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D. V. Osipov

Publications and source records attributed to D. V. Osipov.

24 records · Page 2Linked to original sources
arXiv2006

On some questions related to the Krichever correspondence

We investigate various new properties and examples of one-dimensional and two-dimensional Krichever correspondence developed by Parshin. In particular, we give explicit examples of the Krichever-Parshin map for various plane curves, we introduce analogs of the Schur pairs in a two-dimensional local field and show that they are oft geometrical. At the end we investigate analogs of the KP hierarchy for two-dimensional local skew-fields with arbitrary commutation law instead of the usual law of Weyl algebra. We derive for these hierarchies new partial differential equations, which coincide with the usual KP equation for certain values of parameters.

math.AG↗
arXiv2005

Central extensions and reciprocity laws on algebraic surfaces

We consider an algebraic surface. For an irreducible curve on this surface and for a point on this curve one can associate an artinian ring, which is a sum of two-dimensional local fields. An example of two-dimensional local field is iterated Laurent series. We consider an integer value symbol on a two-dimensional local field. This symbol appears for the description of non-ramified coverings of two-dimensional local fields and for the description of intersection index of divisors on surfaces by adeles. We interpret this symbol as the commutator of lifting of elements in the central extension of the multiplicative group of two-dimensional local field. We prove the reciprocity laws around the points and along the projective curves on the surface. The proof uses the adelic ring on the surface and is similar to the Tate method for the residue of differentials on a projective curve.

math.NT↗
arXiv2000

Krichever correspondence for algebraic varieties

In 70's there was discovered a construction how to attach to some algebraic-geometric data an infinite-dimensional subspace in the space k((z)) of the Laurent power series. Now this construction is called the Krichever map. In e-print math.AG/9911097 A.N. Parshin suggested a generalization of the Krichever map for the case of algebraic surfaces from the point of view of 2-dimensional local fields. In this work we suggest a generalization of the Krichever map to the case of algebraic varieties of arbitrary dimension from the point of view of multidimensional local fields. For surfaces our construction coincides with the Parshin construction. Besides, we obtain new explicit acyclic resolutions of quasicoherent sheaves connected with multidimensional local fields.

math.AG↗
arXiv1999

Stability of torsion free sheaves on curves and infinite-dimensional Grassmanian manifold

There exists the Krichever map from the set of quintets (C,p,F,t,e) (where C is an integral and complete algebraic curve, p a smooth rational point, F a rank 2 torsion free coherent sheaf on C, t a local formal parameter in p and e a formal trivialization of F around p) to the infinite Grassmanian of $k((z)) \oplus k((z))$. We describe the images of quintets with (semi)stable sheaves F in terms of Plucker coordinates and get some analog of GIT Hilbert-Mumford numerical criterion with respect to actions of some 1-parametric subgroups of the group $SL(2,k[[z]])$ on the determinant bundle of the infinite Grassmanian.

math.AG↗
arXiv1998

Adelic constructions of direct images for differentials and symbols

For a projective morphism of an smooth algebraic surface $X$ onto a smooth algebraic curve $S$, both given over a perfect field $k$, we construct the direct image morphism in two cases: from $H^i(X,Ω^2_X)$ to $H^{i-1}(S,Ω^1_S)$ and when $char k =0$ from $H^i(X,K_2(X))$ to $H^{i-1}(S,K_1(S))$. (If i=2, then the last map is the Gysin map from $CH^2(X)$ to $CH^1(S)$.) To do this in the first case we use the known adelic resolution for sheafs $Ω^2_X$ and $Ω^1_S$. In the second case we construct a $K_2$-adelic resolution of a sheaf $K_2(X)$. And thus we reduce the direct image morphism to the construction of some residues and symbols from differentials and symbols of 2-dimensional local fields associated with pairs $x \in C$ ($x$ is a closed point on an irredicuble curve $C \in X$) to 1-dimensional local fields associated with closed points on the curve $S$. We prove reciprocity laws for these maps.

math.AG↗