Linked alternating forms and linked symplectic Grassmannians

TL;DR

This paper introduces linked alternating and symplectic forms on chains of vector bundles, demonstrating that linked symplectic Grassmannians exhibit favorable dimensional properties similar to classical cases, with applications to advanced algebraic geometry.

Contribution

It develops the theory of linked symplectic forms and Grassmannians, extending classical symplectic geometry to linked structures relevant for higher-rank Brill-Noether theory.

Findings

Linked symplectic Grassmannians have expected dimension properties.

The theory applies to higher-rank Brill-Noether problems.

Provides a framework for studying isotropic subbundles in linked contexts.

Abstract

Motivated by applications to higher-rank Brill-Noether theory and the Bertram-Feinberg-Mukai conjecture, we introduce the concepts of linked alternating and linked symplectic forms on a chain of vector bundles, and show that the linked symplectic Grassmannians parametrizing chains of subbundles isotropic for a given linked symplectic form has good dimensional behavior analogous to that of the classical symplectic Grassmannian.