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Results tagged with complex-geometry
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user 622
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
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When is an ample line bundle on an abelian variety base point free?
So, any line bundle $L$ on an abelian variety $X$ determines a type $(d_1,\ldots,d_g)$ where $d_i|d_{i+1}$. It's well known that if $d_1\geq 3$ then $L$ defines an embedding, that if $L$ has no fixed …
- 16k
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Pullback along the Torelli map is an isomorphism
I've been told many times that the Torelli map $J:\mathcal{M}_g\to \mathcal{A}_g$ for ($g\geq 2$, and at least on the level of coarse moduli spaces, over $\mathbb{C}$) gives an isomorphism of Picard g …
- 16k
6
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Ramification formula for orbifolds
It's well known for smooth curves that if $\pi:X\to Y$ is a finite map, $K_X=\pi^*K_Y+Ram(\pi)$, this is just the Riemann-Hurwitz formula at the level of line bundles. I've been told that this formul …
- 16k
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Canonical bundle of compactifications
Let $X$ be a quasi-projective variety. Suppose that we (perhaps partially, if either enough is known) compactify to $\bar{X}$ with $\bar{X}\setminus X=D$ is a divisor. Say that we know the canonical …
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Finding components of a preimage
Let $f:X\to Y$ be a degree $d$ morphism of complex projective varieties, and let $V\subset Y$ an irreducible subvariety, $W$ its preimage under $f$. I want to find all of the components of $W$.
Su …
- 16k
15
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Picard Groups of Moduli Problems
First, yes, I've seen Mumford's paper of this title. I'm actually interested in specific ones, and looking for really the most elementary/elegant proof possible.
I'm told that for $g\geq 2$ it is kn …
- 16k
4
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When is a morphism proper?
A morphism of varieties over $\mathbb{C}$, $f:V\to W$ is proper if it is universally closed and separated. One way to check properness is the valuative criterion.
What other methods do we have for d …
- 16k
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Why do people think that abelian varieties are the hardest case for the Hodge conjecture?
Today, I heard that people think that if you can prove the Hodge conjecture for abelian varieties, then it should be true in general. Apparently this case is important enough (and hard enough) that W …
- 16k
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Exercises in Hodge Theory
I was wondering: is there a good place to find exercises in Hodge theory? Mostly computations and proving small (preferably nifty) theorems, is what I have in mind. Something roughly like the Problem …
- 16k
38
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Intuition for Primitive Cohomology
In complex projective geometry, we have a specified Kähler class $\omega$ and we have a Lefschetz operator $L:H^i(X,\mathbb{C})\to H^{i+2}(X,\mathbb{C})$ given by $L(\eta)=\omega\wedge \eta$. We then …
- 16k
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What do intermediate Jacobians do?
On a smooth complex projective variety of $\dim X=n$, we have $n$ complex tori associated to it via $J^k(X)=F^kH^{2k-1}(X,\mathbb{C})/H_k(X,\mathbb{Z})$ (assuming I've got all the indices right) calle …
- 16k
27
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Stein Manifolds and Affine Varieties
When is a Stein manifold a complex affine variety? I had thought that there was a theorem saying that a variety which is Stein and has finitely generated ring of regular functions implies affine, but …
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