Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes

By D. Huybrechts, Richard P. Thomas. 2008

155 citations recorded by the source corpus.

We give a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family. We recover and generalise the obstruction class of Lowen and Lieblich, and prove that it is a product of Atiyah and Kodaira–Spencer classes. This allows us to obtain deformation-invariant virtual cycles on moduli spaces of objects of the derived category on threefolds.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2dcc3850b925b8a5d0ee3369d684986833b6d8fe

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Citations in the corpus

The integral Hodge conjecture for two-dimensional Calabi–Yau categories (2020, background): “If the extension A → A is trivial, then by the main theorem of [41] we have ω(E) = (idE ⊗ κ(X )) ◦ A(E),”
Stable pairs and BPS invariants (2007, background): “ants, a virtual cycle is required. The usual deformation theory of pairs is problematic, but the ﬁxed-determinant deformation theory of the associated complex (0.1) I• = {O X −→s F}∈Db(X) is shown in [26, 13] to deﬁne a perfect obstruction theory for Pn(X,β) of virtual dimension zero. A virtual cycl”
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].”
Curves on K3 surfaces and modular forms (2010, background): “The rest of the proofs in [15, 45] go through as before.”
A Darboux theorem for derived schemes with shifted symplectic structure (2013, unknown): “Using this, we propose to deﬁne Donaldson–Thomas style invariants counting (semi)stable coherent sheaves on Calabi–Yau 4-folds.”
A theory of generalized Donaldson–Thomas invariants (2008, methodology): “A very similar picture is explained by Huybrechts and Thomas [45], when they show that obstruction class ob for deforming a complex E in D(coh(X)) is the composition of an Atiyah class depending on E, and a Kodaira–Spencer class depending on the square-zero extension.”
Curve counting via stable pairs in the derived category (2007, background): “ck of the canonical bundle of X. We obtain a map (2.13) Rπ∗(RHom(I•,I•) 0 ⊗ωπ)[2] → L• P(X). We claim that (2.13) provides a perfect obstruction theory for P(X) in the sense of [7] via the results of [24, 41, 42]. More precisely, the obstruction classes of [41, 42] are expressed in [24] as a product”
Curve counting theories via stable objects I. DT/PT correspondence (2009, background): “ numerical type v, hence ǫv(σ) = δv(σ) = [[M/G m] → Oi bj(A)], for an algebraic space M with a perfect symmetric obstruction theory and Gm acting trivially, and iis an open immersion of stacks. (cf. [12].) Therefore we have DTn,β(σ) = X i iχ(ν−1 M(i)) = Z [Mvir] 1. The generating series are similarl”
Hodge theory and derived categories of cubic fourfolds (2012, unknown): “The obstruction to deforming O∆S ∈ D(S×S) with this deformation is [23, 24] π∗ 1κS ◦At(O∆S ) ∈ Ext (2)(O∆S ,O∆S ) = HH (2)(S).”
Curve counting theories via stable objects Ⅱ : DT/ncDT flop formula (2009, unknown): “We consider a family of weak stability conditions σξ(t) = (Zξ(t), pBX/Y ) ∈ pUX/Y , which connects σξ and σξ′, where ξ(t) is given by ξ(t) = tξ + (1− t)ξ′.”
Good degeneration of Quot-schemes and coherent systems (2011, methodology): “I δ+,δ0 Y / ⋄ → Aδ+,δ0 ⋄ . Let π+ : Y+ −→ I δ+,δ0 Y+/A⋄ and IZ+ ⊂ OY+ be the universal underlying family and the universal ideal sheaf of Iδ+,δ0 Y+/A⋄ . Proposition 6.3 ([MPT10]). The Atiyah class in [HT10] deﬁnes a perfect relative obstruction theory (6.3) φ+ : E∨ + := Rπ+∗RHom(IZ +,IZ+)0[1] ∨ −→ L”