All Questions
15 questions
7
votes
1
answer
428
views
A geometric characterization of smooth points of a complex algebraic variety
Let $X^m\subset \mathbb{C}^n$ be an irreducible $m$-dimensional complex algebraic subvariety. Let $\mathbb{C}^n$ be equipped with the standard Hermitian metric.
Fix an arbitrary point $p\in X$. Let $...
- 21.8k
7
votes
2
answers
593
views
Whitney stratification and affine grassmanian
Let $G$ a simply connected group over $\mathbb{C}$ and $Gr:=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$ the affine grassmannian. By Cartan decomposition we have a partition of stratas indexed by $\lambda\...
- 3,472
7
votes
1
answer
884
views
Localization of vanishing cycles
Consider a regular holonomic D-module (or a perverse sheaf) $M$ on a smooth variety $X$. Let $f:X\to A^1$ be a polynomial (or holomorphic) function.
Question: Is it true that the $\lambda \in A^1$ ...
- 7,527
6
votes
0
answers
96
views
forms on singular spaces that can be integrated on an LCI
I'd like to characterize rational $k$-forms on a singular scheme $X$ which can be integrated on any (real) bounded submanifold that is locally LCI in $X$ in a real sense (i.e., which is the real ...
- 7,952
6
votes
0
answers
376
views
Topological Singularities in Affine Varieties
Let $X$ be an affine variety over $\mathbb{C}$. Let $x\in X$.
If $x$ is non-singular, then $x$ is locally holomorphic (in the Euclidean topology). See here for a relevant MO post.
By results of ...
- 8,529
5
votes
1
answer
263
views
Measuring contact between algebraic varieties
I have two regular surfaces in three space, both of which are given by an equation. I would like to measure the contact between the two surfaces using only their equations. Usually, one would find a ...
- 126
4
votes
1
answer
312
views
Kähler-Einstein metrics on singular varieties
Let $X$ be a normal projective variety with klt singularities with numerically trivial canonical divisor $K_X$.
Does there always exist a Kähler-Einstein metric on $X$?
user58018
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
2
answers
935
views
Applications of Slope Stability
Ross and Thomas developed slope-stability of $(X,L)$ where $X$ is an $L$-polarised variety and $L$ is an ample line bundle, as an obstruction to K-stability of $(X,L)$.
DISCLAIMER: (Forgive me if I ...
- 2,215
2
votes
1
answer
558
views
quotient singularities
Let $X$ be a relaively compact projective variety and has only quotient singularities then for any n-form $\Omega$ , $$\int_{X_{reg}}\Omega\wedge \bar \Omega$$ is bounded? what about the converse
- 99
2
votes
1
answer
399
views
resolution for the du Val's $(A_3)$-singularity
For the $A_m$-singularity, it can be viewed as the singular part of $\mathbb{C}^2/\mathbb{Z}_m$. The action of $\mathbb{Z}_m$ on $\mathbb{C}^2$ is defined as following
$$
\bar{1} \cdot (z,w) = (z e^{\...
- 167
2
votes
0
answers
510
views
Weil Petersson metric on moduli space of Calabi Yau manifolds
Let $f:(X,D)\to Y$ be a holomorphic fibre space where $D$ is divisor with conic singularities and let fibres $(X_s,D_s)$ are log Calabi-Yau pair .i.e $K_X+D$ is nummerically trivial, then we have ...
user21574
1
vote
1
answer
93
views
Existence of meromorphic 2-forms over normal surface singularities
Let $(X,o)$ be an isolated normal surface singularity. Denote by $U:=X\backslash \{o\}$. I am looking for conditions on $(X,o)$ under which there exists a holomorphic section $\omega \in H^0(U, \Omega^...
- 2,032
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
0
votes
0
answers
228
views
On Whitney's paper on real algebraic varieties
I had previously asked this question on math.stackexchange and did not receive an answer and so I decided to reword it and pose it here.
This question is based on Whitney's paper titled "...
- 821