Daniel Litt "p-curvature and non-abelian cohomology."

Thank you so much for the introduction and the invitation. So everything I'm going to say today is joint work in progress with Josh Lam, and I'll also briefly mention some joint work with Kobi. All right. So for me, the thing that makes algebraic geometry special is that objects of algebraic geometry are infinitary. They're defined by finite functions and polynomials with finitely many coefficients.

Let me move down my microphone. Is that better? Okay. And so they all, at least finally, generate the algebras, and they have arithmetic. And I think we've understood, probably before anyone in this room was born, that there's some interactions between the arithmetic of a variety or other algebraic geometric object and its complex geometry. So typically this is mediated by topological invariance. So for example, when you Grothendieck conjecture, you show there's this realization of the cohomology of an algebraic variety, étale cohomology, which is related to counting points, and then that has some compatibility that seems like Hodge theory. So point counts control something about the cohomology in characteristic zero.

So for this talk, the motivating example will be Katz's proof of the p-curvature conjecture for algebraic varieties. So linear differential equations come from algebraic geometry. But what I'd like to do is talk about some things that go beyond the linear situation. So some results about nonlinear differential equations and algebraic geometry, and how their arithmetic reflects the complex geometry. Okay. So let me start by recalling the linear story. So what's the main conjecture about algebraic differential equations? In my view, it's the Grothendieck-Katz p-curvature conjecture. And here's a simple situation. So take a linear differential equation. So f is a vector of functions, and you want to solve the equation f prime = a * f. And the question the Grothendieck-Katz p-curvature conjecture tries to answer is: when can you explicitly write down the solutions? So when are they algebraic? So in the algebraic closure of the field of rational functions in one variable. And the answer is a simple arithmetic condition. The conjectural answer is a simple arithmetic condition. You take this differential operator to the p power and you ask that it be zero mod p for almost all p.

Okay, so what's known? Well, the forward direction is very easy, and the backward direction is very open. But I just want to stress: I mean, I've written it in a very arithmetic-looking way, but the content of the backwards direction is you want to prove a statement of complex geometry. You want to show that the analytic solutions to this differential equation are finitely branching, that the monodromy of the differential equation is finite. So let's do an example. So here's maybe the simplest differential equation you can write down: f prime = a f over z. And so what are the solutions?

Yeah. So z to the a, or constant times z to the a. And this is algebraic if and only if a is a rational number. And let's just check that that's the same as this differential operator to the p power being zero mod p. Well, here I just computed what it is when you take it to the p power and apply it to a monomial, and you get some kind of falling product. So n minus a times n minus a minus one all the way up to this. So you want that to be zero mod p. And it's easy to see that this is the same as a being an integer mod p, and that happens mod...

Sorry, the Zoom is coming up on the screen. Do you have any idea?

I'm not... um, okay. I'm not sure what to do either. Maybe let's leave it alone for now.

Yeah, so, and it's a consequence of the Chebotarev density theorem that this happens for almost all p if and only if a is a rational number. So this case of the p-curvature conjecture is already related to some deep number theory, the Chebotarev density theorem.

Okay, it'll be convenient for us to have kind of a stronger form, or well, an equivalent form, but maybe more flexible form of the conjecture, which is for linear differential equations in a more general setting. So those parameterized by a more general smooth variety. And here's the conjecture. So you have a flat vector bundle, and the conjecture is that, well, all the solutions to the corresponding differential equation are algebraic, the basis of flat sections, if and only if some invariant, its p-curvature, is zero mod p for almost all p. So let me recall what this means. So first of all, what's a flat vector bundle? You should think it's just a vector bundle with a rule for differentiating its sections. And that rule has the property that mixed partials commute. That's what it means to be flat.

What does it mean to have a basis of algebraic flat sections? It means this flat bundle is trivialized on a finite cover of X. And then what is this p-curvature operation? Well, it's something funny. It measures the difference between applying nabla to V and then taking the p power, or applying nabla to V to the p power. So here in characteristic p, if you take a vector field, its p power gives a new vector field. Okay, but concretely, what does this mean? If you take a vector field like d/dz, so coming from some local coordinate system maybe, so something where its p power is zero, the second term vanishes, and then it just becomes this, which should look familiar. So if you have your differential equation d/dz minus a f is zero, then it's saying that d/dz minus a to the p is zero mod p. We're almost solved.

