Thank you so much for inviting me. It's a real joy to come here on such a wonderful occasion.

Also, since I'm the last one, probably should thank the organizers for their wonderful job. And it was absolutely great to be here. Well, anyway, so let me, I will talk about a tight topology

situation. But let me remind first the theory that was developed by Kashiwara and Shepera already 30 years ago. And their book shifts on manifolds.

So here is the situation. Let X be a complex manifold of dimension n.

And suppose that I have a constructible sheaf, or sheaf for me will mean complex of sheaves, well, with some coefficients.

Then here is the definition. The singular support of F, it is a closed subset in the Cartangent bundle to X. And it is defined as follows.

It is the smallest subset, which is the size

of following condition. You will have any pair U and F, where U is an open subset in X, and F is a holomorphic function on U, such that if I consider the differential of F

and consider its graph, then it does not intersect my closed subset.

So if I have such a pair which satisfies this condition, then F is locally acyclic relative to F.

Locally acyclic means that if, well, I assume that the definition is known, but this means that if I will compute vanishing cycles, there will be no, there will be coefficients in F.

They will be trivial. So that's the definition. And it's more or less, so this is that, well, it's easy to see that this is a conical complex sub-variety in T star X. And moreover, it's

easy to check that the following two completely the check properties that the singular support is empty.

This is the same as F equals to 0, and singular support equals X. And when I write X as a subset of a tangent bundle, this means that it's just a 0 section. Then this means that F is a locally constant and non-zero.

Okay, now a theorem. A theorem they prove is the following, that it is that,

well, it's complex sub-variety. We can look at its irreducible components. And the claim is that all irreducible components have a dimension n.

Okay, so it's evident in these two examples, but it's true in general. And in fact, they show that the singular support is Lagrangian.

Well, it's sub-variety, it can have singularities, but on its open part, it is Lagrangian. And since it's also a conical subset,

this means that actually it is every irreducible component of singular support is the conormal bundle to some closed sub-variety in X. By conormal bundle, I mean that on smooth part,

you take the usual conormal bundle, and then you take the closure.

Well, now, so that is their first main theorem in this situation. And here is another theorem that let me write it maybe on the second board.

Now, suppose that my coefficient, well, is a field. Well, then the claim is that,

well, you have a collection of sub-varieties of middle dimensions, like attention bundles. The claim is that one can naturally assign to every, it's a reducible component, a number, an integer, so which will make it a cycle,

which is called characteristic cycle.

It's denoted in this manner, so that the following properties will hold. So the first property is that suppose that we have a pair U, F as before,

so we have an open subset and a homomorphic function on it. But before, we considered situation when differential of F did not take value in the singular support. But now suppose that it intersects the singular support

at a single point. Well, then we know that by the definition

of singular support, it's true that our function F outside of X is locally acyclic, so there is no vanishing cycles. And so this means that if I will compute the vanishing cycles for F, then it will be skyscraper at X.

And so I can compute as dimension. Well, dimension earlier, it's complex, so it will be earlier characteristic, but let me write dimension. Well, and the claim is that the following formula for this dimension holds. I should put here the sign minus, and then here will be the local intersection index of,

we have DF, which is a section, so we have DF of U. It's a sub-variety in the cotangent bundle. And here, so we consider it as a cycle, certainly with multiplicity one. And here, we put the characteristic cycle,

and we, well, so we have two cycles which intersect by one point, so there is the notion of local intersection index at that point, and the claim, that it will be exactly the dimension of vanishing cycles. Okay?

You don't claim that the multiplicities are, they could be positive, negative, or zero. You don't claim anything. I don't claim anything. Well, I will claim anything, but in a short while. It's an equality or an inequality? It's equality. It's equality. Well, so the second assertion is global.

So assume that X is compact. Then, and let us compute, let's consider the earlier characteristics of X with coefficient in F. So this means that it's,

we compute the cohomology of X and compute the earlier characteristic. And the formula says, the following, that it's just the intersection index of X and the characteristic cycle.

So here is the situation. The cotangent bundle itself, it's certainly non-compact, but since X is, you have zero section, and X itself is compact, then certainly you can intersect X with cycle of complementary dimensions. This is well-defined integer

in the claim that you have this equality. Well, now the last property is the answer to author's question. And let me put it here.

That if F is perverse, if F is perverse sheaf, then a characteristic cycle is effective.

Find it for a usual sheaf, so the same works for a bounded complex? Well, yes. What do you mean the same calls for bounded? You consider the condition that the universally locally acyclic

in the sense of the vanishing cycle are five things. Yes, in the sense of vanishing cycle. Okay, but you can do it for either F to be a single sheaf or an object in the derived category. For me, sheaf means object in the derived category. But is it, but you don't claim that what you get is, okay.

