Email this article Central extensions and the Riemann-Roch theorem on algebraic surfaces
- Authors: Osipov D.V.1,2,3
-
Affiliations:
- Steklov Mathematical Institute of Russian Academy of Sciences
- HSE University
- National University of Science and Technology «MISIS»
- Issue: Vol 213, No 5 (2022)
- Pages: 101-125
- Section: Articles
- URL: https://journals.rcsi.science/0368-8666/article/view/133450
- DOI: https://doi.org/10.4213/sm9623
- ID: 133450
Cite item
Open Access
Access granted
Subscription Access
Abstract
We study canonical central extensions of the general linear group over the ring of adeles on a smooth projective algebraic surface $X$ by means of the group of integers. Via these central extensions and the adelic transition matrices of a rank $n$ locally free sheaf of $\mathcal{O}_X$-modules we obtain a local (adelic) decomposition for the difference of Euler characteristics of this sheaf and the sheaf $\mathcal{O}_X^n$. Two distinct calculations of this difference lead to the Riemann-Roch theorem on $X$ (without Noether's formula). Bibliography: 21 titles.
About the authors
Denis Vasilievich Osipov
Steklov Mathematical Institute of Russian Academy of Sciences; HSE University; National University of Science and Technology «MISIS»
Email: d_osipov@mi-ras.ru
Doctor of physico-mathematical sciences, no status
References
- А. А. Бейлинсон, “Вычеты и адели”, Функц. анализ и его прил., 14:1 (1980), 44–45
- A. A. Beilinson, V. V. Schechtman, “Determinant bundles and Virasoro algebras”, Comm. Math. Phys., 118:4 (1988), 651–701
- J.-L. Brylinski, P. Deligne, “Central extensions of reductive groups by $mathrm K_2$”, Publ. Math. Inst. Hautes Etudes Sci., 94 (2001), 5–85
- B. L. Feigin, B. L. Tsygan, “Riemann–Roch theorem and Lie algebra cohomology. I”, Proceedings of the Winter school on geometry and physics (Srni, 1988), Rend. Circ. Mat. Palermo (2) Suppl., 21, Circ. Mat. Palermo, Palermo, 1989, 15–52
- A. Huber, “On the Parshin–Beilinson adeles for schemes”, Abh. Math. Sem. Univ. Hamburg, 61 (1991), 249–273
- В. Г. Кац, Бесконечномерные алгебры Ли, Мир, М., 1993, 426 с.
- M. Kapranov, Semiinfinite symmetric powers
- D. V. Osipov, “$n$-dimensional local fields and adeles on $n$-dimensional schemes”, Surveys in contemporary mathematics, London Math. Soc. Lecture Note Ser., 347, Cambridge Univ. Press, Cambridge, 2008, 131–164
- D. Osipov, “Adeles on $n$-dimensional schemes and categories $C_n$”, Internat. J. Math., 18:3 (2007), 269–279
- Д. В. Осипов, “Неразветвленное двумерное соответствие Ленглендса”, Изв. РАН. Cер. матем., 77:4 (2013), 73–102
- D. V. Osipov, “Second Chern numbers of vector bundles and higher adeles”, Bull. Korean Math. Soc., 54:5 (2017), 1699–1718
- Д. В. Осипов, А. Н. Паршин, “Гармонический анализ на локальных полях и пространствах аделей. I”, Изв. РАН. Сер. матем., 72:5 (2008), 77–140
- Д. В. Осипов, А. Н. Паршин, “Гармонический анализ и теорема Римана–Роха”, Докл. РАН, 441:4 (2011), 444–448
- А. Н. Паршин, “К арифметике двумерных схем. I. Распределения и вычеты”, Изв. АН СССР. Сер. матем., 40:4 (1976), 736–773
- A. N. Parshin, “Chern classes, adeles and $L$-functions”, J. Reine Angew. Math., 1983:341 (1983), 174–192
- A. N. Parshin, “Representations of higher adelic groups and arithmetic”, Proceedings of the international congress of mathematicians (Hyderabad, 2010), v. 1, Hindustan Book Agency, New Delhi, 2010, 362–392
- V. V. Schechtman, “Riemann–Roch theorem after D. Toledo and Y.-L. Tong”, Proceedings of the Winter School on Geometry and Physics, Srni, 1988, Rend. Circ. Mat. Palermo (2) Suppl., 21, Circ. Mat. Palermo, Palermo, 1989, 53–81
- Ж. Серр, Алгебраические группы и поля классов, Мир, М., 1968, 285 с.
- K. I. Tahara, “On the second cohomology groups of semidirect products”, Math. Z., 129 (1972), 365–379
- J. Tate, “Residues of differentials on curves”, Ann. Sci. Ecole Norm. Sup. (4), 1:1 (1968), 149–159
- A. Yekutieli, An explicit construction of the Grothendieck residue complex, With an appendix by P. Sastry, Asterisque, 208, Soc. Math. France, Paris, 1992, 127 pp.
Supplementary files
