Absolute sets and the Decomposition Theorem
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Budur, Nero
We give a new, more conceptual proof of the Decomposition Theorem for semisimple perverse sheaves of rank-one origin, assuming it for those of constant-sheaf origin, that is, assuming the geometric case proven by Beilinson-Bernstein-Deligne-Gabber.
 Thank you very much for the invitation to come here. It's always a pleasure to come back to Lumini. Let me state what the general theme of the talk is. So you've seen this before. Modular spaces of local systems on complex algebraic varieties should have interesting geometry. So there are thousands of talks given on the subject papers written about it. But I couldn't resist to make a bit of propaganda for myself. Let me give a new result in this direction. That doesn't really have anything to do with the title here. But here's the first theorem. So let's see be a smooth projective curve over the complex numbers of genus greater or equal than 2. Then the space of representations of the fundamental group at the fixed base point into GLRC is a variety with at most rational singularities. So in fact, we know the fundamental group is so we can write this more explicitly. This is the complex points of this. This is an affine variety. The complex points are given by g tuples, g couples of matrices xi, yi in GLRC, such that the product of the multiplicative commutator of each pair is the identity matrix. So the new part here is the rational singularities part. It was proven by Carlos Simpson in the 90s that this is indeed a variety. In fact, it's normal and complete intersection. So it also has this bound on the complexity of the singularities. Now, why is this interesting? This provides an example to this general theme because of the following corollary that is completely group theoretic. So this implies the following theorem. If n is greater or equal than 3, then there is a real constant c positive, a natural number m0, such that for every rank r greater or equal than r, sorry, this is a rank here, r0. For every rank r greater or equal than r0, the number of isomorphism classes of irreducible complex representations of rank of dimension smaller or equal than r of, and this is the surprising part, slnz is smaller or equal than cr squared. So this is completely group theoretic. We know if you replace z here by a finite field, we know by the Linn-Lustig theory how to address the irreducible representations. For slnz, we actually don't know how many there are. We know by Margulis' rigidity that these are finite many, but we don't know exactly how many. So this is currently the best bound. And in fact, the step from here to here is not due to myself, it's due to Eisenbad and Avni, who's so they also done the following bound before, instead of 2 here, 22. So it turns out that representations of fundamental group, even for smooth projective curves, is very much related to group theoretic statements like this. So that's an illustration of the general theme here. Now today's theme is different. Today's theme, and I would like to phrase it as follows. Not only the modular spaces of local systems are interesting, but any other naturally defined subset. So any here is in quotation marks. Any naturally defined subset of modular spaces of local systems should also have interesting geometry. Or perhaps even better, today's theme that also relates with this is try to prove results that are a little bit better than what Simpson has done. So again, this talk is based in trying to generalize some results of Simpson's tool, a more general setup. OK. Whoops. OK. So from now on, this is joint work with Botan Wang. And it's very much related to the talk by Elena Noe earlier. So let me start with absolute sets of local systems. And I'll give a definition for this. So first of all, we're going to start with x being a smooth complex algebraic variety. We have the variety of representations of the fundamental group at some fixed base point, rank R representations. And this is an affine variety, affine scheme actually. By acting via conjugation, we have a group action on this scheme. And we can take the gIT quotient of this and we get this function. We get the gIT quotient of this and we get this so-called betty moduli space of local systems of rank R on the underlying analytic variety manifold. So this guy here quotient by GLRC. So this is an affine variety. So this is the same as the spectrum of the ring of invariants of the coordinate ring here. So this is also an affine scheme. So we'll be looking at this betty moduli space. The complex points of this moduli space are the semi-simple complex local systems of rank R on idianalytic topology. For example, in rank 1, these two spaces are the same because the conjugation doesn't do anything then. And this is an algebraic group. You can tensor two rank 1 local systems and you get another rank 1 local system. And in fact, this is the home from the first homology group of x with coefficients in z to c star. We've seen that earlier today. So it is c star. It is a complex affine torus, c star to the first betty number times if there is some torsion here, you also get the finite abelian group here. So it's finitely many copies of a complex affine torus. So it's easy to understand. Here's the definition. And this is largely inspired by Simpson. I'll tell you later what the difference is with his definition R. So this is now x in addition quasi-projective besides smooth. We would like to define absolute sets in the betty moduli. So I'm looking at subsets here. In fact, I'm looking at subsets of the complex points here. But I'm going to skip writing complex points. So this is absolute. And we are taking something that is constructable. And let's say it's constructable and defined over q bar. So s is a q bar constructable subset here. And we call it absolute if for every automorphism of the complex numbers, there exists s sigma. And s sigma should live in the betty moduli of the conjugation of your variety. And I would like to say that this guy should correspond. But again, as in the talk from before, he can't really start conjugating the topology here. By conjugation, he destroyed completely the topologies. So you have to do something else. And what Simpson did is to look at the Deram version of this. So he constructed some moduli spaces by embedding the variety into something smooth, projective with a complement simple normal crossing divisor. And you look at logarithmic flat connections there. He constructs a moduli space of those. And there's the Riemann Hilbert morphism. So this is all complex points here. You look at this underlying analytic variety here. And there's an analytic morphism to this betty moduli. This happens to be a bihologomorphism if x is projecting. But it's not so necessarily