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Questions tagged [holomorphic-bundles]

A complex vector bundle $\pi \colon E \to M$ is a vector bundle whose fiber bundles $\pi^{-1}(m)$ are a copy of $\mathbb{C}^k$. $\pi$ is a holomorphic vector bundle if it is a holomorphic map between complex manifolds and its transition functions are holomorphic. The simplest example is a holomorphic line bundle, where the fiber is simply a copy of $\mathbb{C}$.

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Classifying space of finite group as a complex manifold

Suppose $G$ be a finite group. Then, is it always possible to construct a classifying space which is a finite dimensional(if not, possibly infinite dimensional) complex manifold? More precisely, are ...
ChoMedit's user avatar
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Complex manifold structure on moduli space of holomorphic bundles on Riemann surface

Consider the space $\mathcal N$ of rank $r$, degree $d$ holomorphic bundles on a compact Riemann surface $X$ of genus $g\geq 2$. Neitzke constructs this by considering $L^2_k$-completions and taking ...
mifrandir's user avatar
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Holomorphic rotational vector field on \PP^1

On p. 45 of the Mirror Symmetry textbook (Hori-Katz-Klemm-Pandharipande-Thomas-Vafa-Vakil-Zaslow), Example 3.5.1 reads: "On $\mathbb{P}^1$ consider the holomorphic vector field $u \frac{\partial}{...
locally trivial's user avatar
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abelian differentials are $L^2$ integrable

Let $C$ be a smooth algebraic curve over $\mathbb{C}$ (we can think for instance of a compact Riemann surface). Let $A$ be the subset of classes of holomorphic forms $\omega$ in $H^1(C,\mathbb{C})$ ...
Conjecture's user avatar
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Holomorphic function of several complex variables and sets of zero Lebesgue measure

Let $f$ be a holomorphic function of $n$ complex variables in region $\Omega \subset \mathbb{C}^n$, with $\Omega \supset \mathbb{R}^n$. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}^n$. Is the ...
HenryYRZ's user avatar
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Lefschetz operator on bundle-valued forms

For a holomorphic vector bundle $V \rightarrow X$ endowed with a Hermitian structure, one may define the corresponding Dolbeault-like operators $\bar{\partial}_V: \Omega^{p,q}(V) \rightarrow \Omega^{p,...
Eweler's user avatar
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Finding a holomorphic representing section

Let $E\rightarrow M$ a holomorphic vector bundle over a complex manifold M, and $S\subset E$ a holomorphic subbundle. Let $\tilde{\eta}$ a local holomorphic section on the quotient bundle $Q=E/S$ (...
kahlerian's user avatar
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Algebraic vector bundles which are analytically but not algebraically isomorphic

I am looking for an example of two algebraic vector bundles on an algebraic complex manifold / smooth complex algebraic variety which are analytically isomorphic, but not algebraically isomorphic. By ...
Earthliŋ's user avatar
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Does a compact Kahler manifolds with negative first Chern class admits any nontrivial holomorphic vector field?

Let $M$ be a compact Kahler manifold with $c_{1}(M)\lt 0$. It seems that I can prove the following claim: there is no nontrivial holomorphic vector field on $M$. Here is my proof: let $T^{1,0}M$ be ...
Holomodric's user avatar
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Equivalent definition to holomorphic vector bundle

Let $\pi:E \rightarrow M$ be a complex vector bundle over a complex manifold $M$. Prove that if there exists a complex structure on $E$ as a manifold, such that the projection $\pi$ is holomorphic ...
inquisitor's user avatar
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What happens when we compactify every fibre of a holomorphic $\Bbb C$ bundle over a compact Riemann surface?

I am a beginner in learning about holomorphic line bundles over Riemann surfaces. In particular, I am a complete naïf as regards sheaf cohomology (so I hope the answer(s) won't involve that too deeply)...
Dan Asimov's user avatar
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4 votes
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Complex vector bundle that does not admit holomorphic structure

Is there any example of a complex vector bundle $E \to M$ whose base space $M$ is also a complex manifold, but where the total space $E$ does not admit a holomorphic structure compatible with the ...
isekaijin's user avatar
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1 answer
315 views

Is flat connection holomorphic?

