Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A 2-systolic inequality on non-rational compact Kähler surfaces with positive scalar curvature

Zehao Sha

Abstract

In this note, we prove a 2-systolic inequality on compact positive scalar curvature Kähler surfaces admitting a nonconstant holomorphic map to a positive-genus compact Riemann surface. According to the classification of positive scalar curvature Kähler surfaces, any such surface must be a ruled surface fibred over a complex curve with positive genus.

A 2-systolic inequality on non-rational compact Kähler surfaces with positive scalar curvature

Abstract

In this note, we prove a 2-systolic inequality on compact positive scalar curvature Kähler surfaces admitting a nonconstant holomorphic map to a positive-genus compact Riemann surface. According to the classification of positive scalar curvature Kähler surfaces, any such surface must be a ruled surface fibred over a complex curve with positive genus.
Paper Structure (3 sections, 7 theorems, 33 equations)

This paper contains 3 sections, 7 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. The Kähler surface fibred over a Riemann surface with positive genus
  3. The 2-systole in Kähler surface

Key Result

Theorem 1.1

Let $(M^3, h)$ be a closed, orientable Riemannian 3-manifold with positive scalar curvature. Then the following inequality holds: Moreover, equality holds if and only if $M^3$ is isometrically covered by $S^2 \times S^1$ with the round metric on $S^2$, product with the flat metric on $S^1$.

Theorems & Definitions (11)

  • Theorem 1.1: Bray-Brendle-Neves,BBN2010
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1: Bochner formula
  • Lemma 2.2: Co-area formula
  • Lemma 2.3
  • proof
  • Definition 3.1: 2-systole in Kähler surfaces
  • Theorem 3.2
  • proof
  • ...and 1 more