All Questions
Tagged with dg.differential-geometry cv.complex-variables
191 questions
37
votes
1
answer
3k
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 ...
- 6,871
33
votes
2
answers
6k
views
Which almost complex manifolds admit a complex structure?
I was reading Yau's list of problems in geometry, and one of them is to prove that any almost complex manifold of complex dimension $n \geq 3$ admits a complex structure. It's been some time since Yau'...
- 4,343
32
votes
2
answers
2k
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 ...
- 6,871
32
votes
0
answers
6k
views
A paper to the question, if the six dimensional sphere is a complex manifold [duplicate]
for a few days a paper was published on arxiv.org with the title "The six dimensional sphere is a complex manifold": http://arxiv.org/PS_cache/math/pdf/0505/0505634v3.pdf
Because I am not able to ...
- 408
29
votes
2
answers
2k
views
Contractibility of the space of Jordan curves
Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.
If the curves are ...
- 7,149
23
votes
2
answers
1k
views
Theta functions on an elliptic curve and Serre duality
Given an elliptic curve $E$ (over $\mathbb{C}$) and line bundle $L$, one can identify $H^0(E,L)$ with a particular space of theta functions.
Serre duality gives a perfect pairing between $H^0(E,L)$ ...
- 411
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
16
votes
3
answers
3k
views
Infinite projective space
Is infinite (say complex) projective space a scheme? More generally, can schemes have infinite cardinal dimension? It seems that infinite dimensional projective space is not a manifold, since it is ...
user2529
16
votes
4
answers
4k
views
Geometric invariant theory for geometers
I am trying to learn "Geometric invariant theory" like it was introduced by Mumford. But I do not have a strong background in algebraic geometry since I work in geometric topology and geometry.
So ...
- 161
16
votes
3
answers
1k
views
Analog of Newlander–Nirenberg theorem for real analytic manifolds
It is well known that one can specify a complex structure on a real $C^\infty$ manifold in two equivalent ways: an atlas with holomorphic transition functions between charts and an integrable almost ...
- 21.5k
14
votes
1
answer
395
views
Regularity of conformal maps
In order to define what it means for a map $f \colon \Omega \subseteq \mathbb R^n \to \mathbb R^n$ to be conformal, it is sufficient to require that $f$ is everywhere differentiable. Does conformality ...
- 1,347
13
votes
5
answers
3k
views
A geometric proof of the Gauss-Lucas theorem
Motivated by a geometric proof of the Fundamental Theorem of Algebra we ask:
Is there a geometric proof for the Gauss-Lucas theorem? Since we are working on a half plane, can one imagine a possible ...
- 356
13
votes
3
answers
1k
views
Do contact and CR structures have corresponding $G$-structures?
For an $n$-dimensional manifold $M$, almost complex and almost symplectic structures on $M$ correspond to reductions on the structure group of the tangent bundle, introducing a $\operatorname{GL}(n/2,\...
- 131
13
votes
1
answer
2k
views
Surgery in complex geometry
I've been thinking about surgery on complex manifolds. Not very seriously, but just to the point that I think it's odd how there's almost no mention of it in the literature. I figure there's something ...
- 4,343
13
votes
1
answer
682
views
How can one "see" the Hopf fibration in the space of lattices in the plane?
This question is inspired from Etienne Ghys's talk on Knots and Dynamics from ICM 2006.
The map $L \mapsto (G_4(L), G_6(L))$ gives a bijection between all lattices $L\subset \mathbb{C}$ (including ...
- 1,821
12
votes
2
answers
754
views
Can the unit complex 1-dimensional disc be embedded isometrically into complex euclidean space?
Let $D$ denote the unit complex 1-dimensional disc, together with the hyperbolic metric $h_D=\frac{4dz\wedge d\bar{z}}{(1-|z|^2)^2}$of curvature $-1$. By Nash's embedding theorem, we can always embed ...
user2529
11
votes
6
answers
3k
views
Explicit Spin Structures on the Torus
Basically, I am trying to build explicit examples of Dirac operators. To this end, I'm looking at the surface E = C/(Z + λZ) - for some λ in H \ SL(2,Z) - with the Euclidean metric and ...
- 22.8k
11
votes
3
answers
1k
views
Can a metric conformal to a Kahler metric be Kahler?
Let $X$ be a non-compact complex manifold of dimension at least 2 equipped with a Kahler metric $\omega$. Take a smooth positive function $f : X \to \mathbb R$, and define a new hermitian metric on $X$...