Singular support is a just subset, no coefficients. Characteristic cycle has coefficients, and you can have singular support huge, but characteristic cycle being zero. For example, take any F, shift it by one,

and take direct sum. Singular support will not change, but vanishing cycle will disappear. Sorry, characteristic cycle will disappear. You wanna say that this cycle is uniquely determined by F?

Yes, uniquely determined by F, so that I will say in a moment, okay? So this first condition actually completely determines characteristic cycle. Well, it's clear that if you have any components, then you can choose, you can find a function

whose differential intersects the thing at its most part, and then this formula will produce you, immediately produce you the multiplicities. What is? And not only effective, but it's strictly positive. Yes, that I did not, yes. And moreover, it's strictly positive,

so the support of CC equals the singular support. Okay, so one comment is that

certainly characteristic cycle depends, well, it depends on the shift in additive manner. So it's a homomorphism from the K group of, K zero group of constructible shifts to the group of cycles.

On the Cartesian bundle, that follows immediately from this formula, formula one. Well, the second thing is that let's consider example when my shift is constant.

Then, as I told you, the singular support, it is a zero section, and by the last property, you see that,

well, equals X, and maybe, let me write the characteristic cycle equals minus one power N times X. Well, and then let us see.

So this first formula then, it is exactly Milner's formula. So what stands to the right, it is exactly the Milner's number of the singularity of F. So this X means that it's critical point. So if it's intersect zero,

and here stands the Milner's number, and here stands the dimensional vanishing cycle, and minus has to do with it because they normalize it to be positive for perversion. Usual one that you can forget. And similarly, the second thing is this, the formula of Euler characteristic.

Well, here it tells you that it's self-intersection of the zero section of the Cartesian bundle, and again, because of this sign thing, this is equivalent to the standard formula that's Euler characteristic of many fault of self-intersection number of the diagonal. So, at least in this situation,

everything is perfectly fine. So one last remark about Kashiwara-Shapiro is that their proofs are very transcendental. For example, they actually, the theory they develop, it works on real analytic manifolds.

And if you consider real analytic manifolds, then you can refine any real analytic stratification to just decomposition by simplices. And in such situation, constructible shifts are easy, and so you can work it.

But it's extremely transcendental operation, and you cannot just push it to a tall situation. Just one question about formula two. The right-hand side is the Hodge collage, right? Well, if you count x, the intersection number of x with a zero cycle,

with itself in the Cartesian bundle, that gives you the Hodge numbers, right? Well, it gives you one number. Pardon? No, it's just usually the correct term. No, it's just usually the correct term. No, it's just usually the correct term. The problem with d-module, it was... Yes, oh, I should...

The first time I heard about this was Walensky. Yes, yes, yeah. Yes, I should certainly have told that their story was completely mitigated. It came not from topology at all, and not from complex geometry, but from the theory of d-models. And the theory of d-models,

the notion of singular support of d-models, that's one of the very first notion, and basic technical notion, how you just develop the theory of d-models from the very beginning. And the notion, well, it is, there is standard, the basic fact is that singular support,

you can define for any d-model, but it will have dimension larger than n, and those d-models which have, for which singular support has dimension n, it is a basic geometric object. Those are Halonomian d-models. And the formula multiplicities, again,

are also defined just from the very definition, like multiplicities and commutative algebra. And the formula, this global Euler characteristic formula in the RAM settings, so for d-models, it is, I think it's, it is due to Brilinsky, Dobson, and Keshavar, if I'm not mistaken.

But I should stress that in this definition, it's, proof is extremely simple. It is one line theorem. And compared to quite complicated proof that you do for constructible shears,

both definition, actually, proof, but it's, again, it's one line proof, but it's extremely non-mattific. It's just the structure that you deal with d-models and not with geometry. And they also proved that when you have, when the coefficients are characteristic zero,

then it corresponds to the d-module characteristic cycle. Yes, they proved that corresponds to d-model characteristic cycle. Sorry, I should not have omitted the history, but anyway, further on, I will omit also other parts. So, well, and not so much of time.

Now, the basic question that the story I will be talking about is the question if you can just, if similar assertion hold in a tight situation. And if you look at this definition,

everything, well, suppose that we work now with an algebraic variety over some field, then you can see that everything in principle makes sense, so the definition. There makes sense and you can, all the statements of the theory make perfect sense.

And in fact, so what I will be talking about is yes, that it's okay, but with minor modifications. So let me do the minor modifications first and then I will discuss the story. So here is the minor modification. So let's consider first the case of constant sheath.

