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

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