lower bound on Calabi functional by the extremal limit λ → −∞.

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