Curve counting via stable pairs in the derived category

By R. Pandharipande, Richard P. Thomas. 2007

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For a nonsingular projective 3-fold X, we define integer invariants virtually enumerating pairs (C,D) where C⊂X is an embedded curve and D⊂C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of X. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category.Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described.The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We prove that our integrality predictions for Gromov-Witten invariants agree with the BPS integrality. Conversely, the BPS geometry imposes strong conditions on the enumeration of pairs.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e0cc8acaf66463b1f3f41e53e925a8cb958a7467

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Citations in the corpus

Cohomological χ–independence for moduli of one-dimensional sheaves and moduli of Higgs bundles (2020, background): “20] on the generalized DT invariants, which in turn implies the strong rationality conjecture for Pandharipande-Thomas invariants [43, 50].”
Curves on K3 surfaces and modular forms (2010, methodology): “Standard obstruction theory As in [45], given a stable pair (F, s) on X, we let I• ∈ D(X) denote the complex of sheaves I• = {OX s −→ F} in degrees 0 and 1.”
The Gopakumar-Vafa formula for symplectic manifolds (2013, methodology): “it, including two ([BP1] and [BP2]) relevant to our approach. For algebraic 3-folds, several BPS-type integer invariants have been dened using holomorphic bundles, including the Pandharipande-Thomas [PT] and Donaldson-Thomas invariants, with conjectural GV-type correspondences GW/DT/PT between them.”
The moduli space of stable quotients (2009, background): “The principal cases (so far) are that (i) stable maps to nonsingular varieties (see Behrend and Fantechi [2], Kim, Kresch and Oh [17] and Li and Tian [22]), (ii) stable sheaves on nonsingular 3–folds (see Pandharipande and Thomas [36] and Thomas [40]), Published: 1 October 2011 DOI: 10.2140/gt.2011.”
Good degeneration of Quot-schemes and coherent systems (2011, methodology): “The PT stable pair invariant was introduced by Pandharipande and Thomas in [PT09].”
Categorification of Donaldson-Thomas invariants via Perverse Sheaves (2012, methodology): “Also by [12] and [35], (3) isknownfor h= 0. Ofcourse, (1)-(3) areconsequencesof(4). Furthermore, establishing the identity (8.1) will equate Deﬁnition 8.3 with that introduced by Pandharipande-Thomas [29] for a large class of CY 3-folds (cf. [28]). 8.3. K3-ﬁbered CY 3-folds. In this last subsection,”
Refined curve counting on complex surfaces (2012, methodology): “q/: Motivated by a circle of ideas relating curve counting, Hilbert schemes on singular curves, and knot invariants (see [25; 26; 43; 50; 58; 61; 62; 67], Katz, Klemm and Vafa [37], Oblomkov and Shende [59], Maulik and Yun [52], Migliorini and Shende [54]), we consider a “zeta function” defined usin”
Gromov-Witten/pairs correspondence for the quintic 3-fold (2012, background): “For Calabi-Yau 3-folds, the formulation in terms of stable pairs [34] was proven to be equivalent in [4, 40].”
Curve Counting on K3 × E, The Igusa Cusp Form χ10, and Descendent Integration (2016, background): “Z, we deﬁne (31) PX n,β,d = Z [Pn(X,(β,d))]red τ0 π∗ 1(β ∨)∪π∗ 2([0]) . We follow here the notation of Section 0 for the projections π1 and π2. The insertions in stable pairs theory are deﬁned in [33]. Deﬁnition (31) is parallel to (2). As in the Gromov-Witten case, deﬁnition (31) is independent o”
Stability conditions and curve counting invariants on Calabi–Yau 3-folds (2011, background): “Also an equivalence between DT and PT theories is conjectured by Pandharipande-Thomas [32].”
Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds (2008, result): “Yau toric 3-fold X in a form closely resembling the topological vertex was obtained in [17]. 4.1.2 Stable pairs The virtual enumeration of stable pairs in the derived category of X was conjectured in [33, 34] to be equivalent to both Gromov-Witten and DonaldsonThomas theory. After the triangle of eq”
A theory of generalized Donaldson–Thomas invariants (2008, methodology): “To study our new invariants D̄T(τ), we find it helpful to introduce another family of invariants PI(τ ), similar to Pandharipande–Thomas invariants [86].”
Gromov-Witten theory and Noether-Lefschetz theory (2007, background): “onjectures 1 and 2 for genus 1 from the genus 0 results. Conjecture 2 for primitive classes β is connected to Euler characteristics of the moduli spaces of stable pairs on K3 by the correspondence of [44, 45]. A proof of Conjecture 2 for primitive classes is given in [40]. 3. Theorem 1 3.1. Result. ”
Curve counting theories via stable objects Ⅱ : DT/ncDT flop formula (2009, background): “ 0-dimensional cokernel. As a convention, the pair (0,0) is also a stable pair. For (n,β) ∈Z⊕N1(X), we denote by Pn(X,β) the moduli space of stable pairs (F,s) with [F] = β, χ(F) = n. It is proved in [26] that Pn(X,β) is a projective scheme with a symmetric perfect obstruction theory, by viewing a s”
Hall algebras and curve-counting invariants (2010, background): “Pandharipande and Thomas [23] introduced new curve-counting invariants by considering stable pairs.”