Harmonic forms on ALF gravitational instantons

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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