And then certainly, then the Milner's formula. Well, it's known that it's true, but you should, in case of, it's true in case when the base field has zero characteristic, but if it has finite characteristic, it should be slightly modified.

And this, namely here instead of dimension, there should stand total dimension. And the word total means that, well, it's a vector space in which the Galois group of the disk down below, for a punctured disk down below X, and in case of finite characteristic,

you can have wide, well, dramatically, and total means that you add to dimension the term, this one conductor. And the formula in, the corresponding formula for constant sheath, it was approved by Dalin in the second volume of SJ7.

There is his talk, which is exactly called Milner's formula. So, well, so that's one modification that should be made. And another modification I should make

on the left board, and this is, okay. So, this assertion is false. And as for the rest assertions, yes, they're all true.

And so what is done on the left blackboard, well, there is my note on archive that proves this theorem, and what is on right blackboard is proved in a preprint of Takayashi's site,

which is probably also in the archive, or it will be the archive in the nearest future. Okay. Now, maybe before I will continue the talk,

let me comment about this situation, why the thing is not Lagrangian. And it is the fact that you really just, the Lagrangian property must disappear in characteristic p. I think it was, well, at least I first heard that from Dalin long ago back in Moscow.

But, and somehow, at that moment, you think that there is no theory, and stop thinking about this. Well, anyway, so let me produce an example. Well, maybe first I need notation,

since it's convenient, and I will also use it afterwards. So suppose that you have a map between smooth algebraic varieties, which is proper.

And suppose that I have a conic sub-variety inside of the cotangent bundle to y. And then it yields, in a pretty standard way,

a conical sub-variety in the cotangent bundle to x, which I will denote in this manner. And by definition, it consists of all points

in the cotangent bundle, so what will be x, and co-vector nu at x, such that there exists y with the property that it lives in the fiber, and such that df at point y applied to nu will lighten c.

There is a standard way to push forward the conical subset by the proper map. And a small remark that it follows directly

from the definition that if I have a shift g on y, well, let me denote by d of y the category of...

Pardon? Ooh, ooh, r, sorry, sorry, sorry. It's r. Thank you, thank you so much. Okay, that if I have a shift on y, then the...

If I consider its direct image, this will be a shift on x. And if I compute its singular support, then it lies inside of the image.

Much in that sense of the singular support of g. So this thing gives you an upper estimate for the singular support. One small remark that this upper estimate can be clever in the sense that if r is closed embedding,

then you have a quality. But on the other hand, it can be extremely stupid. For example, if r is Frobenius map, then this produces you the hook attention bundle. And so it just tells you nothing.

In the case of characteristic zero, it always tells you something. But in characteristic, we know. Well, now example.

Let's consider a map, just r, y and x will be for us just the coordinate planes. And I want that it will be, that the map actually depends only,

well, that the second coordinate would not change. So it will be given by a formula. Well, let's denote the coordinates here as t and y. And here is x and y.

And this will be, say, g of t, y. So let's consider such a transformation. And my shift will be just a direct image of the constant shift on the A2.

Let's consider just the shift here. And even in such a situation, you produced all possible things that cannot happen in characteristic zero. So let's consider the stupidest example

where g of t, y equals t power p plus t, y squared. That just absolutely easiest example. Now what you see? So first you see that if y, now what holds?

So first, what properties? So first, certainly R is finite. A second property, if y is not zero, so outside of y equals zero, the thing is a tile.

Now let's look what happens over the axis y equals to zero. Yeah, you're assuming p is not equal to? Oh, p is correct, I don't care.

I don't care. You mean for a tile thing? Well, you compute the differential and the differential will be this you can forget about and here will be y squared times dt plus something

and there will be dy and so it's invertible. Okay, but now what happens on the axis? On the axis, it is very strange thing happens. You have dr, you have the map dr

and this map, that's the following. If I consider the vector along the axis dt, then it sends it to tangent vector. Dt is sent to zero. Okay, you differentiate.

And then dy, well, dy will be sent to dy. So it is a very strange map on this axis. Again, it's differential along the axis itself equals to zero, but in the normal direction, it is a density. Now, if you apply this estimate,

certainly f itself, it's not local. It's not local system. It's not smooth error simplification and so on. So therefore, it's singular support will look as follows. So there will be zero section. Well, because it's local system outside of the open thing and plus something and there must be something else

because it's not smooth and there's something else come exactly if we apply this estimate. And if you apply this estimate and look at this formula, you immediately see that, so it equals x, so the zero section, times c,

where c is a cone over the axis y equals to zero generated by dt, dx, sorry.

