# Tagged Questions

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### Properness on an open subvariety

I was wondering if the following property holds in general For any morphism $f: X \to Y$ between two varieties over $\mathbb{C}$, there exists a nonempty open subvariety $U \subseteq Y$ such ...
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### Tracing down (or pushing forward) differentials and Abel Jacobi map on principal divisors

Let $X$ be a complete one-dimensional curve over $\Bbb C$ (please add any hypotheses in case I forgot them). Then we have the Abel-Jacobi map $$\mathcal{AJ}: \text{Div}_0 \to \Bbb C^g/ \Lambda$$ ...
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### Elliptic Operators and Continuity

I am reading a book on Hodge theory (Ref: http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/hodge-smf.pdf) or for english ...
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### some fun with holomorphic line bundles

These are probably trivial questions... (for the experts) I'd like to get convinced (perhaps an intuitive/geometric explanation will be more effective than a formal one) of the following facts: i. ...
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### Are the smooth points dense in a projection of a complex variety?

Let $X=V(I)\subset \mathbb{C}^{n}$ be the vanishing set of an ideal of complex polynomials, let $\pi \colon \mathbb{C}^{n} \to \mathbb{C}^{n-1}$ be the projection onto the first $n-1$ coordinates and ...
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### Sheaf cohomology of $\mathbb{P}^3$

Let $\mathbb{P}$ denote the projective space over $\mathbb{C}$. In some lecture notes I found the claim that $$h^0(\mathbb{P}^3, \mathcal{O}(2)) = 10$$ Do you know why this is the case? In ...
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### Meromorphic functions a constant sheaf?

Is the sheaf of meromorphic functions on a (connected) compact Riemann surface constant? I am refering here to meromorphic functions in the sense of complex analysis and not to those of algebraic ...
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### Equations for double etale covers of the hyperelliptic curve $y^2 = x^5+1$

Let $X$ be the (smooth projective model) of the hyperelliptic curve $y^2=x^5+1$ over $\mathbf C$. Can we "easily" write down equations for all double unramified covers of $X$? Topologically, these ...
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### Questions on Chern characters.

Let $X$ be a complex manifold and $\mathcal{O}_p$ a skyscraper sheaf. How can one compute the Chern character $ch(\mathcal{O}_p)$? For vecoto bundles, we have $ch(V)\cup ch(W)=ch(V\otimes W)$, but ...
Let $U$ be a smooth quasi-projective variety over $\mathbf C$, and let $V\subset U$ be a dense open subvariety. Let $X\to U$ and $Y\to U$ be smooth projective morphisms such that their restrictions ...
Let $M$ be a complex $2$-dimensional manifold and $C$ be a compact complex curve in $M$ (possibly singular). Let us suppose that there exists a holomorphic function $f\in\mathcal{O}(M)$ such that ...