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Results tagged with complex-geometry
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user 13441
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.
8
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$c_2$ of Calabi-Yau three-folds
Let $Y$ be a smooth compact Calabi-Yau three-fold (over $\mathbb C$, with $\pi_1(Y)=0$). Is it true that $c_2(Y)$ is Poincare dual to an effective curve? If not, can one construct a counter-example?
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- 14.3k
8
votes
1
answer
282
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Deformations of Hopf manifolds
Recall that a Hopf manifold is a quotient $\mathbb C^n\setminus 0$ by a free action of $\mathbb Z$ where the generator is acting by a holomorphic contraction.
Question 1. Is it true that any deformati …
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9
votes
1
answer
315
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Variety $X$ such that $TX$ is ample on any curve in $X$
Let $X$ be a smooth complex projective variety such that the restriction of $TX$ on any curve $C$ in $X$ is ample. Is true in this case that $X$ is isomorphic to $\mathbb CP^n$?
I guess the above con …
- 14.3k
2
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101
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Smooth complex projective manifolds with a $\mathbb C^*$-action with isolated fixed points
What is minimal $n$, such that there exists a smooth complex projective manifold $M^n$ satisfying the following two conditions:
1) The group of automorphisms of $M^n$ is $\mathbb C^*$.
2) All the f …
- 14.3k
0
votes
1
answer
1k
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Bounding the derivative of a holomorphic function on a disk by its absolute value
Let $f(z)$ be a holomorphic function defined on the disk $|z|\le 2$. Suppose $|f(z)|<1$ for $|z|\le 2$. It looks like there is a constant $c>0$ such that $|f(z)'|<c$ on the disk $|z|\le 1$ (for exampl …
- 14.3k
4
votes
2
answers
210
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Splitting a trivial bundle over punctured $\mathbb C^n$
Suppose I have a trivial rank $k$ bundle $E$ over $\mathbb C^n$. Suppose that on $\mathbb C^n\setminus 0$ I have two algebraic sub-bundles $V_1,V_2\subset E$ of ranks $l$ and $k-l$ such that $V_1\oplu …
- 14.3k
4
votes
1
answer
222
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A normal form of local anti-holomorphic involutions of $\mathbb C^2$?
Suppose an anti-holomorphic involution $\sigma$ is defined in a neighbourhood of $0\in \mathbb C^2$. Suppose that $\sigma$ fixes a real two-dimensional surface $\Sigma$ containing $0$. Is it true tha …
- 14.3k
6
votes
2
answers
306
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Rational maps from $\mathbb CP^n$ to $\mathbb CP^{n-1}$, fixing $\mathbb CP^{n-1}$
Consider $\mathbb CP^n$ and let $H\subset \mathbb CP^n$ be a hyperplane. Suppose $\varphi: \mathbb CP^n\to H$ is a rational map that fixes $H$ pointwise. I believe that $\varphi$ must be a projection …
- 14.3k
8
votes
1
answer
212
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Rational smooth complex projectives three fold with non-rational deformation
This question is prompted by a great talk of Beauville:
http://www.mathnet.ru/php/presentation.phtml?presentid=5821&option_lang=rus
The talk is called "Luroth problem". In this talk Beauville consid …
- 14.3k
4
votes
2
answers
338
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Biholomorphic but not isomorphic complex affine surfaces?
Suppose $X$ and $Y$ are smooth affine surfaces over $\mathbb C$. Suppose there is a biholomorphism $f: X\to Y$. Does it follow that $X$ and $Y$ are isomorphic as affine surfaces (i.e. there exists an …
- 14.3k
3
votes
1
answer
176
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Pull-backs of $\sum dx_i \wedge dy_i$ under radial diffeomorphisms of $\mathbb C^n$
Consider $\mathbb C^n$ with coordinates $(z_1,\dots,z_n)$, $z_j=x_j+iy_j$. Let $\omega=\sum dx_i\wedge dy_i$. Let us call by a radial diffeomorhpism $\varphi$ of $\mathbb C^n$ a diffemorphism of $\var …
- 14.3k
3
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1
answer
843
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Constructing a very ample line bundle on a projective bundle
Let $X$ be a smooth complex projective variety and $p:Y\to X$ be a locally trivial in analytic topology $\mathbb CP^k$-bundle. Suppose we have a line bundle $L$ on $Y$, restricting to $\mathcal O(1)$ …
- 14.3k
4
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342
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A deformation of the second Hirzeburch surface $F_2$ over $\mathbb CP^1$
I would like to know if there exists a smooth complex projective $3$-fold $X$ that admits a fibration $\pi: X\to \mathbb CP^1$ such that all fibers are smooth, $\pi^{-1}(0)$ is the second Hirzebruch s …
- 14.3k
10
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answers
193
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Holomorphic versus algebraic $\mathbb C^*$-actions
I believe that the following is true:
Statement. A holomorphic $\mathbb C^*$-action on a complex projective manifold is algebraic if and only if it has a fixed point.
Where can I find a proof of th …
- 14.3k
1
vote
1
answer
148
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A semi-ampleness criterion for homogeneous bundles on homogeneous spaces?
Let $X$ be a (compact) homogeneous space and $V$ be a homogeneous vector bundle on $X$ of rank $n$, and such that $\operatorname{dim}X\ge n$. Suppose $V$ has a section $s$, whose zeros $s=0$ form a su …
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