arXiv subjects

Denis V. Osipov

Publications and source records attributed to Denis V. Osipov.

3 recordsLinked to original sources

arXiv2026

Contou-Carrère symbol, Deligne pairing and quintets

We prove the reciprocity laws for a family of projective curves over an affine base scheme, where curves can be singular and reducible, and the base can be non-Noetherian. These reciprocity laws generalize the Weil reciprocity law for a projective curve over a field and the reciprocity law for the Contou-Carrère symbol. We apply the proven reciprocity laws to describe the actions of certain central extensions of the group ind-scheme related with the formal punctured disc on certain line bundles on the moduli stacks of quintets. A quintet consists of geometric data including a family of curves and a line bundle on this family. The line bundles on the moduli stacks are constructed using the Deligne pairings of line bundles from quintets.

math.AG↗

arXiv2026

Relative analytic reciprocity laws

We study reciprocity laws involving complex line bundles on fibrations in oriented circles. In particularly, we prove the following reciprocity law. Let $B$ be a complex manifold and $π_i : M_i \to B$ be a fibration in oriented circles, where $i$ runs through a finite set. Let $L_i$ and $N_i$ be complex line bundles on every $M_i$. The reciprocity law states that the sum of all $(π_i)_* \left(c_1(L_i) \cup c_1(N_i) \right)$, where $(π_i)_*$ is the Gysin map and $c_1$ is the first Chern class, equals zero in $H^3(B, {\mathbb Z})$ when the disjoint union of all $M_i$ is embedded into a holomorphic family of compact Riemann surfaces over the base $B$ such that in every fiber of this family the disjoint union of the embedded circles is the boundary of an embedded compact Riemann surface with boundary, and all $L_i$ and all $N_i$ are restrictions of holomorphic line bundles on this family.

math.CV↗

arXiv2025

Analytic diffeomorphisms of the circle and topological Riemann-Roch theorem for circle fibrations

We consider the group $\mathcal G$ which is the semidirect product of the group of analytic functions with values in ${\mathbb C}^*$ on the circle and the group of analytic diffeomorphisms of the circle that preserve the orientation. Then we construct the central extensions of the group $\mathcal G$ by the group ${\mathbb C}^*$. The first central extension, so-called the determinant central extension, is constructed by means of determinants of linear operators acting in infinite-dimensional locally convex topological $\mathbb C$-vector spaces. Other central extensions are constructed by $\cup$-products of group $1$-cocycles with the application to them the map related with algebraic $K$-theory. We prove in the second cohomology group, i.e. modulo of a group $2$-coboundary, the equality of the $12$th power of the $2$-cocycle constructed by the first central extension and the product of integer powers of the $2$-cocycles constructed above by means of \linebreak $\cup$-products (in multiplicative notation). As an application of this result we obtain a new topological Riemann-Roch theorem for a complex line bundle $L$ on a smooth manifold $M$, where $π:M \to B$ is a fibration in oriented circles. More precisely, we prove that in the group $H^3(B, {\mathbb Z})$ the element $12 \, [ {\mathcal Det} (L)]$ is equal to the element $6 \, π_* ( c_1(L) \cup c_1(L))$, where $[{\mathcal Det} (L)]$ is the class of the determinant gerbe on $B$ constructed by $L$ and the determinant central extension.

math.DG↗