Intersection of two quadrics: modular interpretation and Hitchin morphism

TL;DR

This paper reveals that the Lagrangian fibration of the cotangent bundle of a smooth intersection of two quadrics can be interpreted as a Hitchin morphism when viewed through the lens of moduli spaces of twisted Spin-bundles, generalizing classical results.

Contribution

It establishes a new connection between the geometry of intersections of quadrics and Hitchin systems via moduli space structures.

Findings

Identifies the Lagrangian fibration as a Hitchin morphism for certain moduli spaces.

Generalizes classical results from threefolds to higher-dimensional intersections.

Provides a modular interpretation of geometric fibrations in algebraic geometry.

Abstract

The cotangent bundle $T^*X$ of a smooth intersection $X$ of two quadrics admits a Lagrangian fibration determined by the intrinsic geometry of $X$. We show that this fibration is actually the Hitchin morphism if we endow $X$ with a structure of moduli space of twisted Spin-bundles. This generalises the classical result for threefolds, in which case it recovers the Hitchin fibration for the moduli space of rank two bundles with fixed determinant of odd degree on a curve of genus two.