- 4,343
11
votes
1
answer
379
views
Tori in three-space
Recently I was talking to an alien who does not know complex function theory. I was trying to convince her that the set of conformal equivalence classes of smooth embedded tori in $R^3$ is two ...
- 91.8k
10
votes
1
answer
662
views
Hartogs' theorem for real-analytic subvarieties
One version of Hartogs' extension theorem is the following (see, e.g. [1], Theorem 5B, p. 50).
Theorem. Let $U \subset \mathbb{C}^n$ be open and let $X \subset U$ be a complex-analytic subvariety of ...
- 1,383
9
votes
2
answers
1k
views
Is the deformation limit of Ricci-flat Kahler manifolds Kahler?
Let $X$ be a compact complex Kahler manifold with first real Chern class $c_1 = 0$. Consider a family $\pi : \mathcal X \to \Delta$ over the unit disc in $\mathbb C$, where the fibers $X_s$ are ...
- 4,343
9
votes
1
answer
629
views
conformally embedding complex tori into R^3
Let $L$ be a lattice in $\mathbb{C}$ with two fundamental periods, so that $\mathbb{C}/L$ is topologically a torus. Let $p:\mathbb{C}/L \mapsto \mathbb{R}^3$ be an embedding ($C^1$, say). Call $p$ ...
- 1,961
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
9
votes
1
answer
321
views
Notational question about quadratic differentials in Strebel's book "Quadratic differentials"
In Kurt Strebel's book "Quadratic Differentials", in Chapter 2, $\S4$, he begins by saying:
"Every analytic function $\varphi$ is a domain $G$ of the $z$-plane defines, in a natural way, a field of ...
- 7,527
9
votes
1
answer
3k
views
Complex geometry text/research introduction for the analyst
To give some background, I am mainly an analyst trained in harmonic/functional and do work on geometric pde's and spectral multipliers. Of late, I am trying to learn more about (research level) ...
- 141
8
votes
1
answer
824
views
Acyclicity of the sheaf of real analytic differential forms
Let $M$ be a real analytic manifold. In the book "Sheaves on Manifolds" by Kashiwara and Schapira it is claimed on p. 127 (without reference or proof) that
the Poincare lemma holds for the de Rham ...
- 1,645
8
votes
0
answers
315
views
Singularities of a morphism from a smooth projective variety to an abelian variety
Let $f: X\to A$ be a (flat) morphism from a smooth complex projective variety $X$ to an abelian variety $A$. Consider the following natural diagram:
$$T^*X\overset{df}{\longleftarrow}X\times H^0(A, \...
- 1,081
8
votes
0
answers
288
views
Are the Chern numbers of a hyperbolic-type compact complex manifold bounded in terms of the Euler number?
Let $X$ be an $n$-dimensional compact Kahler manifold with negative first Chern class. Are its Chern numbers $\prod_{i=1}^{n-1} c_i^{k_i}$, over $k _i
\geq 0$ with $\sum ik_i = n$, bounded in terms ...
7
votes
2
answers
1k
views
Analog of residue for meromorphic quadratic differentials
Hi I had asked this already on math.stackexchange.com but got no answers.
I was wondering if there was any sort of (natural) analog of the residue of a meromorphic one form that made sense for a ...
- 2,299
7
votes
1
answer
634
views
Hodge diamonds of complex threefolds
There is no closed complex curve or surface with $h^{1, 0}-h^{0, 1}=1$.
Now consider threefolds. Can this condition be satisfied?
Is Serre duality in fact the only restriction on the Hodge diamond?
user164740
7
votes
2
answers
813
views
Criterion for deciding the conformal class of a metric on a complete surface
For orientable closed Riemannian surfaces $(S,g)$, there is a constant curvature metric $\overline{g}$ on $S$ that is conformal to $g$ in the sense that $\overline{g} = e^ug$ for some smooth function $...
- 173
7
votes
1
answer
1k
views
Is a simply connected Ricci-flat Kaehler manifold a Calabi-Yau manifold?
Hi,
I have the following question: Let $(M,\omega, J)$ be a simply connected Kaehler manifold with Ricci-flat Kaehler metric. How can one show that $M$ is a Calabi-Yau manifold. By Calabi-Yau ...
- 71
7
votes
1
answer
535
views
Diffeomorphisms on a real manifold whose derivative are holomorphic maps on the tangent bundle
Edit: According to the answers to the linked MSE question and the comment of Holonomia, I understand that the answer to the second question is that " Every tangent bundles is a complex ...
- 356
7
votes
0
answers
168
views
The relation between Wolf's and Teichmüller's parametrization of the Teichmüller space
Let $\mathcal{T}_g$ be the Teichmüller space of Riemannian surface structures on an oriented 2-dimensional manifold of genus $g$. Fix a point $S \in \mathcal{T}_g$. There are two different ways to ...
