The Dirac-Dolbeault Operator Approach to the Hodge Conjecture

TL;DR

This paper introduces a novel approach using the Dirac-Dolbeault operator to prove the Hodge conjecture for complex projective manifolds by establishing the existence of specific complex submanifolds.

Contribution

It develops a new method leveraging Dirac operators and boundary value techniques to demonstrate the existence of submanifolds satisfying certain PDEs, leading to a proof of the Hodge conjecture.

Findings

Existence of complex submanifolds satisfying a PDE under injectivity assumptions

Submanifolds' fundamental classes span rational Hodge classes

Proof of the Hodge conjecture for complex projective manifolds

Abstract

The Dirac-Dolbeault operator for a compact K\"ahler manifold is a special case of a Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows to express the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash-Moser generalized inverse function theorem we prove the existence of complex submanifolds of a complex projective manifold satisfying globally a certain partial differential equation under a certain injectivity assumption. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for complex projective manifolds.