What are the Hermitian metrics in a trivial line bundle on a Riemann surface X? I read that a Hermitian metric in the trivial line bundle is equivalent to a $\mathcal{C}^{\infty}$ …

Let $C$ be a stable reducible curve. Is there a natural way to define it's canonical model (I guess via the dualizing sheaf)? And does somehow the dualizing sheaf restrict to the ( …

Suppose $X$ is a projective variety over $\mathbb C$. I am happy to entertain more or different adjectives — I'm not looking for the most general statement, but rather to understa …

Let $L$ be a line bundle on an (algebraic) K3 surface over a field $k$. The Riemann-Roch theorem specializes to $$ \chi(X, L)=\frac{1}{2}(L\cdot L)+2 $$ which can be rewritten …

Context: This question arose as I was reading the proof of Application 6.1 in Mumford's Abelian Varieties. However, I have extracted all of the relevant information below so this …

Suppose you have a one parameter family of algebraic varieties over unit disk, such that the central fiber is singular and is a union (normal-crossing) of two varieties and the res …

Can anyone help me with the following question? Let $X$ be a smooth, projective algebraic variety over a field $k$ of characteristic zero. Let $D$ be an effective divisor on $X$ an …

How does the line bundles look like on a proper model (or Néron model) of an abelian variety? Who knows references about this? In particular, let us work over a trait $S=\mathrm{ …

For sure answers to my questions are well known - but I never saw them anywhere. Let $X$ be a smooth projective (or just proper) variety over an algebraically closed field $k$. Le …

When dealing with some lifting problems, I came across the following problem, which probably has a well-known answer, but anyway: Suppose I have a (locally contractible) topologic …

A complex manifold which admits a positive line bundle is automatically Kähler. Furthermore, if the manifold is compact, then it is projective by the Kodaira Embedding Theorem. In …

I have a confusion regarding the line bundles arising in Kahler quantization for the torus. I know of course that the space of holomorphic sections should be isomorphic to a space …

Recall that the Aharonov-Bohm effect can be interpreted mathematically as follows. Consider an electromagnetic field A on some smooth manifold M, i.e., A is an element in the firs …

Let $T_\tau$ be the 2-dimensional torus, with the complex structure induced by the lattice generated by $1$ and $\tau$. Then for a line bundle $L_k$ over $T$ with level $k$, there …

I would like to find some notes (or book), that explains on a very basic level what is a line bundle on a complex projective manifold. Maybe even, what is a line bundle on $\mathb …