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Results tagged with complex-geometry
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user 5952
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
5
votes
Logarithm of complex matrices in holomorphic families
The answer is "no" in general. As Denis suspects, the problem is a global one, and it involves matrices with nontrivial Jordan blocks. These have, in a sense, "fewer" logarithms than the commoners. Co …
- 4,015
4
votes
3
answers
512
views
Logarithm of complex matrices in holomorphic families
Let $n,k\geq 1$ be integers, let $U \subseteq \mathbb C^n$ be a contractible open subset, and let $f:U\to \mathrm{GL}_k(\mathbb C)$ be a holomorphic function. Does there exist a holomorhpic functio …
- 4,015
1
vote
0
answers
580
views
Splitting of vector bundles on a complex torus
Let $X$ be a complex torus (a finite dimensional complex vector space modulo a lattice) and let $E$ be a smooth (not necessarily holomorphic) complex vector bundle over $X$. Is it true $E$ is isomorph …
- 4,015
3
votes
Is there an intrinsic way to define the group law on Abelian varieties?
In dimension 1, the situation is like this: Every smooth proper genus 1 curve with a "marked" rational point has a unique group structure such that the given rational point becomes the neutral element …
- 4,015