Hecke transformation for orthogonal bundles and stability of Picard bundles
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چکیده
Given a holomorphic vector bundle F over a compact Riemann surface X, and a subspace Sx ⊂ Fx in the fiber over a point x, the Hecke transformation produces a new vector bundle E on X [10, 16]. The vector bundle E is the kernel of the natural quotient map F −→ Fx/Sx. Hecke transformation is a very useful tool to study the moduli space. For instance, they are used in computation of cohomologies of coherent sheaves on a moduli space of vector bundles [10]. They are also used in proving stability of various naturally associated bundles on a moduli space [3]. When Sx varies among all subspaces of Fx (the fiber of F at x), with x fixed, we get a family of vector bundles. Under suitable conditions for F , these Hecke transforms are stable vector bundles, so we obtain a morphism from the Grassmannian associated to Fx to the moduli space of vector bundles. The image of this morphism is called a Hecke cycle. An orthogonal bundle is a vector bundle F together with a homomorphism ψ : F ⊗ F −→ M , where M is a line bundle, such that ψ is symmetric and non-degenerate at every fiber. Equivalently, an orthogonal bundle can be thought of as a principal GO(r,C)–bundle. Our aim here is systematically to construct Hecke transformations of orthogonal bundles. If F is an orthogonal bundle over X of rank 2n, and Sx ⊂ Fx is an isotropic subspace of dimension n, then the vector bundle E −→ X defined by the kernel of the homomorphism F −→ Fx/Sx has an induced orthogonal structure. If the orthogonal form on F takes values in a line bundle M , then the orthogonal form on E takes values in M ⊗OX(−x). Summing up, we start with a principal GO(2n,C)–bundle F and a Lagrangian subspace of Fx, and we
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تاریخ انتشار 2011