Okay. So I think the main evidence we have, we have, I would say, kind of meager evidence for this conjecture. The main evidence is the result of Katz, which I'll make precise soon, which is that the conjecture is true for linear differential equations coming from algebraic geometry, Gauss-Manin connections, we saw in Javier's talk. There's some more evidence too. So for example, we know it for differential equations with solvable or virtually solvable monodromy group. Okay. So here's what Katz proved in 1972, more than 50 years ago. So X over S is a smooth proper family over, let's say, the ring of integers of a number field, and we've associated a flat bundle to it, which is, which Javier explained to us, it's the bundle which is given by the homology of the fibers equipped with the Gauss-Manin connection.

Analytically here, it's the bundle whose monodromy is given by the action of pi one of the base on the homology of a fiber. And the miracle proven by Katz and Oda is that this actually is an algebraic flat bundle with an algebraic connection. Okay, so here I guess maybe classically you can think of these differential equations as the ones satisfied by period functions, like in Hodge's talk, and their monodromy controls something about the topology of this map. And what Katz did is he proved the p-curvature conjecture for these bundles. And later in the talk I'll recall something about his proof, but maybe let me just remark the content is to show something geometric, namely that pi one of the base acting on the cohomology of a fiber, that action factors through a finite group if the hypothesis of the p-curvature conjecture are satisfied. So the p-curvature of this differential equation controls something about the topology of this map.

All right. So the question I want to talk about for the rest of the talk is: what about nonlinear differential equations? So I first tell you what people expect to be true about algebraic solutions to nonlinear differential equations. So what's the p-curvature conjecture for those? And then what are nonlinear differential equations coming from algebraic geometry? So what are the analogues of the Gauss-Manin connection? And Ellen briefly alluded to these in her talk. Okay. So what we'll do is instead of taking singular homology of a variety, we'll take the space of representations of pi one into SL_r up to conjugation. And here I'm taking a GIT quotient. So points of this affine variety, complex points of this affine variety, really correspond to semi-simple representations of pi one in SL_r. And I think there's a general philosophy, which I guess I learned by studying work of Simpson and of Laumon, that maybe one should think of this as some analog of, in fact, many of the statements you expect to be true about homology of algebraic varieties should have analogues for this space and its various avatars. Okay. And the goal of this talk is to prove at least an analog of Katz's result where you replace homology with this kind of nonlinear object.

Okay. So I'm going to start the nonlinear story now. Maybe before I do that, are there any questions about the linear story?

Okay. So first I should tell you what is the nonlinear version of the p-curvature conjecture, and I'll give a couple different explanations of that. So here, here's what you expect to be true for a sort of classical nonlinear differential equation. So here I've written the nth derivative of f is equal to some rational function in z and the first n minus one derivatives. So this is maybe not quite the most general kind of nonlinear differential equation one can ask about, but we'll discuss something more general soon. And so here's a conjecture. So this is a sort of concrete form of a conjecture which I think first appeared in unpublished work of Eremenko, Shabat, and Tsikh, and then maybe first appeared in print in beautiful work of Walsh, where he resolved some special cases of it. So what you do is you write down this funny vector field. So here the relationship between this differential equation and this vector field is just that you put this same g in the last coordinate. And the conjecture, so the p-curvature conjecture for this kind of nonlinear differential equation, is the following. So it's that all solutions to this differential equation are algebraic, so algebraic functions, if and only if when you take this vector field v to the p power and you reduce it mod p, that's zero for almost all p.

Okay, so maybe before moving on to a more general version of this conjecture, let me kind of say where this vector field v comes from. So it's actually something you've all seen in like your first calculus course. So what this is is a phase portrait for the differential equation. So it's some vector field whose flow lines are naturally in bijection with solutions for this differential equation. You can write them down. It's like you take the solution, its first derivative, second derivative, and so on. And here I've graphed the phase portrait for some logistic nonlinear differential equation.

