arXiv subjects

A. K. Raina

Publications and source records attributed to A. K. Raina.

3 recordsLinked to original sources

arXiv2012

Rank one connections on abelian varieties, II

Given a holomorphic line bundle $L$ on a compact complex torus $A$, there are two naturally associated holomorphic $Ω_A$--torsors over $A$: one is constructed from the Atiyah exact sequence for $L$, and the other is constructed using the line bundle $(p^*_1 L^*)\otimes (α^*L)$, where $α$ is the addition map on $A\times A$, and $p_1$ is the projection of $A\times A$ to the first factor. In \cite{BHR}, it was shown that these two torsors are isomorphic. The aim here is to produce a canonical isomorphism between them through an explicit construction.

math.AG↗

arXiv2011

Rank one connections on abelian varieties

Let A be a complex abelian variety. The moduli space ${\mathcal M}_C$ of rank one algebraic connections on $A$ is a principal bundle over the dual abelian variety $A^\vee=\text{Pic}^0(A)$ for the group $H^0(A, Ω^1_A)$. Take any line bundle $L$ on $A^\vee$; let ${\mathcal C}(L)$ be the algebraic principal $H^0(A^\vee, Ω^1_{A^\vee})$-bundle over $A^\vee$ given by the sheaf of connections on $L$. The line bundle $L$ produces a homomorphism $H^0(A, Ω^1_A) \rightarrow H^0(A^\vee,\, Ω^1_{A^\vee})$. We prove that ${\mathcal C}(L)$ is isomorphic to the principal $H^0(A^\vee, Ω^1_{A^\vee})$-bundle obtained by extending the structure group of the principal $H^0(A,\, Ω^1_A)$-bundle ${\mathcal M}_C$ using this homomorphism given by $L$. We compute the ring of algebraic functions on ${\mathcal C}(L)$.

math.AG↗

arXiv1996

Projective structures on a Riemann surface

For a compact Riemann surface $X$ of any genus $g$, let $L$denote the line bundle $K_{X\times X}\otimes {\cal O}_{X\times X}(2Δ)$ on $X\times X$, where $K_{X\times X}$ is the canonical bundle of $X\times X$ and $Δ$ is the diagonal divisor. We show that $L$ has a canonical trivialisation over the nonreduced divisor $2Δ$. Our main result is that the space of projective structures on $X$ is canonically identified with the space of all trivialisations of $L$ over $3Δ$ which restrict to the canonical trivialisation of $L$ over $2Δ$ mentioned above. We give a direct identification of this definition of a projective structure with a definition of Deligne.We also describe briefly the origin of this work in the study of the so-called "Sugawara form" of the energy-momentum tensor in a conformal quantum field theory.

alg-geom↗