All Questions
10 questions
1
vote
0
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39
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Currents with logarithmic poles compared with those with no poles
I am learning Deligne homology via U. Jannsen, "Deligne homology, Hodge-$\mathscr{D}$-conjecture, and motives." There, the currents with logarithmic poles are given in Definition 1.4 by
$$
'\...
- 161
0
votes
0
answers
76
views
Constant mean curvature hypersurface
Assume that $f:\mathbb{B}^2\to \mathbb{C}$ is a holomorphic function defined in the unit ball in $\mathbb{C}^2$. Let $u(z)=|f(z)|(1-|z|^2)$ and consider $\Sigma =\{z: u(z)=c\}$. It seems to me that if ...
- 37
17
votes
2
answers
2k
views
Why only $\bar\partial$ but not $\partial$ in Dolbeault cohomology
While I learn about $\partial$ and $\bar{\partial}$ operators, I had some questions about the reason why people prefer $\bar\partial$ over $\partial$. Specifically,
When defining Dolbeault ...
- 419
5
votes
1
answer
395
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Holomorphic Sard's theorem 2
My previous question on this topic had a negative answer, but Tom Goodwillie in the comments suggested a statement, which may be true, and even a strategy of how to prove it. I haven't been able to ...
- 5,529
3
votes
1
answer
177
views
Real solution of a complex equation with complex solution
Assume that $(M, [\lambda, \mu])$ defines an embeddable 3 dimensional CR structure where $\lambda$ is a real form and $\mu$ is a complex 1-form.
Because $M$ is embeddable, $\mu=dz$ for some ...
- 99
9
votes
1
answer
662
views
Holomorphic Sard's theorem?
I have originally posted this question on math.SE, but it received little attention, so I repost it here.
Let $U\subset \mathbb{C}^{n}$ and $V\subset \mathbb{C}^{m}$ be open and connected. Let $\Phi:...
- 5,529
3
votes
0
answers
84
views
Discrete set of critical points of a holomorphic map
I have originally posted this question on math.SE, but it received no attention, so I repost it here.
Let $U$ be an open domain in $\mathbb{C}^{n}$. Let $m\ge n$ and let $F:U\to C^{m}$ be a ...
- 5,529
19
votes
2
answers
1k
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Classification of complex structures on $\mathbb{R}^{2n}$
Is there anything known about classification of complex structures on $\mathbb{R}^{2n}$ up to isomorphism for $n>1$? Say, are there finitely or infinitely many isomorphism classes? If there is a ...
- 21.8k
1
vote
1
answer
164
views
Lifting quadratic forms on the cotangent bundle to higher level forms
Backround
In several complex variables, an essential tool is Hormander's machinery for solving the $\overline{\partial}$ problem with $L^2$ estimates.
If $\alpha$ is a $(p,q+1)$ form on a domain $\...
- 12.1k
3
votes
1
answer
385
views
Explicit form for hermitian structure $h$ with respect to $\omega$
Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on $M$...
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