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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.
2
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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 …
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-2
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Are real-analytic functions in $\mathbb{R}^2$ holomorphic after suitable change of coordinates?
According to existing informative and interesting answers one gets that the local dynamical behavior around fixed points or around periodic orbits may generates some obstructions for being conjuga …
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5
votes
1
answer
224
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Nontrivial extension of the action of complex hyperbolic group $H$ on $\mathbb{C}$
Inspired by this question about conjugation of reql analytic maps to a holomorphic function and with a group action view point we ask the following question.
The complex Lie group $H=\math …
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Einstein metrics on the tangent bundle
Some Particular cases are explained in Papaghiuc - On an Einstein structure on the tanent bundle of a space form.
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4
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2
answers
395
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Is a positive degree self map on a Riemann surface homotopic to a holomorphic self map?
Let $S$ be a compact Riemann surface and $f:S\to S$ be a continuous self map of positive degree. Is $f$ homotopic to a holomorphic map on $S$?
Motivation: I had intention to consider this question f …
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4
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2
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356
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Holomorphic Gauss normal map
Let $S$ be a Riemann surface smoothly embedded in $\mathbb{R}^3$.
Is there necessarily a smooth embedding of $S$ in $\mathbb{R}^3$ such that the Gauss normal map $n:S \to S^2$ would be a holomorphic …
- 356
2
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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
1
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Can a holomorphic vector field have an attractor homoclinic loop?
There is no any kind of, generally speaking, "Charactristic curve" for the vector field $z'=f(z)$ when $f$ is a holomorphic function. By charactristic curve I mean any kind of particular curve wh …
- 356
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A complex limit cycle not intersecting the real plane(2)
This note contains an affirmative answer to the question
https://maco.lu.ac.ir/article-1-86-en.html
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5
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2
answers
254
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Can a holomorphic vector field have an attractor homoclinic loop?
It is well known that a holomorphic vector field $z'=f(z), z\in \mathbb{C}$ does not have any limit cycle.See the last paragraph of this post
Orbits space of real-analytic planar foliations
One can …
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3
votes
0
answers
105
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An algebraic foliation of $\mathbb{C}^2$ which admits a non-algebraic complex limit cycle $L...
Is there a polynomial vector field on $\mathbb{R}^2$ whose corresponding singular complex foliation of $\mathbb{C}^2 $ admits a complex limit cycle $L$ such that $L$ does not intersect the real pl …
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4
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An algebraic foliation of $\mathbb{C}^2$ with real coefficients whose all complex limit cycl...
Is there a polynomial vector field $X$ on $\mathbb{R}^2$ such that every complex limit cycle
of the corresponding foliation of $\mathbb{C}^2$ must necessarily intersect the real plane $\mathbb{R}^2$. …
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3
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1
answer
178
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Analytic or holomorphic extension of the ellipse perimeter function
Let ${\mathbb{R}^2}^+=\{(x,y)\in \mathbb{R}^2\mid x>0, y>0\}$.
Let $P:{\mathbb{R}^2}^+\to \mathbb{R}$ be the function with $P(a,b)=$ $\text{The perimeter of ellipse}\;\; \frac{x^2}{a^2}+\frac{y^2}{b^ …
- 356
12
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1
answer
478
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Holomorphic Urysohn Lemma
Let $M,N$ be two disjoint closed holomorphic submanifolds of $\mathbb{C}^n$. Is there a holomorphic map $f:\mathbb{C}^n\to \mathbb{C}$ with $f(M)=0,\;f(N)=1$.
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4
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1
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Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$...
Let $X=P\partial_x+Q\partial_y$ be a vector field on the plane $\mathbb{R}^2$. Assume that we have :$$P_xP_y+Q_xQ_y=0$$ Does this imply that the vector field $X$ is a divergence-free vector field w …
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