 Okay, so everything I'm going to talk about is joint work with Duzon Park and Polarne. You can read most of the things I'm going to say in our print on the archive. So as Fredrik said, I'm going to say something about motives for logarithmic schemes over a field. So first question is why motives and log schemes together? So why do we want to put these two theories in one? And of course, one layman's where it would be, well, because why not? And then what I'm going to do in this talk is try to convince you that, well, there is a better argument for it, okay? So there are actually good reasons for trying to include some ideas from log geometry into the theory of motives. So before that, let me start with a very quick recollection on logarithmic geometry. So this is a very elementary thing. So I apologize for the experts, but the main idea in log geometry is like what we want to have a tool for studying the analog of manifolds with boundary in algebraic geometry, okay? So the setting that I have in mind is that we fix a field K and we have a scheme S over K, and we fix some open embedding of S into something called S-bar, okay? Typically the S-bar will be proper, but not necessarily. And we want to do it in a way to keep track of the geometry of S in doing so. So of course, let me discuss this baby example. So one can start with something like C minus 0, so GM, and then put it inside C, that's a perfectly valid partial compactification of C minus 0, so locally. And of course, this is not so great because this changes the homotopy type, right? So we pass from something which is the homotopy type of S1 to something which is simply connected. So that's not what log geometry does. Instead, one builds the so-called Kato Nakayama space. So this is the real blow-up at 0 in C. So how does it look like? So you start from, as you can see in the picture, you start from like C minus 0, you have your point, and then you attach an S1, okay? Like you do for a blow-up, your exceptional device or in this setting is going to be like a circle, okay? So in this way, you get something which is locally like a manifold with boundary, and it doesn't change the homotopy type. So that's more or less the idea we want to use in algebraic geometry as well. So what's the main source of log schemes for us? Well, I'm going to talk about, oh, I see that you can't really see that I switched slide. Can you see? Oh, okay, good. So the basic source of log schemes is called SNCD compactifications. So we start from a smooth scheme X, and we consider an open embedding into X bar. You can make X bar normal. In an ideal word, you could make X bar smooth. So for example, if you are overseeing or if you have a resolution of singularities, and you want to make this compactification as nice as possible, and in algebraic geometry, this means that the complement, so this guy here, X minus X, is going to be a divisor, actually a strict norm by crossing divisor on X bar, okay? It means that each reducible component, the I of D, so if I write like this, so my D has some of the components, the I, then each component, the I is actually smooth and intersects a transversely with the other components. Okay, so in coordinate, let me give this example in a bit more detail. So let's pick a point X in the support of this divisor. So the picture you have one point and then two components passing through it. And you can, let's assume that X bar is also smooth over K, okay? So I can choose local parameters, say T1, TD, D is the dimension. And my divisor, big D, is a zero locus of, say, R of M, okay, T1, TR. Then, and here is where the log structure comes in, okay? You can define this map, so this is a morphism of monoids, okay? So you have a monoid N to the R and you map to the multiplicative monoid of OX bar. So you send A1, AR to T1 to the A1, TR to the AR, okay? And then you can, you can build a logarithmic scheme out of this, so which is essentially the datum of an underlying scheme plus a sheaf of monoids like this one, okay? So it's a bit more complicated than this, but it's enough to, I think this is enough for having an idea. So this kind of logarithmic structures are sometimes called deline faulting's log structure, so DF, this is at least the name used in August's book on log geometry. And the pair, Y, so will be the datum of this X bar and our divisor D will produce in a, in a natural way a logarithmic scheme. And then we will call this X bar will be the underlying scheme of a log scheme, okay? So this is just to introduce a bit of notation, okay? So, so far so good. And let me continue. So of course, a very natural question is how does this thing depend of the choice of a compactification, okay? Many compactifications are obviously possible and I would like to stress one specific source of examples of different kind of compactifications. So let's see again what we had in the previous slide. So we have our variety X bar and a point in the boundary. So this sitting in D1 and D2. And of course, what you can do is you can blow up this variety X bar at the point little x. So the picture in the blow up will look like this. So we'll have D1, D2, tilde, the strict transform. So the original components of a divisor. And then you have yet another component. Let's call it E and this is our exceptional divisor. Okay, of course, since X was sitting inside X bar as an open guy and we are blowing up something away from the boundary, both Y and Y prime are equally valid compactifications a priori of X. And there is no, well, there's no uniqueness whatsoever, okay? Now, on the other hand, let me stress that one very special thing happens if you study this map. So for example, let's look at the so-called log differential forms. So omega i of y. So this is the differential forms on X bar. So we have logarithmic poles along D. So this is a pretty familiar object. And you can compare it with a pullback map. You can pull back log forms along pi. So you have an integral map to this pi lower star of the omega i of a blowup. We have log poles on the total transform. So D tilde plus E. And well, you can show that this map is actually an isomorphism, okay? So one way to phrase this fact is that in the eyes of log geometry, the map pi is an etal map. So it's something called log etal. So, and then as it usually happens for etal maps, you don't see a difference in the differentials. So maybe what this is suggesting us is that we should not distinguish between y and y prime. And this idea will be used again later in the talk, okay? And we have just like, see that this is part of a magic of log geometry. So if we put a log structure, like some question by Kirsten, what the log pool is F prime, or yes, so this is generated. Okay, so log poles is like, okay, so Kirsten, to answer your question. So suppose that I have parameters say T1, Td. And as in the previous slide, so D is given by, okay. So in the, say just T1, T2, then you have something like DT1 over T1 and DT2 over T2 and so on, okay? Is this okay, Kirsten? Good, well, just continue. So, okay, great. Okay, so what I was saying is that the magical of geometry is exactly that this kind of thing behaves like an etal map, okay? So this is not even flat. So that's quite cool, okay? So some singular things behave like they were smooth and so on. Okay, but what has this to do with motives? Okay, so in order to answer this question, let me quickly review what I mean by motive and I'm sure that this is absolutely well known in this conference. So for us, motives are motivic complexes in the sense of Wojcicki, Morale, Yub and Szynski-Deglis. So the categories we will focus on are DM or VA. So the construction is well known. So it's a localization in your favorite way. So either as a translated localization as a localization in terms of multiple categories, infinity categories, whatever, of the category of sheaves with or without transfers on the, say, risky Orniznevich or etal site on smooth schemes over K with coefficients in lambda, lambda is some coefficient ring. So what you