Subbundles of symplectic and orthogonal vector bundles over curves

Abstract

We review the notions of symplectic and orthogonal vector bundles over curves, and the connection between principal parts and extensions of vector bundles. We give a criterion for a certain extension of rank 2n to be symplectic or orthogonal. We then describe almost all of its rank n vector subbundles using graphs of sheaf homomorphisms, and give criteria for the isotropy of these subbundles. Finally, we sketch the use of these ideas in moduli questions for symplectic vector bundles.

Irreducibility of Lagrangian Quot schemes over an algebraic curve

“…restricts to a regular symplectic form on W p with respect to which the subsheaf F is Lagrangian. This shows that for each symmetric principal part p ∈ Prin(L −1 ⊗ Sym 2 F) there is a naturally associated symplectic extension of F * ⊗ L by F. We now give a refinement of [11,Criterion 2.1], showing that every symplectic extension can be put into this form.…”

“…for each open set U ⊆ C. It is not hard to see that this is an extension of F * ⊗ L by F. Now there is a canonical pairing , : Rat (F) ⊕ Rat (F * ⊗ L) → Rat (L). By an easy computation (see the proof of [11,Criterion 2.1] for a more general case), the standard symplectic form on Rat (F) ⊕ Rat (F * ⊗ L) defined on sections by…”

“…We take instead a different approach: We exploit the geometry of symplectic extensions, together with deformation arguments, as developed in [4] and [11]. In particular, Proposition 4.14 uses a geometric interpretation for the statement that a nonsaturated Lagrangian subsheaf can be deformed to a subbundle.…”

Maximal isotropic subbundles of orthogonal bundles of odd rank over a curve

“…In this section, we quote some results from [4] and [6] on orthogonal bundles of even rank, which are relevant for our later discussion.…”

Lagrangian subbundles of symplectic bundles over a curve

“…We say that F lifts to W if there is a sheaf injection F → W such that the composition F → W → E * coincides with the elementary transformation µ. We quote two results from [7]: (1) There is a one-to-one correspondence between principal parts p ∈ Prin(E ⊗ E) such that [p] = δ(W ), and elementary transformations F of E * lifting to W as a subbundle, given by p ←→ Ker (p : E * → Prin(E)).…”