Questions tagged [vector-bundles]
A continuously varying family of vector spaces of the same dimension over a topological space. If the vector spaces are one-dimensional, the term line bundle is used and has the associated tag line-bundles.
1,151
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Stable vector bundle and Hitchin map
Let $E$ be a stable vector bundle over a curve $X$. $K_X$ the canonical bundle of $X$. $W$ the base of the Hitchin map.
Is the Hitchin map $H:H^0(E\otimes E^*\otimes K_X)\rightarrow W$ surjective?
...
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2
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Vector bundles on Grassmannians
Let $Gr(k,n)$ be the Grassmannian of $k$-dimensional vector subspaces $H^k$ of an $n$-dimensional vector space $V$.
Let us fix an $h$-dimensional vector subspace $\Gamma\subset V$ with $h\leq k$, and ...
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Natural extension homomorphism and wrong-way maps in K-theory
Let $X \subset Y$ be two smooth manifolds. To the inclusion $I:X \to Y$ corresponds the so called wrong-way map in $K-theory$ $i_!:K(X) \to K(Y)$. It is constructed as follows: to the inclusion $X \...
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The existence of adjoint operator for Sobolev spaces $W^{k,p}(S^2, \mathbb R^n)$
It is known that if $D:H_1 \to H_2$ is a bounded operator between Hilbert spaces, then there exists an adjoint operator $D^* : H_2 \to H_1$ (the field is just $\mathbb R$ rather than $\mathbb C$, so ...
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Vector bundles with symmetric perfect form
Let $X$ be a smooth projective curve, and $E$ a vector bundle on $X$ such that there exist a bilinear perfect symmetric form $$E\otimes E\rightarrow \mathcal O_X$$
When I see $E$ as a $GL_r$ ...
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2
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178
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Name for construction on two vector bundles
I have a construction on two vector bundles and I would like to give it a name and a symbol but I can't find anything.
For two vector bundles $A=\{(x,A_x) : x \in X\}$ and $B=\{(y,B_y):y\in Y\}$ ...
4
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Are rational sections of a vector bundle useful?
Let $X$ be a complex manifold or variety and $L$ a line bundle on it. Given a rational section $s$ of $L$, we get a divisor $D=Div(s)$ and may recover $L$ as $\mathcal{O}(D)$. What about vector ...
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etale homotopy and Adams conjecture
I was reading Quillen's paper on "Some remarks on etale homotopy theory and a conjecture of Adams". Up to some fact about etale version of spherical fibrations which was later proved by Friedlander, ...
4
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2
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Endomorphisms of bundles associated to codimension 2 subvarieties
Preamble
I initially decided to post this question on math.stackexchange a few days ago, as I consider it to be much less of a research question and much more of "I'm learning" question. But there ...
4
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Castelnuovo-Mumford regularity for tensor products of vector bundles
I believe this should be a well known result, but I wasn’t able to prove or find a good reference for it.
Let $E$ and $F$ be $n$-regular, respectively $m$-regular vector bundles in the sense of ...
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Classification of functorial smooth vector fiber bundles
Let $\mathrm{Bundle}$ be the category whose objects are smooth vector fiber bundles over $\mathbb{R}$, and morphisms are fiberwise smooth linear map (that is, the base is not assumed to be fixed).
Let ...
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Relationship between two bundles approaches of spontaneous symmetry breaking
I am trying understand if there is a relation between two formulations of the spontaneous symmetry breaking.
The first is provide by Derdzinski in his book "Geometry of the standard model of ...
4
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$Ext$-algebra of stable vector bundles
Let $X$ be a smooth projective variety over $\mathbb{C}$ and $E$ a slope-stable vector bundle on $X$ with regard to some ample line bundle $H$.
Question: What can we say about the algebra structure of ...
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Extension of stable vector bundles on curves
Let $X$ be a smooth projective curve, let $E',E''$ be stable vector bundles on $X$, with $\mathrm{slope} (E'')>\mathrm{slope} (E')$.
Let $0\neq[E]\in \mathrm{Ext}^1(E'',E')$ be an extension,
$$0\to ...
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Integrability condition for flat connections
I am reading Kobayashi's book "Differential geometry of complex vector bundles". More precisely, I am on section 2 of chapter 1, page 5.
Kobayashi is trying to prove that if $E$ is a vector ...
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$G$-torsor for topological space compared to that for sheaf of groups
I just read about the definitions about torsor of sheaf of groups and get a bit confused.
How does the notion of $G$-torsor for a topological space compared to that of a sheaf of groups? Is there a ...
4
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Nef line bundles over complex analytic spaces
Let $L$ be a line bundle over a compact complex manifold $X$ with a Hermitian metric $\omega$: $L$ is said numerically effective (nef, for short) if for any $\epsilon>0$ there exists a smooth ...
