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.
1,202 questions
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Examples of nontrivial morphism between simple bundles but not isomorphism
We know stable bundles have a good property:
If $f: E\longrightarrow E'$ is a nontrivial morphism where $E,E'$ have the same rank and degree, then $f$ must be an isomorphism.
I'm wondering does this ...
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Kobayashi-Nomizu "Foundations of differential geometry" on page 117 wrong?
$\DeclareMathOperator\GL{GL}$Let $M$ be a smooth manifold, $G$ a Lie group and $P(M,G)$ a principal $G$-bundle and $\rho: G \to \GL(V)$ of $G$ a representation with $V$ finite-dimensional $\mathbb{F}$-...
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Square root of relative Kähler differentials and families of curves
Let $f: X \to S$ be a smooth morphism of schemes of relative dimension $1$. Then $K=\Omega^1_{X/S}$ is a line bundle on $X$. I am interested in the following question:
When does $\Omega_{X/S}$ have a ...
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Almost Complex Structure extending to Complex Structure, aka "Integrable"
Let $M$ be a smooth manifold of (real) dimension $2n$. An almost complex structure $J$ on $M$ is a linear vector bundle isomorphism $J \colon TM\to TM$ on the tangent bundle $TM$ such that $J^2 = − 1 \...
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Intuition on geometry of sections
Premise: I have asked the same question on math.stackexchange about a week ago, without receiving answer. Therefore I've decided to ask it also here. If this violates the rules, I apologize and I'll ...
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Connectedness of degeneracy loci
Let $E, F$ be vector bundles of ranks $e, f$ on a smooth variety $X$ of dimension $n$.
Let $\varphi : E \to F$ be a morphism and let $k$ be such that $n > (e-k)(f-k)$.
Fulton-Lazarsfeld's theorem ...
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Parametrized moduli spaces of semistable bundles by varying Kähler classes
Inspired by Liviu Nicolaescu's answer here, I was eager on trying to come up with examples of parameter spaces $\Lambda$, configuration spaces $\mathscr{C}$ and parametrized moduli spaces $\mathscr{M}$...
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Conceptual understanding of the Néron–Severi group
I'm trying to understand the importance of the Néron–Severi group $\operatorname{NS}(X)$ when $X$ is, say a complex manifold. My background is in the analytic side so I'm much more familiar with line ...
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Is every connection locally flat for an other connection?
Consider a $C^{\infty}$ connection $d_A = d+A$ on the unit ball $B^n\subset \mathbb{R}^n$. Does there exists another connection $d_{\tilde{A}} = d+\tilde{A}$ such that $d_{\tilde{A}} A = 0$? That is ...
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About Chern classes via Atiyah class
I am trying to understand a construction of the Chern classes of a vector bundle $\mathcal{E}$ via the Atiyah class, like is done in this text and here in section 1.4. I am interested in the case ...
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Is the vector bundle over a vector bundle, a vector bundle over the base scheme?
Suppose we have $E\to X$, a vector bundle over $X$ and $E'\to E$ a vector bundle over $E$. Composing the structure maps gives a smooth scheme $E'\to X$ over $X$. My question is, when is this a vector ...
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Turn a coherent subsheaf into a vector subbundle by blowing up
I have some confusions on reading Narasimhan's paper Hermitian-einstein metrics on parabolic stable bundles. The following statements appears between page 103-104.
Let $E$ be a rank 2 holomorphic ...
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Proper smooth pushforward of vector bundle is a vector bundle?
Suppose $X$ and $Y$ are algebraic varieties over a field $k$, and $f:X
\to Y$ is proper smooth. Then for a vector bundle $E$, and any $i\ge 0$, do we have $R^if_*(E)$ locally free? I know we need the ...
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Chern Classes of $\mathcal{O}_E(1)$ on $\mathbb{P}(E)$ for $E = \mathcal{O} \oplus \mathcal{O}(n) \to \mathbb{P}^2$
Let $E =\mathcal{O} \oplus \mathcal{O}(n) \to \mathbb{P}^2$ and denote by $\mathcal{O}_E(1)$ the dual of the tautological bundle.
How can I compute $c_1^2(\mathcal{O}_E(1)), c_1^3(\mathcal{O}_E(1))$, $...
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Pullback of an ample bundle under an embedding is ample
In Example 11.8 on JP Demailly's book on Complex Analytic and Differential Geometry it is being said that
The pullback of a (very) ample line bundle by an embedding is clearly also (very) ample.
I ...
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Irreducible $G$-equivariant vector bundles
Let $G$ be an abelian group, and let $E \to M$ be a $G$-equivariant complex vector bundle. If $E$ is irreducible, meaning it has no nontrivial $G$-equivariant subbundles, can anything special be said ...
