17
votes
Tensor product of vector bundles
Perhaps the fundamental issue here is an unintended consequence of the orthodox set-theoretic foundations of mathematics, which can give the mistaken impression that mathematical objects need be ...
- 109k
12
votes
Is a positive degree self map on a Riemann surface homotopic to a holomorphic self map?
The answer is "yes" if $S$ is the Riemann sphere. This is because a map $f$ of degree $d$ from the sphere to itself is homotopic to $z \mapsto z^d$.
The answer is "basically no" ...
- 26.3k
9
votes
Accepted
Pullback of a vector bundle extension class
For a counterexample, one can take $X$ an elliptic curve over a field of characteristic not $2$, $\phi$ the involution sending each point to its inverse under the group law, $F$ and $G$ both $\mathcal ...
- 137k
9
votes
Tensor product of vector bundles
I guess we should start by speaking somewhat philosophically about what it means to think geometrically about bundles. For me, this usually means looking at their sections: either constructing ...
- 28.8k
8
votes
Obstructions to the existence of a flat connection on a vector bundle
A $d$-dimensional flat real vector bundle $E→M$ is classified by a map $\def\B{{\sf B}}\def\GL{{\rm GL}}M→\B\GL(d)_δ$, where $\GL(d)_δ$ is the orthogonal group equipped with the discrete topology.
...
7
votes
Accepted
Is the subscheme parametrizing the k-th degeneracy loci Cohen-Macaulay?
First, let
$$
M_k \subset \mathrm{Mat}(e,f)
$$
be the variety of matrices of size $e \times f$ and rank at most $k$. This variety is well-known to be Cohen--Macaulay.
Now consider the projection $\pi \...
- 37.2k
6
votes
Is a positive degree self map on a Riemann surface homotopic to a holomorphic self map?
The answer is no. This is a corrected version of Nicolast's comment.
Let $E$ be an elliptic curve, let $f: E \to E$ be an endomorphism and let $H_1(f) : H_1(E) \to H_1(E)$ be the induced map on $H_1$. ...
- 151k
6
votes
Accepted
Identifying the circle bundle of the canonical line bundle $\mathcal{O}(-n-1)$ over a projective space $\mathbb{CP}^n$
Write $X:=\mathbb{CP}^{n}$, the space of lines in $\mathbb C^{n+1}$. For each $k\ge1$ the total space of circle bundle $\mathcal O(-k)\to X$ is $\{\ell\in X,v\in(\mathbb C^{n+1})^{\otimes k}:v\in\ell^{...
- 1,897
6
votes
Accepted
Converging paths implies converging parallel transports along those paths?
The following is a precise version of the "convergence of equations implies convergence of solutions" claim.
Theorem. Let $\Omega\subseteq \mathbb{R}^n$ be open, and $Q\subseteq \mathbb{R}$. ...
- 37.6k
5
votes
Proving that $H^1(X,\mathcal{Hom}(\mathcal{G},\mathcal{E})) \cong \text{Ext}^1(\mathcal{G},\mathcal{E})$ holds for locally free sheaves
This isomorphism holds for any degree of cohomology and it only requires that the first sheaf $\mathcal{G}$ is locally free. This relies on the fact that $\mathcal{H}om(\mathcal{G},-)$ is exact. There ...
- 3,301
5
votes
Accepted
Learning roadmap for holonomy theory
Dominic Joyce has two relevant books: Compact Manifolds with Special Holonomy and Riemannian Holonomy Groups and Calibrated Geometry for the study of holonomy groups of Riemannian manifolds, the ...
- 25.6k
5
votes
Accepted
A smooth family of lattices on the tangent bundle?
A choice of a (smoothly varying) lattice in each tangent space is the same as a reduction of structure group of the tangent bundle from $GL_n(\mathbb R)$ to $GL_n(\mathbb Z)$. For instance, it implies ...
- 4,659
5
votes
A smooth family of lattices on the tangent bundle?
A smooth family of lattices cannot in general be chosen in the tangent bundle of a manifold, even in such a simple case as $S^2$. Indeed, if one could do this it would imply that one can also choose ...