do is you localize respect to A1 and according to whether you have transfers or not, then the resulting category will be denoted by DA effective for topology tau or DM effective for topology tau. Okay, so for us, the focus in this talk will be on the Niznevich or the etal site because something specific happens in the etal setting and I would like to, but I would like to stress. So of course, the basic input is A1 invariance. Okay, the localization we impose that this is satisfied. Okay, and this is great. Yeah, as we all know, but it has some advantages that I would like to, that I would like to stress and hopefully this list of disadvantages will be a source of motivation for why we might be helped by by log geometry in this. Okay, so that's maybe the motivational part of the talk. So hopefully we'll partially answer to the original question. So of course, the comment is that not every invariant that we care about is insensitive to the affine line. Okay, so I think in the summer school, there was already a comment about this at some point during, I think, Tabata series of lectures. But let me just spell this out. So you have, of course, the first invariant that what you might want to look at is the additive group, GA, as a sheet from the big site. And really, GA is not insensitive to the affine line. The global sections of GA on A1 is just K of X. And that's clearly not the same as the global sections of GA on spec of K. So that's easy. And then you have all the relatives of GA, right? So all these coherent guys, or things which have a filtration by coherent guys, or graded pieces by coherent guys, better. So omega j, so the differentials over K, or the absolute differentials over Z. And then you have the drum bit version, so hodge bit chiefs, either over K or over Z. And these are all well-known examples of known A1 invariant chiefs, okay? So in the drum bit case, then you can also see that, for example, that crystalline homology is not A1 homotopy invariant, okay? And this comes from both computations with the drum bit complex. Well, or direct computation, actually, because both things are not revealed on A1. Okay, so let me also stress something very specific about the ethyl topology. So if you are in positive characteristic and if you happen to be interested in Z-mode P to the R coefficients, then you can look at the Arting-Schreyer sequence. So this is the well-known exact sequence in the ethyl setting. It's not exact in the nase-navige setting. So this is a very typical ethyl problem. And of course, the fact that this is exact and the fact that GA, which is represented by A1, is contractable in multiple categories has a very sad consequence, namely that the category DM ethyl with Z-mode P coefficients agrees with DA ethyl with Z-mode P coefficients. And it's a very boring category, right? Because if you put Z-mode P in the coefficients and Z-mode P is just zero here. Okay, too bad. So there is, well, this sort of thing, there is no hope. So, and the question is what are ethyl motifs with say Z-mode P to VR coefficients in positive characteristics? Okay, so of course, you have to figure out something which is not contracting A1 because this would sort of kill everything you're interested in. Okay, so how does logarithmic geometry help in this? Well, let's see. So the way I presented log geometry is a systematic tool for studying compactifications, okay? So in the eyes of logarithmic geometry, then maybe A1 is not really the right thing to look at. But you can look at the compactification of A1. Well, you get P1. That's a unique compactification in this case, fine. So you have a boundary, let's call it infinity, and then this gives rise to very simple logarithmic schemes. Then I will denote this guy alternatively with P1 comma infinity or this annoying notation box one, okay, that is using the theory of motifs with modulus for those who know about these things. Okay, so the idea is that we could try to use P1 infinity now seen as a log scheme instead of A1 to build a meaningful category of motifs. So why is P1 infinity something sort of reasonable to use? Let's look again at the complex realization of this guy. So if you are over the complex numbers, for example, you can look at the Cato Nakayano space associated to P1 minus infinity. So it looks like this, okay? So I hope you can see my picture. So you have, of course, the realization of P1. You remove the point at infinity and you attach an S1. That's your real blow up. And this has the homotopy type of, well, it's homomorphic to the zero times one times zero, one times zero one, okay, is a closed interval. So well, this is something that you can use to parameterize the motopies, okay? So the closed interval zero one is as good as the open interval zero one to parameterize the motopies. So it's not such a crazy idea, okay? So if you, what I'm saying is that if you build a reasonable complex realization, then this thing will be contractable, okay? That's something that we also wanted from A1. And something that you don't have if you, for example, try to contract P1, okay? Good, so then we have this guy, P1 infinity, is an object in this category here that we denote log smooth over K. So this is a category of log smooth, okay? So for the experts, these are fine and saturated log schemes over K. So we don't really need to get into the details. And the main source of examples for objects in this category are things like this, okay? So X bar D in a previous slide, okay? So comment for the experts, so I'm not putting any log structure on the base. So I'm considering K, spec K with trivial log structure. And then so my, in some sense, my log structures will be horizontal, okay? So I will only consider things like compactifications of smooth varieties. So I have an adjunction, yes? There's just a question about someone asking, is there a book of our series of papers that explains the background on log schemes? Yeah, there is. So of course there is a book by Arthur August, I think it's lectures on logarithmic geometry. I don't remember the exact title, but yeah. So that is everything you want to know about log geometry and you never dare asking, okay? August lectures on logarithmic, yes, okay. So I see that the other panelists are providing tons of examples. Okay, so my reference is August's book, of course, okay? Right, so, okay, so let's see, let's go back to my example. So I have this adjunction between smooth schemes and log schemes in this setting. So of course you can start from any smooth scheme and you consider it with a trivial log structure. So you can just see it as a log scheme, it's harmless operation, okay? And on the other hand you can start from a log scheme Y and you can produce out of it a scheme. How do you do this? Well, essentially you have to, okay, so I think the best way to explain it is keeping this analogy of log schemes as being compactifications of things, okay? So if you imagine that a log scheme is essentially the datum of like a chosen compactification of a scheme, then you could just forget the boundary and you get back a scheme, okay? So that's more or less what you do. So you have to look at the points of your log scheme Y where the log structure is not trivial. So this would be the boundary, okay? So in our previous example is exactly this divisor D that I was drawing and you take it out. You get something which is open in Y and is a log scheme with trivial log structure now and so it's just a scheme, okay? So I hope this is clear, but please ask if you need more details about it. But I think it's more or less enough to get an idea of what's going on. Okay, so what I want to do now is to build an analogy between the construction of Wewodski and the word of log geometry, okay? So I've already told you what we are going to use as a replacement for the affine line and also what we are going to use as a replacement for our category of varieties. So it will be log smooth schemes instead of smooth schemes. But I still have to say a few more things before being able to