The Moduli Space of Curves pp 533-563 | Cite as

# On the Cohomology of Moduli Spaces of Rank Two Vector Bundles Over Curves

## Abstract

Let *C* be a Riemann surface, *L* a line bundle over *C*, and *n* a natural number. Then there is a moduli space of stable *n*-dimensional vector bundles *E* over *C* with determinant bundle Λ^{ n }(*E*) ≡ *L*; this moduli space is smooth but in general non-compact and can be compactified by the suitable addition of semi-stable bundles to a projective, but in general singular, variety *N* _{ c,n,L }.The topology of this variety depends only on the genus *g* of *C* and the degree *d* of *L* (in fact, only on *d* modulo *n*, since tensoring *E* with a fixed line bundle *L* _{1} replaces *L* by *L* ⊗ *L* _{1} ^{ n } ), so we will also use the notation *N* _{ g,n,d }. We will be studying only the case *n* = 2, and hence will drop the *n* and replace *d* by ε = (−1)^{d} in the notation. Thus for each *g* we have two moduli spaces of stable 2-dimensional bundles *N* _{g} ^{−} and *N* _{g} ^{+} , both projective varieties of complex dimension 3*g* − 3. We will be looking
mostly at the smooth space *N* _{g} ^{−} and will often denote it simply *N* _{ g }.

## Keywords

Modulus Space Vector Bundle Line Bundle Intersection Number Chern Class## Preview

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