English
Unstable Vector Bundles

Well, I do not know about horrible. There's a lot you can say that's good! I will start rambling and see where I end up.I am going to pretend you said principal GL(n)-bundle instead of rank n vector bundle. Same thing, really, since we have the standard representation. The collection Bun(n,C) of all principal GL(n) bundles P on a smooth curve C is a very nice geometric object: it's an Artin stack. It's not connected; the different components are labelled by topological data, like the Chern class. The tangent "space" (complex, really) to Bun(n,C) at a point P is naturally the derived global sections RGamma(C,ad(P)), where ad(P) is the associated bundle with fiber the adjoint representation of GL(n). The zero-th cohomology gives the infinitesimal automorphisms and the 1st cohomology gives the deformations. So the stabilizer group of any point V in Bun(n,C) is finite-dimensional, and the dimension of the stack is n(g-1) (by Riemann-Roch). Bun(n,C) is smooth, and unobstructed, thanks to the vanishing of H^2(C,ad(P)). Bun(n,C) has a very nice stratification, too. It's an increasing union of quotient stacks [A/G] of projective varieties by finite-dimensional groups. Roughly, A is the stack of pairs (P,t), where t is a trivialization of P in an infinitesimal neighborhood of some point in C. Make the neighborhood large enough, i. e. , r-th order, and you can kill off all the automorphisms of P. Unfortunately, except for n=1, there is no uniform bound on r that works for all bundles. So, Bun(n,C) is not a finite type quotient stack. You can also realize Bun(n,C) (homotopically) as the infinite type quotient stack of U(n)- connections modulo complexified gauge transformations. That's what Atiyah & Bott do in their paper "The Yang-Mills Equations on Riemann Surfaces". (They also have a nice discussion of slope-stability and the stratification. )The top component of the stratification (those bundles where the stabilizer group is as small as possible) is the stack of (semi-)stable vector bundles. If you take the coarse moduli space of this substack, you get the usual moduli space of stable bundles. In summary: If you drop the stability conditions, you get a lot more geometry with a similar flavor, and without the random bits of weirdness that crop up in the theory of moduli spaces. (e. g. , the stack always carries a universal bundle, you do not need the rank and the chern class to be coprime.) OK, I will stop evangelizing now

1. Should we consider our fellow humans as merely expendable bundles of nerve impulses or unique manifestations of a divine spirit?

Neither. You should see them as living, feeling beings with their own wants, that belong to the same species as you.This is different from the choices you wrote. Expendable is a relative term, and thinking of your fellow animals as expendable husks won't do you any good in the long term. You belong to a social species, connecting is a vital need for you. Plus, people don't like to be seen and treated as things to be manipulated. You'd make your life harder.The second choice you wrote is an unfalsifiable but also unprovable thing, but someone shouldn't need a supernatural story to value his fellow beings. Natural is good enough for that, because -again- we have instincts to socialize. That means we have a drive to see value in people. Not necessarily seeing them as "the most important things", that's actually quite narcissistic, and maybe not everyone, but valuable nonetheless. The choice I present differs from these as it's not relative. We all have our own wants and feelings, and we do belong to the same species. Whether you care about these feelings are up to you, but there are practical reasons to do so, at least to an extent, as I've stated in the second paragraph. Should we consider our fellow humans as merely expendable bundles of nerve impulses or unique manifestations of a divine spirit?

2. Jet bundles and partial differential operators

If \$mathcalRsubset J^rX\$ is closed, then there's a smooth function \$f:J^rXtomathbb R\$ with \$mathcalR=f^-1(0)\$. So you can construct a differential operator \$H:J^kXto Mtimes mathbbR\$ by \$H(theta):=(pi_X(pi^r_0(theta)),f(theta))\$ and the equation \$mathcalR\$ will be given by \$H(j^rphi)=0\$.So there is no big difference between the two definitions. If you are only interested in the "space" of solutions of the differential equation, then i would say that the set \$mathcalR\$ is enough, or put differently, you could choose the differential operator which suits you best to represent the equation.Edit In response to Willies comment: Here's a counterexample to what you are asking for: recall that there's no submersion from \$mathbbRP^2\$ to something, which has \$mathbbRP^1subset mathbbRP^2\$ as a fiber. So take \$M=mathbbR\$, \$X=Mtimes mathbbRP^2\$ and \$mathcalR=Mtimes mathbbRP^1subset X\$. Then there's no fiber bundle \$Yto M\$ allowing a submersion \$H:Xto Y\$ with \$mathcalR=H^-1(psi)\$ for any \$psiin Gamma(Y)\$. This is probably a silly example since the PDE is of order zero, but I am sure one can come up with examples in higher order.Anyway: if what you are interested in is an intrinsic notion of overdeterminedness of a PDE you might want to take a look at Bryant and Griffiths Characteristic Cohomology of Differential systems. Roughly the codimension of the characteristic variety serves as such a measure. And the characteristic variety can be defined completely without referring to an operator describing the equation. As Deane says, much of this can be found in the book Exterior differential systems. There are also the books by Vinogradov, Krasil'shchik and Lychagin.

3. Seifert fiberings of zero euler number which are semi-bundles

Let \$Sigma\$ be the orientable generic fiber of the semi-bundle structure on \$M\$. I only consider the case \$chi(Sigma)

get in touch with us
related searches

Copyright © 2020 Coffee bag - Guangzhou tianci packaging industry Co,. Ltd. |﻿ Sitemap