On the motivic class of the stack of bundles

Behrend, Kai; Dhillon, Ajneet

doi:10.1016/j.aim.2006.11.003

Cited by 57 publications

(86 citation statements)

References 18 publications

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“…This ring was studied by Ekedahl in a series of preprint [12][13][14]. Similar constructions have also been studied independently by other authors [3,18,24].…”

Section: Introductionmentioning

confidence: 84%

“…That M 1,(d) is a stack follows from the sheaf property of Pic d C/S . We want to establish an equivalence between the stack quotient [H ns /PGL 3 ] and M 1, (3) . First, we give an explicit description of the pre-stack quotient as a category fibred in groupoids over the category of schemes.…”

Section: Moduli Of Polarized Genus 1 Curvesmentioning

confidence: 99%

“…In particular, it follows that the multiplicativity relation for a universal G-torsor does not imply the multiplicativity relation for arbitrary G-torsors. This gives a negative answer to the question raised in [3,Remark 3.3].…”

Section: Introductionmentioning

confidence: 95%

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Motivic classes of some classifying stacks

Bergh

2015

J. London Math. Soc.

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We prove that the class of the classifying stack BPGLn is the multiplicative inverse of the class of the projective linear group PGL n in the Grothendieck ring of stacks K0(Stack k ) for n = 2 and n = 3 under mild conditions on the base field k. In particular, although it is known that the multiplicativity relation {T } = {S} · {PGL n} does not hold for all PGLn-torsors T → S, it holds for the universal PGL n-torsors for said n.Definition 1.1. Let S be an algebraic stack and let G be an algebraic group over S. We say that G is special, provided that every G-torsor over any field K over S is trivial.Examples of special groups are GL n , SL n , Sp 2n , G a and G m and extensions thereof, whereas PGL n is not special. In particular, split tori are special since they are products of G m . Nonsplit tori need not be special in general, but quasi-split tori are, also when considered over a general base. Proposition 1.1. A quasi-split torus T over any base stack S is special.

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“…This ring was studied by Ekedahl in a series of preprint [12][13][14]. Similar constructions have also been studied independently by other authors [3,18,24].…”

Section: Introductionmentioning

confidence: 84%

Section: Moduli Of Polarized Genus 1 Curvesmentioning

confidence: 99%

See 1 more Smart Citation

Motivic classes of some classifying stacks

Bergh

2015

J. London Math. Soc.

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“…(For generalizations of this result to other groups G, and to the case of smooth reductive group schemes over C, we refer the reader to the work of Behrend and Dhillon [11,12]. )…”

Section: Tamagawa Numbers and Curves Over Finite Fieldsmentioning

confidence: 99%

“…They asked for a deeper understanding of this observation and in particular for a geometric explanation, exploiting the analogy with equivariant cohomology, of the fact that the Tamagawa number of SL n is 1. Contributions since then towards such understanding have included work by Bifet, Ghione and Letizia [14,15], providing another inductive procedure for calculating the Betti numbers of the moduli spaces, which is in some sense intermediate between the arithmetic approach and the Yang-Mills approach, and more recently, work by Teleman, Behrend, Dhillon and others on the moduli stack of bundles over C (see [11,12,24,66]).…”

Section: Introduction and Overviewmentioning

confidence: 99%

Yang-Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics

Asok

Doran

Kirwan

2008

Bulletin of the London Mathematical Society

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Atiyah and Bott used equivariant Morse theory applied to the Yang-Mills functional to calculate the Betti numbers of moduli spaces of vector bundles over a Riemann surface, rederiving inductive formulae obtained from an arithmetic approach which involved the Tamagawa number of SLn. This article attempts to survey and extend our understanding of this link between Yang-Mills theory and Tamagawa numbers, and to explain how methods used over the last three decades to study the singular cohomology of moduli spaces of bundles on a smooth projective curve over C can be adapted to the setting of A 1 -homotopy theory to study the motivic cohomology of these moduli spaces over an algebraically closed field.

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Framed motivic Donaldson–Thomas invariants of small crepant resolutions

Cazzaniga

Ricolfi

2022

Mathematische Nachrichten

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For an arbitrary integer r≥1$r\ge 1$, we compute r‐framed motivic DT and PT invariants of small crepant resolutions of toric Calabi–Yau 3‐folds, establishing a “higher rank” version of the motivic DT/PT wall‐crossing formula. This generalises the work of Morrison and Nagao. Our formulae, in particular their relationship with the r=1$r=1$ theory, fit nicely in the current development of higher rank refined DT invariants.

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On the motivic class of the stack of bundles

Cited by 57 publications

References 18 publications

Motivic classes of some classifying stacks

Motivic classes of some classifying stacks

Yang-Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics

Framed motivic Donaldson–Thomas invariants of small crepant resolutions

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