So I want to start by setting X to be a smooth projective algebraic variety over some field K, which is a subfield of the complex numbers. So in other words, X is, you can think of it as defined by a system of polynomial equations.
Okay, so one learns relatively quickly in algebraic geometry that algebraic varieties live kind of a double life. So of course you can take the space of complex solutions to a system of polynomial equations, and that will give you a geometric object, which in this case will be like a complex manifold inside of projective space. But you can also just think about your system of polynomial equations algebraically, and well, maybe unsurprisingly, algebraic geometry is about algebra and geometry and the connections between them. So you can do things like reduce your polynomial equations mod p and study their solutions over finite fields.
And I think one of the themes of 20th century algebraic geometry was that there was a lot of sort of surprising connections between the geometry of the complex manifold you get by making a maneuver like this and the algebra of the set of solutions. So for example, the Weil conjectures tell you that if you count sets of solutions to these polynomial equations over finite fields, those counts are controlled by the topology of the space.
So we've sort of started to learn that there's some connections between, let's say, arithmetic, maybe some kind of analytic geometry. For example, this is a Kähler manifold and topology.
At least one way of capturing the idea that there should be some relationship is the theory of motives, which tells us that the relationships between these objects are mediated by the cohomology of an algebraic variety. So to a variety X, we associate some object, its cohomology.
Okay, well just like for a manifold, there's many ways of computing cohomology. You can compute using singular cohomology or de Rham cohomology and you'll get the same objects for a manifold, but for a variety you get really different objects. So you get objects which are endowed with extra structure coming from the fact that your topological space here is an algebraic variety. So let me give you some examples of that kind of extra structure.
Well, first of all, you can take the singular cohomology of the complex points of X. And this one doesn't have a lot of extra structure. What it is, is it's a finitely generated abelian group.
You can take the de Rham cohomology of X, the same complex manifold. So this is something you might compute using Dolbeault resolutions, using differential forms. This is now endowed with a couple extra structures. It comes with some kind of filtration, a Hodge filtration, and some kind of quadratic form called a polarization, which plays nicely with the filtration here.
You can also do a more brutal thing. You can just take the associated grading of this filtration, that's what I would call the Hodge cohomology. This is a direct sum of Dolbeault cohomology groups.
And then finally, let me just remark, there's also sort of more algebraic maneuvers one can do, which won't play a large role in this talk, which will be mostly about complex geometry, but I just don't want to give you the impression that I'm exclusively thinking in terms of complex geometry here. You can take, for example, étale cohomology.
This has extra structure given by an action of the absolute Galois group of K, simply because it's an algebraic object and functorial. So the Galois group acts on X_K-bar, so it acts on any functorial invariant. And then there's other things one can do also, which won't play a role in today's talk. So for example, there's p-adic cohomology theories.
And one sees very immediately that this is the existence of these structures. So there's lots of kind of structures here. So here, for example, I'm thinking of a grading, a direct sum decomposition as a structure. And then there's also various compatibilities between them, compatibilities which are the subject of the theory of motives, but they're very powerful.
So let me just give you a very basic example. So for example, the fact that, so let's call this H^{p,q}, the fact that H^{p,q} is conjugate to H^{q,p} tells you that if X is an algebraic variety, the dimension of its first cohomology, its first Betti number is even.
Why? Because this is equal to H^{0,1} plus H^{1,0}, and these are two vector spaces of the same dimension. So this is already a very powerful constraint on the topology of algebraic varieties, and it tells us that these structures impose constraints on the topology or on whatever else you might want to measure about an algebraic variety using these invariants.
So for example, it's arithmetic. Just as an example, I mean, well, if you remember, I told you the Weil conjectures told you the topology of an algebraic variety controls something, but it's arithmetic. It's point counts over finite fields. And so this fact already tells you something about point counts over finite fields. So we made some argument in pure complex geometry, but it in fact tells us something about how many solutions the system of polynomial equations has mod p.
Okay. In families, these objects have even more structure. So now suppose we have some family of algebraic varieties, so π: X → S, a smooth projective morphism. In other words, just think a family of algebraic varieties parameterized by S or a family of systems of polynomials whose coefficients are given by functions on S.
So in this case, we can take, for example, the singular cohomology of X over S, and this is a local system on S. In other words, it corresponds to a representation of π₁ of S on the cohomology of a fiber.
