The tangent complex of an analytic space

Let f : U → A 1 be a regular function on a smooth scheme U over a field K. Pantev, Toën, Vaquié and Vezzosi [30,37] 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 K-scheme, and stable moduli schemes of coherent sheaves on a Calabi-Yau 3-fold over 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 the −1-shifted symplectic derived schemes of [30]. They are simpler than their derived analogues. We also give a complex analytic version of the theory, and an extension to Artin stacks.In the sequels [4-8] we will apply d-critical loci to motivic and categorified Donaldson-Thomas theory, and to intersections of complex Lagrangians in complex symplectic manifolds. We will show that the important structures one wants to associate to a derived critical locusvirtual cycles, perverse sheaves, D-modules, and mixed Hodge modules of vanishing cycles, and motivic Milnor fibres -can be defined for oriented d-critical loci.Pantev, Toën, Vaquié and Vezzosi [30,37] defined a new notion of derived critical locus. It is set in the context of Toën and Vezzosi's theory of derived algebraic geometry [34][35][36], and consists of a quasi-smooth derived scheme X equipped with a −1-shifted symplectic structure ω. In fact Pantev et al. [30] define kshifted symplectic structures on derived stacks for k ∈ Z, but the case relevant to this paper is k = −1, and derived schemes rather than derived stacks.The following are examples of −1-shifted symplectic derived schemes:(a) The critical locus Crit(f ) of a regular function f : U → A 1 on a smooth K-scheme U .

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“…Here in part (b), for (co)tangent complexes of K-schemes see Illusie [15,16], and of complex analytic spaces see Palamodov [25][26][27][28].…”

Section: These Sheaves S X Also Satisfymentioning

confidence: 99%

A classical model for derived critical loci

Joyce¹

2015

J. Differential Geom.

148

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“…Note that our ∂ A differs from that in [10], in that it extends to the last index. We indicate the first few terms of the resolution.…”

Section: The Formula (31)mentioning

confidence: 95%

“…, β m ). The proof of [10,Thm. 1.1] yields that the differential in R A can be written as s A = ∂ A + g A with g A a derivation such that g A (e A,j ) is a section of S k+1 A if ψ(e A,j ) = k.…”

Section: The Formula (31)mentioning

confidence: 99%

“…A good reference, also for the next subsection, is [8]. We actually work with the analytic version of the cotangent complex, which can be constructed from a Tyurina resolvent of the analytic O S -algebra O X , see PALAMODOV's survey [12] and for more details his papers [9], [10] and [11].…”

Section: Cotangent Cohomology (11)mentioning

confidence: 99%

“…(2.1) To describe the relation between cotangent cohomology of a singularity and its first blow-up one has to globalise local constructions. We use the analytic cotangent complex of PALAMODOV; for a general overview see [12], while some technical details are to be found in [10].…”

Section: The Tangent Complex With Values In a Sheafmentioning

confidence: 99%

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