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Twisted and coupled constant scalar curvature Kähler metrics on minimal ruled surfaces

Ramesh Mete

TL;DR

The paper analyzes twisted and coupled constant scalar curvature Kähler metrics on the minimal ruled surface $X=\mathbb{P}(L\oplus\mathcal{O})$ over a genus-$2$ Riemann surface. Using Calabi ansatz and momentum profiles, it establishing explicit curvature and Futaki-invariant computations, showing non-existence of coupled cscK metrics for any pair of Kähler classes while proving the existence of $\chi$-twisted cscK metrics for suitable pairs of classes. It also solves the J-equation under specific conditions and derives a bound for the Chen-Cheng invariant $R([\omega],[\chi])$ along Chen's continuity path on these ruled surfaces. Together, these results delineate the landscape of twisted versus coupled cscK metrics on this non-Fano, non-Weyl-constant setting and quantify the obstructions via Futaki-type invariants and momentum-profile analysis.

Abstract

In this paper, we study the existence of twisted constant scalar curvature Kähler (cscK) metrics and non-existence of coupled cscK metrics on minimal ruled surfaces over a Riemann surface of genus $2$. Moreover, we give a bound for the Chen-Cheng invariant related to Chen's continuity path for cscK problem on these ruled surfaces.

Twisted and coupled constant scalar curvature Kähler metrics on minimal ruled surfaces

TL;DR

The paper analyzes twisted and coupled constant scalar curvature Kähler metrics on the minimal ruled surface over a genus- Riemann surface. Using Calabi ansatz and momentum profiles, it establishing explicit curvature and Futaki-invariant computations, showing non-existence of coupled cscK metrics for any pair of Kähler classes while proving the existence of -twisted cscK metrics for suitable pairs of classes. It also solves the J-equation under specific conditions and derives a bound for the Chen-Cheng invariant along Chen's continuity path on these ruled surfaces. Together, these results delineate the landscape of twisted versus coupled cscK metrics on this non-Fano, non-Weyl-constant setting and quantify the obstructions via Futaki-type invariants and momentum-profile analysis.

Abstract

In this paper, we study the existence of twisted constant scalar curvature Kähler (cscK) metrics and non-existence of coupled cscK metrics on minimal ruled surfaces over a Riemann surface of genus . Moreover, we give a bound for the Chen-Cheng invariant related to Chen's continuity path for cscK problem on these ruled surfaces.

Paper Structure

This paper contains 14 sections, 6 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. (Generalized) Futaki invariant formula for any Kähler manifold
  4. Ruled surfaces over Riemann surfaces of genus 2
  5. Calabi ansatz and momentum profile
  6. Ricci and scalar curvature formula in terms of momentum profile
  7. First Chern class and average scalar curvature
  8. Non-existence of coupled cscK metrics on ruled surfaces
  9. Non-existence of usual cscK metrics
  10. Non-existence of coupled cscK metrics by computing Futaki invariant
  11. Existence of twisted cscK metrics on ruled surfaces
  12. Reduction of the twisted cscK equation on ruled surfaces
  13. Existence of twisted cscK metrics for a choice of momentum profile
  14. Bound of the Chen-Cheng invariant $R([\omega], [\chi])$ on ruled surfaces

Key Result

Theorem 1.1

Suppose $\Omega_a = 2\pi(\mathrm{C} + a D_{\infty})$ and $\Omega_b = 2\pi(\mathrm{C} + b D_{\infty})$ are two normalized Kähler classes on the ruled surface $X = \mathbb{P}(L\oplus \mathcal{O})$ defined above with $b\geq \frac{a(5a+4)}{2(1+a)}$ and $a>0$. Then there exists a Kähler metric $\omega\in such that $\omega$ is a $\chi$-twisted cscK metric for some Kähler metric $\chi$ in the class $\Ome

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Theorem 1.7: ChenCheng21-cscK-exis
  • Proposition 1.8
  • Remark 2.1
  • proof : Proof of Theorem \ref{['thm:solvability-of-J-equ-under-some-condition']}