On the spectrum of the twisted Dolbeault Laplacian over K\"ahler manifolds

We use Dirac operator techniques to a establish sharp lower bound for the first eigenvalue of the Dolbeault Laplacian twisted by Hermitian-Einstein connections on vector bundles of negative degree over compact K\"ahler manifolds.Comment: 14 pages. Completely revised: estimates corrected and shown to be shar

Spectral curves and Nahm transform for doubly-periodic instantons

We explore the role played by the spectral curves associated with Higgs pairs in the context of the Nahm transform of doubly-periodic instantons defined in "Construction of doubly-periodic instantons" (math.DG/9909069) and "Nahm transform for doubly-periodic instantons" (math.DG/9910120). More precisely, we show how to construct a triple consisting of an algebraic curve plus a line bundle with connection over it from a doubly-periodic instanton, and that these coincide with the Hitchin's spectral data associated with the Nahm transformed Higgs bundle.Comment: Completely reformulated, more differential-geometric approach. 12 page

Construction of doubly-periodic instantons

We construct finite energy instanton connection on $R^4$ which are periodic in two directions via an analogue of the Nahm transform for certain singular solutions of Hitchin's equations defined over a 2-torus.Comment: Final version to appear on CM

Trihyperkahler reduction and instanton bundles on CP^3

A trisymplectic structure on a complex 2n-manifold is a triple of holomorphic symplectic forms such that any linear combination of these forms has constant rank 2n, n or 0, and degenerate forms in $\Omega$ belong to a non-degenerate quadric hypersurface. We show that a trisymplectic manifold is equipped with a holomorphic 3-web and the Chern connection of this 3-web is holomorphic, torsion-free, and preserves the three symplectic forms. We construct a trisymplectic structure on the moduli of regular rational curves in the twistor space of a hyperkaehler manifold, and define a trisymplectic reduction of a trisymplectic manifold, which is a complexified form of a hyperkaehler reduction. We prove that the trisymplectic reduction in the space of regular rational curves on the twistor space of a hyperkaehler manifold M is compatible with the hyperkaehler reduction on M. As an application of these geometric ideas, we consider the ADHM construction of instantons and show that the moduli space of rank r, charge c framed instanton bundles on CP^3 is a smooth, connected, trisymplectic manifold of complex dimension 4rc. In particular, it follows that the moduli space of rank 2, charge c instanton bundles on CP^3 is a smooth complex manifold dimension 8c-3, thus settling a 30-year old conjecture.Comment: 42 pages, v. 3.2, changes in section 3.1: the notion of trisymplectic structure stated differently, Clifford algebra action introduce