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Cohomology of desingularization of moduli space of vector bundles. (English) Zbl 0702.14007
Let X be a smooth projective non-hyperelliptic curve of genus \(g\geq 3\) over \({\mathbb{C}}\), and let \(M_ 0\) be the moduli space of rank 2 semistable vector bundles on X with trivial determinant. The aim of this paper is to show that \(H^ 3(M'_ 0,{\mathbb{Z}})\) is a free group of rank \(2g,\) where \(M'_ 0\) is the canonical desingularization of \(M_ 0\) constructed by Narasimhan and Ramanan [cf. M. S. Narasimhan and S. Ramanan, “Geometry of Hecke cycles. I” in C. P. Ramanujam, A tribute, Collect. Publ. of C. P. Ramanujam and Pap. in his Mem., TATA Inst. Fund. Res. Stud. Math. 8, 291-345 (1978; Zbl 0427.14002)].
The motivation of this result comes from the fact that it is known that \(M_ 0\) is unirational, but it is unknown whether \(M_ 0\) is rational or not. On the other hand, the torsion subgroup of \(H^ 3(M'_ 0,{\mathbb{Z}})\) (which is a birational invariant) is of interest to distinguish between the rationality and non-rationality of \(M'_ 0\).
Reviewer: L.Bădescu

14D20 Algebraic moduli problems, moduli of vector bundles
14F45 Topological properties in algebraic geometry
14M20 Rational and unirational varieties
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14H10 Families, moduli of curves (algebraic)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
Zbl 0427.14002
Full Text: Numdam EuDML
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