if x is not projected. However, this is very algebraic here. So everything is defined algebraically. So now you can start conjugating by sigma. And you end up to the Deram version of this moduli space. So the idea now is that this guy that you follow up via this diagram should also be cubar constructible for all such conjugations. So there exists a sigma, cubar constructible, such that s sigma is rh p sigma rh inverse s. So there are some differences with the original definition of Simpson. First of all, the definition of Simpson was done in the case when x was smooth projective. So here we are already quasi-projective. Second of all, in that case, he also assumed that the inverse image by this analytic bihologomorphism in that case should also be constructible. We don't assume this, but we can prove that in the projective case. That's already automatic from these conditions. And the third thing is that Simpson also has a condition on the absoluteness in Simpson's term. Also involves an extra condition in terms of the Higgs moduli space. We do not deal with Higgs moduli spaces at all. So that condition we don't really deal with in this talk. The only thing is, if the rank is 1, then that condition, even in the quasi-projective smooth case, is actually superfluous. And that's already some, more or less, essentially contained in Simpson's paper. All right, so that will be the definition of absolute sets for us. So I'm going to state the theorem now, which illustrates the today's theme of the talk. But the theorem is very hard to make precise. So it will be somewhat vague. I hope I got all my, OK, to keep it in mind with my notes, I should make this definition 3. So x, smooth complex algebraic variety, f, a function from the Betty moduli of x, the complex points of that, I will, most of the time, just leave out the complex points from there. But that's what I'll be talking about, to z. And you can think of z as the Betty moduli of a point, the ranks, or maybe put that into sum by direction with n. Yeah, let me write n. Doesn't really matter. Such that this is a composition of, and here's the vague part, any naturally derived maybe functor defined over q bar, and by the right functor, I mean you look at the derived category with of complexes in constructable cohomologies, the bounded, the right category of complexes to constructable cohomology and complex coefficients. Then for all k, the inverse image of k is an absolute q bar constructable set. So basically, everything that you can lay your hands on, everything you can construct with your hands will be of that type. All right, so this has actually been proven by Simpson in 93 when x is smooth projective and for some functors. So what we do here is to put all these functors in. Now in practice, we cannot really define any. So what we have, we took all the functors that we could think of, and we proved the statement for those functors. So I'll just mention, there's the usual functors, there's the shift by an integer, there's the truncation functors, the cohomology of a complex, the perverse truncation functors, the perverse cohomology of a complex, f inverse, f upper shriek, rf floor star, rf floor shriek, all those guys, nearby cycles, vanishing cycles, derived tensor products, derived r-homes, duality, verdeer duality, intermediate extension functors, all those, anything you can generate via compositions of those, fiber products of those. So yeah, it's everything that we could think of. OK, so how do you try to prove such a thing? I already dropped, if you notice from the statement, the quasi-projective part. That's because the whole thing is to extend constructability and absoluteness beyond moduli. So for example, two subsets. So where s is not necessarily a subset of the complex points of the Betty moduli, but two subsets of the objects of the derived category with the constructable sheaves, up to isomorphisms. Then show that each function you can think of preserves the notion of constructability. Once you have the notion of constructable sets, you also automatically have the notion of constructable functions and the other way around. So then they show each functor you can think of, preserves these two notions by pullback. OK, so let me tell you a bit how we do that. OK. OK, so absoluteness. So remember, now we want to extend the notion of absoluteness to subsets in here. For that is easy. We use instead of the moduli spaces, we use the Riemann-Hilbert correspondence on derived categories. So Riemann-Hilbert correspondence between the bounded derived category of regular hornonomic DX modules to the bounded derived category of complexes with constructable cohomology on X. So we can form a diagram precisely as there, but now with the derived categories instead of modulized. So then that means the only thing that remains is to define the notion of constructability, or q constructability over some smaller fields. So for constructability, here is the crucial definition. It's very naive. So a subset of objects of the derived category with constructable of complexes with constructable cohomology up to isomorphisms is called constructable. Let me skip the definitions of q bar constructable. If for every regular finite type C algebra R, and for every object Fr in the bounded derived category of complex constructable cohomology on X with coefficients in R now, the set of maximal ideals in spec R such that Fr twisted with R mod M belongs to S is constructable in the classical sense. So this played in a naive role of families for us. So let me note that if you want regular functions out of derived categories, out of categories, then you really need to talk about modulized spaces, stacks. But what we are after here, we're after constructable functions, which is less rigid than that. So for constructable functions out of categories, the way we deal with this issue, so you need something much less than that. So we develop a framework for constructing constructable functions. And we have a name for these things, unispaces. It's much less than a stack. It's just that's not very difficult. But the idea is that we have now a way to generate classical constructable functions on varieties like modulized from derived categories, which I don't think has been done before. So we have a moduli free way of generating classical constructable functions. So what's the upshot about these absolute sets? They should satisfy a lot of geometric, nice geometric things. So let me write down a conjecture. Again, this is inspired by Simpson. Simpson. Simpson. So it's pi over Simpson. Simpson. We have the following conjecture. Number five. Every absolute q bar constructable subset of the BITI moduli contains an absolute q bar point. Every absolute q bar point is of geometric origin. So it has to be a semi-simple local system. And yeah, so what