# Tagged Questions

**2**

votes

**1**answer

161 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 ...

**3**

votes

**1**answer

196 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 ...

**0**

votes

**1**answer

76 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 ...

**4**

votes

**1**answer

206 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 ...

**3**

votes

**2**answers

217 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 $M$, ...

**2**

votes

**1**answer

73 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 ...

**0**

votes

**0**answers

212 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

84 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 ...

**2**

votes

**2**answers

151 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

311 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 ...

**2**

votes

**0**answers

97 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 ...

**1**

vote

**0**answers

119 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

347 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

211 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 ...

**1**

vote

**2**answers

271 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 ...

**6**

votes

**2**answers

125 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 ...

**4**

votes

**1**answer

447 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 ...

**1**

vote

**0**answers

122 views

### Distance between probability amplitude functions

Suppose we have two probability measures $P_1$ and $P_2$ on some Riemannian manifold $(\Sigma,g)$. There are many potential distance measures between $P_1$ and $P_2$:
The Wasserstein distance
For ...

**3**

votes

**1**answer

206 views

### Smoothness of solution to a PDE

Let $X$ be a Riemann surface and let $E$ be a smooth complex vector bundle on $X$ with a connection $D$. We can write the connection $D$ as the sum $D'+D''$ where $D'$ is the (1,0) part and $D''$ is ...

**10**

votes

**4**answers

2k 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 ...

**5**

votes

**0**answers

330 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 ...

**3**

votes

**1**answer

290 views

### Special Morse function on a Riemann surface

Let $f: S \to \Bbb R$ be a Morse function on a Riemann surface. Let $x_0$ be a saddle point of $f$. Since $x_0$ is a critical point of $f$, it makes sense to talk about the bilinear forms ...

**2**

votes

**1**answer

336 views

### Are Lefschetz thimbles holomorphic manifolds?

I have a Lefschetz thimble defined by the stable flow of the gradient a holomorphic function
toward a critical point (as defined e.g. in Witten arXiv:1001.2933 and F.Pham "Vanishing homologies and the ...

**23**

votes

**0**answers

923 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

**1**answer

187 views

### What does non-levi flat point mean geometrically

Hello,
$CR$ manifold for example $S^1\times C^{n-1}$ is every where levi flat. Can I have example of $CR$ manifold which has at least one non levi flat point.
I can't see what the happening in ...

**29**

votes

**1**answer

889 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 ...

**3**

votes

**1**answer

319 views

### The Levi form of the distance squared function in a non-positively curved Kaehler manifold

Suppose that if $X$ is a complete, simply connected Kaehler manifold with non-positive sectional curvatures. Let $P \in X$ and $h : X \to \mathbb{R}$ be the function defined by $h(x) = dist(P,X)^2$. ...

**0**

votes

**0**answers

108 views

### Is degree a “strict -transform” birational invariant for surfaces in the complex projective 3-space?

(Edit) My question is as follows: My previous question was about a kind of "strict-transform birational" invariant not birational invariants as usual. So I just delete the question since it ...

**1**

vote

**1**answer

266 views

### skoda el-mir theorem

now i'm studying the skoda el-mir theorem about the extension of a positive closed current $T$.
But if $T$ ed $S$ are two positive closed currents on a manifold $X$ such that are equal on $X\setminus ...

**6**

votes

**2**answers

575 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**

votes

**1**answer

211 views

### Is a certain composition of harmonic forms again harmonic?

Let $(X,\omega)$ be a compact Kahler manifold, and let $\alpha$ and $\beta$ be smooth $(1,1)$-forms on $X$ that are harmonic (with respect to $\omega$). I can consider each of my $(1,1)$-forms as an ...

**2**

votes

**3**answers

453 views

### harmonic 1-form with bounded energy on a strip in $\mathbb{R}^2$

Let $S=[-a,a]\times[b,+\infty] \mod \{(-a,t) =(a,t) \mid t \in [b,+\infty]\}$ be a strip in $\mathbb{R}^2$ with identified sides. Let $w$ be a real harmonic 1-form on $S$, which has a primitive $f$ on ...

