Questions tagged [complex-manifolds]

For questions about or involving complex manifolds.

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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 ...
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1answer
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 ...
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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 ...
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1answer
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 ...
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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, ...
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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_{\...
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1answer
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 ...
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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{...
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1answer
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 ...
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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); $\bullet$ $B$...
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1answer
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 \...
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1answer
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 ...
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0answers
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 ...
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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 ...
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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 ...
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4answers
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 ...
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1answer
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 ...
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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 $...
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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 ...
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2answers
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 ...
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0answers
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 ...
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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 $p \neq q$ ?
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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 ...
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1answer
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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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2answers
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 ...
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1answer
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{\...
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1answer
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 $\mathbb{C}\mathbb{P}^2\# k\overline{\mathbb{C}\mathbb{P}^2}$, ...
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1answer
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 ...
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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 ...
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0answers
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 ...
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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 ...
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1answer
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\...
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2answers
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 ...
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3answers
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
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1answer
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\...
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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 ...
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0answers
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 ...
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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. ...
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1answer
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 ...
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0answers
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 ...
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0answers
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 } ...
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1answer
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 ...
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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 ...
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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: $0 \to \mathbb Z \to \mathcal O_X \to \mathcal O_X^* \to 0$ ...
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1answer
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 $n$-...

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