talk about motives. So this is my table, okay? So on the left side I have what happens in the realm of algebraic geometry and on the right side I have what happens in the realm of log algebraic geometry, okay? So first of all, what is our base? Okay, so base will be a field or any reasonable scheme actually if you want to do algebraic geometry. And in the log setting, as I said before, we are, at least for now, let's confine ourselves to the case where we have k, again, a field, with trivial log structure, okay? Then, so let's just forget about it, okay? So it's just a field. So then, okay, so we have the categories of varieties and in order to construct a DM, we built a category of finite correspondences or rather Susan and Levotsky did. And we are going to generalize this notion to the notion of log correspondences. So there will be a new category in the picture. So I told you already what's the replacement for A1 as an interval object. So this guy, P1, pointed at infinity with its natural log structure. And what about the topology? Okay, so in, for motives, you can use either Watzersky, Orpeniznevich or Vietal topology. And in our log setting, we are going to introduce new topologies. So they're called the dividing topologies, okay? So dividing Niznevich or Etal or whatever. There will be something also called log Etal topologies on log schemes. So I need to explain these two parts, okay? So I have to say something about correspondences and I really need to say something about this topology. So let me start from this, okay? First of all, let's start with a quick reminder of an Niznevich topology on smooth schemes. So, well, we know what this is. So it's a CD topology. So it's a topology associated with a CD structure. And it's generated by so-called elementary Niznevich squares, okay? Like this, okay? So you have an open embedding and a top map. And then you have the isomorphism on the reduced complements of this. And this works perfectly well. But of course, what should we do if now we assume that our ax has a log structure? So again, let's pretend that we are in this setting so that this x is actually given as is written here in the slide. So by some x overline and some d, okay? So d is a strict non-acrossing device over an x overline. And we should try to promote the topology from smooth schemes over k to log schemes. So what should we do? Well, the first attempt is that, well, let's just take Niznevich squares of the underlying schemes and put the induced log structure on, say, u and v, okay? So u and v are just schemes now. They don't have any log structure. We have an open embedding inside x underline, the underlying i, and a top map from v to the underlying i. And let's just say that we can, let's just put an induced log structure on this. So what do I mean by this? So if u is open inside this x overline, this x bar, okay? Then we can just restrict this Cartier divisor, this non-acrossing divisor to u. Well, this will produce another log scheme. That's easy, right? I mean, you have a Cartier divisor. You intersect with an open guy and, well, that's it. Or if you have an ital map, say, f from v to x bar, you can just pull back your non-acrossing divisor. You will get a divisor on v. And again, this will produce a nice log scheme in the same way as I explained before, okay? So this works, okay? So this produces a topology, maybe a bit naive, but let's call it, let's give a name to it. Let's call it strict Nisnevich topology. And of course, you can imagine that you can vary this construction a bit. Instead of starting from Nisnevich squares, you can start from the risky cover and then you get the strict, the risky topology and so on, okay? So that's a perfectly valid topology on log schemes, but it's not quite fine enough for our purposes, okay? So let's consider again the example where I was stressing at the beginning of my lecture, of the change of a compactification, okay? So suppose that you have, that our log scheme is this guy, x bar, so it's a nice smooth scheme equipped with a divisor, say d1 plus d2, and we pick up a point in the intersection and we blow up the point, okay? So as I said before, the morphisms like pi are log et al, so they don't change the differentials, even though they are not even flat, as morphism of the underlined schemes, and we want to give a different name to this maps, let's call them dividing covers. Okay, we're back. We are back, okay? Yeah, everything is fine. Good. Okay, good. So let me just continue. Where is where I stopped? So right, so I don't know where you lost my connection anyway, so I was saying that these maps are called dividing covers in our paper, and this is sort of inspired by what happens in toric geometry, so if you, well, say there is an operation called star subdivision and of monoids, and well, the corresponding maps in algebraic geometry, they look like this, okay? So they look like blowups along special centers. For example, you're allowed to blow up things like the intersection of D1 and D2, but you're not allowed to blow up a point in D1 away from the intersection within the other component of the divisor. Okay, anyway, so this actually turns out to be something that you can do. So you can define your topology if you put together dividing covers and strict Niznevich maps covers and that I discussed in the previous slide and the resulting topology is called the dividing Niznevich topology. Let's call it D knees and symbols. So what is the idea behind this? So that we dividing Niznevich locally, we are allowed to blow up along nice centers in the boundary of our logarithmic schemes, okay? So that's what we are going to do and somehow we are justified in this because if our goal is to represent in a new category of motives, things like differentials and differentials or sheaves or hodgevitch sheaves, then we have already seen that these things, they tend to be pretty invariant to this kind of maps. So that's okay. All right, so let me now say something about finite log correspondences. So the category of smooth schemes over K is embedded in the category of finite correspondences introduced by Suslyn and Wojewski. So finite correspondences and that's building block for DM, okay? So how do we generalize this notion to the case of log geometry? So let's take X and Y log schemes, logs move over K and let's give a bit piece of notation. So if I write X zero as the complement of the boundary, okay? So X minus boundary of X and let's write Y zero as Y minus the boundary of Y. So as I was explaining before, you have an open embedding into X and into Y. So if you, for example, of the underlying schemes and what is the correspondence in our log word between X and Y? Well, first of all, we start from an integral finite correspondence in the usual sense. So we start from some W zero, which is closing X zero times Y zero and this finite and subjective over, say a component of X zero, okay? So W zero is an element of the usual correspondences between finite correspondences between X zero and Y zero. Okay, so how do we promote this to a log thing? Well, let's first of all consider the closure of W zero in the product of the underlying scheme, say X and Y, okay? And let's assume that this is also stays finite and subjective over X, okay? This might seem like a very restrictive condition, but let's just bear with me, okay? And then we do something slightly technical. So we look at the normalization of this disclosure and you look at the composite, so W zero N to Y, okay? And we assume that this gives rise to a morphism in the category of log scheme, okay? Where we keep W zero N with log structure induced from X. So again, in the baby example where all these log schemes are just schemes equipped with a boundary, like a normal crossing divisor, what we do is we pull back the divisor from X to this W zero N and then we have a divisor there, we have a divisor on Y with a map and we compare the two divisors, okay? So if there is a reasonable compatibility, I mean I can get into the details if there are