we think should happen is an analog of the so-called Andre Ort conjectures for Shimura varieties, but now in this more topological setup. And there is one. So the next conjecture, maybe it's a bit speculative, but it's really, well, let me just put it as a question here. So every irreducible component of the Zaricki closure of any possibly infinite set of absolute q bar points an absolute q bar closed subset of some moduli space, BITI moduli space, but perhaps for a different variety, a different rank, which is still isomorphic over q with the one that we started with. So this is not really clear what kind of a conjecture to make in higher ranks for these things yet. Now, the semi-simple local system of geometric origin, what does this mean? So maybe I should write down. Actually, I changed my mind. I will not write down precisely what this is. Let me just say this. Semi-simple local system of geometric origin is something that is obtained from the constant shift on a point via usual operations. And I could list the operations there, but let me not try to do that. We all see not some point this definition, I think. I would like to call it, instead of geometric origin, I would like to call it of constant origin. We'll see why later, because I want to say something about non-constant origin. Now, this conjecture here, if it's true, will give a very easy conceptual explanation why the full decomposition theorem holds if you know it for the geometric local systems. So let me state the composition theorem. So if f from x to y is a proper morphism of complex varieties, l, a semi-simple, let me say a local system on a smooth, locally closed sub-variety z in x, such that, and there's two cases, then there's the first case ever proved semi-simple of geometric origin. So this is in the paper by Bellin, by Bellinson, Bernstein, Bellin, and Gauber also contributed to that. Or the full decomposition theorem is for any semi-simple local system. Then, then, RF lower star of the intersection complex on the closure of z of l has a nice decomposition in terms of the perverse cohomology sheaves of itself appropriately shifted. And moreover, each one of these guys is a semi-simple perversive. So the full decomposition theorem was proved much later after the geometric origin case by Mochizuki, Takuro Mochizuki using complex techniques and also proven by arithmetic techniques by Berkeley, Kare, Dreamfelt, and Gaetzguri. So that's by arithmetic techniques. So what does our conjecture do in this case? What would it do? I forgot which one was which. That's all. So here's an easy result. Our conjecture, in fact, the first two sentences there in the conjecture only plus the decomposition theorem for the geometric local systems implies the full decomposition theorem. And the proof is easy. You can rephrase this condition here by some sort of vanishing between the x between these perverse sheaves. And that is a combination of funtors from our list. So what do you do? The good set of local systems on z such that the dt holds for it is absolute q closed. In fact, it's even defined over q, but that doesn't really matter. Also, q constructive. In fact, it's q closed. And it contains, by the assumption here, and the conjecture, and it contains all geometric points. By the conjecture, the geometric points are the absolute points. So it contains all absolute q bar points. It contains all of them, so it has to be the full space. So yes, it's part of the first. So for this, you only need the first two statements. Yeah, by the conjecture. OK, so you only need actually the first two statements in the conjecture for this. So that's the trick with this. OK, so I have to give some theorem now, because these are only conjectures so far. So the conjecture holds for rank 1. And case is when the rank 1 basically gives all the ranks. So complex affine, tori, or a billion varieties. So this is with or here. So as an immediate corollary, you get a new proof of the full decomposition theorem, let's say, of the z decomposition theorem for all semi-simple local systems of this origin, namely instead of constant shift at the point, take any of the local systems allowed here, for example, rank 1 local system on any smooth variety, and then apply the usual operations as in the definition of geometric points. And for those, you get the new proof of the composition theorem of star origin, assuming only BBDG. Now, the reason why we're able to prove this for rank 1 is because we have a good description of these sets in rank 1. So theorem 7 follows from the following one. If x is a smooth complex algebraic variety, s, a subset of rank 1 local systems on x, that is absolute q bar constructable, then s is obtained from finitely many torsion translated half-mysoptorii via intersection, union, and taking complement. OK, so how many minutes I have left? 0? 15. Great. I can give no examples of absolute sets of rank 1 local systems. So let me see. Enjoy. So by this theorem, anything that I can construct in rank 1 will be of that type, coming from torsion translated sub-torii. So here are examples. So these are examples of absolute sets in rank 1. The first example is set as 0. We do the following. We look at rank 1 local systems. So let's say we're in the situation where we have a function f, a regular function. And we fix some point x here in the 0 locus of f. So we're looking at local system on, let's say, f of x goes to 0. We're looking at rank 1 local systems on the base here. Of course, that's the same as given a complex number, no 0 complex number. Such that, and I'm going to write a bunch of functors here, h, the cohomology of the stock of the nearby cycle of f applied to l to the pullback of l is not 0. Well, this is a composition of functors that I wrote there some condition. So this is, in fact, absolute q closed. I will not write absolute anymore, but this is absolute. So in particular, so these are roots of unity here. So actually, this is the same as the monodromy theorem, which states that the eigenvalues of monodromy on the cohomology of the miller fiber of regular functions are roots of unity. So you see that this statement is some kind of multi-phase of the generalization of the monodromy theorem. So this is equivalent of the monodromy theorem. So what I'm trying to say is that this statement here, together with the one here that I'm erasing, generalizes many of the partial results that were present in the literature up to this point. So q closed means I think some new finite unit forms, or what is it? Yes, well, it is. Why? I mean, how do you do it? It's a closed subset of the modular space, and a closed subset of c star is just a closed. A closed? Most, yeah, so most of, yeah, that's a good question. That's a good question. Most of the functors that we deal with in this list of many functors that we