We can do the same maneuver with the de Rham cohomology, and this is a flat vector bundle. So it's equipped with a Gauss-Manin connection, which captures the same data as this representation here.
So if you were here for the first talk, I told you that the representation of a fundamental group is the same thing as an ODE. And here, this is the ODE corresponding to this representation. And finally, there's an analogous thing one can do with Hodge cohomology in families.
So one can do this maneuver in families and one gets a graded vector bundle on this space, but it's equipped with more data called a Higgs field. And if you don't want to know what that is, don't worry, I will say something about it later. So this gives rise to a Higgs bundle, whatever that is.
But what I want to stress is it's highly not obvious, but this again captures the same data as this flat vector bundle or this local system. So that is a consequence of Simpson's work on non-abelian Hodge theory.
Okay. So we have all these structures and various compatibilities between them, many of which are conjectural. But nonetheless, this is a useful guiding mechanism to mediate between these sort of different areas that algebraic geometry touches on. Okay, but it's not the only way one can mediate between these areas.
And in particular, you know, cohomology is not the only topological invariant in algebraic geometry. So I want to recall a philosophy that I attributed to maybe Simpson and Deligne on Monday. And the goal of today is to kind of elaborate on this philosophy.
So that philosophy is the following. So suppose X... So we're in the setting, X is an algebraic variety, so is a smooth projective variety over some field K inside of C.
Then, well, instead of looking at cohomology, maybe the other most natural, simple and topological invariant you might want to take is the fundamental group of an algebraic variety. That's not quite linear enough to do these sorts of maneuvers, but instead we're going to consider the space of representations of the fundamental group of X.
Let's say into, for today, SL_r, the special linear group, r by r matrices of determinant one up to conjugation. And I'm going to call this M_Betti(X, r). So this is kind of like cohomology, so is kind of like cohomology, is a kind of non-abelian cohomology.
So as stated, this philosophy is a little bit hard to operationalize. So let me try to operationalize it in the most brutal way possible. So here's the slogan I wrote on Monday: Every statement about cohomology should have an analog for this space.
Okay. So this slogan I've found highly motivational in a lot of the things I've tried to understand over the last years, have been trying to make this precise in various ways. So I want to mostly focus on one example of the slogan being realized vigorously, but first I just want to kind of give you some general sort of simple examples of at least conjectural statements that should be familiar from this language.
Okay. So first of all, one thing we can observe about cohomology is that there's many ways to compute it in algebraic geometry. And similarly, there are many ways to compute this space. In fact, I would say that for each way you can think of to compute cohomology, there is a way to compute this space as well, or at least something like it.
So maybe before we had this kind of mysterious cohomology object here, which I did not give a decoration, let's call it H question mark. And then if you insert some decoration like singular or de Rham or Hodge or étale, you get some object with some varied structure. So let me suggest that there should perhaps be a similar mysterious object here.
What's analogous to singular cohomology is the object I've just defined, M_Betti(X, r). This is analogous to singular cohomology. This is like singular cohomology.
I think it's relatively clear what should be analogous to de Rham cohomology. Instead of computing representations as, well, group theoretic objects, one can think of them as ODEs, so as flat vector bundles. And so we can take M_deRham(X, r). This is the space of flat vector bundles of rank r with trivial determinant, up to isomorphism.
Similarly, there's an object called Hodge, which is analogous to the Hodge cohomology, which is maybe less familiar. So let me tell you what it is. M_Dolbeault(X, r). So this is the space of Higgs bundles. That's a vector bundle E with a linear map from E to E tensor omega_X, such that theta squares to zero.
Okay, and then there's various other objects one can consider too, which won't play much of a role in this talk. You should think of as being analogous to those. In particular, one can look at representations that factor through the profinite completion of π₁ into SL_r(Q_p). And that would be some analog of this. And because the profinite completion of π₁ is an algebraic object, it is equipped with Galois action.
Okay, again, one can do this in families. So again, given X → S, one can form the following object. So this is a family of spaces over S whose fiber is just M_Betti of the fiber. And let me write down a formula for it.
So you can take M_Betti of a fiber, multiply it by the universal cover of X, and then quotient out by π₁ here. So if you'd like, this captures some kind of monodromy representation analogous to this. So if you have a fibration like this, π₁ of the base maps to the outer automorphism group of π₁ of the fiber just by dragging loops around, and this induces an action of π₁ of the base on π₁ of the fiber.
