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5
votes
2answers
522 views

On the fundamental group of hypersurfaces

Let $H$ be a smooth projective hypersurface in $\mathbb{P}^n(\mathbb{C})$ where $n\geq 3$. Then by the Lefschetz hyperplane theorem we have that $H^1(H,\mathbb{C})= ...
6
votes
2answers
880 views

Finite unramified analytic coverings vs finite etale coverings

Let $X$ be a smooth quasi-projective variety (so irreducible) over $\mathbf{C}$. We may think of $X$ as a complex manifold which we denote by $X^{an}$. Of course the topology on $X^{an}$ is finer ...
15
votes
2answers
845 views

When can you reverse the orientation of a complex manifold and still get a complex manifold?

I'm told that $\overline{\mathbb{C}P^2}$, i.e. $\mathbb{C}P^2$ with reverse orientation, is not a complex manifold. But for example, $\overline{\mathbb{C}}$ is still a complex manifold and ...
3
votes
2answers
811 views

holomorphy of inverse map

Let $M,N$ be complex manifolds and $f : M \to N$ be a bijective holomorphic map. Is then $f^{-1}$ also holomorphic? The open mapping theorem implies that $f^{-1}$ is continuous. In order to apply the ...
0
votes
2answers
575 views

Diffeomorphism group of the unit sphere of complex n-space

What is the largest subgroup of the diffeomorphism group of a (2n-1)-sphere that only contains members that are holomorphic in the coordinates assigned to the sphere by taking it to be the unit sphere ...
12
votes
2answers
584 views

Highly connected, compact complex manifolds

Here are four remarks about the homology and homotopy type of a compact, complex manifold $M$: If $M$ is Kähler, then it is symplectic and thus $H^2(M,\mathbb{R}) \ne 0$. (Also, as explained in a ...
16
votes
3answers
1k views

Holomorphic vector fields acting on Dolbeault cohomology

The question. Let $(X, J)$ be a complex manifold and $u$ a holomorphic vector field, i.e. $L_uJ = 0$. The holomorphicity of $u$ implies that the Lie derivative $L_u$ on forms preserves the (p,q) ...
12
votes
3answers
1k views

A topological consequence of Riemann-Roch in the almost complex case

This question originated from a conversation with Dmitry that took place here Is there a complex structure on the 6-sphere? The Hirzebruch-Riemann-Roch formula expresses the Euler characteristic of ...
22
votes
3answers
2k views

Topologically distinct Calabi-Yau threefolds

In dimensions 1 and 2 there is only one, respectively 2, compact Kaehler manifolds with zero first Chern class, up to diffeomorphism. However, it is an open problem whether or not the number of ...