Spaces for Stable Pairs
نویسندگان
چکیده
In this paper we shall construct master spaces for certain coupled vector bundle problems over a fixed projective variety X. These master spaces, which are moduli spaces for oriented torsion free coherent sheaves coupled with morphisms into a fixed reference sheaf E 0 , have the structure of polarized projective varieties endowed with a natural C *-action. The fixed point set of this action contains the moduli space of semistable oriented torsion free sheaves; this is the source of the C *-action in the sense of [BS] when the master space happens to be irreducible and normal. Another distinguished subspace of the fixed point set is a certain Quot scheme associated with E 0 ; if non-empty, it is the sink of the C *-action. When X is a curve with trivial reference sheaf E 0 = O ⊕k X , our master space can be considered as a natural compactification of the space described in [BDW]: Their space becomes an open subset of ours whose complement is the Quot scheme alluded to above. Our motivation for the construction of master spaces came originally from the investigation of non-abelian monopole equations on differentiable 4-manifolds [OT1]. In the Kähler case, these equations led to a projective vortex equation and to a corresponding algebro-geometric stability concept for pairs (E, ϕ), consisting of a holomorphic vector bundle E with fixed determinant and a holomorphic section ϕ. Since the moduli spaces of non-abelian monopoles admit an Uhlenbeck type compactification [T], it was natural to try to find a Gieseker compactification of the corresponding algebro-geometric moduli spaces. Such a compactification, besides being of interest in its own right, serves two further purposes: On one hand, it should help to understand the ends of the moduli spaces of non-abelian monopoles in the more difficult differential-geometric situation. Understanding these ends is one of the essential steps in any program trying to relate Donaldson polynomials and Seiberg-Witten invariants.
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