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Results tagged with complex-geometry
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user 943
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.
1
vote
Accepted
Complex fibration over complex torus
Added. Thanks to the comment of abx below I understood that there was a big gap in the reasoning, and $M$ doesn't need to be a fiber bundle.
Example. I'll construct an example $S$ of a complex surface …
- 28.9k
2
votes
Can we construct a dessin of any genus with a cyclic automorphism group of any order?
This is possible for any $g$ and $k$. The idea is as follows:
(1) Start with a dessin embedded in the sphere $S^2$ with symmetry group $C_k$ which has a lot of faces and a lot of triple points.
(2) Ma …
- 28.9k
2
votes
Accepted
Example of a Kähler manifold with certain properties
What about taking a product $X=Y\times \Sigma$, where $Y$ is a fake projective plane and $\Sigma$ is a surface of genus $3$? We have $h^{1,1}(X)=2$, $h^{0,1}(X)=3$, and $h^{0,3}(X)=0$ by Kuneth formu …
- 28.9k
2
votes
Possible number of components of anticanonical sections of projective manifolds
Here is how one can prove that the number of components is at most $2$ in the case of complex projective surfaces. So let $X$ be such a surface and let $D_1\cup \ldots\cup D_k$ be a smooth anti-canoni …
- 28.9k
9
votes
Accepted
Arbitrary torsion in cohomology of Kähler manifolds
The answer is positive and can be deduced from Proposition 15 of "Sur la topologie des varietes algebriques en characteristique p" by Serre. According to this proposition for any finite group $G$ ther …
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19
votes
Accepted
Unique almost complex structure up to diffeomorphism
You were lucky to find the only possible example. If you take any manifold of dimension $\ge 4$ you can pick an almost complex structure that is integrable in some closed ball and make it non-integrab …
- 28.9k
6
votes
Homotoping diffeomorphism to a $J$-holomorphic one
No. Take the torus $T^2=\mathbb R^2/\mathbb Z^2$ and consider the self-map induced by the matrix
$$\begin{pmatrix}
1 & 1 \\
0 & 1
\end{pmatrix}$$
PS. As for simply-connected examples, I think that a …
- 28.9k
10
votes
Accepted
Antiholomorphic involution with a fixed point
No. There exist both non-algebraic and projective counterexamples.
1 Non-algebraic example. Take a flat Euclidean torus $T^4=M$ and let $Z$ be its twistor space. It has an antiholomorphic involution w …
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0
votes
Accepted
Non-algebraic Kähler threefolds with abelian $\pi_1$ of arbitrarily large rank
Let's construct such a Kahler $3$-fold $X$. It will be obtained as an elliptic fibration over a projective surface $S$ with abelian fundamental group $\mathbb Z^{2g}$.
Construction. Recall first that …
- 28.9k
11
votes
Accepted
Chern/Hodge numbers of the conjugate complex manifold
These are all the same.
As for Hodge numbers, you can choose a Kahler metric $g$ on $(M,J)$, and it will also be Kahler for $(M,-J)$. Now we know that $h^{p,q}$ is the dimension of the space of harmon …
- 28.9k
8
votes
Accepted
Fano manifold becoming general type upon conjugation
No. $(M,J)$ and $(M,-J)$ have conjugate pluri-canonical rings, hence have same Kodaira dimension.
Proof. Take a section $\mu$ of $K_{(M, J)}^{\otimes n}$, then $\bar \mu$ is a holomorphic section of $ …
- 28.9k
8
votes
Accepted
Complex projective manifold with an antiholomorphic involution
Corrected. As Robert Bryant points out, it is not enough to show that the manifold can be realised as a submanifold of $\mathbb CP^n$ invariant under some anti-holomorphic involution of $\mathbb CP^n$ …
- 28.9k
10
votes
Accepted
Deformation equivalent vs diffeomorphic to projective manifold
I believe the answer is yes and follows from the combination of Theorem 4.6 here https://arxiv.org/pdf/math/0111245.pdf
and Theorem 1.3 here https://arxiv.org/pdf/math/0111245.pdf
The first result sho …
- 28.9k
10
votes
Symplectic structure on the square of a 3-manifold
Here is a Kahler example. Consider a hyper-elliptic curve $C$ of positive genus with involution $\sigma$. Take $C\times S^1$ and quotient $C\times S^1$ by $\mathbb Z_2$ that rotates $S^1$ by a half-tu …
- 28.9k
2
votes
Accepted
Topological factors of complex projective manifolds
Let $M=\mathbb CP^2\#\mathbb CP^2$ and let $S=T^2$ be the $2$-dimensional torus. I think this gives an example for the original question. As for the symplectic version of the question, I am sure it is …
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