[Paper Review] Derived geometry of the first formal neighborhood of a smooth analytic cycle

Julien Grivaux|arXiv (Cornell University)|May 17, 2015

Homotopy and Cohomology in Algebraic Topology21 references3 citations

TL;DR

This paper computes the derived self-intersection of a smooth analytic cycle in its first-order infinitesimal thickening using derived algebraic geometry. It generalizes the Hochschild-Kostant-Rosenberg isomorphism to arbitrary sheaves on the thickening, establishing formality criteria via cohomological obstructions in H^2(X, Hom(V, I⊗V)), extending Arinkin-Căldăraru's results beyond locally free sheaves.

ABSTRACT

If $X$ is a smooth scheme of characteristic zero or a complex analytic manifold, and $S$ is a locally split infinitesimal thickening of $X$, we compute explicitly the derived self-intersection of $X$ in $S$.

Motivation & Objective

• To generalize the derived self-intersection computation beyond locally free sheaves in first-order thickenings of smooth schemes or complex manifolds.
• To extend the formality criterion of Arinkin and Căldăraru from locally free sheaves to arbitrary coherent sheaves on the thickened ambient space.
• To provide an intrinsic construction of the local HKR class and relate it to obstruction classes in H^2(X, Hom(V, I⊗V))
• To establish a derived equalizer description of the derived pullback via model category techniques in dg-categories.
• To unify the three cohomological obstructions (gerbe, truncation triangle, Yoneda product) in the context of general sheaves, not just locally free ones.

Proposed method

• Uses derived categories and dg-categories to model the derived pullback of sheaves from the thickened space S to the smooth cycle X.
• Applies Toën-Vaquié model structures on dg-modules over C^b(C) ⊗ C^b(C)^op to define fibrant replacements and derived functors.
• Constructs the derived equalizer via the right derived pullback R𝑖_n^* along the diagonal map i_n: X → X^n, modeling the fiber product in derived geometry.
• Employs iterated mapping cones and lax monoidal functors to describe dg-endofunctors on bounded complexes.
• Relies on the octahedral axiom and derived equalizers to identify the truncation triangle and the obstruction class.
• Uses the Yoneda product of the Atiyah class and the Kodaira-Spencer class to define the obstruction in H^2(X, Hom(V, I⊗V)).

Experimental results

Research questions

• RQ1 Under what conditions is the derived pullback Lj^*(j_*V) formal for an arbitrary coherent sheaf V on a first-order thickening?
• RQ2 How do the three cohomological obstructions—gerbe, truncation triangle, and Yoneda product—coincide for general sheaves, not just locally free ones?
• RQ3 Can the derived self-intersection of a smooth analytic cycle in its first formal neighborhood be computed explicitly using model-categorical techniques?
• RQ4 What is the intrinsic geometric meaning of the local HKR class in the context of arbitrary sheaves on the thickening?
• RQ5 How does the derived equalizer construction via R𝑖_n^* recover the derived pullback in the derived intersection setting?

Key findings

• The derived pullback Lj^*(j_*V) is formal if and only if the composition of the Atiyah class of V with the Kodaira-Spencer class of the thickening vanishes in H^2(X, Hom(V, I⊗V)).
• The three cohomological obstructions (gerbe, truncation triangle, Yoneda product) are canonically isomorphic for any coherent sheaf V on X, extending Arinkin-Căldăraru's identification to non-locally free sheaves.
• The derived equalizer of the n-fold diagonal map i_n: X → X^n is isomorphic to the derived pullback R𝑖_n^*(H^n) via the model category structure on dg-modules.
• The natural map from the derived equalizer Δ̃_H^[n] to H^n is a weak equivalence in the model category of dg-modules.
• The construction of the derived self-intersection via R𝑖_n^* provides a canonical resolution of Lj^*(j_*V) in terms of iterated cones and fibrant replacements.
• The result holds in both the algebraic and analytic settings over a field of characteristic zero, generalizing the classical HKR isomorphism to arbitrary sheaves on first-order thickenings.