All Questions
Tagged with kt.k-theory-and-homology complex-geometry
12 questions
4
votes
1
answer
418
views
Definition of Chow quotient
I am reading M. M. Kapranov's paper "Chow quotients of Grassmannians. I." (English) in Sergej Gelfand (ed.) et al., I. M. Gelfand seminar. Part 2: Papers of the Gelfand seminar in ...
- 41
1
vote
0
answers
68
views
Metric and connection on virtual bundles
Let $E=[E^+]-[E^-]$ be an element of the Grothendieck group $K(X)$ of a compact Kahler manifold $X$.
Does it exist a way to define more "geometric" structures on $E\in K(X)$ such that a ...
- 789
1
vote
0
answers
172
views
Calculation about Chern character in a special setting
I'm confused with working out the Chern character in the following special setting.
Let $E$ be a spinor bundle
$$S=P_{Spin(2n)}(S^{2n})\times_\rho \mathbb{C}^{2n}$$
over sphere $S^{2n}$, where $\rho$ ...
- 563
7
votes
3
answers
2k
views
Complex structure on $S^4$
I have heard that there is a proof of non-existence of complex structure on the 4-sphere $S^{4}$ using only the topological K-theory (complex $KU$ and real $KO$). Moreover this argument can not be ...
- 451
6
votes
0
answers
170
views
Does the $K^1$-group of a complete flag variety vanish?
For $U(n)$ the Lie group of $n \times n$ unitary matrices, and $T^n$ its maximal torus subgroup, the homogeneous space
$$
U(n)/T^n
$$
is called the complete flag variety of order $n$. For the special ...
- 141
4
votes
0
answers
81
views
Minimum rank of inverse complex vector bundles
When considering vector bundles (real or complex) over a compact manifold, i know about the existence of inverse bundles. That is, if $\xi$ is a vector bundle over $M$, then there is a bundle $\nu$ ...
- 261
10
votes
0
answers
6k
views
Atiyah's paper "Non-existent complex 6-sphere"
I'm trying to understand the main idea of Atiyah's proof (https://arxiv.org/abs/1610.09366). Although there were discussions on MO year ago I couldn't find answers to my questions.
Consider the ...
- 171
48
votes
0
answers
17k
views
What is the current understanding regarding complex structures on the 6-sphere?
In October 2016, Atiyah famously posted a preprint to the arXiv, "The Non-Existent Complex 6-Sphere" containing a very brief proof $S^6$ admits no complex structure, which I immediately read and ...
- 2,995
50
votes
0
answers
12k
views
Atiyah's paper on complex structures on $S^6$
M. Atiyah has posted a preprint on arXiv on the non-existence of complex structure on the sphere $S^6$.
https://arxiv.org/abs/1610.09366
It relies on the topological $K$-theory $KR$ and in ...
- 9,870
6
votes
0
answers
102
views
On K-theory of blow-ups of compact complex manifolds
Is there a long exact sequence for the K-theory of (coherent sheaves on) blow-ups of compact complex manifolds? Does it split? What can one say on (possibly, singular) complex analytic spaces here?
...
- 16.9k
7
votes
1
answer
485
views
Non trivial rank 2 holomorphic vector bundles in complex dimensions greater than or equal 2
Does every compact complex manifold of complex dimension greater than or equal two
possess a nontrivial rank 2 holomorphic vector bundle?
- 1,236
2
votes
0
answers
338
views
Algebraic K-theory of complex varieties
Maybe this question is trivial, but I was not able to find an answer. The question is this: Consider the algebraic K-theory of smooth complex projective varieties (such that the K-theory and the G-...
- 912