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Results tagged with complex-geometry
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user 24309
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.
12
votes
1
answer
683
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Parameterization of complex analytic subvarieties
Let $V$ be an analytic subvariety of some open set of $\mathbb{C}^n$ (intersection of finitely many zero-locii of analytic functions). Hironaka's desingularization theorem provides a parameterization …
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11
votes
The holomorphic version of Galois theory
Say $a_n=1$. You can obtain this map as a section of the map $\sigma$ sending the $n$-tuple of the roots $(r_1,\ldots,r_n)$ to the coefficients of the polynomial using the symmetric polynomials, corre …
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8
votes
1
answer
729
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Local polynomial form of holomorphic functions
It is well-known that a germ of a holomorphic function of $n\geq 2$ variables, with at most an isolated singularity (i.e. the singularity of the analytic variety defined by the zero locus of the funct …
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8
votes
Accepted
Examples of Stokes data
It helps to understand irregular singularities as the merging of regular singular points, say $$(x^2-a^2)y'+y=0$$ as $a\to0$.
For nonzero $a$ the data is encoded as monodromy (constant) matrices actin …
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5
votes
Are real-analytic functions in $\mathbb{R}^2$ holomorphic after suitable change of coordinates?
To see why the second question cannot have a simple answer, it is sufficient to look at the local context near a fixed-point of a tangent-to-identity mapping, as Alexandre Eremenko suggests. By "a sim …
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4
votes
0
answers
342
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What's the name of this branched covering?
I've come across a double cover of $\mathbb P_1(\mathbb C)$, ramified at $[1:1]$ and $[-1:1]$ in homogeneous coordinates, given as the quotient by the natural $\mathbb Z/2\mathbb Z$-action generated b …
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4
votes
1
answer
178
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Effective bound on "Jacobian rank" for (regular) planar algebraic curves
Let an irreducible, square-free complex polynomial $f\in \mathbb C[x,y]$ be given. It is well known that the curve $\mathcal C:=\{f=0\}$ is nonsingular if and only if $\mathbb C[x,y]=<f,\partial_xf,\p …
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2
votes
Accepted
The entire parametrization of leaves of singular holomorphic foliation of $\mathbb{C}P^2$
For your first question: $dy=y^2$ can be integrated by quadratures. The solutions are homographies.
For your second question: no it is not true. By the uniformization theorem, the universal covering …
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2
votes
Accepted
Can a holomorphic vector field have an attractor homoclinic loop?
The answer is 'no' for much the same reason that the OP indicates: the existence of a homoclinic or heteroclinic connection implies that neighboring trajectories are periodic.
First, one needs to have …
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1
vote
Global vector fields
If the manifold $Y$ admits a global (meromorphic) vector field $W$, then around a regular point $p$ (that is, $W(p)\notin \{0,\infty\}$) one has a local holomorphic rectifying chart as you point out. …
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