Okay. So this is some quite general conjecture for which we have even less evidence, I think, than the linear version of the p-curvature conjecture. Let me tell you a slightly stronger version that we'll need. So here, what is, I think, maybe the most general version of a differential equation where you have analytic local solutions? It's an integrable foliation. So here F is a subsheaf of the tangent sheaf of a smooth variety which satisfies some conditions I'll write down shortly. And the conjecture is that the leaves of this foliation are algebraic if and only if F satisfies some arithmetic property. So it's closed under p power, is going to reduce it mod p for almost all p. Okay. So let me just make explicit what that means. So F is a subsheaf of the tangent sheaf. You can take a section of that. That's a certain class of vector fields, and you can raise them to the p power, that gives you a new vector field, and you want that vector field to still be in F. Okay. So maybe let me just remind you all what a foliation is and slightly expand, you know, define these red words here. So this is a foliation. So I've been teaching in Canada for a while now, and I've learned that I should relate my lectures to my students' lived maple syrup experience. So that's where this picture comes from. Okay. So what's really a foliation? So it's a subsheaf of the tangent sheaf which is closed under the Lie bracket. Right. So the tangent sheaf is a sheaf of Lie algebras, and I would like F to be also closed under that, closed under the Lie bracket. This picture here, oh sorry, is supposed to capture that this gives you a way of writing your space locally as an analytic local product. So in other words, at each point of your space you have a subspace of the tangent space, and you'd like those to all glue together. You'd like them to integrate analytically to leaves. And then here this condition is what you need to do that. It's some nonlinear analog of mixed partials.

Okay. So what is a solution? Well, it's what you get when you glue your local tangent spaces together. And here I've written down the definition of a leaf. So it's just a, well, analytic subspace of X so that the induced map on tangent spaces gives you an isomorphism between the tangent space of your subvariety and your foliation. But really you should just think it's a solution to your differential equation. Okay.

Okay. So this is the conjecture. The goal of the talk is to prove the conjecture for nonlinear differential equations coming from algebraic geometry under what I hope you agree are kind of mild hypotheses. Okay. So next I tell you what are nonlinear differential equations coming from algebraic geometry. But I think when I say linear differential equations coming from algebraic geometry, I mean Gauss-Manin connections. So I have to tell you what is the non-abelian version, the nonlinear version of a Gauss-Manin connection.

Okay. So, well, the Gauss-Manin connection came from taking homology of algebraic varieties. To get something nonlinear, we want to take some kind of non-abelian homology of algebraic varieties. And just like cohomology, non-abelian homology has a lot of different incarnations. So with homology you could take singular homology, Borel-Moore homology, and the Gauss-Manin connection kind of captures something about the relationship between those two objects. So here we'll have a version of non-abelian which is analogous to singular homology and a version which is analogous to Borel-Moore homology. So here X is going to be a smooth projective variety. I already briefly told you what the Betti moduli space of X is, but let me recall it. So it's this: you take maps from pi one, the usual topological pi one of X, into the special linear group, and then you take the... that's some affine variety, and then you take the spec of the invariant functions on that under conjugation by SL_r. So the complex points of this are the same as semi-simple representations of pi one of X into SL_r up to conjugacy. There's another way of thinking about this, which is, well, each local system on X, each representation up to C, gives you a flat vector bundle. So you can take the de Rham moduli space of X, which is the space of flat bundles equipped with an isomorphism between the determinant and O. That's just to make sure the monodromy is in the special linear group. And everything I'm going to do, you could also put another group there, but just for concreteness, let's stick with SL_r for that. Okay. And analytically, it's a kind of famous fact that these, well, the two complex analytic spaces associated to these varieties are biholomorphic. That's called the Riemann-Hilbert correspondence. How does it work? Well, given a local system here, you tensor with O, that gives you a vector bundle, and it's equipped with a flat connection. Given a flat connection, you take its kernel, that gives you a local system. And just to be explicit, what is the analogy here? You're supposed to think this Betti moduli space is the analog of singular homology, the de Rham moduli space is the analog of de Rham cohomology, maybe algebraic de Rham cohomology, and here this is the analog of de Rham's theorem comparing de Rham cohomology and singular cohomology.

