# Questions tagged [complex-manifolds]

For questions about or involving complex manifolds.

245
questions

**0**

votes

**0**answers

10 views

### Opers and global differential operators

This is a follow up question to a previous question of mine and my thought of answer to it.
Given a (compact) Riemann surface $\Sigma$, a $SL(n,\mathbb{C})$-oper is a rank $n$ holomorphic vector ...

**3**

votes

**1**answer

72 views

### Tangent bundle with Sasaki metric is Kähler iff $M$ is locally flat

I'm having a hard time proving the following:
If $M$ is an $n$-dimensional indefinite Riemannian manifold whose metric $g$ has index $s$, then the metric of Sasaki $g^{D}$ is an indefinite metric ...

**2**

votes

**0**answers

147 views

### cohomology classes of complex submanifolds

I was wondering if there were restrictions in what the cohomology classes corresponding to complex submanifolds of a complex manifold could be.
For example, say $T^4$ is regarded as a complex ...

**4**

votes

**1**answer

111 views

### Leaves of stable foliation of holomorphic Anosov diffeomorphism

I'm trying to understand the first half of the paper "Holomorphic Anosov systems" by E. Ghys (the journal reference is Inventiones mathematicae volume 119, pages 585–614(1995)). My question is about a ...

**3**

votes

**0**answers

87 views

### Mixed Hodge structures on (infinite) covers of complex varieties?

Let $X$ be a complex variety, and let $\tilde{X}\to X$ be a covering map. Does the singular cohomology $H_\ast(\tilde{X};\mathbb{Z})$ carry a natural mixed Hodge structure?
If the cover is finite, ...

**4**

votes

**0**answers

128 views

### A metric $w$ on a Kahler manifold is extremal if and only if the gradient vector field of the scalar curvature is holomorphic

I am trying to understand the calculation in An introduction to Extremal kahler metrics. On the fourth line of page 55 the author calculated that $\int_{M} - 2 S R^{\bar k j} \partial_{j} \partial_{\...

**1**

vote

**1**answer

71 views

### Holomorphic local trivialization of a principal toric bundle

Let $G$ be an even-dimensional compact Lie group with Lie algebra $\mathfrak{g}$ and let $T \subset G$ be a maximal torus with Lie algebra $\mathfrak{t}$.
We can construct a left-invariant complex ...

**0**

votes

**0**answers

47 views

### Fourier coefficients of a variation in Teichmuller theory

Prove that for $\dot w[\mu](\zeta)=-\frac{(\zeta-1)(\zeta+1)(\zeta+i)}{\pi}\left\{\iint_{\Delta} \frac{\mu(z) d x d y}{(z-1)(z+1)(z+i)(\zeta-z)}+\iint_{\Delta} \frac{i \overline{\mu(z)} d x d y}{(\bar{...

**1**

vote

**1**answer

130 views

### Are anti-linear maps/semi-linear, such as conjugations, linear in other almost complex structures?

I have asked this on mse, but I did not get any responses even after a bounty.
I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much ...

**1**

vote

**0**answers

77 views

### When is a holomorphic map from a complex surface to a Riemann surface is a holomorphic family of Riemann surfaces?

A holomorphic family of Riemann surfaces of type $(g, n)$ is a triple $(M, \pi,B)$ defined as follows:
$\bullet$ M is a $2$-dimensional complex manifold (topologically, a $4$-manifold);
$\bullet$ $B$...

**2**

votes

**1**answer

214 views

### (1/2) K3 surface or half-K3 surface: Ways to think about it?

I heard from string theorists thinking of the so-called "(1/2) K3 surface" or "half-K3 surface" as follows:
Let $T^2 \times S^1$ be a 3-torus with spin structure periodic in all directions. $T^2 \...