Let $M$ be a complex manifold and $E\to M$ be a holomorphic vector bundle. Let $\nabla:E\to E\otimes A_M^1$ be a flat connection, where $A_M^1$ stands for the sheaf/complex vector bundle of smooth 1-...
Doug's user avatar
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3 votes
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Zero Chern class and trivial line bundle

Let $X$ be a compact complex Kähler manifold and $L$ an holomorphic line bundle over $X$. Let suppose $c_{1}(L) = 0$ in $H^{1, 1}(X, \mathbb{C})$. I would like to show that if $L$ is not the trivial ...
Analyse300's user avatar
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Are there holomorphic vector bundles that do not holomorphically embed in any free one?

Are there any examples of holomorphic vector bundles $E \to M$ of rank $k$ that do not admit a holomorphic classifying map $M \to \mathrm{Gr}_k(\mathbb C^n)$ for any $n \in \mathbb N$? Or, ...
isekaijin's user avatar
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Geometric interpretation of the algebraic multiplicity of an isolated singularity of a holomorphic folation by curves

Let $\mathscr F$ be a singular foliation by curves on a complex manifold $M$, with only isolated singularities. The Milnor number $\mu_p(\mathscr F)$ at a point $p \in M$ is the topological index at $...
isekaijin's user avatar
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Twisting a section of a vector bundle until it only has codimension $\ge 2$ zeros

I am reasonably sure that the following statement is true if by a “space” we mean either a smooth quasi-projective variety or a complex manifold: Let $M$ be a space, $E \to M$ a vector bundle, and $\...
isekaijin's user avatar
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3 votes
1 answer
221 views

Are holomorphic structures on complex vector bundles unique up to isomorphism?

I'm looking for an example of the following situation: $M$ is a complex manifold, $E_1, E_2$ are non-isomorphic holomorphic vector bundles on $M$, but $E_1$ is isomorphic to $E_2$ as a complex smooth ...
red_trumpet's user avatar
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Clarifications on working with holomorphic vector bundles

I am at a loss as to several points made in A. Moroianu's "Lectures on Kähler Geometry", Chapter 9, section 9.2, from where I am trying to understand how to work with holomorphic vector ...
rosecabbage's user avatar
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Complex manifolds $X$ with $TX \to X$ smoothly but not holomorphically trivial

I am aware that there exist holomorphic vector bundles $E \to X$ that are smoothly trivial but not holomorphically so. For example, I think the following example works: let $X$ be a compact Riemann ...
isekaijin's user avatar
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2 votes
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Harmonic sections of $\Omega_X^p\otimes E$ over a Kähler manifold

I am confused by an exercise in Huybrechts Complex geometry: an introduction regarding sections of a certain holomorphic vector bundle over a Kähler manifold, related to Hodge theory. The exercise is ...
topolosaurus's user avatar
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Holomorphic forms on product spaces

Let $C$ be a compact Riemann surface without boundary. In particular, $C$ is a surface of a genus $g$. Then the vector space of holomorphic 1-forms $\mathrm{H}^0(C,\mathrm{K}_C)$, where $\mathrm{K}_C$ ...
KuSi's user avatar
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bundle valued form over $\mathbb{C}P^1$

Suppose $L$ is a line bundle over $\mathbb{C}P^1$, I wonder how to determine whether $H^{0,1}(\mathbb{C}P^1,L)$ is $0$ or not. I want to use kodaira vanishing theorem at first, but in my case the ...
taiat's user avatar
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Topological equivalent vs Holomorphic equivalent of line bundle.

I know there are indeed some examples show that two line bundles over some projective manifold is topological equivalent but not holomorphic equivalent, but I find I give a "proof" shows ...
taiat's user avatar
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1 vote
1 answer
58 views

holomorphic line bundle over germ of complex space

I wonder if holomorphic line bundle over zero of holomorphic function(over $\mathbb{C}^n$) is trivial?(We can assume it to be a manifold if necessary) I Know there is a principle that, for Stein ...
taiat's user avatar
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2 votes
1 answer
193 views

When determinant line bundle is holomorphically trivial

I'm learning the deformation theory of holomorphic structure over given smooth vector bundle by the book Smooth four - manifolds and complex surfaces. However, when talk about holomorphic vector ...
taiat's user avatar
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Chern forms and tensor products

Let $E\to X$ be a rank $r$ holomorphic vector bundle over a Kahler manifold and let $L\to X$ be a holomorphic line bundle. The following relation between Chern classes is (well) known: $$c_2(E\otimes ...
BinAcker's user avatar
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How does the Borel-Weil theorem works? What do we use it for, how does the application look generally?

