All Questions
Tagged with complex-geometry at.algebraic-topology
188
questions
2
votes
0
answers
138
views
Compute the Euler class of tautological $C$-bundle over $CP^1$
$\DeclareMathOperator\SO{SO}$This might be an old question. But since I have not found an explicit answer to this question, I put the question here.
The background is that we need to use a similar ...
- 21
2
votes
1
answer
217
views
Homotopy classes of homeomorphism vs. Homotopy classes of a biholomorphism
This is a more detailed question about my first question Representation theory and topology of Teichmüller space, I asked there how to understand:
$$T_{g}\hookrightarrow Hom(\pi_{1}({S}),PSL_{2}(\...
- 77
6
votes
1
answer
284
views
Is this $\mathbb C$-fibration over compact Riemann surface trivial?
I have a question about a complex manifold $M$ and a holomorphic submersion $p : M \to S$ to a compact complex curve satisfying the following conditions:
$p^{-1}(x)$ is biholomorphic to $\mathbb{C}$ ...
- 651
3
votes
0
answers
151
views
Do the nearby cycle and Beilinson's vanishing cycle functors commute?
Let $X$ be a complex algebraic variety with a pair of regular functions $f_1,f_2$. To these functions we can associate various functors: the nearby cycles functor $\Psi_{f_i}$, the vanishing cycles ...
- 1,061
6
votes
5
answers
943
views
Two arcs in the complement of a disc must intersect?
Let $D=\{z\in \mathbb C:|z|\leq 1\}$ be the unit disc in the complex plane, with interior $U=\{z\in \mathbb C:|z|<1\}$.
Let $A\subset \mathbb C\setminus U$ be an arc intersecting $D$ only at its ...
- 3,105
2
votes
1
answer
229
views
Derived category of local systems of finite type on a $K(\pi,1)$ space: an explicit counterexample
Let $X$ be a nice enough topological space. I am mostly interested in smooth complex algebraic varieties. One may ask whether the bounded derived category of the category $\mathrm{Loc}(X)$ of local ...
- 1,061
6
votes
0
answers
170
views
How exactly does the Kreck-Stolz description of elliptic homology match the one by Totaro?
In
Kreck, Matthias; Stolz, Stephan, $\mathbf H\mathbf P^2$-bundles and elliptic homology, Acta Math. 171, No. 2, 231-261 (1993). ZBL0851.55007.
the $n$th elliptic homology group of a space $X$ is ...
- 17.8k
1
vote
0
answers
57
views
Given a proper submersion $f: X \setminus F_0 \to D \setminus 0$ which extends to $X \to D$, are cycles on $X$ which run around $0$ boundaries in $X$?
I have a proper map of complex manifolds
$$f: X \to D,$$
where $D \subset \mathbb C$ is the unit disc. By assumption, $f$ has connected fibers, is smooth over $D \setminus 0$, and a smooth fiber $F$ ...
- 1,141
6
votes
2
answers
414
views
Presentation of the fundamental group of a complex variety
Let $X$ be a connected smooth complex algebraic variety and $Z=\bigcup_{i=1}^r Z_i$ be a union of smooth connected hypersurfaces, satisfying that each two intersect transversally. Assume for ...
3
votes
1
answer
223
views
Does $H^3\times I$ admit a Kähler metric?
Let $H^3$ be the Heisenberg manifold. It is known that the first betti number of $H^3\times S^1$ is odd and therefore it does not support any Kähler metric. Now let $I=(0,1)$ or $I=[0,1]$, does it ...
- 651
0
votes
1
answer
267
views
Universal covering of symmetric product
Let $C$ be a 1-dimensional complex manifold whose universal covering is provided by the half-plane $\mathcal{H}=\{z \in \mathbb{C} \mid \operatorname{Im}z>0\}$. The symmetric product $C^{(n)} = C^n ...
- 53
13
votes
0
answers
381
views
Does the existence of an almost complex structure solely depend on the topology of the manifold?
To be precise, let $M$ and $N$ be two 2n-dimensional smooth, closed manifolds that are homeomorphic. If $M$ admits an almost complex structure, can we deduce that $N$ also admits an almost complex ...
- 651
2
votes
0
answers
258
views
Topology of Lefschetz pencil
Let $f : X \to D$ be a proper flat holomorphic family of complete algebraic curves over a disk and the central fiber is a nodal curve with a unique singular point $p \in X_0$. Suppose that every fiber ...
- 3,393
2
votes
0
answers
46
views
How to calculate the genus of a 3 dimensional curve (f(kx,ky,kz)) using the Newton polyhedra?
Given a plane affine curve $\sum_{i,j}a_{i,j}k_x^ik_y^j = 0$, its genus can be calculated as the number of integral points of the interior of the convex hull of $\{(i,j) \mid a_{i,j} \neq 0\}$.
How to ...
- 21
13
votes
1
answer
498
views
Local systems arising from higher rational homotopy groups
I should mention I have very little background in algebraic topology and don't really know much about homotopy groups besides the definition.
I am aware that for a topological space $X$ and a point $x ...
- 233