questions otherwise I will just swap these under the carpet, then we save it as an admissible morphism, okay? And so the upshot is that we can define a subgroup of a group of finite correspondences between X zero and Y zero, so it's some condition, okay? It's a slightly restrictive condition, okay? Not every correspondence between X zero and Y zero extends to a correspondence in our sense between X and Y, okay, but anyway, you can do this and this produces easily an embedding of a category of logs move schemes into the category of log correspondences, you can actually show that whatever this thing is, a core then gives rise to a meaningful category so you can compose them, okay? And the composition isn't used by the composition of the underlying correspondences between the open complements of the boundary. Okay, Federico, I missed a question about topology. So the question is, does the DNAs that topology do you define also come coming from a city structure? Yes, yes, absolutely, yes, that's a very, yes, that's a fantastic question. It's a very important thing and it's a difficult result actually, so that you can, well, it's not difficult to show that this is a city structure, it's difficult to show that this city structure has reasonable properties, but it does. It's true, yeah. So it's a, well, it's something called a quasi bounded regular and complete city structure. So quasi bounded is a generalization of the Vox's notion of bounded city structure and we certainly use this properties in order to show, for example, things like compatibility of a transfer structure with a notification. So that's absolutely crucial. Okay, so there's another question by Remy. Why is it reasonable to take normalization rather than working directly with W0 bar? Oh, yeah, well, okay, that's actually a technical condition. So if your underlying scheme is not normal, then the resulting log structure that you put by pulling back from X is not solid, okay? So I mean, log geometry has this problem of having very exotic or exoteric mutations, but, so, and then you cannot compose. I mean, you run into trouble if you try to produce composition. So more specifically, the morphism, what I'm saying is that if you have the morph is from W0 bar to Y by the composition, okay, your example composition like this, okay, might not be an admissible morphism in the category of log schemes. But the outer arrow, so from W0 and to Y might be, okay? So it's a technical condition. I mean, you don't need to be too worried about it. All right, so should I continue? Yes, yeah. I take it as a yes, okay, I don't, because I'm not seeing the questions right now. Okay, good, so fantastic. So now we have a category of spaces. We have topology and then we can finally define what our sheaves are. So let's give a name to this. So the pre-sheaves over the category of log correspondences are called pre-sheaves with log transfer, okay? And thanks to the properties of the dividing is never topology and well, we can prove that sheification with respect to this topology respects transfers, okay? So I want to stress that this is actually a difficult theorem, okay? And in order to maybe explain a bit why this dividing is never if topology is quite peculiar, let me look at the representable guy. So Z log transfer of X, okay? So is it just that you need a guy? Sorry, I missed the question before. So probably in the previous page, I have a question where is the product the FS product? The product, okay. So here, when I write this product X underline times Y underline, this is actually the product of the underlying schemes, okay? So I'm not forming the product of log schemes here. I'm just forming the product of usual schemes. Then I build this W0N and I look at the resulting morphism back to X underline and I put a log structure on W0N so that the resulting morphism is strict, okay? So I put back a log structure on this normalization. Okay, so we can maybe talk about that maybe later, okay? So if you have further questions. Yeah, so. Okay, okay. Right, but that's at this point is just the product of schemes. All right, so what I was saying is that the, if you look at the Z log transfer of X, so you need a guy, this is not a dividing snavage sheath. You can compute the denies sheification of this guy and the sections over say some guy Y. And roughly speaking, this is given by correspondences. So W like in X times Y, such that W becomes finite over X after a log blow up. So in some sense, the very restrictive condition that I put on our correspondences, namely the fact that this closure is already finite over X is softened by the topology because if we simplify, then we are sort of allowing things which are not necessarily finite, but they become finite after some blow up in a very controlled way. Okay, so this, we also have a theorem that allow us to compute the dividing this navage homology of the sheath. Okay, and it's something like this. So it's a collimate of the strict these navage homology groups. So these are very often just the usual these navage homology groups, depends on the sheath F. Okay, but if F sort of comes from algebraic geometry, then this is just the usual is navage, essentially the usual is navage homology. And when you have to take the collimate over this category X smooth div. So this is the category of dividing covers such that the underlying scheme is also smooth. Okay, so it's a fairly reasonable categories in particular this Y, the underlying schemes of all these Y are smooth and this theorem, so this is actually a theorem and this holds under two conditions. So you have a bounded below complex of this navage of the body in these navage sheaths and your scheme X to begin with is in the subcategory that we denote smooth log smooth. Okay, so this means that our schemes, log schemes such that the underlying scheme is smooth and the boundary is a strict non-acrossing divisor. So this is really, these are really guys like X D with X smooth. Okay, so very nice, very simple log smooth schemes. So Federico, there's a question. Yes. Is Dinis subcanonical on the subcategory LSM of K? Snow, meaning without no, it's not, no, no, it's not subcanonical. It's not subcanonical and that's exactly what the point of my remark one, right? So for the schemes without transfers, for the site without transfers, LSM K, okay. For I's K, I don't know on the top of my head I think is not, I need to think a bit about this. But I think it's not, okay? Okay, so let me continue. So at this point we have all the ingredients that we need to build our category of motives. So we start from the category. So of course I will just let me just focus on the DM side. I mean, we have also the DA version of the thing. All right, so we start from the category of sheets with log transfers on log smooth schemes with respect to the dividing snavage topology. You have a canonical functor from the old guides. So the usual sheets with transfers on smooth schemes. So this is the W upper star functor. It's in use by the functor W that I introduced some slides ago from smooth log schemes to usual schemes. Then you can localize respect to the P1 infinity guy and the resulting category is what we denote by log DM, effective, the these K with coefficients in lambda. Okay, so it's, you have, you know, easily an induced functor from DM, which let's still call it W omega upper star. And yeah, that's it. So that's the category of log motives. I was promising you at the beginning of the lecture. Okay, so let me now say something about the properties and well, you can actually prove something with it. Good, so let me list some of the results. So as customary, let me denote by M of X the motive of X. So this will be the image in log DM of the representable guy, lambda transfer of X if lambda is our ring of coefficients. Okay, so first of all some easy, easy stuff. So the easy parts are the monoidal structure. This comes as a day convolution product as it happens for DM from the product on log schemes. You have my inventory squares and the other