A Derived Lagrangian Fibration on the Derived Critical Locus

TL;DR

This paper explores the symplectic structure of derived intersections of Lagrangian morphisms, revealing a natural Lagrangian fibration on the derived critical locus linked to the Hessian quadratic form in non-degenerate cases.

Contribution

It introduces a natural Lagrangian fibration on the derived critical locus for functions, connecting symplectic geometry with derived algebraic geometry.

Findings

Derived critical locus admits a natural Lagrangian fibration.

In non-degenerate cases, the fibration is determined by the Hessian quadratic form.

Provides a geometric framework for understanding derived intersections of Lagrangians.

Abstract

We study the symplectic geometry of derived intersections of Lagrangian morphisms. In particular, we show that for a functional $f : X \rightarrow \mathbb{A}_k^1$, the derived critical locus has a natural Lagrangian fibration $\textbf{Crit}(f) \rightarrow X$. In the case where $f$ is non-degenerate and the strict critical locus is smooth, we show that the Lagrangian fibration on the derived critical locus is determined by the Hessian quadratic form.