Questions tagged [holomorphic-foliations]

For question concerning foliation theory in the holomorphic case: germs of foliation singularities, holomorphic foliations on complex manifolds, Pfaff fields, etc.

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On the Weierstrass Preparation Theorem and Weierstrass Division Theorem

Denote by $R_{n}=\mathbb{C}\{z_{1},\cdots, z_{n}\}$ the ring of convergent power series. We say that $f\in \mathbb{C}\{z_{1},\cdots, z_{n}\}$ is $z_n$-general of order $m$ if there is an $h\in\mathbb{...
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Geometric interpretation of the algebraic multiplicity of an isolated singularity of a holomorphic folation by curves

Let $\mathscr F$ be a singular foliation by curves on a complex manifold $M$, with only isolated singularities. The Milnor number $\mu_p(\mathscr F)$ at a point $p \in M$ is the topological index at $...
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Finding whether a rank-$k$ subbundle of $TM$, specified as a line subbundle of $\bigwedge^k TM$, is integrable

Let $M$ be a complex manifold or smooth algebraic variety, and let $X$ be a global vector field which is nonzero over “most” of $M$. Then I know that $X$ determines a foliation of $M$ by curves, ...
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Is growth type of leaves of foliations invariant under diffeomorphisms?

Let $M$ and $N$ be two complex manifolds of complex dimension $2$. Let $\mathcal{F}_M,\mathcal{F}_N$ be a singular holomorphic one-dimensional foliation on $M,N$; respectively. Thus, the leaves of the ...
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Poincaré-Dulac Normal Form for Holomorphic Foliations of Complex Manifolds of Complex Dimension $2$

Let $\mathcal{F}$ be a singular one-dimensional holomorphic foliation on a complex manifold $M$ of complex dimension $2$ and $p \in M$ a singular point for the foliation $\mathcal{F}$. Assume that the ...
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The degree of a foliation $\mathcal{F}$ on the complex projective plane $\mathbb{C}\mathbb{P}^2$

Let $\mathcal{F}$ be a Singular Holomorphic Foliation on the Complex Projective Plane $\mathbb{C}\mathbb{P}^2$. It is well-known that there are too many different equivalent ways to define the Degree ...
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The Harmonic Measures of Lucy Garnett

The Harmonic Measures of Lucy Garnett, Alberto Candel, Advances in Mathematics $176$ ($2003$) $187-247$. Page $208$, Proposition $5.2$ I have actually two questions: what does he mean by saying "...
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Linearisability of Holomorphic Vector Fields and Poincaré-Dulac Normal Form

Complex Algebraic Foliations, Book by A. Lins Neto and Bruno Scárdua, Chapter 1, Page 36. The last two lines, Don't you think they should have written that the vector field is locally conjugate (...
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Can I find a plane in $\mathbb P^3$ to which a foliation $\mathscr F \subset T\mathbb P^3$ is tangent on a smooth curve?

In this question, we work over an unspecified algebraically closed field, not necessarily of characteristic zero. Let $\mathscr F$ be the $1$-dimensional foliation of $\mathbb P^3$ defined by the ...
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Show that a generic line is not a leaf of an algebraic foliation of $\mathbb P^2$

Consider a foliation of $\mathbb P^2$, whose leaves in the affine part $z = 1$ are the integral curves of a polynomial vector field $$X = P \frac \partial {\partial x} + Q \frac \partial {\partial y}$...
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What are the prerequisites to learning about Lie groupoids, Lie algebroids and holomorphic foliations?

I am a graduate student of Theoretical Physics and intend to take a course titled "Introduction to Lie groupoids, Lie algebroids and holomorphic foliations". The course page doesn't have information ...
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Conormal and Tangent Sheaves of a Distribuition in $\mathbb{P}^{n}$ when $n = 2$ and $n = 3$

Definition 1): A codimension one distribution of degree $d \geq 0$ in $\mathbb{P}^{n}$ is given by an exact sequence: $$\mathscr{F}: 0 \longrightarrow T_{\mathscr{F}} \longrightarrow T\mathbb{P}^{n}\...
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Exact sequence on $\mathbb{P}^3$ obtained from Euler sequence

In the article of title: On the singular scheme of codimension one holomorphic foliations in $\mathbb{P}^3$ the author states that the sequence on $\mathbb{P}^3$ $$0\longrightarrow \mathcal{F}\oplus\...
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Isomorphisms of complex (foliated) n-tori

From here: https://www.encyclopediaofmath.org/index.php/Complex_torus A complex torus is a complex Abelian Lie group obtained from the $n$-dimensional complex space $\mathbb{C}^n$ by factorizing ...
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holomorphic vector fields tangent to a hypersurface

Let $(V,0)\subset (\mathbb{C}^n,0)$, $n\geq 3$ a (germ of) hypersurface given by $\{f=0\}$, $f$ holomorphic. A (germ of) holomorphic vector field $X$ leaves $V$ invariant if ${\rm d}f(X)$ belongs to ...
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