Thank you very much for inviting me to speak here. It's a great pleasure.

Well, we have collaborated with Arthur on one long project, I think, for about three years, and this was, I think, the most fruitful time for me, mathematics. And in addition to that, I learned from Arthur what a log scheme is

and what basic etymology of a log scheme. And in fact, the work I'm going to talk about is, well, grew up also from a question that we discussed a while ago with Arthur. And so the question is the following.

So I have four log schemes, and then I have various homology theories. Well, for example, I can look at a tall homology,

or beta, or say, the RAM. Okay? And so, if my log scheme happens to be defined

over a finite field, so then, so this tall, Atlantic homology of x with QL coefficients,

well, it's a vector space, which is equipped with an action of the Frobenius. And one can, comerital, right. And one can check that for a reasonable scheme x,

the eigenvalues are all the numbers. Of course, they are of different weight. But in particular, because they are real numbers, this gives me the weight filtration. And so the question is,

so let's see whether I should use it. So, okay. So, can one define this filtration geometrically?

Well, so I want a definition which would make sense for all these three theories. Yes. And, well, more precise, well, so you can ask for rational beta homology, or you can ask even whether it's well-defined integrally, as for usual singular homology of algebraic varieties.

So, or, well, so one can ask whether, can one associate a motif, even a Wodzki motif, motif,

or maybe even a finer question, material homotopy type.

Well, I want to tell you, so, a little bit about this question, so what kind of homotopy type you expect to have? So, and this, then for,

so, lies in the construction due to Kata and Nakayama, so, okay, so I will recall it here, Kata, Nakayama, construction. So, they defined basic homology.

So, how, what's, okay. So, that's, beta homology of a Wodzki max, say with z coefficients, any coefficients,

well, to be, by definition, a homology of a certain topological space. So, this is just usual, well, so this is x log, so that's what Arthur explained to me, so. So, this is just a topological space.

Well, in many cases, it's a, yeah, yeah, sorry, this, yeah, my x is now over complex numbers, of course, right? Thank you, thank you. So, x is a log scheme over the complex numbers. And so, this x,

well, it's just general, well, it's a topological spaces, but, well, in many cases, it's a manifold with boundary, it has some additional structure. So, what is it? So,

so this is a topological space which is equipped with a map, it's proper map, to x. And the, so, well, so the construction is the following.

So, well, I have my log scheme, so I will denote by x and the line, then the line scheme, and m is the sheaf of monoids. So, well, so I have the sheaf of monoids,

I have a star that sits inside it, and I have a map to O, okay? So, well, the only thing I care about, in fact, this,

well, it's true for the whole, for the entire talk, is the group completion, m group. So, this is a sheaf of a billion groups, it has O star sitting in it, and it has quotient, which,

well, I will assume, I will look at a fine log scheme, so it's going to be a constructible sheaf for, well, for a tiled topology, or, if you wish, you can consider a version of it for the risky topology.

In fact, in my talk, it will be more convenient to consider all these sheafs for the risky topology, because, at the end, the object that, well, yeah. So, let's stick to this. So, this is, you can think of this,

well, a good example of such lambda is just the constant sheaf supported on a divisor, for example. So, you have such an exact sequence. And, okay, so, to a scheme and to such an extension, I want to associate a topological space. So, how I want to do this? Well, if I have a point,

well, I can sort of restrict my, so that is here, evaluation map, point x. And so, from this extension, I can derive an extension

of the stock of this sheaf lambda at point x, and this will be m by c star.

So, this is just a finitely generated abelian group, and this is c star. So, it's extension. Okay, so, now. M sub x doesn't mean the stock of lambda. Yeah, it doesn't mean the stock. It means pull, push out.

So, and here I have map from c star to s1, which is called the argument. So, well, let's give a name to this map. Let's call it gamma. So, I look at all possible sections, like this.

We'll call it sigma x. So, satisfying the property that if I compose gamma with sigma, I get arguments.

And so, x look. Which is the same exception, so this is splitting of this extension. Well, push out of this extension, right?