have are actually natural transformations between categories. And that is what helps us with the closed condition. I'll tell you what functor is not preserving, is not going to give us closed condition. Is anything that has to do with perverse truncations? For example, the intermediate extension functors that will not preserve, that will not be, that's something I will write down immediately, an example like that. In the genotes, Q closed, why does a theorem use a Q closed? Well, the modular space is just c star. Yeah, but there is no group spatial, no, it's a non-linear. It is absolute Q closed. And absolute, by the main theorem here in rank 1, is coming from finite limited torsion translated sub-tory. There's no sub-tory to talk with, so just a torsion points in c star. Torsion points in c star are roots of unity. So that's the only thing that it sees. So I just wanted to point out that the classical monodormy theorem is a particular case. Another particular case is the following subset of rank 1 local systems such that the dimension of some cohomology is greater or equal than a fixed number k. Again, this is Q closed, absolute Q closed. So this will be a finite union of torsion translated sub-tory. So this was the older, so this is, remember, X is smooth quasi-projective in mic, smooth, not necessarily quasi-projective. So that's a generalization of Simpson's theorem that we've seen before. But we can go beyond this. We can also tensor by something. Let's say L tensor of fixed complex here. In the statement that I erased, I said that any functor defined over Q bar. So by that I meant, for example, if f is a local system that is not defined over Q bar, then that condition will not be satisfied. But let's assume, for example, let me just say what they should be. You get Q closed here if f is, for example, underlying a mixed-hodge module of geometric origin. If f is just a semi-simple perverse sheaf of geometric origin, then you get Q bar closed. But if it's underlying a mixed-hodge module of geometric origin, which means whatever you get from the constant hodge module on a point by all those operations, then you get actually something defined over Q. And that is nice because now you have a Galois of Q bar of over Q action on this. So you get the corollary. You get the following duality kind of statement that is very concrete, that the Ithcohomology on x of a rank on local system tensor of such a complex is the same as the dimensions as the Ithcohomology of the dual of L tensor f. Another statement, and this was something that was not, Simpson's method could not handle the following statement, is intersection cohomology. So jump plus I for intersection cohomology. So L rank once such that dimension of the intersection Ithcohomology on some x bar, and I have to write what that is, of L tensor f greater equal than k. Now again, I'm going to take f as here for simplicity. So this is Q constructible. Here we cannot put close. That's definitely not true. So here x bar, so you have an embedding of x into x bar. This is just some locally closed embedding. So again, by this theorem in the corner, this comes from finite limit torsion translated subtotal, but now you have to take, you have to allow complements as well. But again, you obtain a concrete statement, which I think is new. I have not seen it anywhere. So I have some kind of duality like before. So this generalizes S2 that the dimension of the Ith intersection cohomology group on x bar of L rank on local system tensor f as before is the same as the dimension with the dual of L, or the inverse. So that's a concrete statement. So this statement seems new at the time when we proved it, even for the case when f is the constant shift on, well, in that case, there's not much to say. Anyway, that's still new. Another example is we consider the following setup, x, the complement of some simple normal crossing divisor in a smooth variety. Actually, the complement of any divisor, it doesn't need simple normal crossings. So it's a hyper surface. So this is open embedding. And then you look at the rank on local systems on x, such that the length of the push forward is prescribed, rf lower star. n is the dimension of x. That's a perverse shift, rf lower star of that will also be perverse. And you ask about the length as a perverse shift. That's an notoriously difficult thing to compute the length of a perverse shift. But if we put them all together like this, turns out we can say something about it. This is a q-constructable set, absolute q-constructable set. So again, we have this characterization for this. This was not initially in the long list of functors, length here was not initially in this longest of functors. So that we had to do some work to prove constructability with. So this is a result with, sorry, j. Yeah, this is a result with gatti, lu, and wang. So this is just a sample of what we should dare to write down. But again, any combination of things will work. For example, another set will be a set of rank on local systems such that, let's say you have a map from your variety, smooth variety to another smooth variety. You consider the Leray filtration on the cohomology of your local system here, coming from the Leray spectral sequence. And you put those, again, you prescribe some dimension for the Leray filtration, so for a certain piece of the Leray filtration on a certain cohomology. Again, those will be absolute q bar constructable. And here is another new result for rank on local systems. Let me state it here. With this, I'll finish. With the same proof as for theorem 6, you can get the following. Corollary 9. Let's say now j, x is smooth. j is an open invading to a smooth compactification. So x bar is projective. And now the complement is indeed a simple normal crossing divisor. Then for every rank on local system on x, we have the Leray spectral sequence, which computes the cohomology of L. And that Leray spectral sequence, with E2 term, HP x bar rj lower star L, rjq lower star L, converging to the cohomology of L, degenerates at E3. And the reason is that this statement has been proved by the lean using mixed-hodge structures for the case when L was torsion, meaning a rank on local system of geometric origin. But now, again, by the conjecture 5, which is a statement, it is a theorem for the rank 1 case. And by the same proof here, you get the same statement. Because this condition here of the generation again becomes a mixture of all the factors that were used with. So it seems like this statement is also new according to the lean. And the proof of this, again, does not involve anymore mixed-hodge structures. That's all. Thank you.
