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Analogs of Torelli’s theorem for multidimensional vector bundles over an arbitrary algebraic curve. (Russian) Zbl 0225.14010

Let \(X\) be an algebraic curve of genus \(g\geq 2\), defined over an algebraically closed field \(k\) of characteristic zero, let \(S_{n,d}(X)\) be the variety of all stable vector bundles \(E\) on \(X\) of rank \(n\) such that \(\wedge^nE\) is isomorphic to a fixed line bundle of degree \(d\), \(n\) and \(d\) are relatively prime. The analogue of Torelli’s theorem for \(S_{n,d}\) is the assertion:
\[ \text{If } S_{n,d}(X_1) \cong S_{n,d}(X_2), \text{ then } X_1\cong X_2. \tag{*} \]
For \(n=1\) this is Torelli’s classical theorem. The case \(k=\mathbb C\), \(g=2\), \(n=2\) was studied by P. E. Newstead [Topology 7, 205–215 (1968; Zbl 0174.52901)], who described the complete structure of \(S_{2,d}\), showed a projective embedding of \(S_{2,d}\) in \(\mathbb P_5\) and proved that the variety of lines of \(S_{2,d} \subset \mathbb P_5\) is isomorphic to the Jacobian \(J(X)\) of \(X\). In the case \(k=\mathbb C\), \(g\geq 2\), \(n=2\) D. Mumford and P. E. Newstead [Am. J. Math. 90, 1200–1208 (1968; Zbl 0174.52902)] proved (*) and showed an isomorphism of the intermediate Jacobian \(J^2(S_{2,d}(X)\) to \(J(X)\). The author [Math. USSR, Izv. 3 (1969), 1081–1101 (1971); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 33, 1149–1170 (1969; Zbl 0225.14009)] gave the following geometric reconstruction of \(X\) from \(S_{2,d}(X)\): if \(V\) denotes the base variety of the family of lines of some type on \(S_{n,d}\), \(\varphi: V \to\mathrm{Alb}(V)\) is the Albanese’s application and \(\gamma: J(X)\to J(X)\) is the endomorphism of doubling, then \(\mathrm{Alb}(V)\cong \bar J(X)\) and \(X\cong \gamma(\varphi(V))\).
In the present paper it is proved (*) in the general case and pointed out the geometric way of reconstructing of \(X\) from \(\prod S_{n,d}(X)\). The procedure is the following. Let \(M\) be the set of varieties of form \(\prod S_{n,d}(X)\) \(((n,d)\) runs through a finite set of pairs \((n,d)\), \(n\) and \(d\) are coprime). For \(V\in M\) let \(\bar B_n(V)\) denote a connected component of the base variety of projective spaces \(\mathbb P_n\subset V\) such that \((-K_V)\cdot \mathbb P_n= 2 \mathbb P_{n-1}\), where \(K_V\) is the canonical class on \(V\) and \(\mathbb P_{n-1}\) is a hyperplane in \(\mathbb P_n\) (all connected components of the base variety are actually isomorphic to each other). Let \(B_n(V)\) be a fibre of Albanese’s application \(\bar B_n(V)\to\mathrm{Alb}\bar B_n(V))\) (for \(V\in M\) the fibres are isomorphic to each other). The pair \((n,d)\) defines a sequence of natural numbers \(\{m_1,m_2,\dots,m_N\}\) such that
\[ S_{2,1}(X) =B_{m_N}(B_{m_{N-1}}(\ldots B_{m_1}(S_{n,d}(X))\ldots)). \]
For example the pair \((n,1)\) gives the sequence \(\{(n-1)(g-1), (n-2) (g-1),\dots, 2(g-1)\}\). By examination of projective subspaces of \(S_{n,d}\) were found very useful the new notions of a rigid pair of bundles and rigid bundle: a pair \((E_1,E_2)\) is rigid if \(\text{rk}\,E_1 \deg E_2 - \text{rk}\,E_2 \deg E_1=1\) and a bundle \(E\) is rigid if it is representable as an extension \(0\to E_1\to E\to E_2\to 0\) where \((E_1,E_2)\) is a rigid pair.
Reviewer: M. Kh. Gizatullin

MSC:

14C34 Torelli problem
14H60 Vector bundles on curves and their moduli
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