So, and then this space, as I said, is just set of all pairs, x and c max. Your number is torsion free because you assume f s. It need not be torsion free. I'm going to assume, in a second, I will assume that it's f s,

but for the moment, I don't even need this. So, okay. So, I consider all such splitings. Of course, there is a map to x.

And I want to, so, and the fibers, fiber over point x is a torsion

over home from lambda x to s1. So, in the case of nice log scheme, this is just a real torus, okay?

And so, then there is a topology. So, any section of m group defines a function

on x log is various in s1. Namely, it takes point x sigma to sigma of x applied to m. And I want these functions to be continuous.

So, I consider the weakest topology such that the pullback of any continuous functions on x is continuous as well as these functions are continuous. Okay, so, and that's a topological space, and I want to see whether, well, I want to, the ultimate goal

would be to realize this, the homotopy type. Well, to leave this homotopy type of this log space into a martivistic homotopy category. So, well, the, so that the possibility of such construction

was suggested to me by a long time ago by Maxim and also by Nori, somehow independent at the same time. So, they explained to me one single example, which is kind of really the key example. So, I want to tell you about this example.

So, okay, so what is, so this, the log structure, the log space

that comes from, as a special fiber of a semi-stable degeneration. So, it will be this, it will be x zero, this will be x t, and so semi-stable degeneration.

So, it's steeped, and moreover, so I will assume that it's, there are no triple intersections, very, very simple. So, on my picture, there will be only two divisors.

It can be generalized in the case of many. So, I have these divisors, this is d one, two. And so, then I have two line bundles on x zero, namely I have L of i, which is O of z i,

restricted to f zero. So, this is, so I have i's one, two. And so, this L i comes with trivialization.

If I restrict it, here is another piece of notation. U i is the complement, x zero minus d i. So, and the y's is, when restricted to O i, is canonically trivial, right?

So, these two structures, the two line bundles together with such trivialization can be organized as given such two line bundles versus the same thing as given extension of z, z one, plus z, z two, by O star.

This is everything over x zero. So, I have on x zero, I consider x zero as a log scheme. Well, the log structure is just the pullback

of the log structure and everything given by this divisor. So, but, well, the same queries, this has this very simple description. So, these are the constant shifts on d zero, on d i, extended trivially to everything. So, and I want to, in this case,

I want to look at x look and realize it as a element of the a one homotopy category. Okay? So, well, what you do, so, is the following.

So, well, I need another piece of notation. So, let's call L i circ. This will be g m torso associated to L i.

So, L i is a line bundle, I take the total space, I remove the zero section. Okay? So, and then, so, what I want to do, I want to consider the following complex. Oh, the following complex.

So, I want to construct, well, well, well, in fact, it will be an element of, stable homotopy category, and even unstable after a certain modification. So, okay, so, what is this?

So, let's first define it as a motif. So, well, I want to do something very, very simple. I will take L circ, so, again, this is regarded as a scheme, living over x zero. And I want to restrict it to u one.

Right, so, sorry, I think, I want to restrict it to u two, of course,

because it's stable on u one. And then, the second thing that I want to do, I want to take L circ two, and restrict it to u one.

So, when I write square brackets, it means that I consider this as a motif. Just a, it's a plain scheme, so it makes sense. Well, and what I will have here, is L circ one, times L circ two,

over x zero, restricted to d one two. Okay, so, and this d one two is intersection, okay? So, well, so, my motif is going to be

sort of the corner of this map, this complex. Now, I need to tell you what this is, this arrow. So, and here is a reminder. So, well, this construction is possible, because, well, in the category of Leibovitz schematics,

you have identification between punctured tubular neighborhood of a smooth manifold with punctured normal bundle. And namely, if you have, so, smooth y and z, this is closet, smooth closet,

then there is a canonical map in, from the motif of the punctured normal bundle to y minus z.

This exists in differential geometry. This can be lived to motif category. And, well, the map that I have here is exactly, well, comes from, is precisely that map. So, here is the picture.

So, if I have, hmm, hmm, let's see how we get it. So, okay, so here is my z one, for example.

And here I have this line bundle, which is, which is, right? So, and here I have, this is z one, two. So, I apply this construction.

