for completeness, I created *external tensor product of vector bundles*

I have spelled out the proofs that over a paracompact Hausdorff space every vector sub-bundle is a direct summand, and that over a compact Hausdorff space every topological vector bundle is a direct summand of a trivial bundle, here

]]>created a minimum at *projective bundle*

I have split off a simple entry *sphere bundle* from *spherical fibration* in order to allow more precise linking (for instance from Thom space and from wave front set): “sphere bundle” is about fiber bundles whose fibers are isomorphic to spheres, while “spherical fibrations” is about those whose fibers have the (stable) homotopy type of spheres.

Often when typing “sub-anything” into some $n$Lab entry, I hesitate, wondering if this should come with a hyperlink. Maybe in general this is overkill, but right now, after creating *unit sphere bundle*, I felt like creating a simple entry *sub-bundle*, just for completeness.

I wrote out some elementary details at *basic complex line bundle on the 2-sphere*.

created *inner product of vector bundles* with the construction over paracompact Hausdorff spaces

at *holomorphic vector bundle* I have started a section titled *As complex vector bundles with holomorphically flat connections*.

This deserves much more discussion (and maybe in a dedicated entry), but for the moment I have there the following paragraphs (with lots of room for further improvement):

+– {: .num_theorem #KoszulMalgrangeTheorem}

Holomorphic vector bundles over a complex manifold are equivalently complex vector bundles which are equipped with a holomorphic flat connection. Under this identification the Dolbeault operator $\bar \partial$ acting on the sections of the holomorphic vector bundle is identified with the holomorphic component of the covariant derivative of the given connection.

The analogous statement is true for generalization of vector bundles to chain complexes of module sheaves with coherent cohomology.

=–

For complex vector bundles over complex varieties this statement is due to Alexander Grothendieck and (Koszul-Malgrange 58), recalled for instance as (Pali 06, theorem 1). It may be understood as a special case of the Newlander-Nirenberg theorem, see (Delzant-Py 10, section 6), which also generalises the proof to infinite-dimensional vector bundles. Over Riemann surfaces, see below, the statement was highlighted in (Atiyah-Bott 83) in the context of the Narasimhan–Seshadri theorem.

The generalization from vector bundles to coherent sheaves is due to (Pali 06). In the genrality of (∞,1)-categories of chain complexes (dg-categories) of holomorphic vector bundles the statement is discussed in (Block 05).

+– {: .num_remark}

The equivalence in theorem \ref{KoszulMalgrangeTheorem} serves to relate a fair bit of differential geometry/differential cohomology with constructions in algebraic geometry. For instance intermediate Jacobians arise in differential geometry and quantum field theory as moduli spaces of flat connections equipped with symplectic structure and Kähler polarization, all of which in terms of algebraic geometry directly comes down moduli spaces of abelian sheaf cohomology with coefficients in the structure sheaf (and/or some variants of that, under the exponential exact sequence).

=–

]]>started some minimum at *cubical structure on a line bundle*

So I’m trying to draw some kind of connections here between a lot of really useful stuff that’s written on the nlab. I think it might even deserve its own page, which I would title “Descent Cohomology” (although maybe this is actually really quite trivial and doesn’t need its own page) but I think I need some help from you all to make it make sense in the “$\infty$” case. A lot of this has kind of been inspired by reading “Principal $\infty$-bundles - General Theory” by Urs, Thomas Nikolaus and Danny Stevenson, as well as stuff by Lurie, and a ton of other stuff for the discrete (and 2-categorical) case (Nuss and Wambst, Larry Breen, Knus and Ojanguren, SGA4, etc).

Given a cover in some ($\leq\infty$-) site $C$, $\phi:U\to X$, and some stack (or categorical bifibration?) $\mathcal{F}:C\to \infty-Cat$, I’d like to answer the question, for $M\in \mathcal{F}(X)$, what other $N\in\mathcal{F}(X)$ are there (up to equivalence) such that $\phi^\ast(M)\simeq \phi^\ast(N)$. This is basically asking for “twisted forms” of $M$, and if I’m not mistaken, this should be, at least theoretically, calculable as some kind of “cohomology” in some $\infty$-topos.

In the discrete case, one can compute such a thing by looking at $\check{H}^1(U\overset{\phi}\to X,Aut(\phi^\ast(M)))$, I believe. However, this also seems to compute principal $Aut(M)$-bundles for that cover as well. And in the nice case that the cover is “Galois” for some group $G$, this can be written down in terms of group cohomology of $G$ with coefficients in $Aut(M)$. There is machinery for doing something similar even in the more general scenario of Hopf-Galois extensions by some Hopf-algebra, explained very nicely in the paper by Nuss and Wambst: Non-Abelian Hopf Cohomology. What’s really nice about that scenario is that this same cohomology also classifies descent data for $\phi^\ast(M)$ (continuing with the notation from above). That is, it also classifies isomorphisms between the two different ways to pull back $\phi^\ast(M)$ to $U\times_ U$ (excuse me for skimping on the explanation here, the notation just gets unpleasant), or if we’re in the situation of monadic descent for some monad $T$, it classifies comodule-structures on $\phi^\ast(M)$ for the relevant comonad on the category of $T$-algebras (a relatively nice account of this is given in Mesablishvili’s On Descent Cohomology as well as Menini and Stefan’s Descent Theory and Amitsur Cohomology of Triples . So, this one single cohomology group computes a whole bunch of different things, which are all actually the same thing, and if we have a nice enough cover, we have even nicer ways of computing it.

So I guess my question is the following - Given all that we know about $\infty$-principal bundles (being computed by some $\mathbf{H}(X,BG)$), can we recognize this as some kind of descent cohomology, or higher Amitsur cohomology in the case of descent for either $\infty$-stacks or derived stacks? Now, a descent datum should be, instead of an isomorphism with a cocycle condition, a isomorphism with all higher cocycle conditions and bunch of cells gluing all of this stuff together (or in other words, the category of descent data is the limit of some simplicial $\infty$-category (or colimit and cosimplicial, depending on variance and so forth)). And so the “descent cohomology” in this scenario should be some higher, or derived, mapping space, but it should also depend on the choice of cover.

I’m trying to pick up the $\infty$ -categorical pieces as fast as I can, but I was just wondering if anyone has thought about this particular situation. I’d really really love to chat about it and try to get it ironed out.

Thanks!

]]>I ended up spending some time with expanding *extension of scalars*. Towards the end I had more plans, but I’ll stop now, need to do something else.