All Questions
16 questions
19
votes
2
answers
1k
views
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
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
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
6
votes
1
answer
567
views
How to tell if it's a Moishezon morphism
Suppose that $f \colon X\rightarrow S$ is a proper morphism of reduced and irreducible complex spaces and $f$ is a smooth deformation in the sense of Kodaira and Spencer. If we know each fiber $X_s$, ...
- 1,121
5
votes
1
answer
395
views
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
5
votes
0
answers
543
views
a question on Hodge and Atiyah's paper "integrals of the second kind on an algebraic variety"
I have a question on Hodge and Atiyah's paper "Integrals of the second kind on an algebraic variety". It is about the exact sequence below formula (14) and above formula (15) on page 71:
$$H_{2n-q}(S)...
- 1,121
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$...
user21574
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
3
votes
0
answers
637
views
English reference for Fischer-Grauert theorem and its generalization by Schuster
From this MSE question and its answer, and from this MO question I have learned of the following remarkable theorem of Wolfgang Fischer and Hans Grauert.
Theorem. A proper holomorphic submersion with ...
- 10.5k
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
2
votes
0
answers
120
views
How to get the jet extension over the whole of $X$ in Popovici's article?
Recently, I am reading D. Popovici's article $L^2$ extension for jets of holomorphic sections of a Hermitian line bundle, https://arxiv.org/pdf/math/0409170.pdf where some parts possibly confuse me.
I ...
- 295
2
votes
0
answers
91
views
Automatic plurisubharmonicity for a non-negative function
I feel confused about a point in this very short paper. On the top of page 3, it is claimed that:
If $S$ is a totally real submanifold in a compact almost complex manifold $(X,J)$, then any function ...
- 2,789
2
votes
0
answers
75
views
Notation and geometry facts in a paper on the Diederich-Fornæss index
I am reading this article by Bingyuan Liu on the Diederich-Fornæss index.
I am having some problems with both the notation and the geometrical side.
1)I don't know what kind of objects $N,L$ are ...
- 779
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
1
vote
0
answers
39
views
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 ...
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