@ZhenLin on the last cycle the grad student input was gathered and then discarded as unfairly biased, whatever that means

no wonder no one replied this time

@ClarkBarwick @AaronRoyer @DenisNardin thankyou for having the conversation. sorry i wasn't present for it. :-)

neil's thesis is gr8

hehe yeah i can see why you'd say that. :-)

is there a general result along the lines of the serre spectral sequence of a fiber bundle F -> E -> B of smooth projective algebraic varieties and smooth maps degenerating at the E_2 page?

(over C, let's say)

(i mean the varieties are over C, but feel free to take the cohomology to be over C too)

the motivation is that if everything is also defined over over Z then |E(F_q)| = |B(F_q)| |F(F_q)|, so that suggests a corresponding spectral sequence in l-adic cohomology ought to degenerate, maybe?

so certainly this holds if F and B have algebraic cell decompositions, which is both when their counting functions |F(F_q)| and |B(F_q)| are polynomials and when their cohomology over C is concentrated in even dimensions

oh sweet, it holds for fiber bundles of compact kahler manifolds

maybe?

@QiaochuYuan So if I just have a fiber bundle and the cohomology of the fiber and the base is concentrated in even dimensions (note I am ignoring the potential issue of having to work with local coefficients) then the spectral sequence would collapse for dimension reasons, right? The differentials change one of the bidegrees by an odd number always.

Maybe that is obvious to you though.

@QiaochuYuan there's this result of Deligne's that says for any proper submersion f: X ---> Y of projective (I think even weaker works...) complex varieties, the Leray spectral sequence collapses at E_2. But probably the map had to be holomorphic...

that's a cool result

@QiaochuYuan: The statement about point counting is incorrect: consider the multiplication by 'n' map on an elliptic curve E with fibre E[n] over some finite field F_q with n invertible in F_q. By passing to an extension, you may assume E[n] =~ (Z/n)^2, and then the formula you suggested in clearly incorrect (unless n=1).

@QiaochuYuan: The statement about l-adic cohomology (that the Leray spectral sequence degenerates), however, is indeed correct, and follows from the result of Deligne mentioned by Dylan Wilson (at least in the projective case, and the proper case is OK too, by the Weil conjectures, though presumably there is a direct argument)

@EricPeterson oh, was it? i vaguely remember now that you mention it, maybe students of one of the potential chairs were going around trying to advocate for the other candidates or something like that

@anon: yeah, sorry, I guess I can't assume that the fibers all have the same point counts over F_q. in the example I had in mind F, E, and B all had polynomial point counts

@Dylan: yeah, I found some references stating that result for a proper holomorphic submersion between compact kahler manifolds or something like that

@anon: mmm, wait, that's not the problem, sorry. thanks for the counterexample!

@QiaochuYuan: The problem is that the base is not simply connected, so Leray degeneration does not give the point counting statement. (If the base was simply connected, the local systems coming via pushforward would be trivial, so you'd get the statement you want via Kunneth.)

gotcha

unrelated question: does anyone know any topological obstructions to a compact almost complex manifold being a complex manifold?

So, you can imitate various constructions from algebraic geometry for E_\infty-rings, but it is not clear to me in what kind of geometry we are working with

@user101036: I don't understand the question

Oh, I ddn't finish it, sorry. I was just wondering if there is a nice interpretation of what one does in terms of explicit geometric objects, or if there are good pictures to have in mind

I mean like, say for schemes we have fuzz that corresponds to nilpotents

and one puts the fuzz on top of how one visualizes varieties

But for E_\infty-rings , my intuition is nil

@user101036 If you restrict your attention to connective E_\infty-rings and use the so-called strong topologies, this is pretty close to what you get, there's just somehow more fuzziness. This is the perspective of a lot of the results in the DAG series.

More generally, it's really strange and geometric intuition can fail pretty dramatically, in my experience.

do you have an example of where the intuition fails dramatically?

@AaronMazel-Gee some people did come around to advocate, but it was honest advocation, not 'you should come and blindly tip the vote', and i think not at the candidate's urging

@user101036 Here's two off the top of my head: If you restrict to coconnective E_\infty-rings, your theory looks like unstable homotopy theory, rather than algebraic geometry. This is a consequence of Mandell's work on cochain theories.

Another is if you want to work with different sorts of topologies, i.e. based on THH and TAQ, for example, then you find that sometimes things that you don't really want to be etale have vanishing relative cotangent complexes.

Another issue is that taking symmetric powers of modules is totally screwy, as mentioned in Lurie's thesis.

that is very interesting, I'll look into it. Thanks!

note that different people might have different things they "want" to be etale