**4**

votes

**1**answer

153 views

### Compact complex surface that admits a Kodaira fibration is Kahler

A Kodaira fibration is a compact complex surface X endowed with a holomorphic submersion onto a Riemann surface $\pi: X\to\Sigma$ which has connected fibers and is not isotrivial.
Is there an easy ...

**2**

votes

**0**answers

127 views

### Nonstandard definitions of complexifications [closed]

I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much as possible, but I have been stuck on the concepts of almost complex structures ...

**1**

vote

**0**answers

61 views

### Riemann's bilinear relations

I am reading the paper [1], which states
Haupt showed that a vector with complex entries $(w_1, \cdots, w_g, z_1, \cdots, z_g)$ is the period row of some holomorphic differential with respect to a ...

**2**

votes

**0**answers

50 views

### What's the behavior of the laplacian of dbar operator w.r.t. a singular metric of a holomorphic line bundle?

What's the behavior of the laplacian of dbar operator w.r.t. a singular metric of a holomorphic line bundle or other holomorphic vector bundle over a complex manifold ?Do we have anything similar ...

**2**

votes

**4**answers

288 views

### Learning roadmap for complex geometry

I am interested to pursue my graduate studies in complex geometry but sadly I did not find a lot of references regarding the learning roadmap for complex geometry on the website. (most of them are ...

**8**

votes

**1**answer

272 views

### Holomorphic deformation of complex structure on the real plane

It is known that each complex structures on $\mathbb{R}^2$ is biholomorphic to either $\mathbb{C}$ or the open unit disk $\Delta$.
One can continuously deform one complex structure to the other as is ...

**3**

votes

**0**answers

77 views

### Extension of holomorphic maps to smooth family of holomorphic maps

Let $\pi:X \to B$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and you can think of $B$ as a small disk $...

**5**

votes

**0**answers

149 views

### Does the sheaf $\mathcal{O}^*$ on a complex manifold have an acyclic cover?

Let $X$ be a complex manifold and let $\mathcal{O}^*$ be the sheaf nonvanishing holomorphic functions on it. Does it have an acyclic cover? That is, a cover for which all open sets and all ...

**5**

votes

**2**answers

305 views

### Does Peetre's theorem hold in complex analysis?

Let $E, F$ be two smooth vector bundles over a smooth manifold $M$. Peetre's theorem states that any $\mathbb{R}$-linear morphism $D: \mathcal{E} \to \mathcal{F}$ of the sheaves of sections of $E$ and ...

**1**

vote

**0**answers

76 views

### Embedding representation of fundamental group in functions on universal cover

Let $X$ be a complex manifold, $Y\rightarrow X$ the universal cover, and $\Gamma$ be the Galois group of this cover, ie the fundamental group of $X$.
Finite dimensional $\Gamma$-representations ...

**1**

vote

**0**answers

48 views

### Kodaira Dimension of a Calabi-Eckmann manifold

What is the Kodaira dimension of a Calabi-Eckmann manifold, $X$, when the underlying topological manifold is $S^{2p+1} \times S^{2q+1}$ where $p \geq 1$, $q \geq 1$ and $p \neq q$ ?

**1**

vote

**0**answers

40 views

### Existence of holomorphic coverings having small degree

Let $\Sigma$ be a closed Riemann surface of genus $g$. In the book of Farkas and Kra, they prove that there exists a holomorphic covering map $F : \Sigma \to \mathbb{S}^2$ of degree less than or equal ...

**9**

votes

**1**answer

297 views

### Orientable with respect to complex cobordism?

I have learned that an orientation of a manifold $M$ with respect to ordinary cohomology is an ordinary orientation, that an orientation with respect to complex K-theory is a Spin$^c$ structure, and ...

**3**

votes

**0**answers

71 views

### Period mapping in non-abelian Hodge theory

Given a family of Kaehler manifolds, by looking at the cohomology we can construct the period mapping. What should be the analogue of the period mapping in non-abelian Hodge theory (i.e. if we look at ...

