A classical model for derived critical loci

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 critical Crit$(f)$, an example of a new class of spaces in derived geometry, which they call symplectic derived They show that intersections of 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 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 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 analytic version of the theory, complex analytic loci, and an extension to Artin stacks, d-critical 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 loci or stacks, and we will apply loci to motivic and categorified Donaldson-Thomas theory, and to intersections of (derived) Lagrangians in 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 loci and oriented stacks.