Borel-Weil from Sepanski´s book "Compact Lie Groups": My intuition for Borel-Weil: We have a compact Lie group G. And we want to “describe it nicely” - we want to find its representation to ...
Tereza Tizkova's user avatar
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Producing new connections out of given one

Let $E$ be a holomorphic vector bundle over $M$ which is a complex manifold. Let $D$ be a connection on it. So $D$ maps sections of $E$ to sections of $T^*M \otimes E$. Let $D'$ and $D''$ be the $(1,0)...
Angry_Math_Person's user avatar
5 votes
1 answer
618 views

Meromorphic functions in projective space

It is well known that meromorphic functions in $\mathbb P^n_\mathbb C$ are of the form $p/q$ where $p$, $q$ are homogeneous polynomials in $\mathbb C[x_0 , \ldots , x_n]$ of the same degree, and $q \...
Aitor Iribar Lopez's user avatar
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1 answer
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Representative of the Todd class

Let $E\to X$ be a holomorphic vector bundle. I wonder whether there exists a way to compute a representative of the Todd class $Td(E)\in H^*(X)$ of $E$ (i.e. a proper differential form in $\Omega^*(X)$...
BinAcker's user avatar
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4 votes
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First Chern class coincides with degree of divisor without poincare duality or de rham cohomology

I know there are a lot of references (e.g. Griffiths-Harris page 141), but the issue is that these references always prove the proposition in arbitrary dimensions, using a somewhat contrived ...
nolatos's user avatar
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Tangent fiber space of cone and vector fibered spaces

Suppose that $M$ is a complex analytic space. In the category of analytic spaces over $M$ one can consider cone and vector objects. Which I will call cones or linear fiber spaces over $M$. Every cone ...
Thomas Kurbach's user avatar
2 votes
1 answer
145 views

Question regarding $H^0$ of stable vector bundle

Let $M$ be a compact Riemann surface of genus $g$ and $E$ be a holomorphic vector bundle with the property that for any proper holomorphic subbundle $F$ we have $deg(F)/rank(F) < deg(E)/rank(E)$ ...
Angry_Math_Person's user avatar
1 vote
0 answers
55 views

Expression for the form $F^k\wedge \omega^{n-k}$

Let $E\to X$ be a holomorphic Hermitian vector bundle over a compact Kahler manifold. Let us write the curvature as follows $$F=\sum_{i,j}dz_i\wedge d\bar{z}_j\otimes f_{ij}$$ where $f_{ij}\in End(E)$ ...
BinAcker's user avatar
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0 votes
1 answer
454 views

Holomorphic sections of holomorphic vector bundles.

If $G:E\rightarrow F$ is a holomorphic bundle map over a smooth map $f:M\rightarrow N$ where $M$ and $N$ are two complex manifolds, there is a result that say that every local smooth section can be ...
Uoff's user avatar
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1 answer
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A nonzero holomorphic parallel section never vanishes

I'm reading a vanishing theorem from Kobayashis' differential geometry of complex bundles and I'm stuck with an argument. Let $E\rightarrow M$ a holomorphic vector bundle over a compact Kahler ...
kahlerian's user avatar
1 vote
0 answers
57 views

fiber of holomorphic bundle as pull back sheaf

Let $f:(X,\mathcal{R}_X)\to (Y,\mathcal{R}_Y)$ be the map between two topological space,with the sheaf of ring. Given a sheaf of $\mathcal{R}_Y-$module $\mathcal{N}$,we can define the inverse image ...
yi li's user avatar
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3 votes
1 answer
180 views

Obtaining the Dolbeault operator on the pullbak of the holomorphic tangent bundle.

