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The Quarterly Journal of Mathematics Advance Access originally published online on June 5, 2008
The Quarterly Journal of Mathematics 2009 60(2):235-272; doi:10.1093/qmath/han007
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© 2008. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

HODGE POLYNOMIALS OF THE MODULI SPACES OF TRIPLES OF RANK (2, 2)

Vicente Muñoz{dagger}

Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, Serrano 113bis, 28006 Madrid, Spain
Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA

Daniel Ortega {ddagger}

Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Maria-Jesús Vázquez-Gallo §

Departamento de Ingeniería Civil: Servicios Urbanos, Unidad Docente: Matemáticas, Escuela de Ingenieria de Obras Públicas, Universidad Politécnica de Madrid, Alfonso XII 3 y 5, 28014 Madrid, Spain

{dagger} Corresponding author. E-mail: vicente.munoz{at}imaff.cfmac.csic.es

Received 20 April 2007; revised 31 March 2008
  Abstract

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic triple (E1, E2, {phi}) on X consists of two holomorphic vector bundles E1 and E2 over X and a holomorphic map {phi} : E2 -> E1. There is a concept of stability for triples which depends on a real parameter {sigma}. In this paper, we determine the Hodge polynomials of the moduli spaces of {sigma}-stable triples with rk(E1) = rk(E2) = 2, using the theory of mixed Hodge structures (in the cases that these moduli spaces are smooth and compact). This gives in particular the Poincaré polynomials of these moduli spaces. As a byproduct, we also give the Hodge polynomial of the moduli space of even degree rank 2 stable vector bundles.

{ddagger} E-mail: daniel.ortega{at}uam.es

§ E-mail: mariajesus.vazquez{at}upm.es


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