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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2023, Volume 320, Pages 243–277
DOI: https://doi.org/10.4213/tm4310
(Mi tm4310)
 

This article is cited in 2 scientific papers (total in 2 papers)

Formal Bott–Thurston Cocycle and Part of a Formal Riemann–Roch Theorem

D. V. Osipovabc

a Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
b National University of Science and Technology “MISiS”, Leninskii prosp. 4, Moscow, 119049 Russia
c International Laboratory for Mirror Symmetry and Automorphic Forms, HSE University, ul. Usacheva 6, Moscow, 119048 Russia
Full-text PDF (494 kB) Citations (2)
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Abstract: The Bott–Thurston cocycle is a $2$-cocycle on the group of orientation-preserving diffeomorphisms of the circle. We introduce and study a formal analog of the Bott–Thurston cocycle. The formal Bott–Thurston cocycle is a $2$-cocycle on the group of continuous $A$-automorphisms of the algebra $A((t))$ of Laurent series over a commutative ring $A$ with values in the group $A^*$ of invertible elements of $A$. We prove that the central extension given by the formal Bott–Thurston cocycle is equivalent to the 12-fold Baer sum of the determinantal central extension when $A$ is a $\mathbb Q$-algebra. As a consequence of this result we prove a part of a new formal Riemann–Roch theorem. This Riemann–Roch theorem is applied to a ringed space on a separated scheme $S$ over $\mathbb Q$, where the structure sheaf of the ringed space is locally on $S$ isomorphic to the sheaf $\mathcal O_S((t))$ and the transition automorphisms are continuous. Locally on $S$ this ringed space corresponds to the punctured formal neighborhood of a section of a smooth morphism to $U$ of relative dimension $1$, where $U \subset S$ is an open subset.
Funding agency Grant number
HSE Basic Research Program
The study has been funded within the framework of the HSE University Basic Research Program.
Received: May 5, 2022
Revised: November 14, 2022
Accepted: December 1, 2022
English version:
Proceedings of the Steklov Institute of Mathematics, 2023, Volume 320, Pages 226–257
DOI: https://doi.org/10.1134/S0081543823010108
Bibliographic databases:
Document Type: Article
UDC: 512.667+512.717+512.732.6
Language: Russian
Citation: D. V. Osipov, “Formal Bott–Thurston Cocycle and Part of a Formal Riemann–Roch Theorem”, Algebra and Arithmetic, Algebraic, and Complex Geometry, Collected papers. In memory of Academician Alexey Nikolaevich Parshin, Trudy Mat. Inst. Steklova, 320, Steklov Math. Inst., Moscow, 2023, 243–277; Proc. Steklov Inst. Math., 320 (2023), 226–257
Citation in format AMSBIB
\Bibitem{Osi23}
\by D.~V.~Osipov
\paper Formal Bott--Thurston Cocycle and Part of a Formal Riemann--Roch Theorem
\inbook Algebra and Arithmetic, Algebraic, and Complex Geometry
\bookinfo Collected papers. In memory of Academician Alexey Nikolaevich Parshin
\serial Trudy Mat. Inst. Steklova
\yr 2023
\vol 320
\pages 243--277
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4310}
\crossref{https://doi.org/10.4213/tm4310}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2023
\vol 320
\pages 226--257
\crossref{https://doi.org/10.1134/S0081543823010108}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85161087919}
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  • https://doi.org/10.4213/tm4310
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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