invariance under admissible blowups where the admissible blowups are precisely this. Well, that's actually, we need to work a bit but this comes from the dividing colors, okay? But I introduced before. Okay, so definitely less formal result is two. So as a factor, the motive of X is not just invariant with respect to the guy P one infinity but it's actually invariant with respect to PN comma PN minus one. So this pair, PN PN minus one is a log scheme in the obvious way, okay? So PN is a smooth scheme and then you have a PN minus one sitting inside PN as a hyperplane at infinity and this is a divisor, okay? So you can build a log structure out of it. And yeah, question. So is there a non-effective version of DM? You can form the non-effective version by inverting twists, okay? And the twist is the usual twist. So is whatever. So you start from the motive of the GM or P one with no log structure and you point at one and then you get a number one and then get everything. Okay, so you have a twist and you can form the non-effective version of log M just by formally inverting twist with lambda one. A less formal question would be, well, what's the relationship between this non-effective version and the effective version? And in order to answer to this question, I would need to have a cancellation type result like the boss' cancellation and at the moment is not available but people are working on it, okay? So let's see. Okay, so right. So why is two very, I mean, particularly interesting for me in this picture? Well, because of invariance with respect to AM and in usual DM, it's obvious, right? Because AM is just the end product, oops, sorry, just end copies of A one. And well, if you have motives of invariance with respect to A one, they are automatically invariant with respect to AM. But here, PN is not the end full product of this pair, right, of P one comma infinity. So you actually need to work, okay? And the way you do it is by, well, of course you do induction on N and then you have to sort of cleverly interpolate between P one times P one and P two using dividing colors, okay? So I find this quite amusing. Okay, so what else do we have? So we have Tom space. So you mean you're really using the divided Niznevich topology? Yes, yeah, yeah, yeah. Here I'm really using the dividing Niznevich topology to prove this result, okay? And actually it's, that's another amusing fact that, okay, so you can, you could build a category without imposing dividing invariance. So you can start just with Niznevich, stick Niznevich topology. But then instead of inverting just P one infinity, you can invert all the PN, PN minus one guys. Okay, and then you can ask, well, how far am I from getting my original category log the N? Well, essentially you have it, okay? So in other words, if you have PN, PN minus one invariance for all N, then you can show that you have dividing invariance. So you have invariance under the dividing colors. And on the other hand, if you have dividing covers plus P one infinity invariance when you get the invariance for PN, PN minus one. So in our paper, we put somewhere a table with the implications of a different set of axioms that you can use to build this, to build log PN. Okay, so what else? So we have, so Tom space isomorphisms and a version of Giesen sequences. And these are not quite as you might expect, but I don't think I have time for explaining this in details and I can answer two questions later. But let me first discuss the comparison with Wewelski's category of motives. Okay, so for this, I need to assume that K has a resolution of singularities. Okay, and the first result I can offer is the following. So let's take X and Y and logs move over K and assume moreover that the underlying scheme of X is proper. That's really important over K. Okay, and then I can look at the harm in log DN between M of Y and M of X and the same thing in DM where the open complements of a log structure and turns out that this is an isomorphism. Okay, as long as here you put in ease. And the topology is not irrelevant in this. Okay, so consequence of this is that you can study this adjunction between log DM and DM and you can prove that this is actually a localization. So the right adjoint is fully faithful and you can identify the essential image. So the old DM somehow where the complex is in log DM such that K is A1 local. So A1 local object. So A1 with trivial log structure. Okay, so A1 local objects in log DM are exactly DM. Okay, so if you want these are strictly A1 invariant complexes. Okay. I've got to recall a quick question. Yeah, yes, Mark. What does it mean that the X underline is proper? What does that mean? The X underline is just a scheme. X underline means the underlying scheme of the log scheme. Ah, okay. So, and then books then like you're getting nothing really different. What's the difference then? You do because you have something like the motive of A1 for example, that's not trivial. Okay, thank you. So only if you look at maps to something proper then you get that maps in DM, okay? But for example, you can look at Tom from M of X to I don't know, Omega one as a sheet from the big site. And this would just give you a homology of Omega one, okay? Omega one is not proper. Gotcha. It's like it's not representable by proper thing. Right, very good, thanks. Yeah, sure, but that's actually helpful, Mark, the question because if I, what about if I restrict myself only to the subcategory generated by M of X such that X underline is proper, okay? Then it's true, then you don't get anything new. The category is actually equivalent to DM, okay? So the difference is really in the no proper stuff. If you want, these are two by M of X where X underline is proper. So of course this is suggesting that there is no localization in the usual sense of the category, right? Because otherwise, you know, you could generate everything by the motive of smooth and proper varieties, but that's not the case, okay? And so either as a A1 local gadgets or as the subcategory generated by motives of proper things. And I also want to stress that this is an isnavige phenomenon, okay? So if you replace the isnavige topology, so D is here. If you replace the dividing isnavige topology with the Italian variant, then this would be false, okay? It would not be true that log DM et al proper gives you back DM et al. And that's sort of a useful, actually useful thing because of what I'm going to say next, okay? So, but anyway, so if you can use this comparison and inherit from DM quite a few properties, for example, a nice projective bundle formula and so on. But let's answer again to Marc's question. So what is the difference between log DM and DM, okay? So for example, let's look at the log differentials. So omega j over k, so these are the sheaf on the big site. And you can actually show that these are strictly P1 infinity invariant, D is sheafs with log transfers. It turns out that the transfer structure is a painful part of the story and to deal with is omega i's, okay? So if you are in characteristic zero, you can use a sort of shortcut. There is a work by Florence Lecombe and that where they will be inverse, will work with rational coefficients and then they just put the transfer structure on the differentials, okay? So, but we use the different approach because we wanted to keep working integrally. So we followed some work by Andre, Systematio and Kyru Link. And well, you can work a bit and then starting from their results, you can extend the transfer structures on differentials to the log setting. That's technical, but you can do it. Okay, and then you can prove that you have such an isomorphism, okay? So after is a direct consequence of this theorem that maps in log dm from the motive of x to this guy gives you the comology of x underline. So this is actually, you should be careful with notation here. So this is a comology, the usual the risky code, is never sure whatever, a comology of the underlying schemes and the log structure is completely encapsulated by the