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Do Grassmannians classify numerable vector bundles over arbitrary spaces
The functor $k_{U(d)}$ of the set of isomorphism classes of numerable principal $U(d)$-bundles is represented by $BU(d)$ the Milnor construction of the classifying space. The restriction of $k_{U(d)}$ ...
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Determinant bundle over homogeneous varieties
I am looking for a way to compute the determinant of a homogeneous vector bundle over any homogeous variety. I am awere of how these computations work for the $A_n$ case (i.e., for flag varieties), ...
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Universal picard variety of degree d
Let $M_g$ denote the moduli space of smooth genus $g$ curves over $\mathbb{C}$, where $g \geq 2$. Let $Pic^{d,g}$ denote the universal picard variety over $M_g$, parameterizing pairs $(C,L)$ where $C$...
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Can we find a Jouanolou device for $\mathbb{P}^d$ having dimension $<2d$?
Let us work over an algebraically closed field $k$.
A Jouanolou device for a $k$-variety $X$ is an affine space fiberation $f:Y\to X$ such that $Y$ is an affine scheme. (The condition on $f$ means ...
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Explicit parallelization of an exotic sphere
I asked this question on MathStackExchange a week ago (see here), but, despite a few upvotes, I received no comments or answers. Ideally, I would love a detailed answer, but a yes/no would do the job! ...
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Example of stable bundle whose pullback is polystable
Kempf (1992): "Pulling back bundles" has the following theorem:
Let $f: Y \rightarrow X$ be a finite morphism. If $\mathscr{W}$ is a bundle on $X$ that is stable with respect to an ample ...
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Adelic description of moduli of stable vector bundle of rank n (over finite fields)?
Let $Bun_G$ be the moduli stack of $G$-bundles on a (geometrically irreducible smooth projective) curve $C$ over a finite field $k$, where $G$ is a split reductive group over $k$. Since Weil, we know ...
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${\rm SL}_2(\mathbb C)$-equivariant K-theory of $\mathbb C P^1$
Consider the action of ${\rm SL}_2(\mathbb C)$ on $\mathbb C P^1$ induced by the action of 2×2 matrices on 2-vectors. Is it true that $K^{{\rm SL}_2(\mathbb C)}(\mathbb C P^1)$ is $\mathbb C[t,t^{-1}]$...
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Hodge dual of curvature two-form
I'm trying to compute curvature invariants on a general Kähler manifold $X$. One possibility is taking the norm $I$ of the Riemann curvature two-form $\mathcal{R}$ induced by the Hodge-structure^[1], ...
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Weight space decomposition of smooth representation of complex algebraic torus
Question: Let $T=(\mathbb{C}^{*})^{n}$ and $\pi:\mathcal{E}\to \mathbb{C}^{n}$ a smooth complex $T$-equivariant vector bundle (i.e. $\pi$ is $T$-equivariant and $T$ acts on the fibers linearly). The ...
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Moduli space of stable bundles with fixed Chern class is bounded
This is an excerpt on a set of lecture notes I'm going over:
Classically, a lot of interest in stable vector bundles is due to the fact that stability allows the study of moduli of vector bundles via ...
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Symmetric powers for a short exact sequence of vector bundles
If $0 \to A \to B \to C \to 0$ is a short exact sequence of vector bundles on $X$, is it necessarily true that $[{\rm Sym}^k B] = \sum_{i=0}^k [{\rm Sym}^i A \otimes {\rm Sym}^{k-i} C]$ in the ...
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Unique Hausdorff topology on trivial vector bundle?
Question: Is there a Hausdorff topology other than the product topology on $X\times \mathbb{C}^n$, that turns $(X\times \mathbb{C}^n, \mathrm{pr}_1)$ into a vector bundle, where $\mathrm{pr_1}$ ...
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Vector bundle formed by tangent lines to a quadric curve in $\mathbb P^2$
Let $Q \cong \mathbb P^1$ be a quadric curve in $\mathbb P^2$. Consider the following rank two vector bundle $V$ on $Q$: the fiber of $V$ over a point $p$ of $Q$ is the two-dimensional subspace $V_p$ ...
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$\mathbb P^1$-bundle on a partial flag variety
Let $X$ be the partial flag variety of flags $0 \subset V_k \subset V_{k+2} \subset V$ where $V$ is a fixed vector space of dimension $n$ and ${\rm dim} V_k = k$ and ${\rm dim} V_{k+2} = k+2$. Is it ...
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What textbooks/papers should I read to try to make this rigorous?
Consider a surface of revolution $S$ and an embedding $e:S \hookrightarrow X^3$ for $X^3=[0,1]^3$ with cone points $p,q$ elements of $\partial X^3$ where $\partial X^3=X^3-(0,1)^3$ for $\mathrm {sup}~ ...
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Is there a non-semistable simple sheaf?
Let $C$ be a smooth projective (irreducible) curve over an algebraically closed field $k$.
A sheaf is said to be simple if its endomorphism algebra is isomorphic to $k$.