Okay. This is some analog of the monodromy representation here with the local system. This is kind of clearly endowed with a flat structure because I've written it as a quotient of a product.
Similarly, one can form the relative moduli space, M_Dolbeault(X/S, r). And we saw what this is endowed with last talk. So this thing carries a differential equation, a foliation, which captures the flat structure here. And then the non-obvious fact is that it is in fact an algebraic foliation.
This is equipped with some analog of the Gauss-Manin connection called the isomonodromy foliation, which is defined by the property that its leaves are exactly the horizontal sections to this map in this flat structure.
And then finally, one can form this object M_Dolbeault(X/S, r) in families. And I claim, and much of today's talk I want to explain, that this is endowed with some kind of non-abelian Higgs field.
Okay, so in the interests of... I haven't told you what this means. In the interests of explaining where I'm going, I just want to kind of give you a sense of some kind of theorem about cohomology that one might search for an analog of in this world.
And of course there's many theorems about cohomology, and if you believe this slogan, I think you should search for an analog of all of them. And I and other people are trying to do that, but I, you know, the talk is only an hour long, so I want to do it for one. So here's some theorem, which I don't know who this goes back to, maybe classical antiquity.
And to be honest, it's probably someone in this department 50 years ago or 70 years ago or something. So suppose we have X → S, and suppose we take the following object. So here's a Higgs bundle, can take the direct sum over i plus j equals n of R^i π_* of omega^j(X/S). This will be equipped with some Higgs field, which I'll explain later.
So if theta is zero, then the action of π₁ of S on H^n of the fiber factors through a finite group.
So this is an example of something I told you earlier, which is that in this world, whatever this Higgs bundle associated to cohomology completely controls the monodromy. So the goal of the talk will be to explain an analog of this theorem in this non-abelian setting.
Okay, before I do that, let me explain some analogs of other things that might be more familiar, and then we'll return to this. Goal will be to explain a non-abelian analog of this result.
Okay. Before that, let me explain non-abelian analogs of some other statements you might know and love, and that have been the source of some controversy recently. So let me start by explaining a non-abelian analog of the Hodge conjecture. So just to give you a sense that this slogan is something one can kind of fruitfully pursue other analogs.
How many of you believe the Hodge conjecture? Only one of you? Be brave. I believe it. It's true. Come on. I know Mark doesn't believe it. Okay. This talk is recorded, right? So if it's soon disproved, I will be covered in shame.
Okay, so what does the Hodge conjecture say? It says that the cycle classes of algebraic cycles are the following group. So H^{2i}(X, Q) intersected with... Well, usually people write H^{i,i}. I'm going to write this in kind of a funny way. So the Hodge cohomology of i.
Okay, so here, this i just means I'm going to shift the grading. And a graded vector space is the same thing as a vector space with a C* action. And I claim if you unwind the grading, this is saying that you want to live in the fixed part of the C* action.
Okay. So here's the analog of this statement in non-abelian Hodge theory. This is a conjecture of Simpson. So motivic local systems.
So this just means local systems that arise this way, so that arise from the cohomology of a family of algebraic varieties. So these are the analogs of cycle classes, are exactly the integer points of this space intersected with the C* fixed points of another isomorphic space, this real analytically isomorphic space.
Okay, so let me unwind this a little bit. So I told you that a Higgs bundle, a point of M_Dolbeault is a vector bundle with a linear map like this. There's evidently a scaling action on this space because you can just scale the linear map. So this is the C* action.
It turns out that the C* fixed points are, well, they're evidently graded because they have a C* action, and they turn out to be Higgs bundles associated to what are called complex variations of Hodge structure. So the fancy way of saying this is that a local system comes from a family like this, if and only if it underlies a Z variation of Hodge structure.
Okay, just because I think that if you're paying attention, this might be slightly disturbing. The integral Hodge conjecture is false, but here it looks like there's some kind of integral statement here. I claim this is not really an integral statement. The reason why is that this is about complex motivic local systems, and the same complex local system can have many Z structures.
The map from Z points of this space to complex points is not injective. So if you understand that, really you should say there's a map from here to here, and it's the image that is conjectured to be the same as this.
Okay. At least that's how I would try to... I would say I'm sort of abusing, making an abusive use of the word intersection. There's not really a... This space isn't included in this space.
