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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.
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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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5
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1
answer
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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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4
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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 …
- 366
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 …
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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 …
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5
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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
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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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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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answer
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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^ …
- 366
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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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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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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 …
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Maps between grassmannians with inclusion property
Edit: According to the comment of L. Spice I changed the inclusion sign to the subset sign.
Is there a continuous map $f:\mathbb{C}P^3 \to \textrm{Gr}_{\mathbb{C}}(2,4)$ with $x\subset f(x)$? What ab …
- 366
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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 …
- 366
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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 …
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