- 15.2k
5
votes
Accepted
Isomorphism between tangent bundle of $S^2$ and the kernel of a bundle homomorphism
Since $\pi\circ\iota$ is constant, we have $d\pi\circ d\iota = 0$, so the image of $d\iota$ is contained in the kernel of $d\pi$, i.e. $(d\iota)(T\mathbb{CP}^1) \subseteq L_n$. Since $\iota$ is an ...
- 18.8k
5
votes
Accepted
Bochner Laplacian in coordinates
Example 10.1.32 (which starts on page 456) does not consider $\nabla$ the Levi-Civita for a Riemannian metric. It is considering a general vector bundle $E$ equipped with a Hermitian metric $\langle,\...
- 37.6k
4
votes
Vector bundles on $\mathbb{P}^1$
Consider the exact sequence, $0\to E(-1)\to E\to E_p\to 0$, where the first map is just multiplication by $x$. Then $E_p$ is just the skyscraper sheaf at $p=(0,1)$. Taking cohomologies you get $0\to \...
- 6,117
4
votes
Derived flat bundles
Maybe not exactly what you are looking for, but perhaps of relevance are Flat super-connections. That let $E\to B$ be a $\mathbb{Z}_2$ graded vector bundle, and $A=d_0+d_1+d_2+\ldots$ be an odd ...
- 335
4
votes
Accepted
Self-intersection of zero section of line bundle over elliptic base curve
The result in question is a special case of the self-intersection formula (Fulton, intersection theory, p. 103) which states that the intersection of a smooth subvariety $X$ of a smooth variety $Y$ ...
- 137k
4
votes
Singularities of fibrations in conics
It might be that $X$ is singular along some fiber that is a double line. To arrange this, start with $Y \rightarrow \mathbb{P}^1$ with such a fiber over the point $P$ and then pull back by a double ...
- 3,052
4
votes
Tensor product of vector bundles
A rank-$n$ vector bundle over $X$ consists of a bunch of trivial vector bundles over open subsets $U_i$ covering $X$, patched together on the overlaps. On each overlap $U_{ij}=U_i\cap U_j$, there is ...
- 22.5k
3
votes
Accepted
Orientation bundle and its flat connection
There is a different construction of orientation bundles. One considers the $\{\pm1\}$-principal bundle $o(TM)$ of fibrewise orientations of $TM$. The associated real line bundle $o(TM)\times_{\pm 1}\...
- 6,666
3
votes
Obstructions to the existence of a flat connection on a vector bundle
A slightly different point of view for answering this question is the following one:
First, if $M$ is simply connected, then $E\to M$ admits a flat connection if and only if $E$ is trivial, so in this ...
3
votes
Accepted
Equivariant sheaves on $\mathbb P^1$
Let me explain why the line bundle $\mathcal{O}(1)$ does not admit a $\mathrm{PGL}(2)$-equivariant structure. Indeed, if it does, then the vector space
$$
\mathrm{Hom}(\mathcal{O}, \mathcal{O}(1))
$$
...
- 37.2k
3
votes
Deriving the definition of vector bundle morphisms from Cartan geometry (a.k.a. why are they linear?)
One obvious fact: Cartan geometries are rigid, i.e. the automorphisms of a Cartan geometry, on a manifold with finitely many connected components, form a finite dimensional Lie group acting smoothly. ...
- 25.6k
2
votes
Accepted
Factorization systems for vector bundles
Normally, we consider vector bundles over some base space $X$. The simplest case is if $X$ is a point. Then the category of vector bundles over $X$ is just the category of vector spaces (over whatever ...
- 29.8k
2
votes
Accepted
Normal bundle of veronese as iteration extension of symmetric powers
Your third exact sequence is incorrect --- the correct form is
$$
0 \to S^{d-1}V \otimes \mathcal{O}(d-1)
\to S^{d}V \otimes \mathcal{O}(d)
\to S^dT
\to 0.
$$
- 37.2k
1
vote
Integral mean value property
This is a comment but will be too long for that. The equation given is a (very simple) example of a difference-differential equation (in fact, a difference equation but there are several related ...
- 396
1
vote
Isomorphism between $\operatorname{End}_0(E)$ and $\operatorname{End}_0(E')$ as Lie algebra bundles
With this type of question, it is better to write to one of the authors directly.
$\DeclareMathOperator\Fl{Fl}\DeclareMathOperator\End{End}$First, a vertical vector field is given by a global vector ...
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