So, I have this line bundle, this gm torsion. It leaves over the whole d one, two. And in this, inside this, the total space of l one, I have l one minus, I have this fiber over d one, two. And I apply this construction to y being

this gm torsion over d one. And z being, it's fiber over d one, two. Then you get precisely this, this first map and the second map is constructed similarly. So, you do this. And do you take it, put the sign,

or do you don't think about it? I don't think that there is any sign here. It must be symmetric, at this moment. No, but if you have several things that intersect each other, and then they go in a loop. No, no, in, yeah. Well, so I don't think that there is any sign this big here, so maybe in, so let's,

seems to me that, it seems to me that at least here you don't need sign, but maybe I am. Oh, by sign, it's possible. So, okay, so, in fact, this map that I used here

exists even in a one, just in a one homotopy category. But only after suspension. So, you can kind of do the same gluing except that what you'll get, you cannot get

homotopy type of x log itself, but you can get homotopy type of the suspension of x log. So, at least you'll get something in the stable homotopy category. So, the fact that this, that map exists after suspension is proven by Morel and Vyvost. So, okay, so, and well, one more remark

about this whole picture. So, in fact, this ELI have additional structure if they come from the semi-stable degeneration. Namely, L1 tensor L2 is trivial.

Well, strictly speaking, this trivialization depends on the choice of a parameter here. Oh, I choose a t coordinate, right? Because this is o d one plus o d two, so it's a special fiber, it's lifted from the base. And therefore, well, if I have a map from this,

from everything in this picture to complex numbers, right? So, there is always, there is a map

from these guys to complex numbers, from invertible complex numbers, right? So, and therefore, so let's give a name to this. Let's call it motif of x zero m.

So, it's motif of this log scheme, just a definition. And in fact, in this case, this motif is not, is a motif shift, motif shift over C star.

And it's fibers, for example, well, you can take fibers over, so every scheme here is a scheme over C star, so you can take fibers, you have still this map,

you get what's called the limit motif, right? So, and the picture here is, I cannot really draw a picture of this gluing of this x log, but this x log here,

it actually, it maps to log space associated to the log point, so it maps to the circle. And I can draw a picture of a fiber of this over a point of the circle.

So, imagine that, for example, you have the generation of elliptic curve. So, okay, so here is my, so it generates inter-rational curve with two double points, right? So, this is non-singular elliptic curve,

and this now it degenerates, so what you do, you sort of remove these two points, and then you glue on punctured tubular. And then you get something back, well, homotopical into the original space. Okay, so I want to kind of explain this construction to all log schemes,

and that even if you have, in this semi-stable case, if you have many components and multiple intersections, it becomes really, well, direct, the generalization is really unpleasant, because, well, you can kind of write down

a similar complex, but, well, you have to do it, lift it in somehow in digital level, and then the square of the differential will not be zero, it will be the homotopy equivalent to zero, and these homotopes are given by some double normal cone constructions, and so it's going

to be completely useless, though possible. You can't really do anything with that. So, instead, one should look at the dual thing, one should look at the motivic homology of this motif. And this has a very simple geometric description. I will show it to you at the end in the case of this semi-stable degeneration,

but first I want to formulate the main theorem, in a kind of in abstract form. So, the main theorem is the following. So, while I need a bit, first introduce a bit of,

a certain category, I will call it the category of log-motives, but, well, then you can, well, we can forget about this. I mean, so first I introduce a category

that I will call the category of log-motives. So, and in order to do this, so let me just write down definition of usual category of log-motives, and then I will list two more kind of relations, and get the category of log-motives.

So, well, so I will work with, so log-motives. So, I will assume for simplicity that characteristic of my base field is zero. Well, if it's P, you can do the same thing,

but you have to invert P in the coefficients. And then, so, okay, so for Vyvotsky-motives, you do the following, okay,

motives, so you consider category of schemes, say of finite type over a field. And then, then you form an additive category,

just objects, schemes, all schemes of finite type and morphisms, a linear combination of maps, no correspondences here, and then I want to take

compresses, finite compresses over this, and then I take, I mod it out by certain relations, by sub-category, so this is, well, if you can do it on triangulated level, pass to homotopic category, then take Virgia quotient, or you can regard it

as a differential graded category. So, and what are objects of T? So, objects, well, of T. So, well, I will list them. Let's, let me start, okay, here. So, first object is, well, it's going to be

class of objects, right? So, it's X, so, so objects are accomplices of schemes, right, so, so this is first, this is called A1 homotopy. Okay.

