G\'eom\'etrie \`a l'infini des vari\'et\'es hyperk\"ahl\'eriennes toriques

TL;DR

This paper studies the geometry at infinity of toric hyperKähler manifolds, computes their $L^2$-cohomology, and explores their deformations, providing insights into their asymptotic structure and confirming a conjecture for monopoles.

Contribution

It characterizes the asymptotic geometry of toric hyperKähler manifolds, computes their $L^2$-cohomology, and analyzes Taub-NUT deformations, linking geometry with monopole duality conjectures.

Findings

Simply connected toric hyperKähler metrics are generically quasi-asymptotically conical.

Explicit computation of reduced $L^2$-cohomology groups.

Taub-NUT deformations are of bounded geometry and have uniquely identified tangent cones at infinity.

Abstract

We show that simply connected toric hyperK\"ahler metrics of finite topological type with maximal volume growth are generically quasi-asymptotically conical, which allows us to compute explicitly their reduced $L^2$-cohomology groups. In the asymptotically conical case, we also provide a fine description of the geometry at infinity of their Taub-NUT deformations of order 1 in terms of a compactification by a manifold with corners, which allows us to show that those deformations are of bounded geometry, to estimate their curvature at infinity and their volume growth and to identify uniquely their tangent cone at infinity. In many instances, the dimension of this tangent cone at infinity is strictly smaller than the order of the volume growth. Finally, our methods show that the Taub-NUT deformations of maximal order of the Euclidean metric are quasi-fibered boundary metrics, which allows us to identify uniquely their tangent cone at infinity, to compute their reduced $L^2$ cohomology groups and to prove that Sen's S-duality conjecture holds for centered monopoles of type $(1,1,\ldots,1)$.