2013/04/30 by Dominic Joyce · 2 citations
Mathematics · #math.AG
published as Journal of Differential Geometry 101 (2015), 289-367 · 73 pages. (v3) A lot of new material added on extension to Artin stacks
arxiv created 2013/12/03 · arxiv updated 2015/10/08
Let f:U→\mathbb A1 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.