- 1,170
7
votes
0
answers
769
views
How much differs the category of real-analytic manifolds from $C^\infty$ ones?
I was thinking about the difference between the concept of real-analytic function (for any point the Taylor-series of $f$ converge to the function in a neighborhood of the point) and complex analytic (...
- 395
7
votes
0
answers
202
views
Biholomorphic neighborhoods of the boundary of Stein domains
Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...
- 773
6
votes
2
answers
169
views
Conditions on a unit vector field to be the Gauss map of some surface immersed in R^3?
Let $U$ be a bounded domain in $R^2$ and let $n : U \to S^2$. Which (necessary/sufficient) conditions must $n$ satisfy in order that there exist an immersion $f : U \to R^3$ such that $n(x)$ is the ...
- 63
6
votes
1
answer
1k
views
Holomorphic functions in almost-complex geometry
Maximum principle implies that every holomorphic function on a compact complex manifold is constant. Is this still true if the manifold is only almost complex?
6
votes
3
answers
2k
views
Complex projective space as a $U(1)$ quotient
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}$As is well known, one can view $\mathbb{CP}^n$ as a quotient of the unit $(2n + 1)$-sphere in $\mathbb{C}^{n+1}$ under the action of $\U(1)$, ...
- 751
6
votes
2
answers
755
views
Plurisubharmonic function and complete Kähler metric on certain Kähler manifold
Given a compact Kähler manifold $M$, let $D$ be an effective divisor on $M$.
Is $M\setminus D$ pseudoconvex? That is, can we find a smooth plurisubharmonic function that exhausts $M\setminus D$ ?
...
user40184
6
votes
1
answer
1k
views
Harmonic forms on Ricci-flat Kahler manifolds
Let $X$ be a compact Kahler manifold with $c_1(X) = 0$. Any Kahler metric $\omega$ on $X$ gives a Laplacian $\Delta_\omega$ and the $(1,1)$-form $\omega$ is harmonic with respect to this Laplacian.
...
- 4,343
6
votes
1
answer
409
views
Can the potential of a complete Kahler metric be bounded?
Let $X$ be a complex manifold and $\omega$ a Kahler form on $X$. A smooth function $\rho$ is called a potential of $\omega$ if $i\partial\bar\partial\rho=\omega$. By intuition, it seems that $\rho$ ...
- 285
6
votes
1
answer
219
views
Restriction of holomorphic functions on $G$-invariant subspace
Let $X$ be a complex manifold with a holomorphic action of a complex reductive group $G$. Let $Y \subset X$ be a $G$-invariant reduced complex analytic subspace. Is the restriction
$$
\mathcal{O}_X^G \...
- 1,383
6
votes
1
answer
261
views
The state of art of the singular Levi problem -- and hyperkähler quotients
One of the versions of the classical Levi problem asks the following:
Let $X$ be a complex manifold. Is it true that $X$ is Stein iff
$X$ admits a smooth exhaustion strictly plurisubharmonic ...
- 2,305
6
votes
0
answers
144
views
What does it mean for the torsion to blow up?
Consider the following theorem which is the main result of the Hermitian Curvature Flow paper by Jeffrey Streets and Gang Tian:
Theorem 1.1. Let $(M^{2n}, g_0, J)$ be a complex manifold with Hermitian ...
- 651
6
votes
0
answers
228
views
All complex surfaces embed into a common complex manifold
Is there a closed complex manifold into which every closed complex surface embeds?
user164740
6
votes
0
answers
163
views
Reference request: normal form of k-differentials and flat surfaces at a puncture
Let $f(z)$ be a holomorphic function defined on a punctured neighborhood of $z=0$ with non-essential singularity of degree $d$ at $0$ (namely, $f(z)=z^dh(z)$, where $h(z)$ is a holomorphic function ...
- 1,804
6
votes
0
answers
147
views
What is the meaning of complex values/multiplicities in dimension spectrum?
If we have a manifold $M$ (say smooth, closed) it can be equipped with the Laplace operator $\Delta$. One can consider the function $\textrm{trace}(\Delta^{-s})$ where $s$ is complex parameter and $\...
- 9,330
6
votes
0
answers
286
views
Is the space of holomorphic maps a manifold
To be more specific:
Let $Q\subset\mathbb{C}$ be a Lipschitz bounded domain, and $V$ is a compact complex manifold without boundary. Consider the set of holomorphic maps $f:Q\rightarrow V$, and $f\in ...
- 151