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Quantization of the Kähler-Ricci flow and optimal destabilizer for a Fano manifold

Tomoyuki Hisamoto

Abstract

For a Fano manifold, We consider the geometric quantization of the Kähler-Ricci flow and the associated entropy functional. Convergence to the original flow and entropy is established. It is also possible to formulate the finite-dimensional analogue of the optimal degeneration for the anti-canonical polarization.

Quantization of the Kähler-Ricci flow and optimal destabilizer for a Fano manifold

Abstract

For a Fano manifold, We consider the geometric quantization of the Kähler-Ricci flow and the associated entropy functional. Convergence to the original flow and entropy is established. It is also possible to formulate the finite-dimensional analogue of the optimal degeneration for the anti-canonical polarization.
Paper Structure (12 sections, 15 theorems, 100 equations)

This paper contains 12 sections, 15 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Kähler-Ricci flow and the optimal degeneration
  3. Kähler-Ricci flow
  4. Space of Kähler metrics
  5. Optimal degeneration
  6. Quantization
  7. Anti-canonical settings of Geometric quantization
  8. Quantization of the Kähler-Ricci flow
  9. Quantized entropy functional
  10. Convergence results
  11. Optimal non-Archimedean norm for the quantized entropy
  12. Non-Archimedean norms

Key Result

Theorem 2.3

For an arbitrary Fano manifold one has There exists a unique test configuration $(\mathcal{X}, \mathcal{L})$ which achieves the equality and it coincides with the first one of the 2-step degenerations generated from the Gromov-Hausdorff limit of the Kähler-Ricci flow $(X, \omega(t))$.

Theorems & Definitions (35)

  • Definition 2.1: Don02. See also BHJ17 for the updated terminology.
  • Definition 2.2: Berm16, DS17
  • Theorem 2.3: CSW15, DS17, HL20, and BLXZ21.
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • ...and 25 more