**4**

votes

**0**answers

104 views

### HKT manifolds with non trivial canonical bundle

A compact HKT manifold is a hyperhermitian manifold $(M,I,J,K,g)$ such that either $\partial (\omega_J+i\omega_K)=0$ (if endomorphisms act on the left on the tangent space and $\partial$ is taken with ...

**9**

votes

**0**answers

245 views

### Higher homotopy groups of Calabi-Yaus

Is something known about the higher homotopy groups of Calabi-Yau threefolds? For example, one of the easiest CYs is the quintic, defined as the anticanonical divisor in $\mathbb{CP}_4$. What are its ...

**1**

vote

**0**answers

43 views

### Uniform convergence of holomorphic automorphisms

Let $X$ be a complete Kobayashi hyperbolic complex manifold. It is well-known that the automorphism group of $X$ is a real Lie group where the topology on the automorphim group is the compact-open ...

**4**

votes

**0**answers

155 views

### Constructing new complex manifolds out of old

It is not difficult to build new manifolds out of old in the smooth category, for example
taking the direct product or constructing a fiber bundle,
taking the level set of a regular value of a smooth ...

**5**

votes

**2**answers

361 views

### Compact complex affine Kähler manifold is a torus

Before giving a motivation let me ask the precise question firstly.
By a complex affine manifold I mean a complex manifold $M$ with the property that there exists an holomorphic atlas for which ...

**2**

votes

**1**answer

173 views

### Differences of $\omega$-plurisubharmonic functions

Let $X$ be a complex manifold, and $\omega$ a Kähler form on $X$.
A smooth function $\phi$ on $X$ is $\omega$-plurisubharmonic ($\omega$-psh for short) if the form $\omega+\sqrt{-1}\partial\bar{\...

**3**

votes

**1**answer

160 views

### Surface with Kahler-Einstein metric

Let $3\leq k\leq 8$ be an integer. Suppose $M$ is a complex surface which has a Kahler-Einstein metric and has the same Betti numbers as $\mathbb{C}\mathbb{P}^2\# k\overline{\mathbb{C}\mathbb{P}^2}$, ...

**1**

vote

**1**answer

107 views

### Equalizer of local analytic isomorphisms

Let $a,b : V\to W$ be two morphisms of smooth complex analytic spaces.
Assume $a$ and $b$ are local analytic isomorphisms.
Does the equalizer $U$ of $a,b$ exist as a smooth complex analytic ...

**3**

votes

**0**answers

165 views

### Chern number of projection-Topological magic in physics

I enclosed a computation from a well-known paper in the field of mathematical physics where the Chern number of the first Landau level is computed (the result claimed is $-1$) and the full paper can ...

**2**

votes

**0**answers

159 views

### Examples of certain compact Kaehler manifolds

A Kaehler manifold is a complex manifold which has a Kaehler metric and Ricci curvature tensor $R_{ij}$. The Ricci curvature tensor is a Hermitian matrix having real eigenvalues. My question is: Is ...

**8**

votes

**0**answers

203 views

### Holomorphic structures on vector bundles over $\mathbb C\mathbb P^2$

It is known that every (topological) complex rank $2$ vector bundle over $\mathbb C\mathbb P^2$ admits holomorphic structures. A proof can be found in the book of Okonek, Spindler, Schneider which is ...

**1**

vote

**1**answer

58 views

### On extensions of holomorphic mappings with image in a projective algebraic variety

I am reading the paper "Two Theorems on Extensions of Holomorphic Mappings" by PHILLIP A. GRIFFITHS. In Example 2 of the paper, there is a proposition saying that:
Let $N$ be a complex manifold, $S\...

**3**

votes

**2**answers

218 views

### Integrability of an almost complex structure vs holomorphicity of the section $M\rightarrow \mathcal{J}(M)$

Let's say we have an almost complex manifold $(M, J)$. Consider the complex vector bundle $V\rightarrow M$ whose fiber over $x$ is the space of almost complex structures on $T_x M$.
Is there any ...