 Thank you very much for the invitation to come here.  It's always a pleasure to come back to Lumini.  Let me state what the general theme of the talk is.  So you've seen this before.  Modular spaces of local systems on complex algebraic  varieties should have interesting geometry.  So there are thousands of talks given on the subject papers  written about it.  But I couldn't resist to make a bit of propaganda for myself.  Let me give a new result in this direction.  That doesn't really have anything to do with the title here.  But here's the first theorem.  So let's see be a smooth projective curve  over the complex numbers of genus greater or equal than 2.  Then the space of representations  of the fundamental group at the fixed base point into GLRC  is a variety with at most rational singularities.  So in fact, we know the fundamental group  is so we can write this more explicitly.  This is the complex points of this.  This is an affine variety.  The complex points are given by g tuples, g  couples of matrices xi, yi in GLRC,  such that the product of the multiplicative commutator  of each pair is the identity matrix.  So the new part here is the rational singularities part.  It was proven by Carlos Simpson in the 90s  that this is indeed a variety.  In fact, it's normal and complete intersection.  So it also has this bound on the complexity  of the singularities.  Now, why is this interesting?  This provides an example to this general theme  because of the following corollary that is completely  group theoretic.  So this implies the following theorem.  If n is greater or equal than 3, then there  is a real constant c positive, a natural number m0,  such that for every rank r greater or equal than r,  sorry, this is a rank here, r0.  For every rank r greater or equal than r0,  the number of isomorphism classes  of irreducible complex representations  of rank of dimension smaller or equal than r of,  and this is the surprising part, slnz is smaller  or equal than cr squared.  So this is completely group theoretic.  We know if you replace z here by a finite field,  we know by the Linn-Lustig theory  how to address the irreducible representations.  For slnz, we actually don't know how many there are.  We know by Margulis' rigidity that these are finite many,  but we don't know exactly how many.  So this is currently the best bound.  And in fact, the step from here to here  is not due to myself, it's due to Eisenbad and Avni,  who's so they also done the following bound before,  instead of 2 here, 22.  So it turns out that representations  of fundamental group, even for smooth projective curves,  is very much related to group theoretic statements like this.  So that's an illustration of the general theme here.  Now today's theme is different.  Today's theme, and I would like to phrase it as follows.  Not only the modular spaces of local systems  are interesting, but any other naturally defined subset.  So any here is in quotation marks.  Any naturally defined subset of modular spaces of local systems  should also have interesting geometry.  Or perhaps even better, today's theme that also relates  with this is try to prove results that are a little bit better  than what Simpson has done.  So again, this talk is based in trying  to generalize some results of Simpson's tool,  a more general setup.  OK.  Whoops.  OK.  So from now on, this is joint work with Botan Wang.  And it's very much related to the talk by Elena Noe earlier.  So let me start with absolute sets of local systems.  And I'll give a definition for this.  So first of all, we're going to start  with x being a smooth complex algebraic variety.  We have the variety of representations  of the fundamental group at some fixed base point,  rank R representations.  And this is an affine variety, affine scheme actually.  By acting via conjugation, we have a group action on this scheme.  And we can take the gIT quotient of this  and we get this function.  We get the gIT quotient of this and we get this so-called  betty moduli space of local systems of rank R  on the underlying analytic variety manifold.  So this guy here quotient by GLRC.  So this is an affine variety.  So this is the same as the spectrum  of the ring of invariants of the coordinate ring here.  So this is also an affine scheme.  So we'll be looking at this betty moduli space.  The complex points of this moduli space  are the semi-simple complex local systems of rank R  on idianalytic topology.  For example, in rank 1, these two spaces  are the same because the conjugation doesn't do anything  then.  And this is an algebraic group.  You can tensor two rank 1 local systems  and you get another rank 1 local system.  And in fact, this is the home from the first homology group  of x with coefficients in z to c star.  We've seen that earlier today.  So it is c star.  It is a complex affine torus, c star to the first betty  number times if there is some torsion here,  you also get the finite abelian group here.  So it's finitely many copies of a complex affine torus.  So it's easy to understand.  Here's the definition.  And this is largely inspired by Simpson.  I'll tell you later what the difference is with his definition  R.  So this is now x in addition quasi-projective  besides smooth.  We would like to define absolute sets in the betty  moduli.  So I'm looking at subsets here.  In fact, I'm looking at subsets of the complex points here.  But I'm going to skip writing complex points.  So this is absolute.  And we are taking something that is constructable.  And let's say it's constructable and defined over q bar.  So s is a q bar constructable subset here.  And we call it absolute if for every automorphism  of the complex numbers, there exists s sigma.  And s sigma should live in the betty moduli  of the conjugation of your variety.  And I would like to say that this guy should correspond.  But again, as in the talk from before,  he can't really start conjugating the topology here.  By conjugation, he destroyed completely the topologies.  So you have to do something else.  And what Simpson did is to look at the Deram version of this.  So he constructed some moduli spaces  by embedding the variety into something smooth,  projective with a complement simple normal crossing divisor.  And you look at logarithmic flat connections there.  He constructs a moduli space of those.  And there's the Riemann Hilbert morphism.  So this is all complex points here.  You look at this underlying analytic variety here.  And there's an analytic morphism to this betty moduli.  This happens to be a bihologomorphism if x is projecting.  But it's not so necessarily if x is not projected.  However, this is very algebraic here.  So everything is defined algebraically.  So now you can start conjugating by sigma.  And you end up to the Deram version of this moduli space.  So the idea now is that this guy that you follow up  via this diagram should also be cubar constructible  for all such conjugations.  