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I'm not sure about the general case, but this is at least true if the divisor associated to the curve C is ample: In this case the line bundle L associated to the divisor is ample, so the Kodaira-Nak …

I'll try to sketch a proof of Riemann-Hurwitz using the Leray spectral sequence. It has the feel of a fun exercise. To fix notation, let $X$ and $Y$ be compact Riemann surfaces and let $f : X \to Y$ …

I'm going through a proof of a vanishing theorem by Sommese ($H^{p,q}(X,L) = 0$ for $p+q > n+k$ if $L$ is $k$-ample) and have hit the following brick wall: I've got a complex projective manifold $X$, …

I'm looking for some fairly explicit varieties to use as (counter?-)examples for my thesis and I'd appreciate any suggestions. I need a smooth projective variety $X$ of general type that satisfies: …

Let $X$ be a compact Kahler manifold of complex dimension $n$. Fix a nonzero class $u \in H^1(X,T_X)$. This gives a linear morphism $$ \phi_u : H^0(X,\Omega^n) \to H^{1}(X,\Omega^{n-1}), \quad \sigma …

If $X$ is a compact Kahler manifold then it's well-known that $X$ can be embedded into a projective space if and only if it admits an ample line bundle. Suppose now that we look for other things to em …

Let $S$ be a projective surface and $L$ an ample line bundle on $S$. The Severi variety $\mathcal V_{\mathcal L,\delta}$ parametrizes curves with $\delta$ nodes and no other singularities in the linea …

Let $\pi : X \to B$ be a proper, surjective holomorphic submersion, where both $X$ and $B$ are compact Kahler manifolds. Assume that all the fibers $X_b = \pi^{-1}(b)$ are smooth. Is the family $\pi : …

Let $S$ be a complex manifold and let $p : E \to S$ be a holomorphic vector bundle. Is there any advantage to knowing that $E$ carries a flat Hermitian metric $h$, ie a smooth Hermitian metric with cu …

Let $V$ be a complex vector space of dimension $n$, equipped with a Hermitian inner product whose Kahler form we denote by $\omega$. Let's set $P = \bigwedge^{2p} V^*$ and $Q = \bigwedge^{2q} V^*$ for …

Let $\pi : X \to B$ be a family of compact Kähler manifolds over a smooth base $B$. We then have a local system $\mathcal R^k \pi_* \mathbb Z$ (for your favorite $k$) of abelian groups over $B$, whose …

Let $f : X \to Y$ be a finite morphism between compact complex manifolds of the same dimension $n$. We denote by $R_f \subset X$ the ramification divisor of $f$ and by $J_f \subset X$ the set of point …

I apologize if this question is too basic, but I haven't been able to work this out for myself. Let $X$ and $Y$ be projective schemes, say over the complex numbers. There exists a scheme $Hom(X,Y)$ p …

One of the standard conjectures in algebraic geometry is that an operator $\Lambda$ on the cohomology algebra of a projective variety is algebraic. To my lying eyes it looks like there are two definit …

Let $C$ be an elliptic curve over the complex numbers. Consider a nontrivial extension $$ 0 \to \mathcal O_C \to E \to \mathcal O_C \to 0 $$ of rank 2 of the structure sheaf of $C$. This defines a rul …