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Results tagged with complex-geometry
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user 36688
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.
6
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1
answer
389
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Is polynomial convexity a topological invariant?
Is the property of being polynomially convex a topological invariant?
In other words, let $M$ and $N$ be two homeomorphic, compact subsets of $n$-dimensional complex Euclidean space, and assume in ad …
- 356
2
votes
1
answer
227
views
Real diffeomeorphism preserving the space of Holomorphic vector fields
Assume that $M$ is a complex manifold.
Let $G$ be the group of all (real) smooth diffeomorphisms $\phi$ of $M$ such that $\phi^* (X)$ is a holomorphic vector field for all holomorphic v …
- 356
2
votes
1
answer
151
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Holomorphic manifolds with an Einstein structure and non constant holomorphic sectional curv...
My apology in advance if this question is obvious:
I know that an Einstein manifold need not have a constant sectional curvature example $\mathbb{C}P^n$. But this space has a constan …
- 356
3
votes
2
answers
284
views
Can the "Bisector" be represented by a holomorphic function?
Note:
In this question, a complex number is counted as a vector initiated from the origin.
______________________________________________________________-
Is there a holomorphic function $B:\mat …
- 356
2
votes
0
answers
91
views
Does there exist a leaf of this holomorphic foliation with non trivial holonomy?
Let's $\mathcal{F}$ be the holomorphic foliation of $\mathbb{C}^2$ tangent to the kernel of $\alpha=(sin x) dx -(cos x)dy$.
Are all leaves of $\mathcal{F}$ simply connected? If the answer is no, d …
- 356
2
votes
1
answer
124
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A Singular Foliation of $\mathbb{C}P^2$ which does not admit a global transverse submanifold
Is there a singular holomorphic foliation $F$ of $\mathbb{C}P^2$ which does not admit a global transverse holomorphic curve? More precisely there is no an immersed holomorphic submanifold …
- 356
2
votes
1
answer
116
views
Projection of an invariant almost complex structure to a non-integrable one
My apologies in advance if my question is obvious or elementary.
We identify elements of $S^3$ with their quaternion representation $x_1 + x_2i + x_3j + x_4k$. We consider two independent vector field …
- 356
2
votes
0
answers
218
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Manifold whose symplectic structure of the cotangent bundle is intrinsically different from ...
Inspired by this question Symplectic structure of $TS^{n-1}$ we ask:
What is an example of a manifold $M$ whose cotangent bundle $T^*M$ is an Stein manifold but the canonical symplectic stru …
- 356
10
votes
2
answers
483
views
Riemann surfaces with an atlas all of whose open sets are biholomorphic to $\mathbb{C}$?
Is there a compact Riemann surface other than the sphere with an atlas consisting of open subsets biholomorphic to $\mathbb{C}$? Is there a compact Riemann surface other than the sphere which possess …
- 356
5
votes
0
answers
38
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Complex $2$ manifold $M$ with a given real 2 dimensional submanifold which meets all complex...
Is there a $2$ dimensional holomorphic manifold $M$ with a closed $2$ dimensional real submanifold $A$ of $M$ such that for every singular holomorphic foliation of $M$, all leaf with non trivial ho …
- 356
3
votes
1
answer
323
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Geometric Morse theory ( and its complex analogy)
In the literature are there some concept of geometric version of Morse or Picard Lefschets theory? That is the comparison of level sets as Riemannian submanifold not merely as topologica …
- 356
0
votes
2
answers
538
views
A relation between gradient vector field and Hamiltonian vector field
Assume that $U$ is an open subset of $\mathbb{C}^n$. We fix the standard almost complex structure $J$ on $U$.
Assume that $\omega$ is an arbitrary symplectic structure on $U$.
Is there a Riemannian m …
- 356
2
votes
Vector field with holomorphic flow
Without lose of generality we may work locally in $\mathbb{R}^{2n}$ with its standard complex structure $J$. Let $\phi$ be the flow of the vector field $x'= f(x)$. So the statement in the question is …
- 356
1
vote
0
answers
95
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A singular foliation analogy of the Riemann Hilbert problem
Note:
In this question by $\mathbb{C}P^1 \subset \mathbb{C}P^2$ we mean that we choose the line at infinity in the form $\{[0,y,z]\in \mathbb{C}P^2\} $ which is identified by $\mathbb{C}P^1$.
Assu …
- 356
1
vote
1
answer
87
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The entire parametrization of leaves of singular holomorphic foliation of $\mathbb{C}P^2$
What is an example of an entire non constant holomorphic function $\gamma: \mathbb{C} \to \mathbb{C}P^2$ such that the image of $\gamma$ is a leaf of a singular holomorphic foliation of …
- 356