And so you see that it's absolutely not Lagrangian, okay? So that's all for this example.

Well, now maybe I should, let me formulate a little extension of this theorem which is due to the link. And that's the assumption that if we consider

many faults of dimension two surfaces, then absolutely any conical subset of dimension two in the Cartesian bundle can be realized as singular support of some constructible, as a reducible component of singular support of some constructible shift.

So it's a theorem of non-integrability of characteristics. Well, now I want to discuss, I will not discuss proofs.

So proofs here are not difficult at all. But I want to show a part of the story, and this is a part of the story that explains how you see what singular support is. Certainly, singular support, for example, it's an interesting invariant. It's, well, you have some conical things,

but you don't know how to see them basically because, well, testing by functions, it's not a pleasant thing to do. But now I will describe how one can actually see singular support and, for example, to see that it has right dimension.

Well, so everything, certainly singular support has local origin, so I can assume that, and also, as I told you, that if you embed something by closed embedding, then it transforms in an evident manner.

So I can assume that I live on the project of space. And I will use two, in order to show how the thing looks like, I will use two tools. One is Brininsky's Radon transform,

and the second is Veronese embedding. So let me just recall momentarily what the Radon transform is. So we have my P, we have the dual project of space,

and we have the standard correspondence Q. Well, and the Radon transform, it is, so maybe before Radon transforms,

then you see that in this standard diagram, there is a canonical identification of the projectivizations of a tangent bundle to P and to P check. Namely, both of them canonically identified with Q.

And that's a very classical thing, and essentially evident, because what is a point in, say, in projectivization of the tangent bundle to P?

Well, it is a point in P, and a hyperplane in the tangent space to this point. And certainly, since we live on projective space, a hyperplane in the tangent space

extends uniquely to a hyperplane in the whole space, passing to point. And so we get a point, projective space, and a hyperplane passing through it. And that's an element of Q, that is this identification. Well, same manner here. Well, this thing is called, by the way, Legendre and transform.

And now you have the Radon transform. It is a function R from the category of shifts on P to those in P check, which is given by this correspondence.

Well, it has some wonderful, easy standard properties, but I will not discuss them. Well, they're used in the proof, but I will not discuss them now.

But one thing that is very easy to check is that this identification, in some sense, it's classical approximation to the Radon transform, which means the following, that if I have a shift F here, then I can compute, let's take its singular support.

That's a cone here, so I can consider the corresponding projectivized, its projectivization, which we'll just have a right here. And on the other hand, I can do the same thing for the Radon transform of F, okay?

And the claim is that they're the same. That's a simple fact, but I do not have time to describe it. Instead, let me pass to the story that I want to have.

Now, certainly just playing with Radon transform help you nothing. Well, if you don't know what singular support of a shift is, then you don't know what, just a Radon transform does not help immediately,

but it helps after the very nice embedding. So let's do the following. Let's consider an embedding of my projective space. Let's call it now small projective space. A very nice embedding, well, of some degree, more than one, any degree. And then I will do Radon transform on this larger projective space.

So I have P, let's embed it. So this will be very nice embedding. And then I consider the larger projective space and the Radon transform on this larger space. Okay, well, now notation.

Suppose that I have a cone C inside of T star P. Well, then I can consider its image by I, extend it to a cone here, well.

And then I would like to take its projectivization and just notation will be that I will denote it by C in square bracket.

And this thing leaves here, so here and here. Okay, well, that's the first notation. And second, so we have F now which leaves here.

And what I want to do is to apply all this functors. So I consider the Radon transform of I star F. So this is a sheaf here. And let's look at this ramification divisor.

And ramification divisor means the following that I just restricted to the generic point. So there I have local system. And then local system extends as a local system where it can extend and where it cannot, it's a ramification divisor.

Ramification divisor of, well, so I consider it, so I need to define it, I need to know the thing at the generic point of my projective space. Good.

Now the theorem. So when you say divisor, do you mean multiplicity? Oh, no, no, no, no, just plain subsets. So it's subset of dimension one with no multiplicities.

F is any sheaf on P, on small p. No, but do you take the ramification divisor of the Radon transform? Yes. So I, it is not derived out of A. I take I allow star F, then I take R being Radon and not right derived.

I am sorry, there is no right derived functor in this notation. F is a complex of sheaves. Yes. Well, now the theorem. So in formula, it is the way how you reconstruct

from D, which is somehow a visible invariant of F, that you can reconstruct the singular support of F. And it tells you, this in particular, tells you that it has right dimension. Okay, so the first assertion is that D itself

can be recovered from this thing. Namely, it's just the image of C in square brackets. Oh, sorry, sorry, sorry. So now, from now on, let's put C equal

the singular support of F, okay? Then, D equals image of. Excuse me, I think the slide, the third line from the bottom.

