Deformations of Picard sheaves and moduli of pairs
type: Item
instance of: scholarly article
title: Deformations of Picard sheaves and moduli of pairs
zbMATH Open document ID: 0844.14005
DOI: 10.1215/S0012-7094-94-07632-1
published in: Duke Mathematical Journal
publication date: 1996-08-29T00:00:00Z
review text: Let \(C\) be a smooth complex projective curve of genus \(g > 2\), and denote by \({\mathfrak M} (2, \xi)\) the (fine) moduli space of stable rank-2 vector bundles over \(C\) with determinant isomorphic to a fixed line bundle \(\xi \in \text{Pic} (C)\) of degree \(d\). Then any universal bundle \(U_\xi\) on \(C \times {\mathfrak M} (2, \xi)\) for this moduli problem induces, via the projection map \(p_2 : C \times {\mathfrak M} (2, \xi) \to {\mathfrak M} (2, \xi)\), a so-called Picard sheaf \(W_\xi : = (p_2)_* (U_\xi)\) on \({\mathfrak M} (2, \xi)\). In the case where \(d : = \deg (\xi) \geq 4g - 2\) and an odd number, any \(W_\xi\) is locally free and referred to as a Picard bundle on \({\mathfrak M} (2, \xi)\). In the present paper, the authors study the variation of the Picard bundles \(W_\xi\) under deformations of the universal bundle \(U_\xi\). More precisely, for any \(L \in \text{Jac} (C) = \text{Pic}^0 (C)\), define \(W_\xi (L) : = (p_2)_* (W_\xi \otimes (p_1)^* L)\). This gives a family of deformations of the Picard bundle \(W_\xi\), parametrized by the Jacobian \(\text{Jac} (C)\) of \(C\). The main theorem of the paper says that, for a curve \(C\) without automorphisms, of genus \(g > 2\), the deformations of any Picard bundle \(W_\xi\) defined as above (with \(d = \deg (\xi) \geq 4g - 2\) and \(d\) odd) have the following properties: (i) \(h^i ({\mathfrak M} (2, \xi), \text{ ad} W_\xi) = g\), if \(i = 1\); \(\qquad h^i ({\mathfrak M} (2, \xi), \text{ ad} W_\xi) = 0\), if \(i = 0\) or \(i = 2\); (ii) the family of deformations \(\{W_\xi (L) |L \in \text{Pic}^0 (C)\}\) defined above is injective. This result extends the classical work of \textit{A. Mattuck} [Am. J. Math. 83, 189-206 (1961; Zbl 0225.14020)] and of \textit{G. Kempf} [Ann. Math., II. Ser. 110, 243-273 (1979; Zbl 0452.14011)] on the variations and the cohomology of the Picard bundle on the Jacobian of a curve. In the course of the proof, the authors make extensive use of recent constructions a…
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14D15
Mathematics Subject Classification ID: 14C22
Mathematics Subject Classification ID: 14K30
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 755397
zbMATH Keywords: moduli space of stable rank-2 vector bundles over a smooth complex projective curve
zbMATH Keywords: Picard sheaf
zbMATH Keywords: deformations of the universal bundle
zbMATH Keywords: Torelli-type theorems
zbMATH Keywords: moduli spaces of stable pairs
zbMATH Keywords: intermediate Jacobian
author: V. Balaji
author: Periyapatna A. Vishwanath
reviewed by: Werner Kleinert
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 755397