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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.
4
votes
2
answers
338
views
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
4
votes
0
answers
228
views
An Akbulut cork with a simple equation?
Is there an Akbulut cork that is diffeomorphic to a complex affine surface that can be given by a simple equation? (for example a surface given by zeros of a low-degree polynomial in $\mathbb C^3$)
He …
- 14.3k
8
votes
1
answer
266
views
Self homeomorphism of $\mathbb CP^1$ holomorphic a.e
Suppose $\varphi:\mathbb CP^1\to \mathbb CP^1$ is a homeomorphism holomorphic on a connected open subset $U\subset \mathbb CP^1$ with $\mathbb CP^1\setminus U$ of zero measure.
Is it true that $\varph …
- 14.3k
8
votes
1
answer
269
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Is there a "minimal" Whitney stratification of a complex hypersurface?
Let $X\subset \mathbb C^n$ be a complex hypersurface (given by $F=0$ where $F$ is a polynomial). It is known then that $X$ admits a Whitney stratification. This is a decomposition of $X$ into smooth s …
- 14.3k
2
votes
0
answers
62
views
Whitney stratification of a $\mathbb C^*$-invariant hypersuface in $\mathbb C^n$
Let $H\subset \mathbb C^n$ be an irreducible hypersurface invariant under a diagonal $\mathbb C^*$-action with positive weights ($H$ is given by a quasi-homogeneous polynomial). Consider the Whitney s …
- 14.3k
4
votes
2
answers
210
views
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
6
votes
2
answers
306
views
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
4
votes
1
answer
222
views
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
8
votes
1
answer
282
views
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 …
- 14.3k
3
votes
1
answer
208
views
Holomorphic vector fields with a non-degenerate isolated zero
Let $v$ be a holomorphic vector field defined in a neighbourhood of $0$ on $\mathbb C^n$ with an isolated zero at $0$. Let $\sum_{i,j}{a_{ij}}z_i\frac{\partial}{\partial z_j}$ be the linear term of $ …
- 14.3k
3
votes
1
answer
843
views
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
9
votes
3
answers
1k
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$\mathbb CP^k$ bundles over $\mathbb CP^n$ are projectivisations of vector bundles. Any refe...
Statement. Let $X$ be a smooth complex projective variety that is a $\mathbb CP^k$ bundle over $\mathbb CP^n$ in analtytic topology. It is well known that there exists a rank $k+1$ complex vector bund …
- 14.3k
4
votes
0
answers
342
views
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
votes
0
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
7
votes
1
answer
508
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Rationally connected Kähler manifolds are projective
I would like to find a proof for Remark 0.5 in the following article of Claire Voisin:
https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/fanosymp.pdf
She writes in this remark the following:
R …
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