That? The bottom, yeah. So there is some symbol. So it's I not C, C, P star, P tilde. No, it's contained. It's contained, a subset of P star. Subset, sorry. Ah, subset, sorry. And then, that is P upper star?

No, the next is projectivization. It's projectivization of the cone. So I have C, which leaves over small p, and I extend it in a standard way of the cone, and then I projectivize it. Maybe you write two tilde there on the left.

Ah, yes, absolutely, thank you. Well, here, these notations were without tilde, but okay. Now, so that is the first thing. Second, the second assertion is, moreover,

so my D has different irreducible components. And C has different irreducible components. And the claim is that, in this manner, they correspond one to another. Namely, that for every irreducible component,

D alpha of D, there is a unique irreducible component, C alpha of C, such that D alpha equals,

so basically that this verna of a thing, and then you do Legendre transform, and somehow spreads components that could, here, they could project to something the same and position in P. But when you do the thing,

they will project to absolutely different D. Is this the effect of the verna's embedding? Yes, that's the effect of verna's embedding. That's exactly. In the verna's embedding? Hmm? Any? Pardon? Any of degree more than one. Identity is not allowed. And of course, you shouldn't allow

the zero-dimensional objective space. Then I already pointed out that this, then you don't tell. Probably. Well, you don't acknowledge it much, but then you cannot, there's an embedding, so you cannot talk to it. Zero, yes, okay.

And now three is that actually, this condition two, it uniquely defines C alpha. So that C alpha, in fact,

C alpha is a unique cone of in T star P of dimension, of dimension N with that property,

with property two. Okay, and maybe I should say also, well, maybe I should say also property four,

that the map, this projection from C alpha to D alpha is generically ready to show.

You are in the small projective space of the big one. Pardon? C alpha is in the small or in the? No, no, no, no, no. I am sorry, so here should be bracket. I am sorry. No, here is, here, thank you, okay.

So the thing is generically ready to show. In classical situation, classic, oh sorry, in characteristic theory situation, in fact, this is, the map is birational,

well, certainly it's birational, and also this thing is, well, it sits in the catangent bundle, projectivization of the catangent bundle to P tilde, yes, and D is divisor there, and this is projectivization of the canormal to the alpha but in case of characteristic P,

in case of final characteristics, this absolutely does not need to be, it needn't be true, so since it needn't be Lagrangian, the singular support, well, but somehow you can recover it from the alpha. What about multiplicities? What about multiplicities? About multiplicities, a little bit later.

I, okay, so I will, well, maybe the question is that I would very much want to know how to recover the thing as geometrically as possible, even in case of projection of surfaces, finite projection of one surface

to another direct image of constant sheaf, so here we were lucky that we could recover it by the stupid situation, but even just iteration of those two things of degree P, it will lend you to a situation that you just cannot recover it from geometry, I cannot, but probably some deeper, deeper geometry.

Véronésé, just to avoid linear something, the things which are linear, which will break the... No. If your sheaf at the beginning has nothing, no locus of ramification, which is linear. No, no, no, no, no, no, you see?

You see, if you have identity, identity, identity map, then the thing is just indices of bijection between quadratic cones, so if you have absolutely any quadratic cone, it need not have image, its image need not be divisor,

it need not be radial over its image, and so on. You can just take anything and then go back and produce the corresponding sheaf. So Véronés does something very drastic. Okay, now let's pass to characteristic cycle.

Well, at Takesh's work, it is, well, it's subtle and it uses many other inputs. So this story is pretty rough and elementary, and I cannot discuss it just because of the absence of time,

but instead of it, I will try, well, there is something to be done there yet. It's not all the story. And one thing that comes in Takesh's story is that characteristic cycle in what he can do, it has not integral coefficient,

but there could be powers of P in the denominator. And that's for the reason that components of C can be purely inseparable over its image and the multiplicity in the intersection in Milner's formula will have unavoidable powers of P.

So that's one thing that one would like very much to do. Now, what I would like to say is sort of, well, maybe hoped for formulism that would explain the story. Also, I hope it will provide understanding of things like you can consider for global intersection formula for the characteristic.

You can ask for finer things. So for example, to compute the determinant of cohomology. And I would like to have just the story simultaneously and to have a definition of characteristic, some finer definition of characteristic cycle, which would answer also the second question.

And that would not involve in itself this proof that it does not depend on the choice of functions in Milner's formula and so on, but somehow Milner's formula will be just corollary. So let me try to put this in the remaining minutes.