4
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Vector bundle with a perfect pairing and ($\mathbb Z/2$, $SL_r$)-bundle
I think this is a well knowing result but I can't find any reference,
Let $(E,q)$ be a vector bundle with a non degenerated quadratic form $q:E\rightarrow E^*$ with trivial determinant, suppose ...
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315
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Projectively flat Hermitian curvature proportional to Kähler form?
Is there a classification of the holomorphic Hermitian vector bundles $\pi:E\rightarrow M$, over a given complex Hermitian manifold, which are projectively flat and the curvature is proportional to ...
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A question on $Isom(p_1^*E,p_2^*E) \rightrightarrows X$
I am learning the moduli stacks of vector bundles and have trouble understanding some definitions. Let $E$ be a rank $n$ vector bundle over the scheme $X$. We denote by $p_i$ the $i$th projection $p_i:...
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Metrics with prescribed Levi-Civita connection
My question involves the symmetries of a (pseudo)-Riemannian metric preserving the Levi-Civita connection (LCC), its unique torsion-free metric connection. For a basic example, one notes that the ...
4
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Smoothness of a family of maps induced from isomorphism of trivial bundles
I'm reviewing a theorem from Golubitsky's Stable Mappings and Their Singularities, where it is proved that we can construct new vector bundles from old ones via smooth covariant functors. Anyway, as ...
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Frobenius rank of a manifold
The rank of an smooth manifold M is defined by Milnor, as follows:
"The maximum number of independent commuting vector fields on M"
For example it is well known that the rank of $S^{3}$ is 1 (Lima, ...
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Why do we use the less simple convention for the definition of a vector bundle connection?
For a (smooth) vector bundle $F$ over a manifold $M$, one normally defines a connection to be a linear map
$$
\nabla:\Gamma^{\infty}(V) \to \Omega^1(M) \otimes \Gamma^{\infty}(V),
$$
satisfying $\...
4
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Is there a "simple commutation" relation between $D^{''}$ and $\delta^{'}$, with $D^{''}$ the (0,1) part of the chern connection of a vector bundle and $\delta^{'}$ the adjoint of the (1,0) part?
Hi,
as the title says i'm wondering if there's a "simple" and known commutation relation between the following two differential operators. Let $E$ be a holomorphic vector bundle over a compact kahler ...
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0
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category of vector bundles with connections and its K-theory
For the category of Hermitian vector bundles with unitary connections, an object is (of course) a Hermitian vector bundle with a Hermitian metric and a unitary connection $(E, g^E, \nabla^E)$. For ...
4
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0
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130
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Reference request & more: compute vector bundles for homogeneous $G$-varieties
We work over the field of complex numbers $\mathbb C$.
Let $G$ be a simple linear algebraic group and let $P,Q$ be standard maximal parabolic subgroups of $G$ containing the same Borel subgroup $B$. ...
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105
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About the construction of the associated complex vector bundle of an orbifold one
My question has to do with the general construction that associates to each complex orbifold vector bundle $\mathscr E\rightarrow\mathscr X$ over an orbifold Riemann surface, a complex vector bundle $...
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279
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How to apply theories of line bundles to arbitrary bundles
Many concepts in geometry apply a priori specifically to line bundles. For certain theories like ampleness, nef, positivity I know that in order to generalize to arbitrary ranked vector bundles $E\to ...
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91
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Is the heat semigroup on a manifold the limit of the heat semigroups associated to a compact exhaustion?
Let $M$ be a paracompact Riemannian manifold, and $E \to M$ a Hermitian vector bundle endowed with a Hermitian connection $\nabla$. Write $M$ as an exhaustion $\bigcup _{j \ge 0} U_j$ with relatively ...
4
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Singularities of hypersurfaces in projective bundles
I am doing some calculation on a toy example from the question here.
Let $\mathbb P(E) \rightarrow \mathbb P^1$ be the projectization of the vector bundle $E = \mathcal O \oplus \mathcal O \oplus \...
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Pseudo-tensor- and tensor-densities: Sections of what bundle?
Let $\mathcal{M}$ be a smooth manifold. A tensor field is then usually defined to be a section of the tensor bundle
$$\bigotimes_{i=1}^{p}T\mathcal{M}\otimes\bigotimes_{i=1}^{q}T^{\ast}\mathcal{M}.$$
...
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Coherent sheaves and space filling curves
This paper constructs smooth space filling curves for smooth varieties over finite fields. Let's say we are working in char $p$ on the variety $X$ then this means that there is smooth curve $C_i$ in $...
4
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219
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Relative Thom isomorphism
Let $\tilde{X}$ be a space with an action of the symmetric group $\mathfrak{S}_k$ and define $X:=\tilde{X}/\mathfrak{S}_k$ to be the quotient. On the other hand, $\mathfrak{S}_k$ acts on $(\mathbb{R}^...
