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Variations of moduli of parabolic bundles. (English) Zbl 0821.14007
In this paper, we analyze the effect of varying the weights on the moduli of semistable parabolic bundles. It is observed that for generic weights, the moduli space is a smooth projective variety. Further, the “singular” weights form (real) hyperplanes. The moduli of parabolic bundles with weights $$\alpha$$ and $$\beta$$ on either side of one of these hyperplanes are related by a special type of birational transformation which is similar to a flip in Mori theory, i.e. a generalized “blow- down” followed by a “blow-up” along the singular locus of the singular moduli with weights $$\gamma$$ on this hyperplane.
To prove this, first the singular locus is identified with the product of two moduli of parabolic bundles of lower rank. Then natural inclusion maps are proved to exist on the level of semistable holomorphic structures between the various types of stability (i.e. between $$\alpha$$, $$\beta$$ and $$\gamma$$ stable and semistable bundles). These inclusions descend to the level of moduli to give the necessary birational maps, and it is demonstrated that these maps are isomorphisms off the singular locus. The result is complete by identifying the fiber over a point in the singular locus with the projective space on the nontrivial (parabolic) bundle extensions, and by showing that the dimensions of the two projective fibers (one for $$\alpha$$ and the other for $$\beta)$$ are in a precise sense complementary.
An immediate consequence of this is a formula for the intersection homology of a singular moduli in terms of the homology of either of the two smooth moduli (homology and intersection homology is taken with rational coefficients and middle perversity). For example, one of the smooth moduli is a small resolution of the singular moduli (in the sense of Goresky-MacPherson). Also, as a corollary, one gets a simple formula for the difference of the Poincaré polynomials of the two smooth moduli in terms of the Poincaré polynomial of the product of moduli of lower rank, i.e. the singular locus. Furthermore, this formula holds for integral homology, as a previous paper of the first author proves that the smooth moduli spaces are torsion free.
Further applications of the main theorem are given in the final sections. The case of very singular moduli (i.e. when the weights lie on the intersection of many hyperplanes) are considered and a natural stratification on these singular moduli is introduced. A stratified version of the main theorem is proved with respect to these natural stratifications. For example, when one of the weights is generic, the natural morphism between the moduli is a smooth resolution of the singular moduli. It is conjectured that for every singular moduli, there is a nearby smooth moduli so that the canonical morphism is a small resolution.
For ease of exposition in the earlier sections, we make the simplifying assumptions that there is only one parabolic point and the genus is larger than one. But these assumptions are not essential. In the final section, we outline how to adapt the results to the case of arbitrarily many parabolic points and any genus. Further, it is observed that the same results hold for moduli with fixed determinant. To conclude, rationality of the fixed determinant moduli is proved in the case of genus zero and discussed in the other cases. We close with a question as to the existence of an analogue to the Hecke correspondence in the setting of parabolic bundles.

##### MSC:
 14D20 Algebraic moduli problems, moduli of vector bundles 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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