A classical model for derived critical loci

TL;DR

This paper introduces algebraic d-critical loci as classical truncations of derived critical loci, simplifying the study of spaces with applications in algebraic geometry and Donaldson-Thomas theory.

Contribution

It defines algebraic d-critical loci and extends the theory to complex analytic and stack settings, linking classical and derived geometric structures.

Findings

Defines algebraic d-critical loci as classical truncations of derived schemes

Establishes a framework for d-critical stacks and their applications

Prepares for future work on motivic and categorified Donaldson-Thomas invariants

Abstract

Let $f:U\to{\mathbb A}^1$ be a regular function on a smooth scheme $U$ over a field $\mathbb K$. Pantev, Toen, Vaquie and Vezzosi (arXiv:1111.3209, arXiv:1109.5213) define the "derived critical locus" Crit$(f)$, an example of a new class of spaces in derived algebraic geometry, which they call "$-1$-shifted symplectic derived schemes". They show that intersections of algebraic Lagrangians in a smooth symplectic $\mathbb K$-scheme, and stable moduli schemes of coherent sheaves on a Calabi-Yau 3-fold over $\mathbb K$, are also $-1$-shifted symplectic derived schemes. Thus, their theory may have applications in algebraic symplectic geometry, and in Donaldson-Thomas theory of Calabi-Yau 3-folds. This paper defines and studies a new class of spaces we call "algebraic d-critical loci", which should be regarded as classical truncations of $-1$-shifted symplectic derived schemes. They are simpler than their derived analogues. We also give a complex analytic version of the theory, "complex analytic d-critical loci", and an extension to Artin stacks, "d-critical stacks". In the sequels arXiv:1305.6302, arXiv:1211.3259, arXiv:1305.6428, arXiv:1312.0090 we will define truncation functors from $-1$-shifted symplectic derived schemes or stacks to algebraic d-critical loci or d-critical stacks, and we will apply d-critical loci to motivic and categorified Donaldson-Thomas theory, and to intersections of (derived) complex Lagrangians in complex symplectic manifolds. We will show that the important structures one wants to associate to a derived critical locus -- virtual cycles, perverse sheaves and mixed Hodge modules of vanishing cycles, and motivic Milnor fibres -- can be defined for oriented d-critical loci and oriented d-critical stacks.