Deformations of Picard sheaves and moduli of pairs
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 and results of \textit{M. Thaddeus} [``Stable pairs, linear systems and the Verlinde formula'', Invent. Math. 117, No. 2, 317-353 (1994)], which are summarized at the beginning. As a by-product (and as a basic tool for the proof of their main theorem), they obtain local and global Torelli-type theorems for the Thaddeus moduli spaces of stable pairs. One of them relates the Weil-Griffiths intermediate Jacobian of any Thaddeus moduli space isomorphically to the Jacobian of the base curve, while the other one gives an explicit computation of the cohomology of the moduli spaces of stable pairs with coefficients in the tangent sheaf. Both results provide generalizations of earlier, partial results obtained by \textit{D. Mumford} and \textit{P. Newstead} [Am. J. Math. 90, 1200-1208 (1968; Zbl 0174.52902)] and by \textit{M. S. Narasimhan} and \textit{S. Ramanan} [Ann. Math., II. Ser. 101, 391-417 (1975; Zbl 0314.14004)].
moduli space of stable rank-2 vector bundles over a smooth complex projective curve
scientific article; zbMATH DE number 755397
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