Semi-perfect obstruction theory and Donaldson–Thomas invariants of derived objects

Chang, Huai-Liang; Li, Jun

doi:10.4310/cag.2011.v19.n4.a6

Cited by 16 publications

(29 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Cycles on sheaf stacks. In this section, we recall the notion of cycles on sheaf stacks [7]. Let X be a Deligne-Mumford stack and F be a coherent sheaf on X.…”

Section: 2mentioning

confidence: 99%

“…In this section, we generalize the cosection localization principle in [21] to the setting of semi-perfect obstruction theory in [7].…”

Section: Cosection Localization For Semi-perfect Obstruction Theorymentioning

confidence: 99%

“…Unfortunately it is often hard to find perfect obstruction theories on moduli spaces of derived category objects and hence virtual fundamental classes were not readily available for them. In [7], H.-L. Chang and J. Li constructed the virtual fundamental class under a weaker condition than the existence of a perfect obstruction theory. They require only the local existence of compatible perfect obstruction theories and it was proved in [7] that these local theories are sufficient to give the virtual fundamental class with desired properties.…”

Section: Introductionmentioning

confidence: 99%

“…In [7], H.-L. Chang and J. Li constructed the virtual fundamental class under a weaker condition than the existence of a perfect obstruction theory. They require only the local existence of compatible perfect obstruction theories and it was proved in [7] that these local theories are sufficient to give the virtual fundamental class with desired properties. In fact, this weaker requirement amounts to the theory of tangent-obstruction sheaves of [31].…”

Section: Introductionmentioning

confidence: 99%

“…For many moduli spaces of derived category objects, semi-perfect obstruction theories are easier to construct (cf. [7]) and it was shown in [22] that a critical virtual manifold always has a semi-perfect obstruction theory and hence a virtual fundamental class. The goal of this paper is to generalize and establish the localization theorems in the setting of semi-perfect obstruction theory.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Localizing virtual fundamental cycles for semi-perfect obstruction theories

Kiem

2018

Int. J. Math.

View full text Add to dashboard Cite

Recently H.-L. Chang and J. Li generalized the theory of virtual fundamental class to the setting of semi-perfect obstruction theory. A semi-perfect obstruction theory requires only the local existence of a perfect obstruction theory with compatibility conditions. In this paper, we generalize the torus localization of Graber-Pandharipande [17], the cosection localization [21] and their combination [6], to the setting of semi-perfect obstruction theory. As an application, we show that the Jiang-Thomas theory [20] of virtual signed Euler characteristic works without the technical quasi-smoothness assumption from derived algebraic geometry.A few effective techniques to handle virtual fundamental classes were discovered during the past two decades, such as the torus localization of Graber-Pandharipande [17], the degeneration method of J. Li [30] and the cosection localization [21]. Often combining these techniques turns out to be quite effective. In [6], it was proved that the torus localization works for the cosection localized virtual fundamental classes and this combined localization turned out to be quite useful for the Landau-Ginzburg/Calabi-Yau correspondence [10].

show abstract

“…Cycles on sheaf stacks. In this section, we recall the notion of cycles on sheaf stacks [7]. Let X be a Deligne-Mumford stack and F be a coherent sheaf on X.…”

Section: 2mentioning

confidence: 99%

“…In this section, we generalize the cosection localization principle in [21] to the setting of semi-perfect obstruction theory in [7].…”

Section: Cosection Localization For Semi-perfect Obstruction Theorymentioning

confidence: 99%