And then, the second class of objects is, well, perhaps I have to, let me do it here. So, this will, so, this, the second class of objects,

I will refer to it as CDH topology. So, these are generating kind of sequences, covers. And so, there will be two parts. Well, so, first, suppose I have open,

there is still open subset, and I have in a tilemap, P, this is a tile, and so, what are the conditions?

Condition is that the map P inverse of, from X minus U to U, to X minus U, this map P, the restriction, is an isomorphism of schemes.

So, that means that this is in a tilemap, and on the complement to U, it has a section, it's actually a bijection, right? So, this is the risky open, this is a tile, so, in particular, U and W cover X, but it's much, much, much more than that.

And in this situation, I want relation, and this relation has a form, X, U, W, and here I have the fiber product, U, W.

So, all these maps. And so, this is part A and part B. So, I want the following. I want, I consider maps P from X prime to X,

this are proper, and I assume that there exists closed sub-scheme here, closed sub-scheme.

Such that, over the complement, this map is an isomorphism. So, P induces an isomorphism between P, inverse of X minus Z, and X minus Z.

For example, blowup. So, okay, and in this situation, I want to have the following relation, that motif of X prime plus motif of Z,

and this maps to motif of X, and here I have motif of P, which of Z.

And here you must put the sine sign on those things. Yeah, I have to put sine, so either, it doesn't really matter where. So, yeah, you're right. So, I have to take the difference, otherwise the square will not be zero. So, and, all right, Z is closed, right?

So, and also, A, B, C, D, okay. C is that the map from X is used to X.

Must be nice. That's it. It follows from B, because you can take Z to be the... Okay, okay, so, but, yeah, you're right. So, many things, in fact, follow from others here, so, and, well, you can, you can, so,

so, this is not, that's not, so you can have, so this is not what's called, usually called the category of Vyvotsky motifs, it's also defined and studied by Vyvotsky, and it has a very complicated notation,

this category, this quotient. So, let me give it here. So, it's H, A1, CDH, and then here I take Z of schemes.

I need a Peruvian completion of this. Okay, so, what is the relation of this category with the category of Vyvotsky motifs?

So, there is a, well, at least there is a functor. So, this Z to Vyvotsky category, which is obtained exactly in the same way,

except that you add transfers. So, you start with, take and consider the same relations, but this category is different. You consider schemes, and where maps are finite correspondences. And also only smooth schemes.

Don't think that if you do CDH topology, it makes any difference. You're speaking about the, oh, okay, so you can do it with, yeah, okay. So, if you consider smooth schemes, then CDH topology, then B is,

can be derived from A and other schemes. Okay, so now I want log motifs. So, what I do with log motifs, how I, so let's just consider the same, the same similar category, but let's start with log.

So, so by log schemes, so I consider, I mean, F has finite saturated log schemes.

Okay, and then I want to take, so I want to consider a quotient of Z log

scheme by T. Well, and what is, so what is, what is this T log? Well, so T log consists of,

it's a full subcategory of objects. Well, I need not be objects of type

one and two, two and the following. So, I need to impose a bit more, a few more relations.

So, okay, so what is, what are the relations? So, well, maybe just before we explain the relation, so I have here this,

well, there is homotopy equivalence relation. So, in this homotopy equivalence relation, X is allowed to be any log scheme, as well as here. So, I want everything in this picture, X, well, this must be log schemes.

A1 here is usually A1, but I will have, I will add axiom for log table. And then if you speak about closing merchant, perhaps you want the exact one? Yes, yes. So, okay, so what I want is

the following axiom. So, I have A1 log. So, what is it? It's, the underlying scheme is A1, and the log structure is given by one point. So, my M group is just, corresponds to the line bundle of this point.