shift here, okay? So there's x, there is no underline, but here there is an underline, okay? So for example, for this something like omega x log boundary of x, something like this. That's the sheaf. Okay, so can I like have a couple of more minutes because of my bad connection and then let me comment on this, we conclude this remark. So there's a question about Daniel Diane Grayson. I ask, what if i is negative in the theorem? What if i is negative? What if i is negative? In the previous page, I guess. I mean, the previous page is negative. I'm not sure. This is for every i, that's just a shift. No, no, no, so I get it wrong. It's the other page, the second one. Okay, okay, so the i here, I don't think there is a problem with i negative. I mean, this thing would be zero. Okay, so let me just put i larger than zero to be safe. Okay? Yeah, okay, because in, I guess, I don't know, but in molecular emoji, there's issues with i negative. Yeah, yeah, yeah, yeah, yeah, absolutely, absolutely. So I don't know, he's totally correct. So let me put a positive, I mean, it was hidden in this i, j larger than zero. Both are bigger than zero. Okay, so both are positive, both are non-negative. Okay, good, so let me continue here. And this is a very amusing remark for me. So let's assume the characteristic of k is p. Okay, so it's, all right, larger than zero. And let's look at the constant shift z-mode p, okay? So you have the, again, our art in Schreyer sequence. So zero, z-mode p, g-a, g-a, zero. So this is actually exact for the dividing etal topology on log smooth schemes over k. And okay, so you can look at this guy, log dm, dividing etal effective with z-mode p to the coefficients and that's not zero, right? Because the z-mode p guy is non-zero. g-a survives, everything survives. Okay, so for example, and HOM from n of x to z-mode p is just the homology of z-mode p of the underlying schemes, so h-zero, non-zero. And you can even say more because the z-mode p is actually proper, right? It's just the same thing as the motive of spec k if you have z-mode p coefficients. So it's actually, this guy actually leaves or p to the r version in the subcategory generated by a smooth and proper, well, motive of x such that the x underline is proper. So that's, you can, you see that this, then the comparison that I wrote on the previous page doesn't work in the etal setting, okay? And then, so you have, for every r, you have a category of motives of reason and effectiveness in here with z-mode p to the r coefficients. It has the etal topology in it, and it's a very reasonable question for me to ask whether you have some kind of crystalline integral realization into something like d, d, c, r, z-mode p to the r, where r is something like the ring, okay? And if you invert, so if you speculate a bit, and you imagine doing this for every r, when you pass to the limit in a suitable sense, and you get a periodic version, and then inverting p, the target category would be something like, you know, the drive category of complexes of K, F modules was homology groups are F isocrystals, something like this. So that's of course, well, I know, but I would like this to be true. And yeah, so I was planning to say a couple more things, but let me just end with the to-do list. So more things that we can do in this work. Well, there is a work in progress on log sh, and again, also in progress, some work on the representability of Hodge-Vit homology, and then there are things which are rather questioned. So what is the comparison with the Milner-Ramachandran category of motives, which is built using the Votsky-Zetal motivic complexes away from the characteristic. So in inverting p and exactly the category of, the drive category of modules over the ring at p, and there is a very nice recent result by Shuzhi Saito, comparing reciprocity sheaves with our log motives. So that's also very interesting for me. And we have a very natural question about rigidity. So rigidity in the Susan sense. So what happens if you take Z mode and coefficients and prime to p again in Vietal setting? And then of course, you can try to be more bold and try to put a log structure on the base and then the dividing topology. So the fact that you are inverting blowups, admissible blowups in the boundary suggesting that the resulting category would have something to do with the rigid version of the A and the N, developed by Joseph Ayub and Alberto Bezzan and so on. So let me stop here. Okay. So let's first thank Federico for the nice talk. And so we have already some questions. So I'm starting with one by an anonymous spectator. What's the difference between log dm and dA? So that's log dm and log dA. Log dm and dA. So that's related to your 0.6. Yeah. So, okay. So I don't know if I understood the question. So you want the relationship between log dm and the usual dA? Yeah. Okay. Okay. Well, right. No. Not dA, sorry. Not rigid. Okay. So rigid dA is something that I would like to understand. I mean, is a reasonable, that's a very reasonable question to rigid dA. And, but in order to be able to discuss this, we first need to develop log dm over a log point. Okay. And then, as I said, so I expect this to be, I expect the two categories to be related exactly because in the dividing setting, we are, well, we do something that we do in rigid geometry, right? Well, because we have a close relation. Yeah. So it's something very, very close, right? So that's a very reasonable kind of comparison. But the usual dA, well, I mean, you have to first compare dA with dm and then, I mean, it's a comparison to dm and log dm. Yeah. Yeah. Okay. So we have many questions. I think then Mikhail raised his hand. So I'll try to, sorry. I'll try to give him the hand. Yes. I don't see him. Ah, yeah. Here it is. Okay. Now you can talk, Mikhail. Thank you. So how much do your results depend on the resolution of singularities? And... A lot. Gabbers resolution superities? So it does depend on the resolution of singularities, quite a lot, okay? So... So Gabbers version is not sufficient for a paper. I mean, Gabbers version, okay, so the problem is that I really want to work with integral coefficients, okay? So I don't want to invert P. Okay, but if you... Then will Gabbers version help or no? If I do invert P, then I suspect that nothing very interesting happens. So if I invert P, for example, I suspect that the category... Okay, so that's a conjecture, okay? So, okay, so let me write this. So for example, let me use log dm and the tile setting and ask me invert P, okay? Then we strongly suspect that this is, at least if I restrict to the... Yeah, let's say that I more or restrict to the proper category, then I strongly suspect this is just dm. So the point is that every time... Dm, et al, et al, et al, et al. I'm working in et al, I'm et al here, okay? So, well, essentially the point is that, okay, so all the examples that I know of things which are in log dm, but not in dm, they are all killed by a power of P, okay? So if I invert P, then I strongly suspect that I get back dm. So I actually, yeah, so that's why I don't want to invert P. But if you want, that's actually a good point, right? Because towards the comparison with Milner-Ramachandran categories, then this will be actually very good, right? It would mean exactly that we are very close to getting a category which is built in a geometric way and that inverting P gives you the usual motifs, et al motifs, and then it has new information exactly at P. Okay, so I'm trying to, so we have questions from many sources, so I'm trying to ask them. So there's, do the dividing topology contain cumia log et al? If not, why not? Okay, so... Yeah, et al. Okay, so first... Right, so log et al and first of all, the first question about log et al. So proper log et al monomorphisms are precisely the dividing colors, okay? So log et al maps are in the dividing topology. For cumer et al, well, why not? Well, essentially because we have another version which is the log et al version of our category. So in some