All Questions
Tagged with complex-geometry gt.geometric-topology
97
questions
2
votes
0
answers
233
views
Smooth compactification of complex varieties and uniqueness
Since I'm working in differential geometry, for the following I'm strictly interested in the smooth setting over $\mathbb{C}$ and its relation to the setting over $\mathbb{R}$.
Here are a few useful ...
- 1,755
7
votes
1
answer
117
views
Picture of the isotopy class of a degree $d$ smooth complex curve
All smooth complex curves of degree $d$ in $\mathbb{C}P^2$ are isotopic. Let $C$ be such a curve. I often picture $\mathbb{C}P^2$ as a 2-dimensional disk bundle over $S^2$ (of Euler class 1) which ...
- 1,065
3
votes
0
answers
321
views
Fundamental group of blow-ups
Let $M$ be a simply-connected compact complex manifold of dimension three and $C$ is a smooth complex curve in $M$.
Let $M'$ be the blow-up of $M$ along $C$.
My question is:
Is $M'$ also simply-...
- 1,831
3
votes
0
answers
131
views
Diffeomorphism problem for complex surfaces?
I'm sure the following is well known by the right people, I'm just hoping for some pointers. I know about Markov's theorem that the diffeomorphism problem for general 4-manifolds is undecidable.
Let $...
- 1,065
1
vote
0
answers
89
views
Sufficient condition for moduli space of slope-stable bundles to be non-empty
I'm trying to find to a statement concerning the non-emptiness of the moduli space of slope stable vector bundles over a Kähler surface in the literature.
Let $X$ be a Kähler surface. Let $\mathscr{M}(...
- 11
3
votes
1
answer
143
views
Existence of covering isomorphism
Let $C,D$ be two non-compact complex algebraic smooth curves. Suppose that two unramified regular finite maps $p_1, p_2: C \rightarrow D$ are given and have the same degree. Is there always an ...
- 73
9
votes
0
answers
261
views
Fundamental group of the complement to a plane curve with unramified normalization
Suppose that $C\subset\mathbb P^2$ is an irreducible projective curve over $\mathbb C$ such that the normalization morphism $\bar C\to C$ is unramified (i.e., the induced morphism $\bar C\to\mathbb P^...
- 1,839
2
votes
1
answer
283
views
Vanishing cycles exact sequence for degeneration of curves
Let $f : X\rightarrow D$ be a family of smooth projective curves over the complex unit disk $D$, degenerating to a nodal curve $X_0$ above $0\in D$.
Let $\eta\in D - \{0\}$ be a general point, and let ...
- 7,313
1
vote
0
answers
150
views
Decomposing the homology of a connected sum of surfaces in a way which highlights the combinatorics of gluing
For each $i = 1,2$ and $j = 1,\ldots,n$, let $C_{i,j}$ be a connected compact oriented surface with $k$ boundary components. For $i = 1,2$, let $C_i = \sqcup_{j=1}^n C_{i,j}$, and let $C$ be the ...
- 7,313
13
votes
2
answers
1k
views
Can we define Whitney stratification algebraically?
For a subset $S$ of a smooth manifold $M$, a locally finite decomposition
$$S = \bigsqcup_{\alpha} S_\alpha$$
into smooth submanifolds (strata) is called a Whitney stratification of $S$ if each pair $(...
- 793
2
votes
0
answers
122
views
Single theorem for hybrid of winding number and rotation number?
I am trying to make mathematical sense of some observations from my physics research, so I hope that you will bear with me.
For a complex-valued function $z(t)$ dependent on parameter $t$, I calculate ...
- 121
4
votes
0
answers
226
views
An Akbulut cork with a simple equation?
Is there an Akbulut cork that is diffeomorphic to a complex affine surface that can be given by a simple equation? (for example a surface given by zeros of a low-degree polynomial in $\mathbb C^3$)
...
- 14.1k
18
votes
4
answers
2k
views
How would a topologist explain "every Riemann surface of genus $g$ is hyperelliptic if and only if $g=2$"?
Recall that a compact Riemann surface/algebraic curve $C$ is hyperelliptic if it admits a branched double cover $C \to \mathbb P^1$, where $\mathbb P^1$ is the complex projective line/Riemann sphere. ...
- 355
4
votes
0
answers
170
views
How to judge whether an orbifold is good
My own case comes from dynamic system on compact complex manifolds. To be precise, let $M$ be a compact complex 3-dimentional manifold, $W^c$ a holomorphic foliation of M with 1-dimentional uniformly ...
- 651
6
votes
1
answer
414
views
Comparison of two monodromies
Let us consider a smooth projective curve $\Sigma_b$ of genus $b$, and assume that there is a surjective group homomorphism $$\varphi \colon \pi_1 (\Sigma_b \times \Sigma_b - \Delta) \to G,$$ where $\...
- 65.7k