**1**

vote

**0**answers

142 views

### Holomorphic vector fields with growth conditions on $X_\mathrm{reg}$

Let $M$ be a complex manifold with a hermitian metric (volumes and distances will be wrt this metric). Let $X\subset M$ be a complex analytic subspace of $M$ and $Y\subset X$ an analytic set ...

**0**

votes

**1**answer

412 views

### is the differential of the distance function holomorphic?

i have the following problem. Let $M$ be a complex n-dim manifold and $X \subset M$ be a n-dim real analytic submanifold. Consider $d_{X}(z)$ be the squared distance from $z \in M$ to $X$. For $z$ ...

**1**

vote

**0**answers

199 views

### Non-realizable CR structures?

Hill, Penrose, and Sparling have an example of a non-realizable CR structure, a 5-manifold $M^5$ that comes equipped with a "twisted version" of the Lewy operator for the quadric $Q^2$,
$v = ...

**9**

votes

**3**answers

532 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 ...

**2**

votes

**0**answers

244 views

### Is there asymptotic expansion of heat kernel of complex laplacian?

On real Riemannian manifold , the heat kernel of the laplacian have an asymptotic expansion . But on complex manifold , i haven't seen a result like this , i.e. the heat kernel of the Kodaira ...

**0**

votes

**0**answers

479 views

### Relative tangent bundle of the pullback of a vector bundle splits?

Let $S$ be a complex manifold, which for our purposes we can take to be a small ball in $\mathbb C^n$. Let $p : E \to S$ be a holomorphic vector bundle over $S$, and let $\pi : X \to S$ be a family of ...

**3**

votes

**1**answer

160 views

### If X fails to be holmorphic, what is Lie_X \bar\d - \bar\d Lie_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 ...

**5**

votes

**2**answers

522 views

### Conformal structure determined by principal curvatures

On a smooth surface of positive Gauss curvature that is embedded in Euclidean 3 space the principal curvatures seem to define a smooth field of ellipses whose major and minor axes are lined up with ...

**5**

votes

**0**answers

493 views

### Computing the Chern class for a flat line bundle using the holonomy group?

Let $X$ be a Riemann surface of genus $g \geq 2$. I would like to consider flat line bundles on $X$. Flat line bundles can be identified with representations $\pi_1(X) \rightarrow U(1)$. From ...

**1**

vote

**1**answer

281 views

### metrics compatible with conformal structures

I have three related questions:
(1) How does one describe the possible Riemannian metrics that are compatible with a conformal structure on a two dimensional surface?
(2) Can all conformable ...

**4**

votes

**3**answers

338 views

### Examples of non-Kahler surfaces with explicit non-Kahler metric

Hi,everyone.Can someone give me some examples of non-Kahler surfaces whose complex structure and metric structure are all clear?

**1**

vote

**2**answers

274 views

### Families of Fuchsian models

A Fuchsian model for a Riemann surface $X$ is a discrete subgroup $G$ of $PSL_2(\mathbb{R})$ such that there is a biholomorphic map from $U/G$ to $X$.
For a fixed genus $g \geq 2$ one knows from Bers ...

**6**

votes

**1**answer

688 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 ...

**2**

votes

**1**answer

246 views

### Extending holomorphic connections

Let $D$ denote the disk $|z|<1$ in the complex plane and $U=D\0$(punctured disk). Define a holomorphic connection $\nabla$ on $\mathscr{O}_U$ by $\nabla(1)=\exp{(-1/z)}$. Does this extend to a ...

**21**

votes

**3**answers

2k 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 ...

**25**

votes

**0**answers

3k views

### A paper to the question, if the six dimensional sphere is a complex manifold

Hi,
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 ...

**2**

votes

**0**answers

285 views

### Boundary behavior of Kähler cone with curvature restriction

Let $(M,\omega)$ be a compact Kähler manifold. The boundary behavior of Kähler cone is very interesting; however,it's hard to understand.
A fundamental result is due to Demailly and Paun: they ...

**8**

votes

**1**answer

830 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 ...