Spin^c Structures and Scalar Curvature Estimates

TL;DR

This paper investigates scalar curvature estimates on Riemannian manifolds related to spin^c Dirac operators, demonstrating limitations on metric deformations and providing explicit bounds with sharpness examples.

Contribution

It establishes new constraints on scalar curvature modifications for Kaehler metrics and derives explicit bounds for submanifolds of complex projective space.

Findings

Enlarging a Kaehler metric with positive Ricci curvature cannot increase scalar curvature everywhere.

Explicit upper bounds for minimum scalar curvature are provided for certain submanifolds.

Examples demonstrate cases where these bounds are sharp and attained.

Abstract

In this note, we look at estimates for the scalar curvature k of a Riemannian manifold M which are related to spin^c Dirac operators: We show that one may not enlarge a Kaehler metric with positive Ricci curvature without making k smaller somewhere on M. We also give explicit upper bounds for min(k) for arbitrary Riemannian metrics on certain submanifolds of complex projective space. In certain cases, these estimates are sharp: we give examples where equality is obtained.