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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 $\...
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 ...
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 $M\setminus D$ ? ...
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 ...
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 ...
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 ...
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 $...
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 $\rho$ ...
1 vote
1 answer
190 views

What is the Fano index for Hermitian symmetric spaces of compact type?

As we know Hermitian Symmetric spaces of compact type are all Fano picard number one, we can talk about his Fano index. Suppose $X$ is one of those Hermitian symmetric spaces, $L$ is the generator of ...
5 votes
1 answer
342 views

harmonic extension of a curve by different parametrization

Let us consider a curve $\gamma :S^1 \rightarrow \mathbb{R}^3$ (or even a planar convex one if it simplifies). Then I look to the harmonic extension to the disc $h:\mathbb{D}\rightarrow \mathbb{R}^3$ (...
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 ...
2 votes
2 answers
191 views

Comparision theorem for distance function

Assume that $\rho$ and $\rho'$ are conformal metrics on the unit disk which is a geodesic disk of radius $1$ w.r.t. both metrics $\rho$ and $\rho'$, and assume that $\rho'$ has a constant Gauss ...
0 votes
1 answer
502 views

Analytic Chern classes

I have two questions on Chern classes, following Huybrechts' Complex Geometry. Are the analytic Chern forms just the elementary symmetric polynomials of the eigenvalues of the curvature? I googled ...
1 vote
0 answers
300 views

Steepest descent path and Picard-Lefschetz theory

Assume that an ordinary integral of the form $$I=\int_{-\infty}^{\infty}dx e^{-f(x)} $$ for some real function $f(x)$ is given where $f(x)$ is well defined over all $\mathbb{R}$ and the integral is ...
2 votes
0 answers
179 views

Questions about transformation or integral transformation

I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...
3 votes
2 answers
459 views

Smooth paths on affine varieties

I have the following question which is in some way related to an application of Randell Isotopy Theorem to complex hyperplane arrangements. Let $h,k\geq1$ be integer numbers and let $F_{1},\ldots,F_{...
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 ...
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 ...
3 votes
2 answers
517 views

Connected complement manifold

I'm working on some problem in algebraic geometry. I need a reference to the following result: Let $h\in\mathbb{N}$ with $h\geq1$ and let $F\in\mathbb{C}\left[x_{1},\ldots,x_{h}\right]$ be a non ...
3 votes
1 answer
371 views

If $X$ fails to be holomorphic, what is $\mathcal{L}_X \bar\partial - \bar\partial \mathcal{L}_X$?

I present a simplified version of the problem, and if that's easy to answer then someone can try answering the more complicated version. Simplified version Suppose $X$ is a tangent vector field on a ...
1 vote
1 answer
868 views

Triviality of holomorphic vector bundles over contractible Stein manifolds

If I have correctly undrestood,it is a result of the so called Grauert-Oka principle that all holomorphic vector bundles over contractible Stein manifolds are holomorhically trivial.Does any one knows ...
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 $\...
1 vote
1 answer
191 views

Sequence of smooth maps converging to the identity [closed]

Let $\{g_{n}:B_{1}(0) \rightarrow \mathbb{R}^{2}\}_{n\in \mathbb{N}}$ be a sequence of smooth maps where $B_{1}(0)=\{x\in \mathbb{R}^{2} \mid |x|<1\}$ is the unit ball in $\mathbb{R}^{2}$. Assume ...
1 vote
0 answers
215 views

Coordinate charts on converging Riemann surfaces

Let $S$ be a $2-$dim manifold and $q \in S$. Furthermore, let $j_{n}$ be a sequence of complex structures on $S$ converging in $C^{\infty}_{\text{loc}}$ to a complex structure $j$ on $S$ as $n\...
1 vote
1 answer
386 views

existence of positive curved line bundles on a compact Riemann surface

Can anyone suggest a proof of the existence of positive line bundles on a compact Riemann surface, avoiding Hodge decomposition. (I am aware of the method in Dror Varolin's book, but I consider that ...
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 ...
2 votes
0 answers
221 views

why is this result about Gaussian analytic functions equivalent to the Crofton formula

I am reading Zeros of Gaussian Analytic Functions by Mikhail Sodin and he gives an much-too-easy proof of density of zeros of a Gaussian Analytic function. Definition A Gaussian analytic function $...
9 votes
1 answer
4k 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) ...
3 votes
2 answers
322 views

The Szego projector, the dual disc bundle $\overline{D}$ and representation of $S^1$ on $H^2$($\overline{D}$)

This construction arises when constructing the Szego projector. Let's consider the dual disc bundle $\overline{D}$ of a positive Hermitian line bundle ($L$,$h$) over a compact Kahler manifold $M$, i....
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 $M$...
2 votes
1 answer
2k views

Definition of a complex structure on a vector bundle

Given a Riemann surface $S$, e.g. $\mathbb{P}^1(\mathbb{C})$, with complex conjugation on the coordinates and a holomorphic vector bundle $E$ over $S$. The complex conjugation $f$ is not holomorphic, ...
2 votes
1 answer
333 views

almost holomorphic line bundles

Let $(L,\omega_L) \to (M,\omega_M)$ be a symplectic line bundle (symplectic line=real dimension 2) over a symplectic manifold $M$. Each of these objects can be equipped with an almost complex ...
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'...
0 votes
1 answer
253 views

