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Izv. RAN. Ser. Mat., 2001, Volume 65, Issue 5, Pages 91–128 (Mi izv358)  

This article is cited in 20 scientific papers (total in 20 papers)

Krichever correspondence for algebraic varieties

D. V. Osipov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We construct new acyclic resolutions of quasicoherent sheaves. These resolutions are connected with multidimensional local fields. The resolutions obtained are applied to construct a generalization of the Krichever map to algebraic varieties of any dimension.
This map canonically produces two $k$-subspaces $B\subset k((z_1))…((z_n))$ and $W\subset k((z_1))…((z_n))^{\oplus r}$ from the following data: an arbitrary algebraic $n$-dimensional Cohen–Macaulay projective integral scheme $X$ over a field $k$, a flag of closed integral subschemes $X=Y_0 \supset Y_1 \supset…\supset Y_n$ such that $Y_i$ is an ample Cartier divisor on $Y_{i-1}$ and $Y_n$ is a smooth point on all $Y_i$, formal local parameters of this flag at the point $Y_n$, a rank $r$ vector bundle $\mathscr F$ on $X$, and a trivialization of $\mathscr F$ in the formal neighbourhood of the point $Y_n$ where the $n$-dimensional local field $B\subset k((z_1))…((z_n))$ is associated with the flag $Y_0\supset Y_1\supset…\supset Y_n$. In addition, the map constructed is injective, that is, one can uniquely reconstruct all the original geometric data. Moreover, given the subspace $B$, we can explicitly write down a complex which calculates the cohomology of the sheaf $\mathscr O_X$ on $X$ and, given the subspace $W$, we can explicitly write down a complex which calculates the cohomology of $\mathscr F$ on $X$.

DOI: https://doi.org/10.4213/im358

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English version:
Izvestiya: Mathematics, 2001, 65:5, 941–975

Bibliographic databases:

MSC: 14F05
Received: 21.03.2000

Citation: D. V. Osipov, “Krichever correspondence for algebraic varieties”, Izv. RAN. Ser. Mat., 65:5 (2001), 91–128; Izv. Math., 65:5 (2001), 941–975

Citation in format AMSBIB
\Bibitem{Osi01}
\by D.~V.~Osipov
\paper Krichever correspondence for algebraic varieties
\jour Izv. RAN. Ser. Mat.
\yr 2001
\vol 65
\issue 5
\pages 91--128
\mathnet{http://mi.mathnet.ru/izv358}
\crossref{https://doi.org/10.4213/im358}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1874355}
\zmath{https://zbmath.org/?q=an:1068.14053}
\transl
\jour Izv. Math.
\yr 2001
\vol 65
\issue 5
\pages 941--975
\crossref{https://doi.org/10.1070/IM2001v065n05ABEH000358}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0042220544}


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    This publication is cited in the following articles:
    1. Parshin A.N., “Integrable systems and local fields”, Communications in Algebra, 29:9 (2001), 4157–4181  crossref  mathscinet  zmath  isi  elib  scopus
    2. Lee M.H., “Tau functions associated to pseudodifferential operators of several variables”, Journal of Nonlinear Mathematical Physics, 9:4 (2002), 517–529  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Lee M.H., “Pseudodifferential operators of several variables and Baker functions”, Letters in Mathematical Physics, 60:1 (2002), 1–8  crossref  mathscinet  zmath  isi  elib  scopus
    4. Chalykh O., Etingof P., Oblomkov A., “Generalized Lamé operators”, Comm. Math. Phys., 239:1–2 (2003), 115–153  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. Rothstein M., “Dynamics of the Krichever construction in several variables”, J. Reine Angew. Math., 572 (2004), 111–138  crossref  mathscinet  zmath  isi  elib
    6. D. V. Osipov, “Central extensions and reciprocity laws on algebraic surfaces”, Sb. Math., 196:10 (2005), 1503–1527  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. A. B. Zheglov, D. V. Osipov, “On Some Questions Related to the Krichever Correspondence”, Math. Notes, 81:4 (2007), 467–476  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. Chalykh O., “Algebro-geometric Schrodinger operators in many dimensions”, Philosophical Transactions of the Royal Society A-Mathematical Physical and Engineering Sciences, 366:1867 (2008), 947–971  crossref  mathscinet  zmath  adsnasa  isi  scopus
    9. Previato E., “Multivariable Burchnall-Chaundy theory”, Philosophical Transactions of the Royal Society A-Mathematical Physical and Engineering Sciences, 366:1867 (2008), 1155–1177  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    10. Kurke H., Osipov D., Zheglov A., “Formal punctured ribbons and two-dimensional local fields”, J. Reine Angew. Math., 629 (2009), 133–170  crossref  mathscinet  zmath  isi  elib  scopus
    11. Kurke H., Osipov D.V., Zheglov A.B., “Formal Groups Arising From Formal Punctured Ribbons”, International Journal of Mathematics, 21:6 (2010), 755–797  crossref  mathscinet  zmath  isi  elib  scopus
    12. A. B. Zheglov, A. E. Mironov, “Moduli Beikera – Akhiezera, puchki Krichevera i kommutativnye koltsa differentsialnykh operatorov v chastnykh proizvodnykh”, Dalnevost. matem. zhurn., 12:1 (2012), 20–34  mathnet
    13. R. Ya. Budylin, S. O. Gorchinskiy, “Intersections of adelic groups on a surface”, Sb. Math., 204:12 (2013), 1701–1711  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    14. A. B. Zheglov, “On rings of commuting partial differential operators”, St. Petersburg Math. J., 25:5 (2014), 775–814  mathnet  crossref  mathscinet  zmath  isi  elib
    15. Kurke H., Osipov D., Zheglov A., “Commuting Differential Operators and Higher-Dimensional Algebraic Varieties”, Sel. Math.-New Ser., 20:4 (2014), 1159–1195  crossref  mathscinet  zmath  isi  scopus
    16. A. B. Zheglov, H. Kurke, “Geometric properties of commutative subalgebras of partial differential operators”, Sb. Math., 206:5 (2015), 676–717  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    17. Mazel-Gee A., Peterson E., Stapleton N., “a Relative Lubin-Tate Theorem Via Higher Formal Geometry”, Algebr. Geom. Topol., 15:4 (2015), 2239–2268  crossref  mathscinet  zmath  isi  scopus
    18. D. V. Osipov, “Adelic quotient group for algebraic surfaces”, St. Petersburg Math. J., 30 (2019), 111–122  mathnet  crossref  isi  elib
    19. A. B. Zheglov, “Surprising examples of nonrational smooth spectral surfaces”, Sb. Math., 209:8 (2018), 1131–1154  mathnet  crossref  crossref  adsnasa  isi  elib
    20. Vik. S. Kulikov, “On divisors of small canonical degree on Godeaux surfaces”, Sb. Math., 209:8 (2018), 1155–1163  mathnet  crossref  crossref  adsnasa  isi  elib
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