I have ran into a question while reading this paper by Witten. The question is mostly mathematical, so I thought it would be better posed here instead of on Physics SE. Let $X$ be a Kahler manifold, $\...
CoffeeCrow's user avatar
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1 vote
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Example of splitting vector bundle over projective line

Birkhoff-Grothendieck theorem states that any holomorphic vector bundle over $\mathbb P^1$ splits as a direct sum $E = \bigoplus \mathcal O_{\mathbb P^1}(a_i)$. I am trying to work out an explicit ...
Troshkin Michael's user avatar
1 vote
0 answers
69 views

Proving that a quasi-isomorphism between complexes of trivial vector bundles is a cochain isomorphism.

Let $$ \cdots\rightarrow \mathcal{E}^{j-1}\xrightarrow{d_{j-1}}\mathcal{E}^{j}\xrightarrow{d_{j}}\mathcal{E}^{j+1}\rightarrow\cdots $$ be a complex of holomorphic vector bundles over a manifold $X$, ...
CoffeeCrow's user avatar
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3 votes
0 answers
101 views

Structure of the space of holomorphic structure on a vector bundle

It is well known that the space of connections on a vector bundle $E\rightarrow X$ is an affine space modeled on $\Omega^1(X,End(E))$. Let $Dol(E)$ denote the space of holomorphic structures $\bar{\...
BinAcker's user avatar
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0 votes
1 answer
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Determinant of the vertical tangent bundle of a projectivised bundle

Let $\pi : PE \to B$ be a projectivised rank $n$ vector bundle. In this MathOverflow answer, Michael Thaddeus says that for such a bundle, $T_{\pi} = \operatorname{Hom}(O(-1), E/O(-1)) \cong E(1)/O$, ...
Analyse300's user avatar
1 vote
1 answer
332 views

Confusion about Chern connection one form formula

Let $(E,h)$ be a holomorphic Hermitian vector bundle with local frame $\{e_i\}$. Denote by $H$ the matrix of $h$ in $\{e_i\}$. I found two different expression for the Chern connection 1-form $\omega$...
BinAcker's user avatar
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1 vote
1 answer
280 views

Why is this bundle positive?

Let $E$ be a holomorphic bundle (with rank $r$) over a compact complex manifold $X$. $E^{*}$ will denote its dual. $\mathbb{P}(E^{*})$ will denote the projective bundle associated to the dual of $E$ ...
Analyse300's user avatar
2 votes
0 answers
99 views

The identification of $iX$ with $JX$

I am reading the book, Lectures on Kahler Geometry, by Andrei Moroianu. Here a link, https://books.google.com.hk/books?id=oqmroUc9E8YC&printsec=frontcover&hl=zh-CN&source=gbs_ge_summary_r&...
Kai Xu's user avatar
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1 vote
2 answers
135 views

Decision problem for a bundle in a line bundle extension of $\dim Ext^1=1$.

Let $E$ be a vector bundle over a nonsingular irreducible surface $X$ given by an extension $$0 \rightarrow A \rightarrow E \rightarrow B \rightarrow 0,$$ where both $A$ and $B$ are a line bundle over ...
nida08's user avatar
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1 vote
1 answer
205 views

Does the natural exact sequence of the holomorphic jet bundle spilt?

Let $X$ be a complex manifold. Let us consider the jet bundle of the trivial line bundle on $X$. We denote it as $J_{1}(\mathbb{C})$. We have the short exact sequence: $$0\rightarrow\Omega_{X}\...
Chenxi Yin's user avatar
1 vote
1 answer
37 views

Why $\dfrac{\partial v}{\partial y}=0$ which also gives us $\dfrac{\partial v}{\partial x}=0$?

I passed through this answer here and I don't understand the following. We are at the moment: If $u^2+v^2 \neq 0$, then $\dfrac{\partial v}{\partial y}=0$ which also gives us $\dfrac{\partial v}{\...
Frederick Manfred's user avatar
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0 answers
62 views

Non-degenerate pairing in the context of holomorphic vector bundles

I'm trying to do exercise 2.2.3 on Huybrechts, Complex Geometry: Show that for any holomorphic vector bundle $E$ of rank $r$ there exists a non-degenerate pairing $$\Lambda^k E\times \Lambda^{r-k} E\...
Alejandro Tolcachier's user avatar