Vector bundles on Fano threefolds and K3 surfaces
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Vector bundles on Fano threefolds and K3 surfaces Arnaud Beauville1 Received: 30 June 2020 / Accepted: 1 October 2020 © Unione Matematica Italiana 2020
Abstract Let X be a Fano threefold, and let S ⊂ X be a K3 surface. Any moduli space M S of simple vector bundles on S carries a holomorphic symplectic structure. Following an idea of Tyurin, we show that in some cases, those vector bundles which come from X form a Lagrangian subvariety of M S . We illustrate this with a number of concrete examples.
1 Introduction Let X be a Fano threefold, and let S be a smooth surface in the anticanonical system of X , so that S is a K3 surface. Suppose we have a nice moduli space of vector bundles M X on X , such that their restriction to S belongs to a moduli space M S . Under mild hypotheses M S has a natural (holomorphic) symplectic structure. What can we say of the restriction map res : M X → M S , in particular with respect to this symplectic structure? In a 1990 preprint [14], Tyurin made a remarkable observation: if H 2 (X , E nd(E)) = 0 for all E in M X , res is a local isomorphism to a Lagrangian subvariety of M S . The proof is quite simple (see Sect. 2). However Tyurin does not give any example where this result can be applied. We will show that under appropriate hypotheses, the Serre construction provides such a situation (in rank 2). This will give us a number of examples of Lagrangian subvarieties, in particular inside the O’Grady hyperkähler manifold OG10 .
2 Tyurin’s theorem Throughout the paper, we will denote by X a Fano threefold (over C), and by S a smooth surface in the anticanonical system |K X−1 |; thus S is a K3 surface. Recall that the moduli space of simple vector bundles on X or S exists as an algebraic space [2] (equivalently, as an analytic Moishezon space). Theorem (Tyurin) Let M X be a component of the moduli space of simple vector bundles on X . Assume H 2 (X , E nd(E)) = 0 for every E in M X . Then:
To Fabrizio, on his 70th birthday.
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Arnaud Beauville [email protected] CNRS-Laboratoire J.-A. Dieudonné, Université Côte d’Azur, Parc Valrose, 06108 Nice Cedex 2, France
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(1) M X is smooth; for each E in M X , the vector bundle E |S is simple. Let M S be the component of the moduli space of simple vector bundles on S containing the vector bundles E |S . By [11] M S is smooth and carries a symplectic structure. (2) The restriction map res : M X → M S is a Lagrangian immersion—that is, a local isomorphism into a Lagrangian subvariety of M S . (We say that a subvariety of M is Lagrangian if its smooth part is Lagrangian—we will see in Sect. 7 an example where the image is singular.) Proof Let E be a vector bundle in M X , and let E S be its restriction to S. The condition H 2 (X , E nd(E)) = 0 implies that M X is smooth at [E]. Let s be a section of K X−1 defining s
S. Tensoring the exact sequence 0 → K X −→ O X → O S → 0 with E nd(E) gives an exact sequence 0 → E nd(E) ⊗ K X → E nd(E) → E nd(E S ) → 0. Consider the associated long exact sequence. Since H 2 (X , E nd(E))
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