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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.

\[ \text{If } S_{n,d}(X_1) \cong S_{n,d}(X_2), \text{ then } X_1\cong X_2. \tag{*} \]

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