sense, the cumer story is really an et al story, okay? And so that's why you don't see this in the dividing is nevish construction. We do have a cumer version. You can see the discussion in this in our paper. For example, I formulated the conjecture of like the rigidity version of sort of 0.5 here. So our log version of a Gabel's rigidity using the cumer et al site. And so let me specify this. So for example, okay, so the category of sheets on the small cumer et al site, okay, k is the field and then k is also for cumer. So k, k et with say z-mode and coefficients, this should be compared with log dm et al, say log et al, which includes cumer et al stuff. Okay, z-mode, okay? So this will be a comparison between the small site and the big site in our setting. This is a conjecture, it's not a theorem. I hope this answers to the question. And so you need finite coefficients for this conjecture. Yeah, that's finite coefficients, that's prime to p, that's NP1. Or in general, like find the coefficients prime to the characteristic of k. Okay. So maybe no questions. So Remy is asking, does remark one page one suggest that it's more natural to work with a more general definition of correspondences? Remark one at page one. 11, 11. Mark one at page one. Remy, you- 11, page 11. Remark one at page 11. Here. Well, yes and no. I mean, in some sense, the remark one tells you that you can't work with a restricted, like a smaller class of maps, which is better because you have better control. And then if you, when you're not losing, dividing the information, sorry, then you're not losing information. So we, okay, so but, okay, all right. So let me explain this better. We actually have another version of a category of correspondences in our paper. We call them dividing correspondences. And these are exactly what Remy was suggesting somehow, okay? So you start from your correspondences, but you don't ask the closure to be finite over X. You ask this to happen only after a log blow up. Then you sort of build a category of motives out of this and we show that the resulting category is the same. So you could start from a more relaxed category of correspondences and you would not produce a different category of motives. So let's discuss those in our paper. Okay, so Brian Sheen is asking, is it clear which objects need to be inverted to obtain log SH spheres with some sort of log structure? That's not, it's not clear. No, it's not clear. I mean, I cannot comment too much on this. Okay. So yeah, I will not say anything. So Mark Levin has a question. Maybe you can ask it directly Mark, if you hear me. Sure, yes. So I wondered, I mean, a lot of this started out with trying to understand wild ramification in class field theory, say over a finite field. So do you have some kind of uniform description using log DMA tall? Okay. Okay, that's a very good question. So I would say that we cannot see in right now the bound on the ramification out of our category. Okay. So in some sense, okay. So you have two types of objects here. Okay. So you have things like a motive of A1. And if you think about the motive of A1 as being the motive of P1 without infinity, okay. Then you are somehow allowing ramification without any restriction at infinity. And P1 comma infinity is sort of corresponds to the tame case as to tame ramification. So in some sense here, we see only the two extremes. We see the tame ramification. So with the tame part, and this gives you back Vm in some sense. And then we do see the wild part, okay. So by, for example, mapping into known proper things. But we cannot quite bound precisely ramification at the motivic level. So let me write, okay, let me write a statement, okay. So here. So let me write a following theorem, okay. So there exists a fully faithful and exact functor. Let's call it log. So this is here, this is due to Shuzhin. It's from a category of reciprocity in his Navy sheaves to sheaves with log transfers. Okay, such that log F is strictly cubing variant in our setting, in our sense, okay. And the comology of any reciprocity sheaves in their sense is on in log Dm. So here on this side, you have the filtration. That's sort of what we learn from ramification theory from the theory of reciprocity sheaves and so on, okay. But I don't know what to put here. So I can control the old comology of any reciprocity sheaves. Okay. So for example, also of known A1 invariant stuff, that's okay. I mean, they do leave in our category that, but we cannot write any finite level of ramification filtration in. So even by just sort of adjusting the log structure. Okay, so that's not quite, okay. So that's another thing that we are trying to do. So the answer is no, if you don't put a log structure on the base. So if you insist on using K with trivial log structure, then it doesn't help putting, adjusting the log structure, it doesn't help. Okay. It's just the categories insensitive to this operation. For example, changing, replacing a divisor with multiple. So you don't get like cycles with modulus then somehow in this category or do you? Sorry. Well, no, I mean, that's the best thing we can obtain is the formula I wrote here. So for example, you can write, so if f of x is like the, okay. So for example, if you can use as f of x, the sheaf, the Nisnevich sheaf, sending you to h1, etal, u, q mod z. Okay. That's the dual to the etal fundamental group, which has like, you know, no bound on the, whatever ramification, like a wider ramification without any restriction at infinity. Okay. So the comology of this guy is, can be captured in our category, but you cannot capture something like pi one x, d for this, no, not this guy. Okay. Thanks very much. Yeah. Sure. Okay. But on the other hand, if you put some, just one last comment. So if you put a log structure on the base, so if you're using K, you look like you use something like a log point. So K and then you take, okay, so something like N to K, and then you put a zero here. Okay. Then you can play with a multiplicity, okay? Because you have an extra sort of piece of information that can help you. And that's something that we are trying to understand right now. Oh, nice. Okay. Okay. Next question by Feng Zujin. Is there a psychoclass map with values in the log et alco, log et alco homoji? Log et alco, what does actually mean? Log et almoji, maybe I am not sure. Log et almoji that you define my log et alco homoji. Tell me, Feng Zujin, if I reformulate correctly. So what I expect is that there is, so what I want to, what I do believe it's possible to construct is some kind of crystalline realization from the et al version of log em to the category of coherent complexes of graded R modules, okay? R being the renewal ring, okay? And that's something that has to be, has to be written down, okay? So at the moment, I think that, okay, so for example, we need to complete our theorem that has to be written, but the mathematics should be okay, that we can represent also homology of hodge-vit sheaves, not just of hodge-sheaves. And then if you have things like log smooth scheme, such as the underlying scheme is also smooth, then this, you can use log-dram-vit complexes to compute log crystalline homology. And this would be representable in our category and presumably you can promote this assignment to a realization factor from all motives as you do in the usual way, right? I mean, you define it on representable guys, on the motive of x, and then you try to extend it, yeah, to every motive complex. But this is sort of all, I mean, in the air at the moment. Okay. Fangzu wants to add something, so maybe I'll give him a right to speak. Fangzu, can you? It's just, I mean, by log attack homology, I mean, the version defined by Nakayama, which... Ah, that's a good thing. Okay. Sorry. Okay, so Nakayama, okay. Well, I suppose yes, but I haven't worked this out, so I don't know. So the answer is probably yes. Okay, thanks. Okay, so next question, Shantilson, ask, do we know the synod algebra in this log setting? No, we don't. So there's also... Quick