Let me try to put it on the blackboard. So first, there is a notion of, the moment you have the notion of singular support, you have the notion of micro-local sheaves.

Well, usual sheaves, they live on our space and the category of sheaves, the triangulated category, they form a sheave of triangulated categories on my space. I can consider for every open and consider the corresponding category, and this will be.

So we have D sheave triangulated categories on X. Well, the moment we have the notions of singular support, we can do the following thing. We can consider the Catanjan bundle. Sorry, can you write that again?

Oh well. I'll write it down. I can't read what you wrote. We have our manifold X, and D is, well, it's a sheave of triangulated categories on X. So I signed to open set the category of sheaves, the triangulated category of sheaves on it. But it is not a sheave, right?

Well, it's, in modern parlance, the triangulated category means whatever, whatever infinity index you will put there. Okay, now, the moment I have the notion

of singular support, you can micro-localize D over the Catanjan bundle. Well, so consider the Catanjan bundle, but I will consider it not with planes. This is a risky topology, but only those open subsets which are conical.

Well, conical. Well, how do you define it? Well, so if you have an open conical subset

in T star X, then we can put D mu of U. This will be the quotient of D of X, modula, the thick subcategory of those sheaves whose singular support lies in the complement of U.

So this is, well, this is a pre-sheave of triangulated categories that has nature

of T structure which has to do with perverse structure. Here, and you can ask about things like co-dimension three conjecture, but that I don't want to discuss, but just let's consider this data. Now, what I want to consider,

what sort of a question I want to try to ask is the following. So suppose that my X, I will assume that X is compact from now on, and then we have considered the functor gamma

from D of X to just the lambda modules. So let the point be K be algebraically closed. And then I can pass to the corresponding map between K theory and spectra.

And what I want to do, to know, is to find this map, this homotopy map of spectra.

No, K is K theory, it's equivalent K spectrum. So in particular, if I have a sheath, then I have actual sheath, then I have, it defines your point here. And so if I know the thing, I will know the homotopy, its image, it will be homotopy point here. And such a homotopy point defines you,

defines you whatever you want. It defines your characteristic if you pass to connected components. If you look it as an element in Poincare groupoid, it will define you that are gamma, and all the things. So that's actually what we want to have. Well, now let me try to put on the blackboard

what I want to, oh yes I will, I will.

Okay, so I want to, basically I want to

have a localization of, so we have this map, and I want to localize it twice. I want to localize it with respect to X, and then I want to micro-localize it to the Cartangent bundle. And the claim is that it's all that is needed for the theory of characteristic cycle.

So let's see. So we have point X and the Cartangent bundle X, and let me denote this by pi, and this will be P.

Well, and here, on this zero level, we have this story, this map, that I want to understand. What does it mean to localize?

to localize the thing to x. Well, as I told you, d itself, it forms sheaf of categories, of triangulated categories on x, and then I can apply to it k, and it says that I will get a sheaf of spectra. Let's call it k of d.

That's sheaf of spectra over x, okay? Now, what I want to do is to find first a map. Now, it's a map of sheaf of spectra to the following thing. So this is, again, this is a spectrum over the point,

and I want to consider its upper shriek pullback to x. So I will say in a moment what it means. Well, such things, well, let's consider usual spectra as part of a mativik spectra of,

ah, there is... Here, spectra in the sense of topology. Yes, yes, yes, yes. Spectra for me is always in the sense of topology. Mativik spectra in the sense of A1 homotopy, A1 of homotopy theory. So the thing is, the thing is

sheaf of mativik spectrum over x, and it looks as follows. So if instead of k of lambda, this will be z, and for example, if lambda is a field you have mapped to z, then upper shriek pullback looks as follows. So we should take tate, tate motif,

and then we should shift it by the end, and then put it to x, okay? So that will be situation in case of z. So this will be that thing. Well, now, so what I want to have is to get this basically, basically should come by a junction. So what sits here is essentially

the map from direct image, so x is compact from direct image of this, this is a part of direct image, and you have a map to k of lambda, and I want to have this to get such a map by junction. By the way, it is not,

well, in usual, such thing exists in classical, usual topology, but in, in algebraic geometry it's much more interesting. Maybe I will give, ah, do I have two minutes? Yes. Okay. Okay, so we have this picture. Now, so here I will put small question mark,

and here there will be larger question mark, okay? The larger question mark is this. So let's consider this arrow, so I assume that it exists, and now consider it's just plain pullback by p. So here we'll get, okay, now, ah,

well, so this shift, ah, well, this shift of spectra,

it has natural, ah, natural map. I recall that there was this d mu, and there is the corresponding k shift. Well, there is a map from pullback of d to d mu, and that's a map on the corresponding k spectra.