So there exists a sigma, cubar constructible,  such that s sigma is rh p sigma rh inverse s.  So there are some differences with the original definition  of Simpson.  First of all, the definition of Simpson  was done in the case when x was smooth projective.  So here we are already quasi-projective.  Second of all, in that case, he also  assumed that the inverse image by this analytic bihologomorphism  in that case should also be constructible.  We don't assume this, but we can prove that in the projective  case.  That's already automatic from these conditions.  And the third thing is that Simpson also  has a condition on the absoluteness in Simpson's term.  Also involves an extra condition in terms  of the Higgs moduli space.  We do not deal with Higgs moduli spaces at all.  So that condition we don't really deal with in this talk.  The only thing is, if the rank is 1,  then that condition, even in the quasi-projective smooth case,  is actually superfluous.  And that's already some, more or less, essentially  contained in Simpson's paper.  All right, so that will be the definition of absolute sets  for us.  So I'm going to state the theorem now,  which illustrates the today's theme of the talk.  But the theorem is very hard to make precise.  So it will be somewhat vague.  I hope I got all my, OK, to keep it in mind with my notes,  I should make this definition 3.  So x, smooth complex algebraic variety,  f, a function from the Betty moduli of x,  the complex points of that, I will, most of the time,  just leave out the complex points from there.  But that's what I'll be talking about, to z.  And you can think of z as the Betty moduli of a point,  the ranks, or maybe put that into sum by direction with n.  Yeah, let me write n.  Doesn't really matter.  Such that this is a composition of,  and here's the vague part, any naturally derived maybe  functor defined over q bar, and by the right functor,  I mean you look at the derived category  with of complexes in constructable cohomologies,  the bounded, the right category of complexes  to constructable cohomology and complex coefficients.  Then for all k, the inverse image of k  is an absolute q bar constructable set.  So basically, everything that you can lay your hands on,  everything you can construct with your hands  will be of that type.  All right, so this has actually been proven by Simpson in 93  when x is smooth projective and for some functors.  So what we do here is to put all these functors in.  Now in practice, we cannot really define any.  So what we have, we took all the functors  that we could think of, and we proved the statement  for those functors.  So I'll just mention, there's the usual functors,  there's the shift by an integer, there's the truncation  functors, the cohomology of a complex,  the perverse truncation functors,  the perverse cohomology of a complex,  f inverse, f upper shriek, rf floor star,  rf floor shriek, all those guys, nearby cycles,  vanishing cycles, derived tensor products,  derived r-homes, duality, verdeer duality,  intermediate extension functors, all those,  anything you can generate via compositions of those,  fiber products of those.  So yeah, it's everything that we could think of.  OK, so how do you try to prove such a thing?  I already dropped, if you notice from the statement,  the quasi-projective part.  That's because the whole thing is  to extend constructability and absoluteness beyond moduli.  So for example, two subsets.  So where s is not necessarily a subset  of the complex points of the Betty moduli,  but two subsets of the objects of the derived category  with the constructable sheaves, up to isomorphisms.  Then show that each function you can think of  preserves the notion of constructability.  Once you have the notion of constructable sets,  you also automatically have the notion of constructable functions  and the other way around.  So then they show each functor you can think of,  preserves these two notions by pullback.  OK, so let me tell you a bit how we do that.  OK.  OK, so absoluteness.  So remember, now we want to extend  the notion of absoluteness to subsets in here.  For that is easy.  We use instead of the moduli spaces,  we use the Riemann-Hilbert correspondence  on derived categories.  So Riemann-Hilbert correspondence  between the bounded derived category  of regular hornonomic DX modules to the bounded derived  category of complexes with constructable cohomology on X.  So we can form a diagram precisely as there,  but now with the derived categories instead of modulized.  So then that means the only thing that remains  is to define the notion of constructability,  or q constructability over some smaller fields.  So for constructability, here is the crucial definition.  It's very naive.  So a subset of objects of the derived category  with constructable of complexes with constructable cohomology  up to isomorphisms is called constructable.  Let me skip the definitions of q bar constructable.  If for every regular finite type C algebra R,  and for every object Fr in the bounded derived category  of complex constructable cohomology  on X with coefficients in R now, the set of maximal ideals  in spec R such that Fr twisted with R mod M belongs to S  is constructable in the classical sense.  So this played in a naive role of families for us.  So let me note that if you want regular functions out  of derived categories, out of categories,  then you really need to talk about modulized spaces, stacks.  But what we are after here, we're  after constructable functions, which is less rigid than that.  So for constructable functions out of categories,  the way we deal with this issue, so you  need something much less than that.  So we develop a framework for constructing  constructable functions.  And we have a name for these things, unispaces.  It's much less than a stack.  It's just that's not very difficult.  But the idea is that we have now a way  to generate classical constructable functions  on varieties like modulized from derived categories, which  I don't think has been done before.  So we have a moduli free way of generating  classical constructable functions.  So what's the upshot about these absolute sets?  They should satisfy a lot of geometric,  nice geometric things.  So let me write down a conjecture.  Again, this is inspired by Simpson.  Simpson.  Simpson.  So it's pi over Simpson.  Simpson.  We have the following conjecture.  Number five.  Every absolute q bar constructable subset  of the BITI moduli contains an absolute q bar point.  Every absolute q bar point is of geometric origin.  So it has to be a semi-simple local system.  And yeah, so what we think should happen  is an analog of the so-called Andre Ort conjectures  for Shimura varieties, but now in this more topological setup.  And there is one.  