Geometric flow, Multiplier ideal sheaves and Optimal destabilizer for a Fano manifold

Tomoyuki Hisamoto

Paper · Jan 24, 2019

Geometric flow, Multiplier ideal sheaves and Optimal destabilizer for a Fano manifold

Tomoyuki Hisamoto

math.DGarXiv:1901.08480v3

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Abstract

S. K. Donaldson asked whether the lower bound of the Calabi functional is achieved by a sequence the normalized Donaldson-Futaki invariants. We answer the question for the Ricci curvature formalism, in place of the scalar curvature. The principle is that the stability indicator is optimized by the multiplier ideal sheaves of certain weak geodesic ray asymptotic to the geometric flow. We actually prove it in the two case: the inverse Monge-Ampere flow and the Kahler-Ricci flow.

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Submitted: Jan 24, 2019
Last revised: Jan 21, 2020 (v3)
Subjects: math.DG, math.CV

Comments: We modified the proof of the main theorem using monotonicity of the K-energy along the inverse Monge-Ampére flow

Ricci flow Kähler–Ricci flow Kähler geometry Scalar curvature Stability & uniqueness Kähler–Einstein & K-stability

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

5A.6
Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry
The Kähler–Ricci flow, and when to come back to it.
6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.
9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.

[1] 5A.6
Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry
The Kähler–Ricci flow, and when to come back to it.

[2] 6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

[3] 9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.

Geometric flow, Multiplier ideal sheaves and Optimal destabilizer for a Fano manifold

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