Okay, so before I continue, just a warning: these are spaces, they're typically highly singular, and I'm going to make a lot of statements that only make sense a priori on the smooth locus of these spaces. So you have two choices: you can just trust me that one can make sense of these statements on singular loci as well. Many people, including Simpson, have figured out how to do so. Or every time I make a statement, you can just replace these varieties with the smooth locus, and you'll still get true statements. Although the proofs of those statements, especially later in the talk, will have to go through some possibly singular locus. So in the proofs, you and I will have to replace these spaces with this smooth locus.

So okay. So what is the non-abelian Gauss-Manin connection? So recall that if you had a smooth proper family, the Gauss-Manin connection was a differential equation whose monodromy only captured the action of pi one of the base on the cohomology of the fiber. And there's a non-abelian version of that exact same setup. So if you have a fibration like this, there's an outer action of pi one of the base on pi one of the fiber. If you like, you can think of, there's a, you know, there's a map from S to BDiff of the fiber, for example, that gives you this data. Okay, and that's enough to get an action of pi one of the base on the Betti moduli space of the fiber. So the outer automorphism group of a group acts on conjugacy classes of representations. So if you have a pi one equipped with a map to the outer automorphism group, that gives you an action on the Betti moduli space. Okay. Okay, and I claim that this is enough to define some kind of foliation on a relative moduli space of flat connections or representations. So to define that, I'm going to set S tilde to be the universal cover of S. And here I'm going to make a definition. I'm going to twist the Betti moduli space of the fiber by the action of pi one. Okay. So here, for now, ignore the right-hand side of this diagram. So all I've done is taken the sort of fiber bundle over S whose fiber is given by the Betti moduli space of a fiber, so representations of pi one of a fiber, except I twist it out by this pi one action on the Betti moduli space. So this is some map whose fibers are all isomorphic to the Betti moduli space of the fiber of pi, but that isomorphism is not, you can't make, well, you've twisted it by the pi one action. Okay, anytime you make a product construction like this, you get a foliation for free. Well, why is that? I told you a foliation is a description of your space locally as a product. And well, here I took a product and I quotiented by some group acting, you know, freely discontinuously. And so that doesn't destroy the local product behavior.

Okay. So this Betti moduli space is a family. It parameterizes kind of families of representations, and this foliation, well, leaves of it correspond to families of representations which are locally constant up to conjugacy, and that makes sense because all the fibers of this map are homotopic to it. So I have a representation of pi one of two nearby fibers. Well, that's representations of the same group. You can ask, are they conjugate?

The Riemann-Hilbert correspondence works in families. So you can take the relative moduli space of flat bundles on X over S. I guess I should say with trivialized determinant. And again, something naturally obvious happens. So I think this was written down first explicitly by Simpson. This foliation, which I described in a purely analytic way over here, becomes algebraic. Okay. And again, you can understand what its leaves mean complex analytically. The leaves are families of flat bundles whose monodromy representations are locally constant. And this is the foliation where I want to study this nonlinear version of the p-curvature for the rest of this talk.

And maybe just to make things a little bit more concrete. So it's not too hard to see that this foliation is horizontal over S. So it kind of maps isomorphically onto the pullback of the tangent bundle of S to the de Rham moduli space. So what you can do is you can think of it as a rule for lifting tangent vectors on S. So each, sorry, each vector field on S is equipped with a canonical lift here. And well, let's just say pretend S was an open subset of the affine line and you took the vector field d/dz. So this would give you a canonical vector field upstairs corresponding to d/dz. So this looks just like our picture, our phase portrait that we did from a couple slides ago. Okay. Concretely, what does this mean? It means if you take a point here, so a flat vector bundle on a fiber pi, and you take a little deformation of the point of S it lives over, you have a canonical recipe for deforming your flat bundles in nearby fiber.

So just to, I mean, I think the way I've described it, it looks like this really fancy object, but it's really, really classical. So I think maybe the...

Yeah.

