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Results tagged with complex-geometry
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user 605
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.
2
votes
resolution of singularities on surfaces
Over $\mathbb C$ at least, If the surface is non-singular then a finite number of blow-ups at points suffices to resolve the linear system. Indeed any birational morphism between smooth surfaces is …
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7
votes
Accepted
Divisors, extensions of functions
The very same result holds in arbitrary dimensions: a locally bounded holomorphic function
defined in the complement of a divisor extends. In many textbooks (like Gunning's) this is also called Riemm …
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11
votes
The Relationship between Complex and Algebraic Geomety
While every algebraic manifold is complex analytic the converse is far from true.
If $M$ is a compact complex manifold then $a(M)$, the algebraic dimension of $M$, is defined
as the transcendence deg …
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3
votes
Accepted
Holomorphic vector fields on $\mathbb{P}^n$ that extend to the blow up
1) No. There are many more vector fields. The vector fields you are looking for are precisely those which vanish at $p_0$. Since $h^0( \mathbb P^n, T \mathbb P^n) = (n+1)^2 -1$ and you are imposing $n …
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6
votes
2
answers
917
views
Rational Hilbert modular surfaces
Background.
Let $\mathbb H= \lbrace z \in \mathbb C \; | \; Im(z)>0 \rbrace$ be the upper half-plane, and let
$\Gamma \subset PSL(2,\mathbb R)\times PSL(2,\mathbb R)$ be an irreducible lattice. More …
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5
votes
Accepted
Stein manifolds isomorphic at infinity
I believe the answer is positive in dimension at least two. Stein manifolds admit proper embeddings on vector spaces. An isomorphism from $M$ to $N$ can be represented by a collection of holomorphic f …
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40
votes
10
answers
6k
views
Algebraic Geometry versus Complex Geometry
This question is motivated by this one.
I would like to hear about results concerning complex projective varieties which
have a complex analytic proof but no known algebraic proof; or
have an al …
6
votes
Accepted
Algebraicity and non-algebraicity of leaves of the characteristic foliation
Suppose $X$ is projective manifold endowed with holomorphic symplectic form. Let $D$ be smooth divisor on $X$.
Characteristic foliation with algebraic and non-algebraic leaves.
Suppose that $X$ is …
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7
votes
Accepted
Holomorphic Foliations having transverse sections
If a smooth curve $C$ has zero self-intersection and is everywhere transverse to a foliation on a surface then there are also strong restrictions.
If $C$ is rational then the foliation is a Riccati …
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3
votes
Kähler manifold which is not algebraic
You might want to take a look at this previous MO question.
There, I mentioned Voisin's results disproving Kodaira's conjecture (every Kahler manifold
is deformation equivalent to a projective manif …
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2
votes
Nonalgebraic complex manifolds
These issues are discussed to some extent at this previous MO question.
For instance, if a compact complex manifold is algebraic then its field of meromorphic functions has transcendence degree over …
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2
votes
Kahler manifolds with special submanifolds
If your $X$ is projective and $E$ is an ample vector bundle over $M$ then the field of meromorphic functions of $X$ is a finite extension of the field of meromorphic functions of $\mathbb P(E \oplus …
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5
votes
Complex manifold with boundary
If $M$ is real analytic then Élie Cartan proved that, in suitable holomorphic coordinates, $M$ is cut out by the imaginary part of $z$. I learned this from the paper https://hal.archives-ouvertes.fr/h …
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6
votes
0
answers
320
views
When a proper smooth fibration is isotrivial?
Let $\pi : X \to Y$ be a proper smooth morphism between complex quasi-projective varieties. Assume there exists a connected and Zariski dense analytic subvariety $Z$ of $Y$ such that for any two point …
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13
votes
Complex hypersurface in complex projective space
A more general version of the question made by the OP is the following
Question. Let $H_1$ and $H_2$ be two smooth connect hypersurfaces in a projective manifold $X$. Suppose $H_1$ is homologous to …
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