A classical model for derived critical loci

By D. Joyce. 2013

89 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/05e5e3729dfd6c080f14fdcc908458dc35219ef6

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Citations in the corpus

Sheaves of categories and the notion of 1-affineness (2013, background): “became popular because of its role in Donaldson-Thomas theory ([19], [20]). Derived stacks are expected to play an important role in a deeper understanding of Donaldson–Thomas theory, see for example [3, 4, 5, 8, 16, 7, 10, 11, 9] and the references therein. One type of examples come from Calabi-Yau”
Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds (2015, background): “In the same way, the authors expect that there should exist a new geometric structure which is a classical truncation of −2-shifted symplectic derived Cschemes, similar in spirit to [24], and that our construction of Xdm factors through this classical truncation.”
A Darboux theorem for derived schemes with shifted symplectic structure (2013, unknown): “Then [16, Def.”
Symmetries and stabilization for sheaves of vanishing cycles (2012, background): “In the fourth paper Bussi, Joyce and Meinhardt [11] will generalize the ideas of this paper to motivic Milnor fibres, and deduce new results on motivic Donaldson–Thomas invariants using [23, 41].”
A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications (2013, methodology): “When K = C, a related result for coherent sheaves only, with U a complex manifold and f a holomorphic function, was proved by Joyce and Song [16, Th. 5.5] using gauge theory and transcendental complex methods.”