Harmonic forms on ALF gravitational instantons

Abstract

We study the space of square-integrable harmonic forms over ALF gravitational instantons of type $A_{K-1}$ and of type $D_K$. We first calculate its dimension making use of a result by Hausel, Hunsicker and Mazzeo which relates the Hodge cohomology of a gravitational instanton $M$ to the singular cohomology of a particular compactification $X_M$ of $M$. We then exhibit an explicit basis, exact for $A_{K-1}$ and approximate for $D_K$, and interpret geometrically the relations between $M$, $X_M$ and their cohomologies.

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