Okay, let me give you one more example. So this is a conjecture which is called the relative Fontaine-Mazur conjecture. Well, so first let me recall the Tate conjecture. The Tate conjecture is the version of this, the Hodge conjecture for... Sorry, I should have said this is the Hodge conjecture, is the version of this for ℓ-adic cohomology.
It says that cycle classes, at least the Q_ℓ linear span of such, are the same as you take the ℓ-adic cohomology of your variety and you take the Galois invariants if K is finitely generated. And again, there is an analog of this conjecture in the non-abelian world called the relative Fontaine-Mazur conjecture.
Yes, sorry, thank you. Which is that motivic local systems are exactly... One can take the space of representations π₁(X-hat) into SL_r(Q_p) up to conjugacy.
And okay, one maybe doesn't quite want to take the fixed points, but one can take the finite elements. So the union of Gal(K-bar/K') where here K' over K is finite. So maybe this is more closely analogous to what you get if you try to formulate the Tate conjecture over K-bar. So if you try to understand cycles over K-bar, you'd get exactly the same kind of union.
Okay, great. All right. So my goal is to explain for the rest of the talk how to understand something about an analog of this statement. Before I do that, let me just recall something with how one proves the statement, so in particular what this object is and in particular what this data is and how to get at it. There's a few different ways of thinking about it.
There's conjectures. Yeah, so you can look at... Yeah, let me say, this is a great question. So Katz, one of the strongest pieces of evidence we have for the Hodge conjecture is that the locus where a certain cohomology class is Hodge, so it lives in this intersection, is an algebraic subvariety of, if you'd like, let's say the base of a family.
So this was proved by Katz, Kapanali. It's a prediction of the Hodge conjecture. One can ask the same analogous question in this... Sorry, I covered it up... In this world where you have, let's say, the space of flat bundles on a family.
So there you want to know is the set of points where a given local system underlies a Z variation of Hodge structure an algebraic subvariety. So what's known is that it's a closed analytic subvariety, so that's already not totally obvious. If S is compact, then of course you win. But yeah, we do not understand these objects in general.
So here's a very basic question that is open. So just look at the universal curve. We know from work of Deligne that each curve carries finitely many Z variations of Hodge structure of a given rank. One prediction algebraicity of these loci would make is that there's a bound independent of the curve, and that we do not know. It's a very interesting question, I think.
There's been some work on it by Bill Engel and Slim Yau, but I think it's still broadly open. Okay. So I want to just briefly review this classical theorem before explaining something about its non-abelian analog.
Okay. Probably some of this will be a little bit familiar from Monday. So this is section two of the talk, classical antiquity, which I mean the Kodaira-Spencer map.
So I want to explain how to construct some object, which I think non-obviously captures the monodromy of a family of algebraic varieties. So again, we have X → S, π smooth proper. And well, I can first tell you just what vector bundle we're going to study. So E will be the direct sum of H^p of omega^q(X/S).
So this is the vector bundle whose fiber over a point of little s of S is the p-th cohomology of the q-forms on the fiber. So I claim... So theta, I claim that there's a natural map from the tangent space of S to... Write it this way, direct sum over p plus q.
So here, this should be over p plus q equals n, H^p omega^q(X/S), the same thing where you shift the indices by one.
Okay, so there are a few ways to get this. I'll say in words one, and then I'll write down an explicit formula a different way. So this is the associated graded of a filtration on de Rham cohomology. That filtration is not preserved by the Gauss-Manin connection, but it's preserved up to a shift by one. So this is what you get when you take the associated graded.
So maybe a simpler description, you can write down a formula for it. A point is that if you take a first order deformation of your point little s, so that's the tangent space, that maps to first order deformations of the complex structure on the fiber. And this is identified via the Beltrami differential with a cohomology group, namely the cohomology of the tangent bundle.
And then this maps into this space as follows. So this maps to direct sum H^p omega^q(X/S) to H^{p+1} omega^q(X/S) minus one, X/S, just v maps to cupping with v.
Okay, so this is some explicit formula. You're not supposed to get anything out of it except that it's sort of, I think highly not obviously related to the topology of this map. It seems to be like a purely analytic object. So there's another way of understanding this. This is the so-called derivative of the period map.
In other words, it controls something about how the Hodge structure, how this direct sum decomposition on the cohomology of the fibers varies as one moves around in S.
