All Questions
Tagged with cv.complex-variables singularity-theory
31 questions
2
votes
1
answer
199
views
Section 3 of Atiyah's "On analytic surfaces with double points" — some questions
I have some questions about section 3 of Atiyah's "On analytic surfaces with double points," a short 9 page paper. Section 3 is all dedicated to proving lemma 4.
Near the end of section 3, ...
- 129
1
vote
0
answers
42
views
Concerning the definition of a class of functions introduced by Nilsson
In the paper 'Some growth and ramification properties of certain integrals on algebraic manifolds' by Nilsson we find the following definitions:
My question is how does one prove the remark "It ...
- 360
6
votes
0
answers
200
views
Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$
In my research I encountered automorphisms of the ring of convergent power series
$$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$
which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...
- 1,286
0
votes
0
answers
121
views
Is any singularity a subgerm of $(\mathbb{C}^n, 0)$?
I am studying singularity theory. I have often come across, in the literature, the sentence which says "let $(X,0) \subset (\mathbb{C}^n,0)$ be a singularity". Here a singularity is a ...
- 369
6
votes
0
answers
219
views
Is the Taylor map continuous?
(Skip to the bolded theorem below for my question, if you'd like)
Some context on asymptotic expansions and the Taylor map
In the setting of irregular singularities of meromorphic connections on the ...
- 370
1
vote
0
answers
116
views
Converse of transfer theorem : does asymptotic behaviour of coefficients describe the singularity?
I'm interested in Singularity Analysis after reading this Math.Stack post. In the "Analytic Combinatorics" book by P. Flajolet and R. Sedgewick, at p.390, the following transfer theorem (...
- 233
0
votes
0
answers
74
views
When a strictly positive log pluriharmonic function $g$ is equal to the norm of holomorphic function?
Suppose $V$ is a local analytic variety (singular). Suppose $g$ a strictly positive log pluriharmonic function on $V$, i.e. $\log g$ is pluriharmonic. I wonder when $g=|f|^2$, where $f$ is a ...
- 623
0
votes
0
answers
253
views
Singularity of inverse exponential integral function
The exponential integral function is defined by
$$ Ei(z) = \int_{-\infty}^z dw \frac{e^w}{w} \,.$$
Away from the negative real axis the exponential integral function has a Taylor series about $z=0$:
$$...
- 81
6
votes
1
answer
963
views
Is there a, in depth, classification of branch points in complex analysis?
Disclaimer: This question was originally posted in math.stackexchange.com and, after 30 days with no answers, I followed the instructions of this topic.
In complex analysis we have well known results ...
- 480
6
votes
1
answer
276
views
How to solve the following ODE with a parameter?
I am considering the following ODE
\begin{equation}
\begin{split}
&\frac{d^2}{dy^2}u + \frac{\alpha}{(1+y^2)^{\frac{r}{2}}}u = \delta(y)\\
&\lim_{|y|\to \infty}u(y) = 0.
\end{split}
\end{...
- 903
3
votes
1
answer
214
views
Holomorphic vector fields with a non-degenerate isolated zero
Let $v$ be a holomorphic vector field defined in a neighbourhood of $0$ on $\mathbb C^n$ with an isolated zero at $0$. Let $\sum_{i,j}{a_{ij}}z_i\frac{\partial}{\partial z_j}$ be the linear term of $...
- 14.3k
4
votes
0
answers
72
views
Asymptotics of a certain integral in singularity theory
Let $f:\mathbb{C}^2\to \mathbb{C}$ be an isolated plane curve singularity. Consider the versal deformation space $\mathbb{C}^\mu$ parameterizing deformations $f_\lambda$ for $\lambda \in \mathbb C^\mu$...
- 2,830
1
vote
1
answer
125
views
Applying analytic coordinate changes to singular function germs [closed]
Suppose we are given a function germ
\begin{align}
f = \sum a_{ijk}x^iy^jz^k
\end{align}
such that $f\in \mathfrak{m}^2$, where $\mathfrak{m}$ is the ideal in $\mathbb{C}\{x,y,z\}$ of holomorphic ...
- 19
1
vote
0
answers
925
views
canonical divisor on singular curves with nodal point
What's the definition of canonical divisor(or whatever related concept) on singular curve with nodal point. More generally, what the definition of canonical divisor on a singular variety X, which is ...
- 623
2
votes
1
answer
1k
views
Essential singularity [closed]
In shaum's outline complex analysis,definition of essential point is:
An isolated singularity that is not pole or removable singularity is called essential singularity
Now in the same book there is an ...