So, and here I have, so, log point, it's kind of the origin. And here I have GM, it has trivial log structure.

So, and I want these maps to be isomorphism in my quotient category, but not only this, I want multiply this by,

and again, X is any scheme, log scheme. So, and also this GM times X is embedded.

Well, there is one additional axiom. It's strange. It just has to do, we already observed that, well, all the construction we have made so far depend only on M group, not on M. M is irrelevant,

and I could consider the category of our scheme just by the category of extensions. What do you say about that? The M in my category T. So, you require it to be zero? That's a log. That's a log. Yes. You wanted that A1 log is an isomorphic for you to that is a log. Yes. A1 log.

Yes, yeah. So, these are objects in T. Like, for example, this is this. And finally, four is very, very strange. Well, so it's very, very, is the following axiom. So, suppose I have any map P from X prime to X. These are log schemes,

and the underlying map between usual schemes is an isomorphism. And suppose that it induces, the mapping is used on M group, P of a star to M prime group is an isomorphism.

Then, I want the following relation that X prime goes to X must be in my category. So, this kind of way to say that everything depends on the M group. But again, so I could consider just the category of pairs X plus this extension of star, this star. So,

now, very good. So, here is the theorem. Now, when you do block blow ups, what seems, when you do the operation of block blow up, I think that the space, I don't remember now, the space X log doesn't change, or? No, it changes. Oh, the space X log, yeah,

changes, but it's the same homoid. So, here is the theorem. well, I have category A one,

CGH. This is my category. And here is the name for this. So, this is Z of smooth schemes,

of schemes. So, there is, of course, a functor to this larger category where I can see that A one, CGH, and here I have Z log, okay?

So, right, so this functor is an occurrence of categories. So, what does it mean? So, what are, for example, functors from this category to, say, the category of complexes? These are homologous theories, right,

that satisfy these basic properties. The claim is that any homologous theory can be extended uniquely to log schemes, provided it has all these properties. Okay, so, so let's see.

So, in particular, well, of course, here you have map to functor to usual Weivotsky category, and then you can compose the inverse and get Weivotsky motif.

Now, so, the construction, the proof is, has very little, uses very little geometry, in fact. It's more or less linear algebra, and so, well, first of all, you have to prove,

so, very sketchy, so you have to prove two things, that this functor that I have here, the obvious functor, let's give a name to this functor, let's call it, is fully faithful, or homotopically fully faithful

or if it's, you think of this as G functor. So, fully faithful. And B, so essentially subjective, subjective.

Well, and in fact, so easy, easy step is that A implies B, and therefore, so all you have to do is to prove A. So, and that's because,

and you can do it by induction on dimension. So, you need to show that any motif of any log scheme is in the image of this functor. So, have enough relations to express a motif of any log scheme in terms of motifs of usual scheme.

So, but that's easier. So, you do it again by dimension of your log scheme. So, if it's zero dimensional, then, well, it's fine saturated, so it means that my monoid M group is just Z to the N, and so the corresponding,

so if I have point and take its, well, its motif, for example, its motif of GM, and if I consider Cartesian product of any scheme, of log point with itself,

then of course, it will be GM times GM. This follows from this axiom, and so any, and using this property, any zero dimensional log scheme has the same motif as product of this. So, now, what you do in one dimensional case,

so you have some curve, well, now, genetically over an open set, the log structure is trivial. It just corresponds to the trivial extension. M group is just trivial extension of Z to the N

by your star, and therefore, okay, so you know what to do over an open set, and then you can use this CDH axiom to see that this log motif of this curve

is also in the image. So, what, what, you have this for a paper, have you? No, I used, because, so let's see, well, if you look at this CDH axiom, how it looks like, so you have some T inverse of Z,

and then you have Z plus X prime, and this maps to X. So, what, so I want to show that this object,

just, sorry, that this object, so knowing that this object and this object are in the image, and also this, I want to conclude that this object is in the image, but then I need a map from X, from this square bracket X to this guy,

and I need to show that this map exists in the usual category of Vojvotsky, usual Vojvotsky category, it exists by definition, so because it's exact triangle, there is a map from square bracket X to this guy, shifted by one, and I need to show that this map actually exists.