Okay. So I did a little maneuver here. I wrote theta as mapping from the tangent sheaf to the endomorphisms of this bundle, but you can just do some adjunction to get what I call a Higgs bundle from before, which would be a map from E to E tensor Omega. Just move this to the other side.
Okay. So the theorem whose proof I want to recall is that if theta is zero, then π₁ of the base acts on the cohomology of the fiber through a finite group.
Okay. And the proof we've in fact already seen, but I just want to recall it. So step one is that, well, I gave you in words the description of this map theta. I told you it measured the failure of the Gauss-Manin connection to preserve the Hodge filtration. So in other words, if theta equals zero, then the simplest way to say this is that π₁ of the base acting on the cohomology of the fiber preserves the Hodge decomposition.
Okay. Then step two was that, well, there's a natural polarization on the space coming from the cup product, and it plays nicely with the Hodge decomposition on each piece of the Hodge decomposition up to some triangle sign changes, it's positive definite. So if you preserve the Hodge decomposition, you preserve this inner product, this representation is unitary. So this implies this representation is unitary.
And then finally, oh, we have this representation and it factors through a unitary group and also the automorphisms of integral singular cohomology, a finitely generated abelian group. And I told you if you were here on the first day that there was one thing you should take away from that talk. Does anyone remember what it was?
Yeah, discrete and compact implies finite. So here this is compact plus discrete, hence finite.
Okay. So I would like to explain there's the talk and analog of this theorem for non-abelian cohomology, which is really just an excuse to develop some of these structures on these spaces of representations.
Okay. So I think there are a few mysteries. Maybe the first one is, well, what even is the analog of this theta? And then, so I have to explain that. But even if you want to start trying to understand the proof of this theorem, you need to make sense of what is the Hodge decomposition, what is polarization? So the Hodge index theorem and so on and so forth. So I want to try to give you some brief cartoon of what I think those objects are.
Three is the Hodge decomposition, the Hodge filtration at least, and Hodge index theorem in the non-abelian world.
All right. Okay, so first, let me kind of tell you a formalism. So when I began this talk, I told you that the cohomology of algebraic varieties is endowed with all sorts of extra structure. And I think from a modern point of view, it's useful to think of that extra structure as being characterized by being a sheaf on a certain space. So outputs of cohomology are coherent sheaves on certain spaces.
Maybe I shouldn't call this a formalism, this is some kind of philosophy. So just as an example, well, one thing you might get out of cohomology is a filtered vector space. And these are the same as vector bundles on the affine line, which are equivariant for the action of scaling. So maybe let me denote by C*, the scaling action.
Okay, so this is called the Arinkin reconstruction. It's a fun exercise if you haven't seen before to go back and forth between these two objects. Taking the associated graded of a filtration is the same as taking the fiber over zero here, for example.
But this suggests to us that maybe if we want to understand the analogs of these structures for these spaces of representations, we should look for spaces living over the same spaces to capture the structures on cohomology. So let me give you, I think, the simplest example of this.
So for example, one of the structures we saw on cohomology was a filtration, the Hodge filtration, and that suggests that there should be some analogous structure on the space of flat connections on an algebraic variety. If that filtration is the same data as a sheaf on A¹ mod C*, so scaling equivariant sheaf, we should find some space living over A¹ with a scaling action.
And that space was discovered by Laumon and Simpson. It's called the Hodge moduli space. And it's the space of lambda connections on X.
It's equipped with a map, a function, lambda to A¹. And let me just briefly tell you what a lambda connection is to give you a sense of what's going on. So it's a pair E, gamma where E, gamma is a map from E to E tensor omega¹_X, which looks like a connection. So gamma squares to zero.
But then it also, it's twisted by this notion lambda. So gamma of a function times a section to your vector bundle doesn't quite satisfy the product rule for derivatives, but it almost does. So it's gamma of f times s plus lambda s tensor df.
So if you stare at this for a second, you can convince yourself that when lambda is one, this is the same data as a flat vector bundle. And when lambda is zero, this is the same data as a Higgs bundle.
Okay. So moreover, there's a scaling action on this. The t times gamma is just, sorry, t times E, gamma is just E, t gamma. You can convince yourself that this is a family, a C*-equivariant family over A¹, which is supposed to be like a filtration.