8
votes
0
answers
178
views
Padé Approximants of Power Series with Natural Boundaries
Consider a power series $\sum_{n=0}^{\infty}c_{n}z^{n}$ for which $c_{n}\in\left\{ 0,1\right\}$ for all $n$. One can write this as: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}...
- 1,284
1
vote
0
answers
152
views
Is the normalized derivative of a holomorphic function Sobolev?
This question is a cross-post from MSE. it is also a special case of this question.
Let $B=\{z\in \mathbb C \,|\,|z|\le 1\}$, and let $f:B \to \mathbb{C}$ be holomorphic on the interior $B^o$, and ...
- 6,741
0
votes
0
answers
225
views
Generating function with essential singularities
I was recently introduced to analytic combinatorics, and found the method of removing poles astonishing. More precisely, I was reading the last chapter of the popular "generatingfunctionology", in ...
- 5,230
18
votes
1
answer
830
views
Cohomology of real analytic coherent sheaves
Let $M$ be a real analytic variety
(if someone is concerned about distinction between
"real analytic spaces" and "real analytic varieties"
in real analytic geometry, let's assume that $M$
is both "...
- 9,237
2
votes
0
answers
76
views
Equations needed to define a normal complex surface singularity
This questions is highly related with this other question of mine: Irreducible surface singularity that is not a local set-theoretical complete intersection I just thought that a different look at the ...
- 1,409
2
votes
0
answers
34
views
Smoothings over a real interval
I am asking if somebody knows if the following kind of objects are studied somewhere or if there is some kind of obvious obstruction for them to appear.
Let $(X,0) \subset \mathbb{C}^n$ be a germ of ...
- 83
2
votes
1
answer
89
views
Smoothings of isolated non-irreducible surface singularities
Let $(X,0)$ be a normal surface singularity. Suppose that it does not admit a smoothing.
Is it possible that there exists an isolated surface singularity $(Y,0)$ reduced near $0$ which is not ...
- 83
4
votes
0
answers
86
views
Action of the monodromy on the cycle made of the real points
Let $f : \Bbb C^n \to \Bbb C$ be a polynomial function with real coefficients.
Let $X_t = f^{-1}(t)$ denote the fiber above some $t \in \Bbb C$. Let assume that the set of real points of $X_t$, for $t ...
- 1,044
9
votes
2
answers
873
views
Implicit Function Theorem on Singular Varieties
Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function
$$f\colon X \to Y $$
and assume that it is bijective at the level of $...
- 2,384
3
votes
0
answers
324
views
Implicit function theorem for singularities
I am looking for an implicit function theorem which holds also on singular spaces, at least if the singularities are "mild".
For example, let $0 = z^2 - x y + z w + w^2 + \epsilon w$ define a ...
0
votes
1
answer
286
views
A condition on isolated singularity
Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = 0$...
- 1,207
3
votes
1
answer
609
views
Normal form for a holomorphic Morse function
Similarly to Morse lemma, a holomorphic Morse function can be written, near a critical point, as $W_1^2+W_2^2+...+W_n^2+C$ , for well chosen coordinates $W_1,W_2,...,W_n$. I want to cite this result ...
- 31
2
votes
0
answers
98
views
Is it obvious that the defining conditions to obtain a particular singularity are well-defined on the quotient space?
Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function
vanishing at the origin, with
the following properties:
$$ f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 \...
- 3,245
1
vote
1
answer
128
views
Finite construction of lacunary functions using algebraic and certain analytic operations
Algebraic functions have a discrete set of singularities. Lacunary functions, e.g. $f(z)=\sum_{n=0}^\infty z^{2^n}$, have a continuum of singularities at every point of the boundary of their disk of ...
- 13
2
votes
1
answer
583
views
Brieskorn's proof of a theorem by Milnor about the Milnor number
I am looking for a reference or short explanation of a proof by E. Brieskorn.
In his famous work "Singularities of complex hypersurfaces" Milnor proves that the (nowadays called) Milnor Number (in ...
- 1,124
4
votes
2
answers
627
views
The link of a singular quintic hypersurface in CP^4
Given a family of quintic hypersurfaces in $\mathbb{CP}^4$ by
$x_1^5+x_2^5+x_3^5+x_4^5+x_5^5+(5+\epsilon)x_1x_2x_3x_4x_5$
we get a singular variety for $\epsilon=0$ with 125 singular points.
I know ...
- 93