Symmetric Obstruction Theories and Hilbert Schemes of Points on Threefolds

By K. Behrend, Barbara Fantechi. 2005

187 citations recorded by the source corpus.

Recall that in an earlier paper by one of the authors DonaldsonThomas type invariants were expressed as certain weighted Euler characteristics of the moduli space. The Euler characteristic is weighted by a certain canonical Z-valued constructible function on the moduli space. This constructible function associates to any point of the moduli space a certain invariant of the singularity of the space at the point. In the present paper, we evaluate this invariant for the case of a singularity which is an isolated point of a C ∗ -action and which admits a symmetric obstruction theory compatible with the C ∗ -action. The answer is (−1) d , where d is the dimension of the Zariski tangent space. We use this result to prove that for any threefold, proper or not, the weighted Euler characteristic of the Hilbert scheme of n points on the threefold is, up to sign, equal to the usual Euler characteristic. For the case of a projective Calabi-Yau threefold, we deduce that the DonaldsonThomas invariant of the Hilbert scheme of n points is, up to sign, equal to the Euler characteristic. This proves a conjecture of Maulik-NekrasovOkounkov-Pandharipande.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bfb5cb9824fdbc73cb817dda3a2e7e83df387608

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Citations in the corpus

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”
Wall crossing from Boltzmann black hole halos (2010, unknown): “8) where M(q) = ∏∞ k=1(1− qk)−k is the Mac-Mahon function [53, 54, 55, 56].”
Non-commutative Donaldson-Thomas theory and the conifold (2007, background): “ength one empty room” F 1. Proof. The rank-2 torus T W acts on all moduli spaces M v. By construction, the symmetric obstruction theory constructed in Theorem 1.3.1 is T W-equivariant in the sense of [4]. Since D v is a weighted Euler characteristic, with weights which are constant on T W-orbits, no”
A theory of generalized Donaldson–Thomas invariants (2008, methodology): “Behrend and Fantechi prove the following theorem, which both explains the term obstruction theory and provides a criterion for verification in practice: Theorem 12.7 (Behrend and Fantechi [5, Th. 4.5]).”
Curve counting theories via stable objects Ⅱ : DT/ncDT flop formula (2009, background): “Here DT0(X) is given by [2], [18], [16], DT0(X) := ∑”
Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (2008, methodology): “istic. By Thom-Sebastiani theorem this number does not change if we add to fa function with a quadratic singularity at x(stable equivalence). 111 Let Mbe a scheme with perfect obstruction theory (see [4]). Thus Mis locally represented as a scheme of critical points of a function fon a manifold X. Th”
Motivic degree zero Donaldson–Thomas invariants (2009, background): “A virtual motive of Z is an element [Z]vir in a suitably augmented Grothendieck group of varieties (the “ring of motivic weights ”, see §1.1 MC) such that χ ([Z]vir) = χvir(Z).”
Hall algebras and curve-counting invariants (2010, background): “It is immediate from the definition that the functor D = RHomOM (−,OM )[2] : D Coh(M) → D Coh(M) is a contravariant equivalence and satisfies D(2) ∼= id.”
Calabi-Yau algebras (2006, unknown): “ a smooth G-variety and φa Ginvariant regular function on X. Then, there is a well deﬁned virtual fundamental class crit(φ) in the G-equivariant Borel-Moore homology of X. By the main result of [Be], [BF], one has ZG [crit(φ)]vir 1 = χG(Van(φ)). (2.10.6) Problem 2.10.7. Let a reductive group Gact li”
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]?”
On the noncommutative Donaldson-Thomas invariants arising from brane tilings (2008, unknown): “utative Donaldson-Thomas theory in the case of conifold to the case of quiver potentials arising from arbitrary brane tilings (see Section 3). That is, we compute the Donaldson-Thomas type invariants [1] of the moduli spaces of framed cyclic modules over the quiver potential algebra. The Donaldson-T”
Stability conditions and curve counting invariants on Calabi–Yau 3-folds (2011, methodology): “(i) In the case of β = 0, the generating series of In,0 is computed by Li [27], Behrend-Fantechi [5] and Levine-Pandharipande [26], ∑”
Curve counting via stable pairs in the derived category (2007, methodology): “unction is deﬁned [43] by dividing out by the degree 0 series, Z′ DT,β(q) = ZDT,β(q) ZDT,0(q) . The degree 0 series is evaluated by the formula ZDT,0(q) = M(−q)χ(X), conjectured in [43] and proved in [8, 34, 38]. Here, M(q) = Y n≥1 (1−qn)−n is the MacMahon function. The reduced series Z′ DT,β(q) is ”
Derived categories of small toric Calabi-Yau 3-folds and counting invariants (2008, methodology): “c to the moduli space of ﬁnite dimensional quotients of a module over the quiver with the superpotential µkr ◦···◦µk1(A). As a consequence, we can apply Behrend-Fantechi’s torus localization theorem ([BF08]) to show that the weighted Euler characteristics coincide with the Euler characteristics up t”