# Tagged Questions

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.

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### Transferring connection information to associated bundles and back

This might not be research level but I've tried more than once to ask about this in MSE and it got nowhere. So I thought It's fair to at least try. At the risk of repeating well known stuff I tried ...
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### Stable vector bundles in Weil's parametrization

Let $C$ be a smooth projective algebraic curve. Isomorphism classes of vector bundles on $C$ are in bijection with $GL_n(F) \backslash GL_n (\mathbb{A}) / GL_n(\mathcal{O})$, I think (one trivalizes ...
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### the pfaffian-adjugate and its counterparts for matrices odd size

Let $R$ be a commutative ring (zero characteristic). Take a skew-symmetric matrix $A\in Mat^{skew-sym}(n,R)$. If $n$ is even, then $\det(A)=Pf^2(A)$ and there exists the "Pfaffian adjugate/adjoint" ...
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### Cohomology and deformations of moduli of vector bundles

I believe that the following is well-known, but I cannot find a reference in the literature... Let $X$ be a smooth variety (in our case $X = M^s(r)$ coarse moduli space of stable rank $r$ vector ...
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### Horizontal vector fields and the push forward of the differential of the projection map

I am not very familiar with differential geometry but need to understand some aspects of it for my research. This includes in particular the notion of horizontal vector fields and I would like to ...
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### Triviality of certain vector bundles

Let $M$ be a smooth manifold and let $SM$ be the bundle of symmetric bi-linear forms on $TM.$ Riemannian metrics are a particular kind of sections in this bundle. Since any manifold admits a global ...
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### If the restriction of a vector bundle to a divisor is semi stable, then is the vector bundle itself semistable?

Let $X$ be a smooth projective variety of dimension $n$. Let $D$ be a smooth divisor of $X$. Let $i:D\hookrightarrow X$ be the inclusion. Let $H$ be an ample line bundle on $X$. Let $E$ be a vector ...
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### Principal bundles that can't be detected by spheres

The question I'm trying to answer is the following: Let $P \to X$ be a principal $G$-bundle (over a connected CW complex) satisfying that all pullbacks to spheres (of arbitrary dimension) are ...
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### Sheaf of Sections of Cone

Fulton's intersection theory book at Chapter 4 makes the following claim: If $\mathcal{E}$ is a locally free sheaf (on $X$), and $E:=Spec(Sym(\mathcal{E}))$ a total space of some cone/bundle on $X$, ...
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### Maximal trivialising subspace for a vector bundle

Let $X$ be a locally compact Hausdorff space. Given a vector bundle $p: E\to X$, a subspace $Y$ of $X$ is called trivialising (for this bundle), if after restricting this bundle to $Y$, it is a ...
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### Vector bundles, finitely generated projective module?

Let $B$ be a Tychonoff space and let $\mathbb{R}(B)$ denote the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $S(\xi)$ denote the $\mathbb{R}(B)$-module ...
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### Determine the representation given by space of sections of symmetric products of cotangent bundle of projective plane

In a recent project, it was interesting for me to determine the $PGL(3)$ representation given by $H^0(S^2(\Omega(1)) \otimes \mathcal O(4))$ on $\mathbb P^2$. I did this by using the Euler sequence, ...
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### Vector bundle $\gamma^1$ over $\textbf{P}^\infty$ does not have finite type? [closed]

Using Stiefel-Whitney classes, how do I see that the vector bundle $\gamma^1$ over $\textbf{P}^\infty$ does not have finite type?
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### Relative Characteristic classes

A pair of vector bundles over a base space $X$ is a pair $(E,F)$ where $E$ is a vector bundle over $X$ and $F$ is a sub-bundle of $E$. Two pairs $(E_{1},F_{1})$ and $(E_{2}, F_{2})$ are ...
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### “High-concept” explanation for proof of a theorem of Ochanine?

See Akhil Mathew's notes on Ochanine's theorem for elliptic genera here and here. Let $\phi: \Omega_{SO} \to \Lambda$ be a genus. We might ask when $\phi$ satisfies the following multiplicative ...
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### Isometry theorem, exists homeomorphism that carries each fiber isomorphically onto itself

Let $\mu$ and $\mu'$ be two different Euclidean metrics on the same vector bundle $\xi$. How do I see there exists a homeomorphism $f: E(\xi) \to E(\xi)$ which carries each fiber isomorphically onto ...
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### Tangent bundle of $S^2 \times S^1$ trivial or not [closed]

Is the tangent bundle of $S^2 \times S^1$ trivial or not?
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### Are “vector spaces” over a smooth scheme with constant fiber dimension locally free?

I've got the follow question which drives me almost crazy as the answer seems to be simple. Given a morphism $p:V\to S$ of schemes of finite type over some base field. Assume that $p$ has all the ...
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### Higher tangent bundles of manifolds with non integer dimension

