Scalar curvature, Kodaira dimension and $\hat A$-genus

TL;DR

This paper links scalar curvature conditions on compact complex manifolds to their algebraic and topological properties, showing that positive scalar curvature implies negative Kodaira dimension and vanishing of the $ ext{A}$-hat genus under certain conditions.

Contribution

It establishes new connections between scalar curvature, Kodaira dimension, and topological invariants like the $ ext{A}$-hat genus for complex manifolds.

Findings

Positive scalar curvature implies negative Kodaira dimension.

Introduction of the complex Yamabe number $ ext{λ}_c(X)$ and its implications.

Vanishing of the $ ext{A}$-hat genus for spin manifolds with positive $ ext{λ}_c(X)$.

Abstract

Let $(X,g)$ be a compact Riemannian manifold with quasi-positive Riemannian scalar curvature. If there exists a complex structure $J$ compatible with $g$, then the canonical bundle $K_X$ is not pseudo-effective and the Kodaira dimension $\kappa(X,J)=-\infty$. We also introduce the complex Yamabe number $\lambda_c(X)$ for compact complex manifold $X$, and show that if $\lambda_c(X)>0$, then $\kappa(X)=-\infty$; moreover, if $X$ is also spin, then the Hirzebruch $A$-hat genus $\hat A(X)=0$.