Donaldson-Thomas invariants via microlocal geometry

By K. Behrend. 2005

145 citations recorded by the source corpus.

We prove that Donaldson-Thomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory used to define them. We also introduce new invariants generalizing Donaldson-Thomas type invariants to moduli problems with open moduli space. These are useful for computing Donaldson-Thomas type invariants over stratifications.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/78a99f4cfca16a2ea38c95690be093bf3f759c77

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In Mathematical Physics

Citations in the corpus

Curve counting via stable pairs in the derived category (2007, background): “13In the Calabi-Yau case, the obstruction theory on Pn(X, β) is self-dual [4, 15] and the virtual count Pn,β is the virtual Euler characteristic, up to sign.”
Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (2008, methodology): “collection (Ω(γ))γ∈Γ seems to be the correct mathematical deﬁnition of the counting of BPS states in String Theory. Finally, we make a comment about the relationship with the work of Kai Behrend (see [2]). Recall that he deﬁned a Z-valued invariant of a critical point xof a function fon Xwhich is eq”
Stability conditions and curve counting invariants on Calabi–Yau 3-folds (2011, background): “In fact for any C-scheme M , Behrend [2] constructs a canonical constructible function, νM : M → Z, satisfying the following properties.”
Curves on K3 surfaces and modular forms (2010, background): “Is the answer to Question 1 or 2 positive if the obstruction theory E• is symmetric [2, 4]?”
Motivic degree zero Donaldson–Thomas invariants (2009, background): “The virtual Euler characteristic of Z is defined to be the topological Euler characteristic, weighted by the integervalued constructible function νZ introduced by the first author [1]: χvir(Z) = ∑ k∈Z kχ ( ν−1 Z (k) ) .”
Polynomial Bridgeland stability conditions and the large volume limit (2007, methodology): “e pairs and one-dimensional torsion sheaves. Let X be a Calabi-Yau threefold, and β,nas before. By a counting invariant we will always denote the signed weighted Euler characteristic (in the sense of [Beh05]) of a moduli space of stable objects of some ﬁxed numerical class, and with trivialized d et”
Hilbert schemes and stable pairs: GIT and derived category wall crossings (2009, unknown): “ual cycle: Ivir m,β = Z [Im(X,β)]vir 1, and Pvir m,β = Z [Pm(X,β)]vir 1. In this case the deformation-obstruction theories [Th, PT1, HT] used to deﬁne the virtual cycles are self dual in the sense of [Be]. This means that Ivir m,β, P vir m,β are in fact weighted Euler characteristics: Ivir m,β = e(I”
Calabi-Yau algebras (2006, unknown): “d symmetric obstruction theory. Let φbe a regular function on a smooth variety, or a smooth stack X . The substack crit(φ) of critical points 13 of φhas virtual dimension zero. Moreover, according to [Be], such a substack automatically comes with a symmetric perfect obstruction theory provided by th”
Derived categories of small toric Calabi-Yau 3-folds and counting invariants (2008, methodology): “pace of ideal sheaves IZ of one dimensional subschemes OZ ⊂ OY such that χ(OZ) = nand [Z] = β. We deﬁne the Donaldson-Thomas invariants In,β from In(Y,β) using Behrend’s function as is §2.3 ([Tho00], [Beh]), and their generating function by ZDT(Y;q,t) := X n,β In,β ·qntβ where tβ = Q k∈I t βk k and ”
An introduction to motivic Hall algebras (2010, background): “Recall [1] that Behrend associates to any scheme S of finite type over C a constructible function νS : S → Z with the property that if S is a proper moduli scheme with a symmetric obstruction theory then the associated Donaldson–Thomas virtual count coincides with the weighted Euler characteristic:”
Localizing virtual cycles by cosections (2010, unknown): “we have the composition E −→ τLM/S −→ π ∗ SΩS [1]”
Stable pairs and BPS invariants (2007, background): “ ofanynonzero ssection is 0-dimensional and (F,s) is a stable pair. Since Xis Calabi-Yau, both Pn(X,β) [26] and Mn(X,β) [31] have self-dual obstruction theories. We can therefore apply the results of [3]. Lemma 1.3. The obstruction theory of Pn(X,β) obtained from ﬁxed determinant deformations in the”
Curve counting theories via stable objects Ⅱ : DT/ncDT flop formula (2009, methodology): “[7] There are vector bundles pE on X for p = 0,−1, which admit derived equivalences, pΦ = Rf∗RHom( pE , ∗) : Db(Coh(X)) ∼= −→ Db(Coh(pAY )).”
Algebraic cobordism revisited (2006, methodology): “is then obtained from the toric calculations of [23] via the complex cobordism class. A proof of Conjecture 1 in case X is a Calabi-Yau 3-fold via a study of self-dual obstruction theories appears in [1, 2]. Our proof is direct and algebraic, but depends upon the construction of relative Donaldson-T”