Hyperkähler manifolds (Q2715827)

A hyperkähler manifold is a Riemannian manifold \(M^{4n}\) equipped with two anticommuting Kähler structures. Equivalently, \(M^{4n}\) is hyperkähler if and only if its holonomy group is contained in the symplectic group \(Sp(n)\). The present book consists of two independent chapters.NEWLINENEWLINENEWLINEChapter I: M. Verbitsky, Hyperholomorphic sheaves and new examples of hyperkähler manifolds.NEWLINENEWLINENEWLINEThis chapter deals with compact hyperkähler manifolds. Beauville constructed two series of compact hyperkähler manifolds based on the Hilbert scheme of points on a torus or on a K3 surface, respectively. It was conjectured that any compact hyperkähler manifold \(M\) with \(H^1(M)=0\) and \(H^{2,0}(M)= \mathbb{C}\) is deformationally equivalent to one of these examples. The author develops a new approach to the construction of compact hyperkähler manifolds in the hope to find counterexamples to this conjecture.NEWLINENEWLINENEWLINEThe moduli spaces of stable holomorphic vector bundles with fixed Chern classes over compact 4-dimensional hyperkähler manifolds are known to be smooth and hyperkähler. In higher dimensions such moduli spaces are no longer smooth, but provide singular hyperkähler varieties. The author introduces hyperholomorphic bundles over compact hyperkähler manifolds \(M^{4n}\), \(n\geq 2\). A hyperholomorphic bundle over \(M\) is a holomorphic vector bundle equipped with a Hermitian connection which is compatible with all complex structures on \(M\) induced by the hyperkähler structure. The deformation space of hyper-holomorphic bundles is a non-compact singular hyperkähler variety. In previous work the author developed a method for desingularizing a singular hyperkähler variety to a smooth hyperkähler manifold. In the present chapter of the book the author is dealing with the compactification problem.NEWLINENEWLINENEWLINEIn order to be able to apply the desingularization procedure the compactification should provide a singular hyperkähler variety. For this purpose the author introduces hyperholomorphic sheaves, that are coherent sheaves which are compatible with the hyperkähler structure in a certain manner. The major part of this chapter consists of a thorough study of the deformations of hyperholomorphic sheaves. This culminates in the construction of compact hyperkähler manifolds arising as desingularizations of deformation spaces of hyperholomorphic sheaves. At the end of the chapter the author outlines a strategy for finding counterexamples to the above mentioned conjecture.NEWLINENEWLINENEWLINEChapter II: D. Kaledin, Hyperkähler structures on total spaces of holomorphic cotangent bundles.NEWLINENEWLINENEWLINEThis chapter deals with non-compact hyperkähler manifolds. The motivation for the author is the conjecture that the total space \(T^*M\) of the cotangent bundle of any Kähler manifold \(M\) admits a hyperkähler structure. The main purpose of this chapter is to give an affirmative answer to a slight modification of this conjecture, and to prove an analogon which does not involve metrics.NEWLINENEWLINENEWLINELet \(M\) be a complex manifold equipped with a Kähler metric. This metric can be extended to a hyperkähler metric \(h\) on the formal neighborhood of the zero section in the total space \(T^*M\) of the holomorphic cotangent bundle of \(M\). The dilations along the fibers of the projection \(T^*M\to M\) induce an action of \(U(1)\) on \(T^*M\) leaving \(h\) invariant. Any \(U(1)\)-invariant hyperkähler metric on the holomorphically symplectic manifold \(T^*M\) differs from \(h\) by a holomorphic symplectic \(U(1)\)-equivariant automorphism of \(T^*M\). If the Kähler metric on \(M\) is real analytic, then the formal hyperkähler metric \(h\) converges to a real analytic metric in an open neighborhood of the zero section in \(T^*M\).NEWLINENEWLINENEWLINEThe version without a metric is as follows. Let \(M\) be a complex manifold and denote by \(\overline TM\) the total space of the complex-conjugate of the tangent bundle of \(M\). Consider \(\overline TM\) equipped with a \(U(1)\)-action arising from dilations along the fibers of the projection \(\overline TM\to M\). Then there exists a one-to-one correspondence between the set of all Kähler connections on the cotangent bundle of \(M\) and the set of all isomorphism classes of \(U(1)\)-equivariant hypercomplex structures on the formal neighborhood of the zero section in \(\overline TM\) such that the projection \(\overline TM\to M\) is holomorphic. If the Kähler connection on \(M\) is real analytic, then the corresponding hypercomplex structure is defined on an open neighborhood of the zero section in \(\overline TM\).NEWLINENEWLINENEWLINEThe main tool for proving these results is the relation between \(U(1)\)-equivariant hyperkähler manifolds and the theory of \(\mathbb{R}\)-Hodge structures discovered by Deligne and Simpson. Apart from the proofs of these results, this chapter contains a thorough introduction to Hodge manifolds, \(\mathbb{R}\)-Hodge structures, and the necessary concepts for understanding the details of the proof.