No, no, no, no, no, definitely not. So I mean, this subsheaf of the tangent sheaf is along the smooth locus. What I really mean is that this is a crystal. That's the precise thing to say. And the question we're trying to ask is when are the leaves algebraic? Right? So in general, if you have a differential algebraic differential equation, its solutions are not algebraic, and the question the p-curvature conjecture tries to answer is when are the leaves, when are the solutions algebraic? And so in general these leaves are very far, they're typically infinitely branching and quite wild. Okay, and maybe just to stress that. So okay, I think these are the first examples of non-Gauss-Manin isomonodromy deformations that people wrote down. So the Painlevé VI equation was not written down first by Painlevé, it was written down first by Luchs in 1905, more or less from this point of view. So he was studying connections on P1 minus 4 points and then how to isomonodromically deform them as you move the points around. And then more generally, Schlesinger in 1912 studied the sort of general connections on P1 minus N points, and this is the differential equation that controls how to deform them.

And here, I guess this doesn't quite fit into the story I've told because these are isomonodromy problems corresponding to non-proper varieties, but in fact for any family you get an isomonodromy differential equation, and you can always put it in this form where you put some constraints on the a's and b's. So this is a fairly typical example of an isomonodromy differential equation that might actually come up, and you can see that they're quite classical.

Great. So our goal is to try to understand something about the p-curvature conjecture for this class of differential equations, and in particular how the arithmetic of these differential equations reflects something about the geometry of your map pi, your X over S.

Okay. So let me tell you the results. Start by recalling the result which motivates all this, which was Katz's result. So we have X over S. You take the cohomology of the fibers. This gives you some flat vector bundle, and Katz proved the p-curvature conjecture for this flat vector bundle. He says that if the p-curvature vanishes mod p, well, actually only for infinitely many p in this case, we don't need almost all p, then, well, then the way I've written it is the action of pi one on the integral cohomology of the fiber factors through a finite group. So it turns out this is the same because this connection, as Eremenko has regular singularities, as having all the solutions being algebraic. So this is a reformulation of what Katz proved, and I'll recall the proof in a few slides. Okay, so the first theorem I want to tell you about is, I think, a fairly strong non-abelian analog of this theorem. So this is joint work with Josh. It's not yet available, but hopefully quite soon. And it's the exact same statement where you replace with the moduli space of flat connections. Okay. So you have X over S smooth proper. You take this moduli space of flat connections on X over S equipped with this non-abelian Gauss-Manin connection, the isomonodromy foliation. And then you assume that this foliation is closed under p powers for almost all p. So here I wrote the p-curvature of the foliation vanishes mod p, that just means that the foliation is closed under taking powers. And then the theorem is that the action of pi one of the base on the Z points of the Betti moduli space of the fiber, so representations of pi one into, well, at least with traces valued in Z, factors through a finite group.

Okay. So you could ask, does this resolve this p-curvature conjecture for the isomonodromy foliation? And the answer is no. Okay. So why is that? Well, for example, this is some random algebraic variety. It could have very few Z points. So knowing what happens when you act on the Z points, it doesn't necessarily tell you much about the geometry of this foliation. So in order to prove this p-curvature conjecture, you have to replace this Z with a C. If you could actually show that the pi one of the base action on the complex points of this Betti moduli space factors through a finite group, you would have proven this p-curvature conjecture for isomonodromy. So it's not totally obvious it's true.

Yeah. So this is just like in Katz's theorem, you only need infinitely many p. And in this case, because there's sort of... Yeah. So maybe let me make a remark about sort of an important open question in the p-curvature conjecture. So you could ask, is the p-curvature conjecture true for some Gauss-Manin connections?

And that we don't know. So Katz's result doesn't know why they know that.

Yeah. So we know it if there's a five-motivic group by André. We don't know it in general. And it's not hard to see that if you just want to prove it for some Gauss-Manin connections, then you need density one, a density one set of primes. So here also, just like in Katz, when you take the whole moduli space, you only need the foliation to be closed under p powers mod p for infinitely many p. But I think if you try, you can formulate versions of this for subvarieties of the moduli space, and then you would almost certainly need density one.

Well, I'll tell you an improvement very soon. Yeah, even without... Yeah. So maybe let me just say, so here was Katz's theorem, right? So why does this imply the p-curvature conjecture for Gauss-Manin connections? Well, you want to show that the action of pi one on the singular homology with complex coefficients factors through a finite group. That's just, you take this finitely generated group and you tensor it with C, right? So if you have a, you know, Z to...

the N and an action on it which factors through a finite group, then that action also factors through a finite group when you look at the induced action on C to the N. But here this is a kind of nonlinear object. So just knowing what happens on the Z points is not enough to tell you what happens on the C point. So we're really kind of what makes this challenging is this is not a distance.

