Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Torelli theorem for the moduli space of framed bundles

I. BISWASa1, T. GÓMEZa2 and V. MUÑOZa2

a1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India. e-mail: indranil@math.tifr.res.in

a2 Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Serrano 113bis, 28006 Madrid, Spain and Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain. e-mail: tomas.gomez@mat.csic.es, vicente.munoz@mat.ucm.es

Abstract

Let X be an irreducible smooth complex projective curve of genus g ≥ 2, and let x xs2208 X be a fixed point. Fix r > 1, and assume that g > 2 if r = 2. A framed bundle is a pair (E, φ), where E is coherent sheaf on X of rank r and fixed determinant ξ, and φ: Exr is a non–zero homomorphism. There is a notion of (semi)stability for framed bundles depending on a parameter τ > 0, which gives rise to the moduli space of τ–semistable framed bundles τ. We prove a Torelli theorem for τ, for τ > 0 small enough, meaning, the isomorphism class of the one–pointed curve (X, x), and also the integer r, are uniquely determined by the isomorphism class of the variety τ.

(Received December 04 2008)

(Revised September 28 2009)

(Online publication November 26 2009)