# Tagged Questions

**23**

votes

**0**answers

922 views

### Is S^2 x S^4 a complex manifold?

As observed by Calabi a long time ago, the manifold $S^2\times S^4$ admits an almost-complex structure (obtained by embedding it in $\mathbb{R}^7$ and using the octonionic product), which however is ...

**29**

votes

**1**answer

889 views

### Example of a compact Kähler manifold with non-finitely generated canonical ring?

A celebrated recent theorem of Birkar-Cascini-Hacon-McKernan and Siu says that the canonical ring $R(X)=\oplus_{m\geq 0}H^0(X,mK_X)$ of any smooth algebraic variety $X$ over $\mathbb{C}$ is a finitely ...

**21**

votes

**1**answer

1k views

### Complex vector bundles that are not holomorphic

Is there an example of a complex bundle on $CP^n$ or on a Fano variety (defined over complex numbers), that does not admit a holomorphic structure? We impose of course that the Chern classes of the ...

**6**

votes

**1**answer

522 views

### Infinite dimensional Newlander-Nirenberg theorem

The Newlander-Nirenberg theorem states that an almost complex structure is integrable if and only if the Nijenhuis tensor vanishes.
I heard that this statement is not true in infinite dimensions, ...

**9**

votes

**2**answers

497 views

### A group action of the Heisenberg group with special symmetries

Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural ...

**28**

votes

**6**answers

5k views

### Is there a complex structure on the 6-sphere?

I don't know who first asked this question, but it's a question that I think many differential and complex geometers have tried to answer because it sounds so simple and fundamental. There are even a ...