So, and now, so what you do for, do you assume that the log structure is a risky locally traded? I use it, it's a risky locally traded, right. So I could, yeah. So you don't have motives for a tile?

Then I need to work with a tile topology, rather than with the risky, and well, at the end I will get an Atal-Vojvotsky motive. Well, some applications that I had in mind have to do with this integral weight iteration,

which exists only for realization of usual Vojvotsky motives, not Atal. So, okay. So what you do for A,

so what you have to, it's enough just by formal nonsense, it's enough to prove the theorem, it's enough to do the following, so given any object of this category, H, usual category, A one, C D H,

maybe I will just say the following. So let me, because my time is short,

so I claim that it's enough, it suffices. I will sort of suppress the completely formal part, so it only uses the fact that the category of Vojvotsky motives is rigid, duality. It suffices to show that

the functor, home, Z of N,

from this usual category of C D H schemes

extends to a larger category, this category, S D H, Z.

And what you do here, so you just sort of, you want kind of to define motivic homology. It's not quite motivic, it's home in this category with no transfer, so I don't want to call it

motivic homology, it would be motivic homology if I consider this category, this transfer. So you want to do it for any log scheme. And the idea is very simple. So you have the same group. It can be evaluated, it can be considered

as a motivic shift over this motivic shift, shift over the underlying scheme.

That means simply that you can evaluate it over any scheme which is smooth over this guy. You simply pull the log structure. If I have a math like this, then you can pull log structure, this log structure here, and take the same, there.

So, and then what you do, you just compute, you take a homology of, of x, these coefficients in symmetric powers. These are operations in the category of matrixial symmetric powers. Well, they're defined by Weivotsky.

They're characterized by the properties that if you take x, that motif of a scheme x, is the symmetric power of, motif of a scheme x is the nth, the motif of the symmetric power of x. Are you using nth bar group or nth group? Just nth group.

So, for example, what happens if your n is one? Well, when you pull it back, you don't make it a log structure of y, you just take, you don't enlarge the units. I mean, usually when you pull back n. Yeah, n large units. You argue large units. Yes. So, for example, what is first,

well, my typical homology is n equal one. This is just homology of n group. It's kind of analog of the picard. So, okay. And now I want to finish by. Homology which category? This is, no, this is the.

Homology of what? So, well, you. From the big, so this is homology of x. Yeah, this is Nieszniecki homology of this motif, of this motif. So, well, I want to finish by a very explicit formula that I promised at the beginning.

Just very geometric. For the material homology of the limit motif, in the case of semi-stable degeneration, those are for the tubular new book. It will be just in the case of semi-stable degeneration. And it comes from this, I just, well, this formula is here. I just want to, I will make it explicit.

Which is trivial from this definition. So, the formula is the following. So, here is my setting. Well, now I will have, again, I have the log structure coming from semi-stable degeneration. But now I can have multiple intersections.

Well, but it's still semi-stable. Maybe that's not a good picture. So, this should be planes rather than lines. So, okay, probably I should not put it here, yeah. So, it's confusing.

Okay, semi-stable degeneration. And it's strict, right? So, I want really, that's, I want log structure for the risky topology. So, strict, strict, strictly semi-stable, semi-stable. Degeneration.

So, I have components, d1, d2. Well, so, di. I, I belongs to some set. I have this gm torso.

So, first of all, I have li, which is of di. I restrict it to x zero. And then I have the corresponding gm torsers. And I have ui, which is a complement of x zero. So, okay, so now I will produce,

construct the material homology. So, well, first I will produce a complex of smooth varieties over x zero. So, the fibers will be algebraic trie. So, what are these varieties?

So, so step one, complex, very simple. Complex of smooth varieties. So, this will be product of all this li.

So, this is product over zero. It's a torso over an algebraic trie of dimension. Now, what is the next term?

So, you take a sum. Let's consider it as a, kind of, I want to, it's not really, in fact it's a, I want to consider.

kind of an additive category, I want to make sense of sum. So it means that it's, you can think of this the joint union, a joint union. So, of products of i in i, i is not equal to j. And here I will have l circ i

times ui over f0. So what is this map? Remember that li is trivial over ui. Therefore I have such a map. It's just given by, well, it's a section. Now, it's a long complex, and here is its last term.

