Related papers: A classical model for derived critical loci

We prove a 'Darboux theorem' for derived schemes with symplectic forms of degree $k<0$, in the sense of Pantev, Toen, Vaquie and Vezzosi arXiv:1111.3209. More precisely, we show that a derived scheme $X$ with symplectic form $\omega$ of…

The notion of a d-critical locus is an ingredient in the definition of motivic Donaldson-Thomas invariants by [BJM19]. There is a canonical d-critical locus structure on the Hilbert scheme of dimension zero subschemes on local toric…

Given a regular function $\phi$ on a smooth stack, and a $(-1)$-shifted Lagrangian $M$ on the derived critical locus of $\phi$, under fairly general hypotheses, we construct a pullback map from the Grothendieck group of coherent matrix…

The notion of a d-critical locus is an ingredient in the definition of motivic Donaldson-Thomas invariants by [BJM19]. In this paper we show that there is a d-critical locus structure on the Hilbert scheme of dimension zero subschemes on…

We show that any $(-2)$-shifted symplectic derived scheme $\textbf{X}$ (of finite type over an algebraically closed field of characteristic zero) is locally equivalent to the derived intersection of two Lagrangian morphisms to a…

For a $(-1)$-shifted Lagrangian in a critical locus, we construct a homomorphism from the $K$-group of matrix factorisations of the critical locus to the $K$-group of the Lagrangian, partially answering the Joyce-Safronov conjecture. The…

This is the first of a series of papers about \emph{quantization} in the context of \emph{derived algebraic geometry}. In this first part, we introduce the notion of \emph{$n$-shifted symplectic structures}, a generalization of the notion…

This is a continuation of prior work of the author on cosection localization for d-manifolds. We construct reduced virtual fundamental classes for derived manifolds with surjective cosections and cosection localized virtual fundamental…

This is the fifth in a series arXiv:1304.4508, arXiv:1305,6302, arXiv:1211.3259, arXiv:1305.6428 on the '$k$-shifted symplectic derived algebraic geometry' of Pantev, Toen, Vaquie and Vezzosi, arXiv:1111.3209. This paper extends the…

The notion of d-critical loci was introduced by Joyce in order to give classical shadows of $(-1)$-shifted symplectic derived schemes. In this paper, we discuss birational geometry for d-critical loci, by introducing notions such as…

In this paper, we investigate Keller's deformed Calabi--Yau completion of the derived category of coherent sheaves on a smooth variety. In particular, for an $n$-dimensional smooth variety $Y$, we describe the derived category of the total…

We introduce how to pushforward shifted symplectic fibrations along base changes. This is achieved by considering symplectic forms that are closed in a stronger sense. Examples include: symplectic zero loci and symplectic quotients.…

Let $({\bf X},\omega_{\bf X}^*)$ be a separated, $-2$-shifted symplectic derived $\mathbb C$-scheme, in the sense of Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209, of complex virtual dimension ${\rm vdim}_{\mathbb C}{\bf X}=n\in\mathbb…

The purpose of this paper is to shed a new light on classical constructions in enumerative geometry from the view point of derived algebraic geometry. We first prove that the cosection localized virtual cycle of a quasi-smooth derived…

We prove that the derived critical locus of a $G$-invariant function $S:X\to\mathbb{A}^1$ carries a shifted moment map, and that its derived symplectic reduction is the derived critical locus of the induced function…

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…

We study intersections of complex Lagrangian in complex symplectic manifolds, proving two main results. First, we construct global canonical perverse sheaves on complex Lagrangian intersections in complex symplectic manifolds for any pair…

We show that the moduli spaces of Thaddeus pairs on smooth projective curves and those of dual pairs are related by d-critical flips, which are virtual birational transformations introduced by the second author. We then prove the existence…

We extend a recent result of Pantev-Toen-Vaquie-Vezzosi, who constructed shifted symplectic structures on derived mapping stacks having a Calabi-Yau source and a shifted symplectic target. Their construction gives a clear conceptual…

This is a survey paper on derived symplectic geometry, that will appear as a chapter contribution to the book "New Spaces for Mathematics and Physics", edited by Mathieu Anel and Gabriel Catren. Our goal is to explain how derived stacks can…