Krichever correspondence for algebraic varieties

D. V. Osipov

Izvestiya. Mathematics , Tome 65 (2001) no. 5, pp. 941-975.

Résumé

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 $B\subset k((z_1))\dots((z_n))$ and $W\subset k((z_1))\dots((z_n))^{\oplus r}$ 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 $X=Y_0 \supset Y_1 \supset\dots\supset Y_n$ 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 $B\subset k((z_1))\dots((z_n))$ is associated with the flag $Y_0\supset Y_1\supset\dots\supset Y_n$. 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 $X$.

Bibliographie

[1] Atya M., Makdonald I., Vvedenie v kommutativnuyu algebru, Mir, M., 1972 | MR

[2] Beilinson A. A., “Vychety i adeli”, Funkts. analiz i ego prilozh., 14:1 (1980), 44–45 | MR | Zbl

[3] Godeman R., Algebraicheskaya topologiya i teoriya puchkov, IL, M., 1961

[4] Demidov E. E., Ierarkhiya Kadomtseva–Petviashvili i problema Shotki, Izd-vo MK NMU, M., 1995 | MR

[5] Krichever I. M., “Metody algebraicheskoi geometrii v teorii nelineinykh uravnenii”, UMN, 32 (1977), 185–214

[6] Parshin A. N., “K arifmetike dvumernykh skhem. I: Raspredeleniya i vychety”, Izv. AN SSSR. Ser. matem., 40:4 (1976), 736–773 | MR | Zbl

[7] Parshin A. N., “O koltse formalnykh psevdodifferentsialnykh operatorov”, Tr. MI RAN, 224 (1999), 291–305 ; e-print math.AG/9911098 | MR | Zbl

[8] Sigal G., Vilson Dzh., “Gruppa petel i uravneniya tipa KdF”, V kn.: E. Pressli, G. Sigal, Gruppy petel, Mir, M., 1990, 379–442 | MR

[9] Khartskhorn R., Algebraicheskaya geometriya, Mir, M., 1981 | MR | Zbl

[10] Arbarello E., De Concini C., Kac V. G., Procesi C., “Moduli Spaces of Curves and Representation Theory”, Comm. Math. Phys., 117 (1988), 1–36 | DOI | MR | Zbl

[11] Beilinson A. A., Schechtman V. V., “Determinant bundles and Virasoro algebra”, Comm. Math. Phys., 118 (1988), 651–701 | DOI | MR | Zbl

[12] Ben-Zvi D., Frenkel E., Spectral curves, opers and integrable systems, E-print math.AG/9902068 | MR

[13] Fimmel T., Parshin A. N., An introduction to the higher adelic theory, Book in preparation

[14] Huber A., “Adele für Schemata und Zariski-Kohomologie”, Schriftenreihe des Math. Inst. der Univ. Muenster. 3 Serie, Heft 3, 1991 | MR

[15] Huber A., “On the Parshin–Beilinson Adeles for Schemas”, Abh. Math. Sem. Univ. Hamburg, no. 61, 1991 | MR

[16] Mulase M., “Category of vector bundles on algebraic curves and infinite dimensional grassmanians”, Intern. J. of Math., 1:3 (1990), 293–342 | DOI | MR | Zbl

[17] Parshin A. N., “Chern classes, adeles and L-functions”, J. für die reine und angewandte Mathematik, 341 (1983), 174–192 | MR | Zbl

[18] Parshin A. N., Krichever Correspondence for Algebraic Surfaces, Preprint Bericht 99/17, TU Braunschweig | MR | Zbl

[19] Quandt I., “On a relative version of the Krichever correspondence”, Bayreuther Mathematische Schriften, 52 (1997), 1–74 ; E-print alg-geom/9606015 | MR | Zbl