3
votes
1answer
254 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 ...
1
vote
0answers
54 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 ...
1
vote
1answer
73 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 ...
1
vote
1answer
364 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 ...
3
votes
3answers
406 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 ...