All Questions

For a non-compact Riemann surface $X$ there is an isomorphism: $$\Omega(X)/\mathrm d \mathcal O(X)\simeq H^1(X,\mathbb C)$$ where $\Omega$ is the sheaf of holomorphic forms on $X$. The group on the ...

Motivation: For a given rank 2 vector bundle we want to know how many theta-characteristic valued twisted endomorphisms it has. Setting: Let $C$ be a smooth algebraic curve over a field of ...

Let $K$ be the canonical line bundle of a compact Riemann surface $M$ of genus $g$. Consider the pull back of $K$ on $M \times M$ via projection on the first factor. What is the dimension of the space ...

Let $A$ be a sheaf such that $$A(U) = \{ f \in \mathbb M(U): f \in \mathbb{O}(U \backslash\{p_1,\ldots, p_n\}) \ \mbox{with at worst a simple pole at}\ p_i \} $$ where $\mathbb M(U)$ means the set of ...

Suppose $X$ is a Riemann surface. If $X$ is compact, then Serre duality tells us that we have an isomorphism in sheaf cohomology $$ H^1(X,E) \cong H^0(X,\Omega\otimes E^\ast)^\ast $$ Can we say ...