One way to define the tangent space of a manifold at a point $p\in M$ is the following: We define an equivalent relation on the space of curves passing $p$ as follows: Two curves $\alpha, \beta$ are ...
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Let $S$ be a surface. $C$ be a curve on $S$ and $i:C\hookrightarrow S$ is the inclusion. Let $E$ be a rank 2 vector bundle on $S$ and $F$ a line bundle on $C$. Suppose we have a surjection $E\... 4answers 573 views ### What does “higher monodromy” tell us about a principal bundle Let$P \to X$be a principal$G-$bundle and let$f: X \to BG$be its classifying map. As I understand there's some way to associate a monodromy representation$\pi_1(X) \to G$to it. I know how to ... 1answer 61 views ### Spaces of Killing spinors for different orientation Simply put, I want to understand how a change of orientation on a Riemannian spin manifold can change the space of Killing spinors. To be more precise: Let$M$be a spin manifold (i.e. the first and ... 0answers 670 views ### Vector bundle$L$admits connection if and only if degree of every direct summand of$L$divisible by$\text{char}\,k$, intuition Consider the following theorem of Atiyah. Let$X$be a connected smooth projective curve over an algebraically closed field$k$. Then a vector bundle$L$on$X$admits a connection if and only if ... 0answers 95 views ### Pull back of a semistable vector bundle to a product is semistable? Let$X$be a smooth projective surface over$\mathbb{C}$. Let$L$be an ample line bundle on$X$. Let$F$be a$\mu_L$semistable rank 2 vector bundle on$X$(semistability in the sense of Mumford-... 2answers 369 views ### Pullback along Frobenius morphism Let$X$be a scheme over a finite field$\mathbb{F}_q$and let$F : X \to X$be the absolute Frobenius morphism. If$\mathcal{L}$is an invertible$\mathcal{O}_X$-module, then there is a natural ... 1answer 144 views ### covering map from spheres to projective spaces and the associated vector bundle Let$S^n$be the$n$-sphere and consider a$2$-sheeted covering $$S^n\longrightarrow\mathbb{R}P^n.$$ We have an associated vector bundle $$\xi: \mathbb{R}^2\longrightarrow S^n\times_{\mathbb{Z}/2}\... 0answers 126 views ### Extend a vector bundle on a flat family Let f: X\to T be a flat family, and \mathcal{F}_t is a vector bundle on X_t for some t\in T. Can this \mathcal{F}_t be extended to a vector bundle \mathcal{F} on f^{-1}(U) for some open ... 1answer 165 views ### classifying maps of Whitney sums of vector bundles For an n-dimensional vector bundle \xi with structure group G\leq O(n) over a CW-complex B, we have a classifying map up to homotopy$$ f(\xi): B\longrightarrow BG, $$f(\xi)\in [B;BG], ... 0answers 172 views ### Which map realizes the isomorphism KO_n(X)\otimes \mathbb{Q}\to \bigoplus_{i\in\mathbb{Z}}H_{n-4i}(X;\mathbb{Q})? The description of the real KO-homology groups KO_n(X) can be given abstractly as maps to the real K-theory spectrum KO smash X, or via triples (M,x,\phi) where M is a closed manifold, \... 1answer 118 views ### Vector bundles with symmetric perfect form Let X be a smooth projective curve, and E a vector bundle on X such that there exist a bilinear perfect symmetric form$$E\otimes E\rightarrow \mathcal O_X$$When I see E as a GL_r ... 1answer 275 views ### Why do we need ampleness in the definition of stability/semistability This is a general question that I have. Let X be a projective variety over an algebraically closed field k. Let L be an ample line bundle over X. Let F be a vector bundle on X. We say that ... 1answer 363 views ### Zero scheme of global sections of vector bundles on affine varieties I want to understand better the notion of zero scheme of a section of a vector bundle. For simplicity I will consider the case of affine varieties. Let \mathbb{K} be an algebraically closed field, ... 1answer 205 views ### Smooth manifolds as idempotent splitting completion The nlab has a particularly interesting thing to say about the category of smooth manifolds: it is the idempotent-splitting completion of the category of open sets of Euclidean spaces and smooth maps. ... 1answer 143 views ### Semistability of principal bundle vs vector bundle Ramanathan has defined the semistability of a principal G-bundle E over a curve X as follows: E is semistable iff for any parabolic subgroup P\subset G, for any reduction of the ... 2answers 870 views ### Is every vector bundle over a noncompact finite-dimensional manifold a summand of a trivial bundle? In the notes of Vector Bundles and K-theory by Prof Allen Hatcher, on page 12 he proved a Proposition that for each vector bundle E\to B with B compact hausdorff there exists a vector bundle E'\... 1answer 102 views ### Dimension of Quot scheme of zero dimensional quotients of a locally free sheaf Given a locally free sheaf E of rank r on a (smooth, projective, algebraic) surface, I want to know the dimension of the scheme parametrizing the zero-dimensional (meaning they have zero ... 1answer 230 views ### A question on complex line bundle over S^{2} Consider the trivial bundle \epsilon_{2}=S^{2}\times \mathbb{C}^{2} with the standard Hermitian inner product <(a,b), (c,d)>=a\bar{c}+b\bar{d}. Assume that \ell is a sub line bundle of ... 1answer 119 views ### triviality of a 2-sheeted covering map and the triviality of the associated vector bundle Let X be a space with a (free and properly discontinuous) \mathbb{Z}/2-action and$$p: X\to X/(\mathbb{Z}/2) $$be a 2-sheeted covering map. Then we have an associated vector bundle$$ \xi: \... 0answers 110 views ### line bundle on affine grassmannian and central extension Let$G$be a connected reductive group over$\mathbb{C}$, let$Gr$be the affine grassmannian of$G$. On$Gr$, we know that there is a canonical line bundle$L$(the generator of$Pic(Gr)$). Now$G(\...
Let $AG_k(\mathbb{R}^N)$ be the "affine Grassmannian" consisting of $k$-dimensional hyperplanes (i.e. affine subspaces) in $\mathbb{R}^N$. Is there any relation between $AG_k(\mathbb{R}^N)$ and the ...