So not so in fact, I mean, I've stated something less general. So if you replace C with any ring of integers here, we win. The problem is that we don't get a uniform finite group. So in principle, yeah.

Yeah, so you could do it uniformly in the in the yeah, then you oh sorry, not not you would have to do it with S inverse I think because you don't know, I mean this could have like no Zbar points, right? This this could have no Zbar points like except for the identity or something, right? So even if you know it just for rings of integers.

Zar points? No, no, so so this we don't know this. I mean, I guess you've proven times we'll talk about this, this variety has some Zar points, but we don't if you just have a statement about the actual Zar points, you might not win. Yeah. Okay. So so here's here's what we prove. So here's the cases under which we can resolve the p-curvature conjecture for these foliations. So the same hypothesis as before. So suppose the p-curvature of the foliation vanishes mod p for infinitely many p, meaning the foliation is closed under powers when you reduce mod p. And suppose further that each component of this space contains a smooth Zbar.

So then the statement is that the p-curvature conjecture is true in this case. Okay, so let's so in other words, the action of pi one on the base on the set of semi-simple complex r-dimensional representations into into SLRC.

Okay. So this tells us that something the arithmetic of this foliation controls something about the geometry of this map. Okay, so let's try to understand how strong a hypothesis is this this red condition. So when do we know that this moduli space contains a smooth Zbar point? Okay, so I would conjecture it's basically always true. So so the conjecture I would even conjecture that Zbar points are are Zariski dense in the complex points of the Betti moduli space and that at least if the Betti moduli space is reduced, that would imply there's always a smooth Zbar point. Okay, so when do we know that? So as far as I'm aware there are two cases where we know this is true. So the simplest case is when r is two.

So we're looking at two-dimensional representations and this is joint work with Simon and Koch who will I think talk about it tomorrow in one of the contributed talks. And then the other case which is is quite a bit deeper and relies on the Langlands program and theory of companions is due to Dong and Ao in the case where the Betti moduli space is irreducible. So in this case, sorry, combining their work with the with older work of Romagny, one finds that Zbar points are actually even analytically dense.

Yeah, I I think I think I got it. Okay. Yeah. Phil, did you have a question?

Yeah. So can the action of pi one on the Betti moduli space have infinite kernel? Is that the question or on this just to the Z points of the Betti moduli?

I would oh it's the Z points as opposed to the Zbar points.

I don't know an example. Oh, I know an example if you allow me to take relative character varieties. I would have to think a little bit if you if you want if you want the smooth proper setting. But I think the answer is morally it shouldn't be huge. I mean in like well the the in the case I'm in where Y is smooth and proper there's always a Z point because there's the trivial representation.

Yeah. But probably there are lots of examples with basically no Z points. So I would think yeah so so if you take P1 minus four points with with sort of some choices of local monodromy this could have no Z points at all. So yeah.

Yeah. Okay. And then so just combining this theorem and and theorem two, you get an immediate corollary which is is you can prove the p-curvature conjecture for this isoperimetric foliation, the nonabelian Gauss-Manin connection, well under the conditions that that that that we know of the conjecture. So if r is two we have it in complete generality and if if if the Betti moduli space is integral, then we know I expect this integral can probably be weakened to irreducible.

Only a little bit of work. Okay. Are there any questions about these statements?

Okay. So so this is the main result at the top. And I want to explain I for me I think it's a fun result but for me it's really an excuse to think about kind of what structures exist on the nonabelian cohomology of a variety and like to what extent can Katz's argument in the abelian setting be imitated. So for the rest of the talk I want to kind of recall what Katz did and then say how we how we copy it.

Okay, so what does Katz's proof use? So well he's studying the cohomology of an algebraic variety and cohomology has a huge amount of structure. So I'll here you can see a list of things some of which we already understand with the Gauss-Manin connection, maybe we understand what the nonabelian version is, but there's all this structure. So in particular like the Hodge index theorem that when I started thinking about this problem I didn't really know how to think about it in a nonabelian way but we really need nonabelian versions of both structures to prove the theorems that I I've enunciated here. Okay. So so let me recall what Katz did. So I guess Plato said that all knowledge is is recalling things. So maybe I'm only recalling it in that sense but I don't I don't assume that you've actually thought about this before. So what did Katz prove? So he you have a smooth proper map associated to it is some some differential equation and you assume the p-curvature of that differential equation vanishes mod p for different p you want to show it it's finite monodromy.