It is known that a stable ...
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Existence and injectivity of a map from $\mathrm{Num}(X) \otimes_\mathbb{Z} \mathbb{C}$ to $H^1(X,\Omega^1_X)$
I am currently reading a paper by Mustata and Popa on the Van de Ven theorem, you can read the paper here.
On page 51 there is the following map
$$\alpha_\mathbb{C} : \mathrm{Num}(X) \otimes_\mathbb{Z}...
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Surjectivity of a restricted Hitchin map
Let $C$ be a smooth projective algebraic curve over the complex numbers and let $V$ be a stable rank 2 vector bundle on $C$. Then consider the vector bundle $\mathcal{E}nd_0(V) $ of tracefree ...
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When do quotients of $G$-vector bundles exist?
Let's work in the category of smooth (paracompact, Hausdorff) manifolds. Let $M$ be a manifold and $G$ a Lie group acting on $M$. Suppose $E$ is a $G$-vector bundle on $M$ (that is, $G$ acts on $E$ by ...
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Reality of connection or meromorphic function
Let's considering a family of connections: $\nabla^{\lambda}:\mathbb{C}^{*}\rightarrow \Omega^{1}(sl(2,C))$ of trivial rank2 bundle on $\mathbb{P}^{1}-\{ 0,1,\infty \}$ with simple pole. In this case, ...
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Smooth version of the splitting principle
Inspried by this MO question A manifold whose tangent space is a sum of line bundles and higher rank vector bundles we pose the following question as a possible smooth version of the splitting ...
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Dual of slope semistable vector bundle on higher dimensional variety
Recall the definition of slope semistability, taken from section 1.2 of Huybrechts and Lehn's "Geometry of Moduli Spaces of Sheaves" book. Let $X$ be a projective $\mathbb{C}$-scheme and $E \...
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A manifold whose tangent space is a sum of line bundles and higher rank vector bundles
I am looking for an example of the following situation. Let $M$ be a connected (if possible compact) manifold such that its tangent bundle $T(M)$ admits a vector bundle decomposition
$$
T(M) = A \...
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Let $\phi : A \to B$ be a surjective $*$-homomorphism of star algebras, is there any good notion of "normal bundle of $B$ in $A$"?
Let $\phi : A \to B$ be a surjective $*$-homomorphism of star algebras (maybe more restricted kind of star algebra), is there any good notion of "normal bundle of $B$ in $A$"? By a "...
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What is the adjoint bundle of groups $P\times_{G}G$?
It is said that G acts on itself by conjugation. I am familiar with another type of adjoint bundle in which a representation of G on a vector space is given. Can someone explain the differences and ...
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Question on notation for definition of symbol of differential operator
I was looking at this definition of the symbol of a differential operator, and am unsure what "$T^*X\otimes_XE$" means. I couldn’t find an explanation anywhere on nlab either. My main ...
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Stable homotopy of vector bundles
Consider the category $\mathsf{VectBun}$ of real vector bundles over topological spaces, where
the morphisms are bundle maps that are fiberwise isomorphisms.
This category has a stabilization functor
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Holomorphic fibre bundles over noncompact Riemann surfaces
Some days ago I came across the paper "Holomorphic fiber bundles over Riemann surfaces", by H. Rohrl.
At the beginning of Section 1, the following theorem is quoted:
Theorem. Every fiber ...
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Can we get a connection on the principal bundle from a connection on the associated vector bundle?
Assume $G$ is a Lie group, $P \to M$ is a smooth principal $G$-bundle, and $\rho \colon G \to GL(V)$ is a smooth representation of $G$. We can define a connection on the associated vector bundle $E := ...
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Origin of the name Trace resp Integral symbol for the trace map of Dualizing Sheaf
Let $X \subset \mathbb{P}^n_k$ be a normal projective subscheme over $k$ of dimension $n$. The dualizing sheaf is in context of Serre duality a pair $(\omega_X,t)$ (which exists in that case) ...
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Extending the natural thom form of a vector bundle from the boundary of a manifold
(Edited after taking into account Tom Goodwillie's answer.)
Let $E \rightarrow X$ be an orientable vector bundle.
In this MO answer it is explained how to obtain a representative of the Thom class (...
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Is the subscheme parametrizing the k-th degeneracy loci Cohen-Macaulay?
Let $X$ be a Cohen-Macaulay projective variety (say over an algebraically closed field of characteristic 0), and $E,F$ two vector bundles such that $E^*\otimes F$ is globally generated. Consider the ...
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Identifying the circle bundle of the canonical line bundle $\mathcal{O}(-n-1)$ over a projective space $\mathbb{CP}^n$
It's not hard to see the following fact: the circle bundle of the tautological line bundle $\mathcal{O}(-1)\rightarrow \mathbb{CP}^n$ is $S^{2n+1}$, the unit sphere inside $C^{n+1}.$ I want to see ...
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