So the next conjecture, maybe it's a bit speculative,  but it's really, well, let me just put it as a question here.  So every irreducible component of the Zaricki closure  of any possibly infinite set of absolute q bar points  an absolute q bar closed subset of some moduli space,  BITI moduli space, but perhaps for a different variety,  a different rank, which is still isomorphic over q  with the one that we started with.  So this is not really clear what kind of a conjecture  to make in higher ranks for these things yet.  Now, the semi-simple local system of geometric origin,  what does this mean?  So maybe I should write down.  Actually, I changed my mind.  I will not write down precisely what this is.  Let me just say this.  Semi-simple local system of geometric origin  is something that is obtained from the constant shift  on a point via usual operations.  And I could list the operations there,  but let me not try to do that.  We all see not some point this definition, I think.  I would like to call it, instead of geometric origin,  I would like to call it of constant origin.  We'll see why later, because I want  to say something about non-constant origin.  Now, this conjecture here, if it's true,  will give a very easy conceptual explanation  why the full decomposition theorem  holds if you know it for the geometric local systems.  So let me state the composition theorem.  So if f from x to y is a proper morphism  of complex varieties, l, a semi-simple,  let me say a local system on a smooth, locally closed  sub-variety z in x, such that, and there's two cases,  then there's the first case ever proved semi-simple  of geometric origin.  So this is in the paper by Bellin,  by Bellinson, Bernstein, Bellin, and Gauber also  contributed to that.  Or the full decomposition theorem  is for any semi-simple local system.  Then, then, RF lower star of the intersection complex  on the closure of z of l has a nice decomposition  in terms of the perverse cohomology  sheaves of itself appropriately shifted.  And moreover, each one of these guys  is a semi-simple perversive.  So the full decomposition theorem  was proved much later after the geometric origin case  by Mochizuki, Takuro Mochizuki using complex techniques  and also proven by arithmetic techniques  by Berkeley, Kare, Dreamfelt, and Gaetzguri.  So that's by arithmetic techniques.  So what does our conjecture do in this case?  What would it do?  I forgot which one was which.  That's all.  So here's an easy result.  Our conjecture, in fact, the first two sentences  there in the conjecture only plus the decomposition theorem  for the geometric local systems implies the full decomposition  theorem.  And the proof is easy.  You can rephrase this condition here  by some sort of vanishing between the x between these perverse  sheaves.  And that is a combination of funtors from our list.  So what do you do?  The good set of local systems on z such that the dt  holds for it is absolute q closed.  In fact, it's even defined over q, but that doesn't really  matter.  Also, q constructive.  In fact, it's q closed.  And it contains, by the assumption here, and the conjecture,  and it contains all geometric points.  By the conjecture, the geometric points  are the absolute points.  So it contains all absolute q bar points.  It contains all of them, so it has to be the full space.  So yes, it's part of the first.  So for this, you only need the first two statements.  Yeah, by the conjecture.  OK, so you only need actually the first two statements  in the conjecture for this.  So that's the trick with this.  OK, so I have to give some theorem now,  because these are only conjectures so far.  So the conjecture holds for rank 1.  And case is when the rank 1 basically gives all the ranks.  So complex affine, tori, or a billion varieties.  So this is with or here.  So as an immediate corollary, you  get a new proof of the full decomposition theorem,  let's say, of the z decomposition theorem  for all semi-simple local systems of this origin,  namely instead of constant shift at the point,  take any of the local systems allowed here,  for example, rank 1 local system on any smooth variety,  and then apply the usual operations  as in the definition of geometric points.  And for those, you get the new proof of the composition  theorem of star origin, assuming only BBDG.  Now, the reason why we're able to prove this for rank 1  is because we have a good description of these sets  in rank 1.  So theorem 7 follows from the following one.  If x is a smooth complex algebraic variety,  s, a subset of rank 1 local systems on x,  that is absolute q bar constructable,  then s is obtained from finitely many torsion  translated half-mysoptorii via intersection, union,  and taking complement.  OK, so how many minutes I have left?  0?  15.  Great.  I can give no examples of absolute sets  of rank 1 local systems.  So let me see.  Enjoy.  So by this theorem, anything that I can construct in rank 1  will be of that type, coming from torsion  translated sub-torii.  So here are examples.  So these are examples of absolute sets in rank 1.  The first example is set as 0.  We do the following.  We look at rank 1 local systems.  So let's say we're in the situation  where we have a function f, a regular function.  And we fix some point x here in the 0 locus of f.  So we're looking at local system on, let's say,  f of x goes to 0.  We're looking at rank 1 local systems on the base here.  Of course, that's the same as given a complex number,  no 0 complex number.  Such that, and I'm going to write a bunch of functors here,  h, the cohomology of the stock of the nearby cycle of f  applied to l to the pullback of l is not 0.  Well, this is a composition of functors  that I wrote there some condition.  So this is, in fact, absolute q closed.  I will not write absolute anymore, but this is absolute.  So in particular, so these are roots of unity here.  So actually, this is the same as the monodromy theorem,  which states that the eigenvalues of monodromy  on the cohomology of the miller fiber of regular functions  are roots of unity.  So you see that this statement is some kind of multi-phase  of the generalization of the monodromy theorem.  So this is equivalent of the monodromy theorem.  So what I'm trying to say is that this statement here,  together with the one here that I'm erasing,  generalizes many of the partial results  that were present in the literature up to this point.  So q closed means I think some new finite unit  forms, or what is it?  Yes, well, it is.  Why?  I mean, how do you do it?  It's a closed subset of the modular space,  and a closed subset of c star is just a closed.  A closed?  Most, yeah, so most of, yeah, that's a good question.  That's a good question.  Most of the functors that we deal with in this list  of many functors that we have are actually  natural transformations between categories.  