Okay. So let me record the relevant facts. So M_Hodge(X, r, 1) is what we called M_deRham before, the space of flat connections, and M_Hodge(X, r, 0) is what we called M_Dolbeault before, the space of Higgs bundles.
Okay, and I told you that this was sort of supposed to be analogous to de Rham cohomology. It had something like the Gauss-Manin connection, which was this isomonodromy. This is supposed to be like the de Rham cohomology of an object. Sorry, if you look at this in families, it has... So this is supposed to be the de Rham cohomology of a family.
And that's just maybe this should be something like a Higgs bundle. So the question is what is the analog of a Higgs field? So let me make a definition. So a Higgs scheme.
So suppose we have Y → S, pi is a smooth morphism. So A, the structure of a Higgs scheme or Higgs field on Y over S is a map of the algebras from the pullback of the tangent bundle of S to the relative tangent bundle of Y over S.
Okay. And the claim is that... So a Higgs scheme will be a scheme like this equipped with a Higgs field. And the claim is that this is a Higgs scheme. So it's equipped with a Higgs field.
I'll give you a formula for it too. But I mean in this sense, okay, there's some things to unwind here. In particular, this thing is not smooth, but if you know the right things to say to make this definition in the non-smooth thing, you will be able to guess... In the non-smooth setting, you'll be able to guess the right definition.
So before I tell you the formula for this theta, let me just state a theorem to give you a sense that whatever definition I'm going to make is actually the right one. Here, this theorem, like everything I'm saying, is joint work with Josh Lam.
So suppose X → S, pi is a smooth projective morphism. And to this data, we can associate this relative moduli space of Higgs bundles equipped with its Higgs field capital theta. And the theorem is that if theta is zero, then the action of pi one of the base on the space of representations of pi one of the fiber factors through a finite group.
Okay. So whatever this theta is, it controls the non-linear monodromy operator we're studying. Okay. So I want to give you a sense of what goes into this. Before I do that, let me just give you a definition for theta, so I'm not giving you an empty theorem.
So here's some facts. So first of all, there's a natural map from the space of Higgs bundles to some vector space. So sum equals two to R of the space of symmetric differentials. I guess this is a vector bundle over S, Sym^i omega^1 X over S.
Okay, so this is called the Hitchin map. It starts at two because I'm working with SLR objects. This map is easy to define. Given a Higgs bundle E theta, you send it to the characteristic polynomial of theta. And if you unwind definitions, the coefficients of the characteristic polynomial lie in this vector space or lie in this vector bundle.
Okay. So the second fact is that M Dolbeault X over S R has a natural symplectic structure. This comes from the fact that its tangent spaces are defined in terms of cohomology groups and there's a cup product in cohomology groups.
And then finally, there's a natural map from the space of deformations of our fiber, which we saw before, to functions on this space. So this maps to H0 X S M2 omega 1 X dual, just given by the cup product.
So I guess I'm going to write it down for relative dimension one. There's a way of doing this in general, writing down a symbol formula. So this maps to cupping with V, which if you work it out lives in H1 of Omega 1, which for a curve is just the complex numbers.
Okay, so here's a formula for theta in this case of curves. So this theta is one half of the... Sorry, it's... I'm going to call this map KS. So it's sigma maps to one half of the Hamiltonian vector field associated to the pullback of this.
Sorry, let me call this, give this a name. This is going to be V dual, H of KS of sigma dual. Okay. This is some weird formula you're supposed to think it's analogous to this formula I wrote down over there.
So in other words, given a first order deformation of X, when it's a functional on this space, you can pull it back through this Hitchin map H that gives you a functional on the Dolbeault moduli space, and I'm taking the associated Hamiltonian vector field.
There's no semi-stability in this theorem. This is the full Higgs moduli space. Yeah, they're sort of disjoint unions. And this is not a finite type thing. They're disjoint union. I'll say something about this at the end, but yeah, they're disjoint unions of varieties typically. The general fiber is a disjoint union of infinitely many abelian varieties.
In general, as long as the special variety is smooth. Yeah. And then of course there are many bad fibers. I guess the general fiber of this map is empty if X is not a family of curves.
Yeah, so there's a logarithmic, you can talk about logarithmic bundles, but I don't know if anyone has really worked out the theory well. But yes, if you have a semi-stable thing, then there's something you can write down at least. Okay.