A question on the area of the unit disc w.r.t. a complete conformal metric

Question: Let $\Delta$ be the unit disc in $\mathbb{C}$ and $\rho(z)|dz|^2$ be a complete conformal metric on $\Delta$ where $\rho(z)$ is continuous on $\Delta$. Let $a$ be the infimum of $p (p>0)...
1 vote
1 answer
102 views

A subspace of the algebra of infinitesimal CR automorphisms

Let $(M^{2n+1}, D, J)$ be a strictly pseudoconvex CR sturcture on a compact $2n+1$-dimensional manifold, where $D$ is a nonintegrable distribution of codimension 1. The algebra of the infinitesimal CR ...
3 votes
1 answer
609 views

Normal form for a holomorphic Morse function

Similarly to Morse lemma, a holomorphic Morse function can be written, near a critical point, as $W_1^2+W_2^2+...+W_n^2+C$ , for well chosen coordinates $W_1,W_2,...,W_n$. I want to cite this result ...
2 votes
1 answer
90 views

How a matrix of C^1 functions on a domain Ω in Cn generates a C^1 distribution in Ω

I am reading the paper Complex foliation generated by (1,1)-forms by M. Klimek. I have a problem understanding why is true a detail in one of the theorems: Let $\Omega$ be an open connected subset ...
1 vote
0 answers
554 views

Local System and Gauss-Manin connection

Fix a complex manifold $X$. Then if we have a line bundle $L=\mathcal{O}(D)$ together with Gauss-Manin connection $\nabla: L \rightarrow L \otimes \Omega^{1}_X$, we get the locally constant sheaf $F$ ...
2 votes
0 answers
122 views

A parametrix for the $\bar\partial$ operator adapted to a holomorphic foliation

Let $X$ be a compact complex manifold with (regular) holomorphic foliation given by a holomorphic subbundle $\mathcal F$ of the tangent bundle. The foliation induces a filtration on differential forms....
2 votes
2 answers
175 views

What is the moduli space of germs of one-sided complex structures near the circle?

Consider a one-sided ( say, internal) neighborhood $U$ of the unit circle $S$ ( $U$ contains $S$) on the plane with a choice of smooth complex stricture $\tau$ on $U$. By smoothness of $\tau$ on $U$ ...
0 votes
1 answer
715 views

Dolbeault cohomology

Let $X=S^{2n-1} \times S^1$. I have to compute $H^{(1,0)}_{\bar{\partial}} (X)$ and $H^{(0,1)}_{\bar{\partial}}(X)$ . I don't know how to do this but if we use Kunnet formula we have that $dim(H^{...
1 vote
0 answers
130 views

Question about a oscillatory integrals on manifold

Let $M$ be a compact oriented Riemannian manifold without boundary. Set $f(x)=a(x)+\sqrt{-1}b(x)$ be a complex-valued function on $M$, where $a(x),b(x)$ are real-valued function on $M$. Then, how to ...
6 votes
0 answers
457 views

Jet differentials and hyperbolicity: possible mistake in the literature?

I was reading this note by Jingzhou Sun http://arxiv.org/abs/1109.1329 about Demailly's approach to hyperbolicity using jet differentials. The author seems to claim that there is a mistake in one of ...
2 votes
1 answer
364 views

Variant of the Riemann Mapping Theorem for $Conf(\mathbb H^2)$?

According to the Riemann mapping theorem it is possible to map a simply connected open subset $B \subset \mathbb C$ into any other $B' \subset \mathbb C$ by a (bi-)holomorphic mapping. Moreover, such ...
5 votes
1 answer
466 views

infimum of the Calabi energy in a given Kahler class

Given a compact Kahler manifold $M^n$ and a Kahler class $\Omega$. We have the Calabi energy (or Calabi functional) $$\textrm{Ca}(\omega)=\int_Ms^2(\omega)\omega^n,\qquad \forall \omega\in\Omega.$$ ...
1 vote
0 answers
175 views

The shape operator and an almost contact structure of a real hypersurface in $\mathbb{C}^n$

Let $S$ be an immersed real hypersurface in the Euclidean $\mathbb{C}^n$ with the standard complex structure $J$. Let $A:T(S)\rightarrow T(S)$ be the shape operator of $S$ (e.g. w.r.t. the outer ...
2 votes
2 answers
422 views

Bolza curve admits no anticonformal fixedpointfree involution

The Bolza curve B double covers the Riemann sphere with branching at the vertices of a regular octahedron. An affine model is given by the locus of $y^2=x^5-x$. How does one show that B does not ...
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 ...
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 $X$...
2 votes
2 answers
685 views

Automorphy Factors and Bundles

The question I'm considering is the following: given an 1-cocycle in of the modular group in Hom$(H;\textrm{GL}_{r}(C))$ call it $f$ when does it induce a vector bundle structure on the corresponding ...