Scalar-flat K\"ahler surfaces whose Weyl tensor annihilates the Ricci form

TL;DR

This paper investigates scalar-flat Kähler surfaces with specific Weyl tensor properties, conjecturing they are either Ricci-flat or product surfaces, and proves this in three special cases.

Contribution

It formulates a conjecture about the structure of such surfaces and proves it under three specific conditions.

Findings

Confirmed the conjecture for compact manifolds.

Proved the conjecture when a Ricci eigendistribution is integrable.

Established the conjecture when Ricci and Weyl tensor norms are functionally dependent.

Abstract

We conjecture that any scalar-flat K\"ahler surface in which the Weyl tensor acting on 2-forms annihilates the Ricci form must be either Ricci-flat or locally isometric to a Riemannian product of two real surfaces with mutually opposite nonzero constant Gaussian curvatures. This amounts to the nonexistence of proper weakly Einstein anti-self-dual K\"ahler surfaces. We prove the above conjecture in three special cases: when the manifold is compact, when one of the Ricci eigendistributions is integrable, and when the norms of the Ricci and Weyl tensors are functionally dependent