And that is what helps us with the closed condition.  I'll tell you what functor is not preserving,  is not going to give us closed condition.  Is anything that has to do with perverse truncations?  For example, the intermediate extension  functors that will not preserve, that will not be,  that's something I will write down immediately,  an example like that.  In the genotes, Q closed, why does a theorem  use a Q closed?  Well, the modular space is just c star.  Yeah, but there is no group spatial, no,  it's a non-linear.  It is absolute Q closed.  And absolute, by the main theorem here in rank 1,  is coming from finite limited torsion  translated sub-tory.  There's no sub-tory to talk with,  so just a torsion points in c star.  Torsion points in c star are roots of unity.  So that's the only thing that it sees.  So I just wanted to point out that the classical monodormy  theorem is a particular case.  Another particular case is the following subset  of rank 1 local systems such that the dimension of some  cohomology is greater or equal than a fixed number k.  Again, this is Q closed, absolute Q closed.  So this will be a finite union of torsion  translated sub-tory.  So this was the older, so this is,  remember, X is smooth quasi-projective in mic,  smooth, not necessarily quasi-projective.  So that's a generalization of Simpson's theorem  that we've seen before.  But we can go beyond this.  We can also tensor by something.  Let's say L tensor of fixed complex here.  In the statement that I erased, I  said that any functor defined over Q bar.  So by that I meant, for example, if f  is a local system that is not defined over Q bar,  then that condition will not be satisfied.  But let's assume, for example, let me just say what they should  be.  You get Q closed here if f is, for example,  underlying a mixed-hodge module of geometric origin.  If f is just a semi-simple perverse  sheaf of geometric origin, then you get Q bar closed.  But if it's underlying a mixed-hodge module  of geometric origin, which means whatever  you get from the constant hodge module on a point  by all those operations, then you get actually  something defined over Q. And that  is nice because now you have a Galois of Q bar of over Q action  on this.  So you get the corollary.  You get the following duality kind of statement  that is very concrete, that the Ithcohomology on x  of a rank on local system tensor of such a complex  is the same as the dimensions as the Ithcohomology  of the dual of L tensor f.  Another statement, and this was something that was not,  Simpson's method could not handle the following statement,  is intersection cohomology.  So jump plus I for intersection cohomology.  So L rank once such that dimension  of the intersection Ithcohomology on some x bar,  and I have to write what that is, of L tensor f greater  equal than k.  Now again, I'm going to take f as here for simplicity.  So this is Q constructible.  Here we cannot put close.  That's definitely not true.  So here x bar, so you have an embedding of x into x bar.  This is just some locally closed embedding.  So again, by this theorem in the corner,  this comes from finite limit torsion  translated subtotal, but now you have  to take, you have to allow complements as well.  But again, you obtain a concrete statement,  which I think is new.  I have not seen it anywhere.  So I have some kind of duality like before.  So this generalizes S2 that the dimension  of the Ith intersection cohomology group on x bar of L  rank on local system tensor f as before  is the same as the dimension with the dual of L,  or the inverse.  So that's a concrete statement.  So this statement seems new at the time when we proved it,  even for the case when f is the constant shift on,  well, in that case, there's not much to say.  Anyway, that's still new.  Another example is we consider the following setup, x,  the complement of some simple normal crossing divisor  in a smooth variety.  Actually, the complement of any divisor,  it doesn't need simple normal crossings.  So it's a hyper surface.  So this is open embedding.  And then you look at the rank on local systems on x,  such that the length of the push forward  is prescribed, rf lower star.  n is the dimension of x.  That's a perverse shift, rf lower star of that  will also be perverse.  And you ask about the length as a perverse shift.  That's an notoriously difficult thing  to compute the length of a perverse shift.  But if we put them all together like this,  turns out we can say something about it.  This is a q-constructable set, absolute q-constructable set.  So again, we have this characterization for this.  This was not initially in the long list of functors,  length here was not initially in this longest of functors.  So that we had to do some work to prove constructability with.  So this is a result with, sorry, j.  Yeah, this is a result with gatti, lu, and wang.  So this is just a sample of what we should dare to write down.  But again, any combination of things will work.  For example, another set will be a set of rank on local systems  such that, let's say you have a map from your variety,  smooth variety to another smooth variety.  You consider the Leray filtration  on the cohomology of your local system here,  coming from the Leray spectral sequence.  And you put those, again, you prescribe some dimension  for the Leray filtration, so for a certain piece of the Leray  filtration on a certain cohomology.  Again, those will be absolute q bar constructable.  And here is another new result for rank on local systems.  Let me state it here.  With this, I'll finish.  With the same proof as for theorem 6,  you can get the following.  Corollary 9.  Let's say now j, x is smooth.  j is an open invading to a smooth compactification.  So x bar is projective.  And now the complement is indeed a simple normal crossing  divisor.  Then for every rank on local system on x,  we have the Leray spectral sequence,  which computes the cohomology of L.  And that Leray spectral sequence,  with E2 term, HP x bar rj lower star L, rjq lower star L,  converging to the cohomology of L, degenerates at E3.  And the reason is that this statement has been proved  by the lean using mixed-hodge structures for the case  when L was torsion, meaning a rank on local system  of geometric origin.  But now, again, by the conjecture 5, which is a statement,  it is a theorem for the rank 1 case.  And by the same proof here, you get the same statement.  Because this condition here of the generation  again becomes a mixture of all the factors that were used with.  So it seems like this statement is also new according  to the lean.  And the proof of this, again, does not involve anymore  mixed-hodge structures.  That's all.  Thank you.
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