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Push forward of Chern character and index theorem
I have some trouble understanding a proposition in Leung's paper "Symplectic Structures on Gauge Theory" published in Commun. Math. Phys. 193, 47 – 67 (1998).
I expose here the setup for my ...
4
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0
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127
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Cancellation property of vector bundles on non-proper varieties
Krull-Schmidt theorem for proper varieties over a field implies that given an isomorphism of vector bundles between $E\oplus F$ and $G\oplus F$ we can deduce that $E$ and $G$ are isomorphic. My ...
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139
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Linearly dependent vector fields not spanned by fewer continuous vector fields
This is a follow-up to the question https://math.stackexchange.com/questions/3845080/linearly-dependent-vector-fields-that-are-not-spanned-by-fewer-continuous-vect, which attracted very little ...
4
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Global algebraic function over the moduli space of semistable higgs bundles $\mathcal{M}$
Let $X$ be a smooth projective curve over $\mathbb{C}$, $\mathcal{M}$ be the moduli space of semistable higgs bundles, and $h: \mathcal{M}\rightarrow W=\bigoplus H^0(X,K_X^{\otimes{i}})$ be the ...
4
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211
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Real part of a holomorphic section of a vector bundle
Let $F\to M$ be a holomorphic vector bundle over a complex manifold $M$ and let $s:M\to F$ be a no-zero section. Let $E$ be the complexification of $F$, and suppose that $E$ admits a holomorphic ...
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Serre vanishing on one-point blow-ups
This is basically the last step of problem 5.3.7 in Huybrechts' Complex Geometry.
Let $X$ be a complex manifold, $x \in X$, $E$ a holomorphic vector bundle on $X$ and $L$ a positive line bundle. ...
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Deformations of vector bundles and tubular neighborhood
I had a number of questions that are somewhat related to each other. I decided to post them altogether instead of separately. I'd appreciate any kinds of answers, ideas or sources regarding any of ...
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Sheaf of smooth functions and restriction to a divisor
My question is targeted towards a very particular detail in my research that I am trying to understand. I will therefore break it down into some more general questions.
Let $X$ be a smooth variety, $i:...
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How big is the complement of stable locus $\operatorname{Bun}G$
Let $\Sigma$ be a smooth projective curve, and $G$ a reductive group. Let $\operatorname{Bun}G$ be the stack of principal $G$ bundles on $\Sigma$ (with a fixed topological type).
What is the ...
4
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0
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Is the determinant map $det:\mathcal{M}(r,d)\rightarrow Pic^d(X)$ on moduli space of semistable vector bundles a fibration?
Let $X$ smooth projective curve over $\mathbb{C}$, fix a line bundle $L$ of degree $d$, and let $\mathcal{M}(r,d)$ denote the moduli space of semistable vector bundles of rank $r$ and degree $d$. It ...
4
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107
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Generalized de Rham cohomology on product bundle giving specified cohomology
Given a compact, smooth manifold $M$ and a real vector bundle $E \to M$ (in general not flat). There already have been numerous questions about how to equip the space $\bigoplus_k \Gamma(\Lambda^k T^* ...
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118
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Flatness equivalence
Let $\pi:E\rightarrow M$ be a complex vector bundle and $H$ a hermitian metric over it. If $D$ is a connection over $E$, using the metric $H$, we can decompose it as:
$$
D=D_H+\phi
$$
Where $D_H$ is ...
4
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0
answers
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Minimum rank of inverse complex vector bundles
When considering vector bundles (real or complex) over a compact manifold, i know about the existence of inverse bundles. That is, if $\xi$ is a vector bundle over $M$, then there is a bundle $\nu$ ...
4
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168
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Harder-Narasimhan over arbitrary coefficients
Let $X$ be an $n$ dimensional smooth projective variety over $k$. Let $H$ be a hyperplane section. Define the slope $\mu(E)=\frac{c_1(E).H^{n-1}}{rank(E)}$ for vector bundles on $X$. Does the Harder-...
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When is the Chern class of a principal $G$-bundle same to the Chern class of the associated complex vector bundle?
Let $P\rightarrow M$ be a principal bundle, with structure group $G$. If $G$ has a representation $\rho:G\rightarrow GL(n,\mathbb{C})$, then we can define its associated vector bundle $E=P\times_{\rho}...
4
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0
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254
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Endomorphism of involution invariant vector bundles
Let $C$ be an hyperelliptic curve with an involution $\sigma$, and let $E$ be a rank $2$ stable involution invariant vector bundle on $C$, that is $\sigma^{*}E \cong E$. Let $(End(E) \otimes K_C)_{+}$ ...