Okay, so the idea is just to copy the proof of this theorem. And so maybe it shouldn't be totally surprising that this is sort of the complex geometry that goes into what I talked about on Monday, but there's some arithmetic which I'm not going to talk about.
But I want to try to tell you a more precise statement, which kind of hints at some of the extra structure that's hiding behind this theorem. So maybe let me call this theorem A, and I'll write a kind of more precise statement, which is theorem A prime.
I think there's something I find a little shocking about this result, which is that it shows something happening that you don't really see in the classical setting that is over here. So here's the setup. So again, X to S I is smooth and projective.
So for the complex numbers. Now K is any local field. B, QP, FP, double parentheses T, the local field.
So suppose this nonlinear Higgs field is zero on M Dolbeault X over S R. This is the moduli of all Higgs bundles. Then two things happen.
So one, the action of Pi one of the base on space of representations of Pi one of the fiber valued in SLRK has orbits with compact closure using the kind of analytic topology from K. And two, for KC, the action of Pi one on the complex points preserves a natural Kähler metric.
So I think there's something that was surprising to me at least here, which is that in this non-abelian world, the representations into fields other than C all seem to be on the same footing. So when K here is C, this seems like some analog of the Hodge index theorem. So this unitarity appeared here, but there's also a unitarity statement over other fields.
I think this is the source of many interesting open questions that I'll try to allude to in the last couple minutes. Let me briefly indicate how to deduce from this what I called theorem A.
Oh, I see. Yes, yes, that's right. For an honest Higgs bundle, that's right. Yes. Okay. Yeah, so let me briefly explain why this implies the other theorem.
So suppose now that L is a number field, a finite extension of the rational numbers, and one can consider the embedding of the representations valued in the special linear group over L into the product over the completions of L of the space of representations valued in those completions.
So this embedding is discrete image. This is a general fact from number theory. It's analogous to the fact that the integers are discrete in the real numbers. And then moreover, sorry, there should be an S here. The action of Pi one of the base on this product has orbits with compact closure.
And then can anyone say why at least the action here on any point here factors through a finite group? Yeah, exactly right.
So we're not quite done. What this shows is that the action of this group on any point of this space factors through a finite group. And then there's a little argument using Riemannian geometry from two to... There's a general fact if you act on a Riemannian manifold by isometries and a dense set of points are periodic, then the whole action factors through a finite group. There's a little bit of Riemannian geometry.
Okay. So in the last four minutes, or maybe I'll try to end two minutes early. In the last two minutes, I just want to point out that I think something kind of weird happened here, which is that it looks like there's some kind of p-adic version of the Hodge index theorem.
In other words, we also have some statement that is analogous to the Hodge index theorem for representations valued in a non-Archimedean local field. So observe, there seems to be some version of the Hodge index theorem for p-adic fields.
I think the analog of this statement, the abelian analog of this is probably just that there is an integral structure on singular cohomology, but I think it's a very fruitful thing to study. So let me just conclude by making some kind of technical remark, but I think it points in some interesting direction.
So there's a deficiency of this theorem for the experts in the room, which is that it really requires the Higgs map theta to be zero on the whole Dolbeault moduli stack. And you might ask them maybe should be restricted to some semi-stable locus, Simpson, and training classes, blah, blah, blah.
And there's some interesting extra structure that appears kind of relevant to that question that I think has not been so well studied. So one way of understanding this question is that there's a... Well, there's one is sort of studying the Hitchin map to the space of symmetric differentials.
And what ends up being relevant to proving this theorem in the case of non-Archimedean local field is understanding something about the image of this map. So there's a parallel theory due to Grochow and Shapiro, which starts with a p-adic representation of Pi one or a representation valued in some non-Archimedean local field.
And they've constructed a natural continuous map to the same complex vector space. So this is a construction due to Grochow and Shapiro. I'd be happy to tell you about it after the talk. So let me call this the complex Hitchin map and this is the p-adic Hitchin map.
And I think this result is suggestive of some compatibility between these two stories. So I think the natural question is, is the image of this p-adic object contained in the image of the complex object? And let me just conclude with a little kind of some work in progress.
So a point of this gives rise to something called the spectral variety. It's a polynomial on the cotangent bundle, so it cuts out a variety of the cotangent bundle. And the theorem is that if the spectral variety is smooth, then this comes from a complex representation too.
Okay. So I think there's much to study here and I'm happy to discuss this more if I'm around the whole semester.