Okay, so there's three steps. So the first step is algebraic. He shows that if the p-curvature vanishes for infinitely many p then the Gauss-Manin connection preserves the Hodge filtration.

Okay, I'll say a little bit more about this shortly. Then there's an analytic step which is that if you have a Z-variation of Hodge structure where the the connection preserves the Hodge filtration, well then it it also preserves the polarization. Sorry, I should have said a polarizable Z-variation of structure, but but we understand how the polarization on the cohomology of a family of algebraic varieties interacts with the Hodge filtration and so what you can do is you can modify it to get a unitary inner product. So so pi one actually acts on the cohomology of the fiber in a unitary way.

And then finally, well, now you know the action of pi one on the cohomology of the fiber factors through GLrZ intersect the unitary group. Oh, well Z because this singular cohomology has a Z structure on it and the intersection of these two groups is finite. So you okay so so I I think probably most of you have have this being a Hodge theory session, probably most of you have seen the definition of a polarization so I won't expand too much on on like the the input from the Hodge theorem here but let me just say a brief word about this algebraic step. So somehow you have to connect the p-curvature to the obstruction to the Gauss-Manin connection preserving the Hodge filtration and what Katz shows is that there's some filtration on de Rham cohomology mod p so that the associated graded of the p-curvature with respect to that filtration is identified after Frobenius twist with the obstruction to the Gauss-Manin connection preserving the Hodge filtration. So that's that's captured in the commutativity of this kind of complicated diagram. I'm not going to say what the conjugate filtration is which is what Katz uses but so so really all you need is that that a filtration with these properties exists.

Okay, so this is what Katz did. So somehow he finds a connection between the associated graded of the Gauss-Manin connection, the Deligne-Spencer map, and p-curvature. Then he uses the Hodge index theorem to get some unitarity and then he uses the Z structure and unitarity to get finiteness. Okay, so I've kind of highlighted all the things that maybe seem problematic in the nonabelian setting. You have to figure out what is a filtration, what is the associated graded, what is the associated graded of the p-curvature, what is the associated graded of the Gauss-Manin connection. What does it mean to be unitary? What is the Hodge index theorem? Okay. And so in the last 14 minutes I want to explain that.

Are there any questions before I do so?

Okay. So let's first of all talk about what a filtration is. Katz's argument you need two filtrations: the Hodge filtration and this thing called the conjugate filtration. And here Simpson tells us what to do. So let me tell you what a filtration on the DR moduli space is. So it's a space Q with a GM action, a GM-equivariant map to A1 where GM acts on A1 by scaling, and an identification between the fiber over one and the DR moduli space. Okay. And then we're just going to make a definition. I'll call the associated graded of the DR moduli space with respect to such a filtration the fiber over zero.

And the motivation I think for this is that it's also a fun exercise if you get bored, is that there's some equivalence of categories between GM-equivariant vector bundles on A1 and filtered vector spaces or p-dimensional filtered vector spaces with an exhaustive filtration.

Okay, so what I want to do is I want to just recall for you definitions of two filtrations on the DR moduli space analogous to the Hodge filtration of the... Okay, so first of all what's the nonabelian version of the Hodge filtration? So I think this is due to Simpson. So what it is is the moduli space of lambda-connections. So here you should think lambda is a number. And what is a lambda-connection? It's a vector bundle with a map that looks kind of like a connection except the Leibniz rule is twisted by by this number lambda. So if lambda is one you get a connection, if lambda is zero this is something O-linear. And then there's also a flatness condition. So this is the nonabelian version of the Hodge filtration.

And what's this associated graded? Well, it's something that's probably familiar to many of you. So what's the fiber over zero? Well, when lambda is zero, this just says this map is linear. So here gamma of f* f of s is f * gamma of s is what it means to be linear. And here so we have things with an O-linear map whose square is zero and that's a Higgs field. So what this says is that the associated graded of the moduli space of flat connections with respect to the Hodge filtration is the moduli space of Higgs bundles.