And now? We are on t star x now. Yes, we are leaving now here, so here, so here we leave over the point, here we leave over x, and here we leave over t star x. Well, maybe, let me put question mark on the right. So small, and here there will be large, okay?

And the larger is that there is an absolutely canonical map here. Well, now, ah, ah, what I know is that if you live in classical, in the situation

of Keshavar and Shipara, and work with classical topology instead of motifs, then such a construction exists. Well, now, ah, ah, I believe that, ah, ah, if, ah, ah, well, that it should come if you, if one understands how this story with singular support

actually related to the story with, ah, vanishing cycles over multidimensional bases, then that map should come by itself. Just for the reasons that in Keshavar or Shipara situation it's essentially playing with vanishing cycles over multidimensional bases. Well, ah, but some version of it for usual topology.

Well, now, ah, I would, let me just, ah, so the claim is that whenever you have such a formulism, then you have, ah, then you have all things that you wish to have. So, for example, you have, so let me just say why vanishing cycles come as a rough, rough, rough,

yeah, well, how it comes from this picture. And it comes like this. It's a characteristic cycle. Hmm? Characteristic cycle. Characteristic cycle, sorry. So, we have a ship, so we have a point here, so, hmm, so we have a section here,

and then, ah, and then we have, it comes from a section of this story. And what does it mean to have a section of the thing? So, if we have, if we have a ship that crisps, so the thing, and this element, if I restrict it to the complement

of the singular support of my ship, it's trivialized there, the section just vanishes as an element of the quotient category. And so, given such an arrow, it produces you for every ship, so consider the corresponding section here, and it produces a trivialization of the section

when you pull it back to the cotangent bundle, and then restrict to the complement of the singular support. Okay, now, let's project the story from K-theory to Z, just by the Euler characteristic map,

then here we'll have Z of N, and then we'll pull it back here, and then you know that sections of such things are the same as cycles of co-dimension N, actually as a chow group. But then we consider the section supported on some sub-varieties, this will be the chow group of N cycles on this sub-variety, and if the sub-variety, our singular support

has dimension N, this means exactly that we know multiplicities at the generic point, such. So, in this manner, the thing immediately, if you replace K-theory by Z, it produces you in particular as a cycle, and believe me, it also produces you all formulas

as a global Euler characteristic formula, and so on. So, again, that is my hope, but I need to stop now.

You're saying it's very hard to recognize the singular support, so probably I give you a closed, irreducible sub-variety of predictions. Can you tell me what's the singular support and what's the characteristic cycle? I wouldn't even close, I need to have a sheaf. A sheaf, okay, the trivial constant sheaf. Well, if it's, well, nobody,

if your sub-variety is smooth, then it is a conormal bundle, not like that. If it's not smooth, then nobody knows, it depends on the singularity. What about the singularity, can you tell what this is? No, I cannot, I cannot, I cannot, and it's, well, if it's not complete intersection,

then even you don't know if it would be a perversion sheaf or not, so. Hello, and so, we have briefly explained

this construction of the characteristic cycle with multiplicity is going to be why the conjunctural formula for it? Ah, what do you mean? This is, yes, from this triangle.

From this triangle, okay. Conjectural formula for it. Okay, okay, so if I have, if I have a sheaf, sheaf on x, so I have a point here, so I will have a section here,

and just consider its image, its image here. Then, since it is the same as if I pass first here to here, this will be a section of the sheaf of spectrum which is trivialized on the complement of the singular support. Just because the image of the section here is trivialized on the complement of the singular support

because micro-local sheaf just vanishes outside. Okay, and so you will have a section of this fellow which is equipped, so you replace this by z, yes? Equipped with trivialization on the complement. This means that you have cohomology class with coefficient in the thing

with support in the singular support. And this is absolutely the same as giving multiplicities. So you do actually need the trivialization on the complement to get that? Yes, sure. Or just the fact that it's trivializable there? No, no, to get, otherwise it will be element of the chow group, of the chow group of the cycles on.

And if you consider cohomology of the thing supported on sub-variety, it is the same as chow group of the sub-variety. And since it has same co-dimension ends, it is just a cycle, it's a generic point, nothing else. So another thing, I think, one thing is clear

on this question that if you take f and d f, so the cc is the same. If I take f and dual? Because phi is in particular the duality. Okay, wonderful, yes.

And maybe you can see it here also. I hope, yes, and actually look somehow there are things that I should say that there should be relative picture for morphisms of varieties, which is very much possible. And I would love to know at least how to spell it out.

