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A DERIVED LAGRANGIAN FIBRATION ON THE DERIVED CRITICAL LOCUS

Part of: Symplectic geometry, contact geometry Algebraic geometry: Foundations Categories and geometry

Published online by Cambridge University Press:  31 August 2022

Albin Grataloup Open the ORCID record for Albin Grataloup [Opens in a new window]
Albin Grataloup*
Affiliation:
Institut Montpelliérain Alexander Grothendieck, University of Montpellier, Centre National de la Recherche Scientifique, 34090 Montpellier, France
*
[email protected]
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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 nondegenerate 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.

Keywords

derived geometryshifted symplectic structuresLagrangianLagrangian fibrationsderived intersectionsderived critical locus

MSC classification

Primary: 53D99: None of the above, but in this section
Secondary: 14A15: Schemes and morphisms 18F99: None of the above, but in this section
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 1 , January 2024 , pp. 311 - 345
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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