There's some stability conditions that I've omitted from the slide probably. Okay, so what's the nonabelian version of the conjugate filtration? So as far as I know this is maybe first written down by Mochizuki although it needs to be modified slightly. And maybe I should say I think this question of studying the p-curvature conjecture for isomonodromic foliations was probably first thought about by Pison and then I think a number of his students have thought about it so I don't know of anonymous related work but also Pappas and Mochizuki.

So okay, so here's what what I think the nonabelian version of the conjugate filtration should be. So you take triples (E, nabla, phi) where E now is a flat bundle and phi is something which equals the p-curvature up to your scalar lambda. Okay. And and has to satisfy some conditions.

Okay. And now it's a little exercise to work out what the associated graded is. But this is this is an application of Cartan theory. So what's the fiber of this thing over zero? Well, all it says if this is if lambda is zero, it just says that the p-curvature is zero and then phi is any horizontal map whose square is zero. And if you work this out, you see that this is the moduli space of Higgs bundles on the Frobenius twist of X.

Okay, so another way of saying that is that the associated graded of the Hodge filtration is up to twist the associated graded of the conjugate filtration. And in the abelian setting, that's what's called an excision structure. In this commutative diagram of facts, it's the two horizontal arrows in the square.

Okay. So now I have to tell you what the vertical arrows are and I I won't actually do that. I'll just tell you that that they exist. So so there is a nonabelian version of Katz's formula. If you ask me about it after the talk I can tell you what the analog of the associated graded of the p-curvature and the associated graded of the Gauss-Manin connection are. But we show that they're the same up to twist under this identification from this previous slide. And in particular a consequence of this is that if the p-curvature of the isomonodromic foliation vanishes mod p for infinitely many p, the nonabelian Gauss-Manin connection preserves the Hodge filtration.

Okay, what does that mean in the nonabelian setting? I think it's kind of clear what it means if you have a filtered vector bundle for a connection to preserve the filtration, but here remember a filtration was a family over A1, so a family over the DR moduli space mapping to the moduli of Higgs bundles. And what it means is well on the fiber over one and hence on every fiber away from zero this thing carries a foliation and what it means is that the foliation extends across the central fiber.

So what this theorem says is that if the p-curvature of the isomonodromic foliation vanishes mod p for infinitely many p then the foliation extends to the moduli space.

Okay. And just some remarks. So the proof relies on the proof of this last step relies on Obuski's work on nonabelian Hodge theory in positive characteristic and it has a precursor. So Mochizuki proved this last statement under the assumption that X over S has a Frobenius lift.

All right. So this is the nonabelian version of Katz's formula. Okay. So let's just take it as given now and try to prove the rest of the theorems. So the other thing we needed was the Hodge index theorem.

Right. So what is the nonabelian Hodge index theorem? So let me just say what the so I don't think what I'm going to say is the last word on a nonabelian Hodge index theorem. I think there's more one would want but what we do is we prove the consequence of the Hodge index theorem that Katz uses. So let me remind you what Katz used. So if you have a variation of Hodge structure, a polarizable variation of Hodge structure where the connection preserves the Hodge filtration, then the associated monodromy representation is unitary. But that's what Katz used and in fact we found kind of two analogs of this statement which turned out to be useful to us.

So one consequence of a representation being unitary is that if you take any vector and you look at its orbit under your group action it has compact closure. Okay. And the first part of this theorem says that if the nonabelian Gauss-Manin connection preserves the Hodge filtration, so meaning it extends across the central fiber of the moduli space, it gives you a foliation on the moduli space. And so in particular, if the p-curvature vanishes mod p for infinitely many p by Katz's theorem, then the orbits of pi one of the base on the complex points of the Betti moduli space all have compact closure. So you should think this is some kind of nonabelian version of being a unitary representation.

Okay. So that's the first analog of the of the Hodge index theorem. The second one is that under the same hypothesis this action preserves the Kähler metric on the smooth locus, this is the Betti moduli space.

And so the proof of the theorem I'm not going to say much about but I'll say it's an it's a theorem of analysis.