**1**

vote

**3**answers

158 views

### Almost complex structure corresponds to unique complex structure up to biholomorphism [closed]

Assume I have a smooth manifold which has two different holomorphic atlases that induce the same almost complex structure on it. How to show that two complex manifolds (corresponding to the two ...

**2**

votes

**1**answer

132 views

### Pushing forward a complex structure by submersion

I have a surjective smooth map with surjective differential between two balls $\phi:B^{2n}\rightarrow B^{2k}$. Fix an integrable almost complex structure $J$ on $B^{2n}$. Assume that $\mathrm{Ker}\:d\...

**4**

votes

**0**answers

59 views

### Extension of holomorphic function on family of relatively compact strictly pseudoconvex domains

Let $Y \to M$ be a (proper and locally trivial) family of relatively compact strictly pseudoconvex domains which are smooth (not neccesarily Stein spaces). So $Y$ and $M$ are a complex manifold and ...

**1**

vote

**0**answers

97 views

### Interpretation of deformation of complex structure

Let $X$ be a smooth complex analytic space and let $D$ be the unit disk in $\mathbb{C}$. Let $\omega:Y \to D$ be a deformation of complex structures of $X$ in the sense that (1) $\omega^{-1}(0) \simeq ...

**1**

vote

**0**answers

125 views

### Example of open manifold with no free integer homology non-homeomorphic to a ball

I would like to state that if an open oriented even-dimensional (complex) manifold $M$ is such that $dim(H_k(M,\mathbb{Z}))=0$ for $k>0$, and 1 for $k=0$, then $M$ is homeomorphic to an open ball.
...

**6**

votes

**1**answer

284 views

### Cohomology of neighborhood of $\mathbb{C}\mathbb{P}^1$ in $\mathbb{C}\mathbb{P}^n$

Let $\mathbb{C}\mathbb{P}^1$ be embedded linearly to $\mathbb{C}\mathbb{P}^n$ with $n>1$. (Such an embedding is given in coordinates by $[x:y]\mapsto [x:y:0:\dots: 0]$.)
Is it true that for any ...

**3**

votes

**0**answers

136 views

### Cohomology of a tubular neiborhood of submanifold vs cohomology of the formal neighborhood

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $\hat Z$ be the formal neighborhood of $Z$. Let $U$ be an open neighborhood of $Z$.
Is ...

**8**

votes

**0**answers

250 views

### Cohomology of complex manifold vs cohomology of its complex submanifold

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that
$$H^i(Z, A|_Z)=0 \mbox{ for any } ...

**1**

vote

**1**answer

58 views

### Minimal complex surfaces with pseudo-effective canonical bundles

A complex line bundle $L$ over a complex surface $X$ is said to be pseudo-effective if it admits a (possibly singular) Hermitian metric $h$ whose curvature is positive semi-definite in the sense of ...

**1**

vote

**0**answers

308 views

### Path lifting property of holomorphic unbranched map

Suppose $X$ is a Riemann surface and $ a\in\ X $ suppose $ \phi\in\mathcal O_a $ is a holomorphic function germ at $a.$ According to the theorem 7.8 of Forster's book Lectures on Riemann surfaces on ...

**2**

votes

**0**answers

180 views

### Finding the torsion of the Neron Severi group in the first homology group

Let X be a variety over $\mathbb C$. I will implicitly identify this with us complex analytification. Consider the exponential sequence:
$0 \to \mathbb Z \to \mathcal O_X \to \mathcal O_X^* \to 0$
...

**7**

votes

**1**answer

374 views

### How does one complexify a real $n$-dimensional Riemannian manifold $(M,g)$?

If $V$ is a real vector space, then the complexification of $V$ is formally defined as $V^{\mathbb{C}}=V\otimes_{\mathbb{R}}\mathbb{C}$. Is there an analogous complexification operation for a real $n$-...