Hodge polynomials of the moduli spaces of triples of rank (2,2)

Details Hodge polynomials of the moduli spaces of triples of rank (2,2) Hodge polynomials of the moduli spaces of triples of rank (2,2)

2007-01-23

arxiv.org/abs/math/0701642v1

Mathematics

Algebraic Geometry

28 pages

Scientific paper

Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. A holomorphic triple $(E_1,E_2,\phi)$ on $X$ consists of two holomorphic vector bundles $E_{1}$ and $E_{2}$ over $X$ and a holomorphic map $\phi \colon E_{2} \to E_{1}$. 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 $\mathrm{rk}(E_1)=\mathrm{rk}(E_2)=2$, using the theory of mixed Hodge structures (in the cases that they are smooth and compact). This gives in particular the Poincar{\'e} 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.

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