It's sum i li over times intersection

j not equal to i, uj. It's again over zero. And so all I use here, all the differentials come from the sections. Remember that li is trivial over ui. So this gives me such a, again,

this should be usually in here. So this is step one. It's very simple complex. Now, form, define a complex of brackets on x zero.

Well, what you do, so you call this z dot, and then you take the following.

So if you have a scheme over x zero, then, well, the value of my complex of pressures on x zero and y is the following. You take correspondences over x zero

from y to this complex. So, well, informally, you just take sections of this complex of varieties over y. And then you make it homotopy invariant by applying the system construction here.

So let's call this f, and my material homology of x zero m z

is just CDH homology of this x zero, these coefficients in f. That's it. That's all I have to say. So there is one, well, there is, as you see, you can ask whether the log geometry here,

and in fact, there is no log geometry. You see, it's just linear algebra. And the log geometry appears if you want to prove, for example, that if you have a smooth log scheme, then it's motif. It's isomorphic to the motif of the open part where the log structure is trivial. This I don't know how to do.

I know how to do it in the case of normal crossing. Normal crossing situation, but in general, it really requires some geometry. Thank you very much. Was there any case in notation of the motifical homology

did you say you have several, you have zn for different n? Yeah, so here, I defined only with one index, right? This index is absolute value of this set of indices, i. So if you want, define it for larger n, right?

You have to just add in the whole construction, empty divisor formally. And to define it for smaller, you don't? Yeah, yeah. You don't? You don't need to define it for small by cancellation. It suffices to define this for sufficiently large numbers.

This follows on the proofs? Yes. No, no, just motifical homology, yeah. So if you want it for, you have cancellation, so you can express motifical homology of something with coefficients in z of n

in terms of homology of the twist, these coefficients in z of n plus one. Which is the product of this gm? Yeah, yeah, product of this gm, for example. Yes? So I just was curious, what is n? Where is n? Shouldn't we have it, if we have motifical homology?

Yeah, yeah, this twister here, right? No, no, n is the log of monotony. Yeah, that's a very good question. It's this whole complex of schemes. It's leafs over gm.

If you have, it's exactly as in the picture I started with. This gives you the tubular neighborhood. If you want the limit, the vanishing cycles, you have to take the fiber over.

So it's, yeah, it's unipotent motif over gm. But how can you define the map to gm if you're given just, you're in the fiber, also?

No, there is a, the map to gm comes from the trivialization of the tensor product of this line bundles. So each of these schemes over x0 admits a map to gm.

For example? Because the other one have a table. The product is, yeah. This is implicit, when I said that you were gonna, you started talking about how to define the white filtration. Does this come out easily from this? No, Betty, yes, so there is a,

yes, it comes thanks to work of Bandarka. So if Betty homology or the Ram homology or Tal homology of any wey-wotsky motif is equipped with integral wey filtration.

Yes, but after you have to show that it's equivalent to. Rational, yeah, to what? Yes, he has defined the white filtration from Bandarka, but after, you have to show that it's equivalent to the others. Well, but here you don't have, well, after you worry, you don't have the others.

What you have to show is that the Betty homology of this log motif that I, of this wey-wotsky motif I defined, coincides with homology of this, as defined by Kata Nakayama, but this is obvious because Kata Nakayama homology theory

satisfies the list of all my axioms, and therefore this functor from log motifs is uniquely determined by its restriction to usual motifs, and for usual schemes there is nothing to prove. And, sorry, last question, characteristic P? Yes, yes, yes, yes, yes. Yeah, you have to invert P.

And always with Zariski? Yeah, for Zariski, again, so the only reason why I work with Zariski is that I want wey-wotsky motif really for Nisnevich topology, as opposed to wey-wotsky motifs for et al topology. And so it is enough to have Nisnevich locally, Nisnevich charts for the log stack? Yes, yes, yes, yes, though I don't really know,

no, it may be that way, there are. Yes. Any more questions?