Scalar curvature and holomorphy potentials

TL;DR

This paper explores a generalization of extremal Kähler metrics involving products of scalar curvature and holomorphy potentials, identifying conditions for critical metrics and examining their existence on specific manifolds.

Contribution

It introduces a new class of metrics characterized by holomorphy potential products and analyzes their criticality conditions, extending Calabi's extremal metric theory.

Findings

Critical metrics are characterized by a specific functional involving scalar curvature and holomorphy potentials.

Existence of such metrics is established in certain special cases and examples.

In higher dimensions, only one type of nontrivial criticality is found for specific manifolds.

Abstract

A holomorphy potential is a complex valued function whose complex gradient, with respect to some K\"ahler metric, is a holomorphic vector field. Given $k$ holomorphic vector fields on a compact complex manifold, form, for a given K\"ahler metric, a product of the following type: a function of the scalar curvature multiplied by functions of the holomorphy potentials of each of the vector fields. It is shown that the stipulation that such a product be itself a holomorphy potential for yet another vector field singles out critical metrics for a particular functional. This may be regarded as a generalization of the extremal metric variation of Calabi, where $k=0$ and the functional is the square of the $L^2$-norm of the scalar curvature. The existence question for such metrics is examined in a number of special cases. Examples are constructed in the case of certain multifactored product manifolds. For the \sk metrics investigated by Derdzinski and Maschler and residing in the complex projective space, it is shown that only one type of nontrivial criticality holds in dimension three and above.