At least conjecture. But again, the support for this is that you have absolutely canonical picture in the setting of real analytic varieties, which is even nice in case of circle.

That means that in some sense you are looking for some complex of sheave and the cotangent space supported by the? By the singular support. Which reflects the vanishing cycle? It's not complex of sheaves. It is a section of the spec. No, no, okay, but you are looking for a substitute? Yes, for a substitute, yes, exactly.

But on the smooth part of the? No, on the whole thing. On the whole thing. On the smooth part, it is just the local system that you can in some sense see with looking the vanishing cycle. Yes, K class, it's K class, very probably, yes. But with the micro-local things,

the Japanese team in Kashiwara, they could not produce something? Some module micro-local, I don't know. Ah, probably they produce, but I believe that they don't understand that at singular points.

And I would like that the theory will be as rough as you don't, yes, yes, I wanted to leave everywhere since probably you need to know it everywhere to have actual picture on the level of whole K theory, probably. Not for Euler characteristic, you need to know it to the generic points only, but for Sattler.

Any more questions? So when you consider this E after mu on the cotangent of this direct smooth GM,

this was for the zariski topology. So several related questions. One thing is that you used the word shape of triangular categories. So imagine that this means that there is a patching result when you work in some higher context that you... Yes, often it's in stable infinity category, whatever.

Then it's a shift, yes. And then in the analytic case, what do you do? You don't have, if you just do analytic topology, how, what do you... Same, absolutely same thing. So you don't have enough singular support

in the, that is, do you do it for complex analytic or real analytic? Real analytic, you do it for real analytic thing. Ah, and then you have enough things with... Then you have enough, yes. I mean, all of them sit in some canormals.

And it's sort of a funny shift. Every section, it's supported on, well, it's supported in co-dimension N. So...

And what was the conjecture that you mentioned there where you spoke about this, you said there is a T-structure and some conjecture that it didn't stay? Ah, okay. So that's just the words. So as I told you, that every object here, just if you start with a shift,

then it's a section of D mu. It's a microlocally, it supports it on its singular support, yes? And now, what... Now, suppose that you play not with triangulated categories but with perversives, with a heart.

So let's consider a perversive. Now, the following, so the thing, again, so it supports it in co-dimension N. Now, what you want to do is that the functor of restriction outside of co-dimension N plus one. Of these categories will be a faithful restriction

to co-dimension... Yes, N plus two, it will be fully faithful. And then one dimension less,

it's an equivalence of categories. So this means that if I have a microlocal perversive, then on the whole co-tangent bundle or on some domain, that it is the same as microlocal perversive on its open subset obtained by removing as many points

of co-dimension N plus three as you want. N plus three, I think, or N plus four, I don't. This is for the analytic topology? Yes, now this is a very old conjecture in the context of demodules, or maybe demodules with regular singularities.

That was proved fairly recently, maybe three years ago, by Kashiwara and Vilainen. They have, in the archive, it's called something called dimension-free conjecture and so on. But amazingly, they cannot prove, they prove it actually for regular demodules, or perversives, but with coefficients

in the field of characteristic zero for perversives, the metal coefficients, they don't know how to prove it. I mean, the proof is analytic. No, but what is the statement? Because you define this, what do you call it, shift of primary integrable value. So you just take the quotient of dx by something,

and since the singular suppose is only so purely of dimension something, it does not make sense. I mean, it didn't. Look, so what happened? So suppose that you have perverse shift, yes? Look at it as a generic point of the singular support. There you have some data, some categories

that it would be nice to describe. Now your shift, if your shift is non-zero, then this data is non-zero, definitely, that you know. Now you want to reconstruct your shift from this data. What you should do, you should add to it some data at dimension one more, some sort of a gluing data.

If two components intersect by something called dimension n minus one, then you should add there some sort of a gluing data. Now, and this gluing data, the thing plus gluing data uniquely defines your shift. But in order to reconstruct the shift just from this gluing data,

you should have compatibility, which will sit in one dimension more, a generic point in one co-dimension more. And the moment you have it, you don't need to go any deeper. No, but once you define this d mu of u, the dx modulo, the six sub-category of singular support in the complement of u, when the complement of u

has large co-dimension, more than the middle dimension. No, no, the thing is that you shift the phi. You shift the phi as a story. So when you define it using this quadrants, you get a pre-shift of categories there. Now you consider the true shift.

And for this true shift, in principle, it could have sort of a gluing data which lives deeper and deeper. Ah, you did not say that this is, ah, you have to shift phi? Yes, yes. Sorry, that was over precision, so I'm saying so vague things.