Virtual pullbacks in K-theory

Qu, Fufa

doi:10.5802/aif.3194

Cited by 33 publications

(57 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We study K-theoretic invariants using virtual equivariant localization as developed in [FG10,GP99], [CK09] and [Qu18]; a concise exposition of the application we use can be found in [Tho20]. In this subsection, let M be a quasi-projective scheme equipped with the action of a torus T and a T-equivariant perfect obstruction theory.…”

Section: Equivariant Localizationmentioning

confidence: 99%

K-theoretic Donaldson–Thomas theory and the Hilbert scheme of points on a surface

Arbesfeld¹

2021

Alg. Geom.

View full text Add to dashboard Cite

Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface frequently arise in enumerative problems. We use the K-theoretic Donaldson-Thomas theory of certain toric Calabi-Yau threefolds to study K-theoretic variants of such expressions.We study limits of the K-theoretic Donaldson-Thomas partition function of a toric Calabi-Yau threefold under certain one-parameter subgroups called slopes and formulate a condition under which two such limits coincide. We then explicitly compute the limits of components of the partition function under so-called preferred slopes, obtaining explicit combinatorial expressions related to the refined topological vertex of Iqbal, Kosçaz and Vafa.Applying these results to specific Calabi-Yau threefolds, we deduce dualities satisfied by a generating function built from tautological bundles on the Hilbert scheme of points on C 2 . We then use this duality to study holomorphic Euler characteristics of exterior and symmetric powers of tautological bundles on the Hilbert scheme of points on a general surface.

show abstract

Section: Equivariant Localizationmentioning

confidence: 99%

K-theoretic Donaldson–Thomas theory and the Hilbert scheme of points on a surface

Arbesfeld¹

2021

Alg. Geom.

View full text Add to dashboard Cite

show abstract

“…In the subsequent sections, we will generalize the torus localization theorem [7,26], the cosection localization theorem [12,14] and the wall-crossing formula [5] to the virtual structure sheaves associated with almost perfect obstruction theories.…”

Section: Almost Perfect Obstruction Theory and Virtual Structure Sheafmentioning

confidence: 99%

“…A virtual torus localization formula has been established at the level of intersection theory for virtual fundamental cycles in the cases of perfect [7] and semi-perfect obstruction theory [11] and at the level of -theory for virtual structure sheaves for perfect obstruction theory [26]. In this section, we generalize the formula to the setting of virtual structure sheaves in -theory obtained by an almost perfect obstruction theory.…”

Section: Virtual Torus Localizationmentioning

confidence: 99%

“…Now we can prove the following. Finally, the following standard equality for virtual pullbacks holds in our setting (see also [26,Proposition 2.14]).…”

Section: Refined Intersection With the Fixed Locusmentioning

confidence: 99%

“…Several techniques have been developed to handle virtual fundamental cycles and virtual structure sheaves arising from perfect obstruction theories on Deligne-Mumford stacks, such as the virtual torus localization of Graber-Pandharipande [7,26], the cosection localization of Kiem-Li [12,14], virtual pullback [22,26] and wall-crossing formulas [13]. Often, combining these (see also, for example, [5]) can be quite effective.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Localizing virtual structure sheaves for almost perfect obstruction theories

Kiem

Savvas

2020

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and K-theoretic invariants for many moduli stacks of interest, including K-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the K-theory and Gysin maps of sheaf stacks. In this paper, we generalize the virtual torus localization and cosection localization formulas and their combination to the setting of almost perfect obstruction theory. To this end, we further investigate the K-theory of sheaf stacks and its functoriality properties. As applications of the localization formulas, we establish a K-theoretic wall-crossing formula for simple $\mathbb{C} ^\ast $ -wall crossings and define K-theoretic invariants refining the Jiang-Thomas virtual signed Euler characteristics.

show abstract

Vertical Vafa–Witten invariants

Laarakker¹

2021

Sel. Math. New Ser.

View full text Add to dashboard Cite

Virtual pullbacks in K-theory

Abstract: We consider virtual pullbacks in K-theory, and show that they are bivariant classes and satisfy certain functoriality. As applications to K-theoretic counting invariants, we include proofs of a virtual localization formula for schemes and a degeneration formula in Donaldson-Thomas theory.

Cited by 33 publications

References 21 publications

K-theoretic Donaldson–Thomas theory and the Hilbert scheme of points on a surface

K-theoretic Donaldson–Thomas theory and the Hilbert scheme of points on a surface

Localizing virtual structure sheaves for almost perfect obstruction theories

Vertical Vafa–Witten invariants

Contact Info

Product

Resources

About