answer. Hola, Sande, ask. So he says that DM et al effective of the field with finite coefficient is isomorphic to the derived category of art in motive, say. Is there something analog to our log DM et al? No, but be my guest. You can try. That's a sensible question. I don't know. Maybe... Yeah, there are many questions. I have also some questions, but okay. Kirsten, as a question, maybe she can ask directly to you. This was related to Brian Shin's question, and I know you said you didn't know what to invert or maybe you said it wasn't obvious what to invert, but you knew it. But one reason to invert things is for a T, a duality, and you also have this log version of boundary. So the question is, is there a formula like the dual of M mod its boundary being the Tom class at the negative tangent space? Oh, that's a, okay. So that's a great, okay, that's a great question. So I don't know, but what I can offer you is something that I prepared in case I had more time. So let me just move to another page and I can at least tell you what the Tom motive in our setting is, okay? So in, all right. So what we... Good, that was one of my questions. I can, maybe let me start from, maybe let me... Just ignore what I wrote on the top of the slide and just let's look at the gizine sequence, okay? So let's just to try to understand what the problem is, okay? So in DM, okay, what you do have is that if you have something like a smooth embedding of say co-dimension C of say Z inside X, I'm here, okay? So smooth of co-dimension C, then the co-fiber of the motive of the X minus Z into X is the motive of Z twist C shift to C, right? That's a usual gizine isomorphism, okay? So that's not true in log DM. We don't have such a formula. So why we don't have such formula? Well, in some sense, the reason is because X minus Z is a bad object because X minus Z would be, okay, if we think of our gadgets as being compactifications of schemes, so this X minus Z has to be compactified in a suitable sense, okay? So we need to, in order to get a reasonable formula, we have to replace X minus Z with something which is more reasonable in our setting. So what is it? Well, that's what we, that's the answer. So we replace X minus Z with the following guy. So we look at the blow up of X along Z and we turn this thing into a log scheme by adding the exceptional divisor as a log structure. So let's suppose that X and Z have trivial log structure here, right? Just put X and Z inside log DM, but no log structure on that, okay? Then we can just produce a new object in which like genuinely leaves in our category that would be this BZX comma E. That's our, and then our theorem is that the natural isomorphism in log DM is this gizine map, okay? So you have the motive of the cone of this natural morphism, so from this guy to X. And this is, I now, equivalent to the top motive of the normal bundle of X in, of Z in X, okay? Which you can define and in some sense is the, it's probably what Kirsten was, or something like what Kirsten was asking about. So let me go back to this slide and look at the topological picture. So you have something like a trivial bundle and then you have a zero section. And then what we do is we take the blow up of AN at zero. We take the exceptional divisor and the log materialization would be this thing. So some S to N minus one. Then you have a natural map to AN with trivial log structure. The materialization would be a disk. And then the realization of our top motive for AN would be, would be this thing. So with this bundle, modular sphere bundle, okay? So we do have a theory of top spaces in this thing and it's, but it's a bit tricky, you see, right? So for example, you have something, formula like this one, okay? So the top motive of X times A1 is like something like Mx1, two. Does Kirsten, is this answering? Seven, seven, eight. Yes, I see what you're saying. Okay, great. So over question. So Tony Annala is again anticipating log S H, I think. So he asked, what about algebraic K theory, not K H? Yeah. What about algebraic K theory? What about it? So is it presentable in log S H? Okay, so yeah. That's something that we are trying to understand. Okay, so the real question is, what is K theory for a log scheme? Okay, so you can cook up a definition. I think the definition would not give you, yeah, right. So she defined it, but it's hard to sort of put her definition in our setting, okay? And we hope, okay, so we hope to be able to define a reasonable K theory for logarithmic schemes, and which is maybe a bit more naive version of K theory than what Nidziol did. And we expect this to be representable in log S H. Okay, we are working on this right now. So we are working on it. But the honest answer is, I don't know. Okay, but of course it is a problem that we, I mean, somehow it's a question that we ask ourselves and we don't quite know what to do. So Remy has a question from Boban de Bruyne, a related question to Mark Leving's question. Do the et al fundamental groups with brackets in this sense look more like 10 fundamental groups or a full et al fundamental groups? No, I mean, somehow you get the full guy, but you don't get the verification filtration. So it's like, I'm sort of back to this slide, okay? So here the, at least the ability, I mean, of course I'm talking about the ability and I is version of it, so not the full et al. Like, I mean, I'm talking about classic theory. So the ability and quotient of by one. So here also I should have written a billion. So this guy, f of X, so this is a Niznevich sheet which assigns you to this guy to H1UQ mod Z et al. So this would be the dual. So like by one et al, U, a billion and then dual. So this is a reciprocity Niznevich sheet, okay? And if you believe in Shuzhi Saito's theorem, then this guy produces an object in our category. So produces a motive in our sense, okay? And it's full homology can be recovered as a home group in our category. Of course, this is not like the optimal result because you would like to be able to control the ramification, right? So in some sense, we are really in the extreme, we have two extreme cases. We have the tame guy, which sort of lives already in the Voskis sense, in the Voskis world. And then we have the wide ramified guy, but we cannot control its ramification. So yeah, that's sort of the drawback. Remy, does this answer to your question? Maybe wait, I can give him the right to speak, if he wants to. Yeah, I was probably asking something more naive, but this is fine, thanks. Okay, great. Okay, so of course there are so many questions that we can ask because we have a classical setting and what things, so Mirail Bondarko is asking, do you hope to obtain some six-functional formalism in this log setting? I do, but before that I need to, we need to develop a theory over Bayes. So we need to define DDA over any long scheme. That's difficult, okay? So, and it's also difficult to think about the six-functional formalism because, well, for example, we don't expect to have a localization sequence, right? So, sure, many of the tools, kind of a modification of this localization, I don't know. Well, you need to modify it suitably. Like for example, using the, I would say, using as input the Gizin isomorphism that I showed before. Yeah, okay. So yeah, I mean, that's of course, I mean, it's a reasonable thing that we, well, it's not been written down yet, and we are certainly not working on it right now. But it's, of course, I would like to have it. I would like to have a full six-functional formalism in this setting. Four years to come then. Are you still there? Sorry, sorry for this. I said four years to come. Four years to come. Yeah, four years to come, yeah, four years to come. Okay, so now maybe I can ask, so one of my questions, just one. So in Wawiewski's theory, there is a comparison between DM Niznevich and DM CDH overfield and over for SH. So you explained that you have some environments on the admissible blow-up, say, but what about a more complete comparison between Niznevich and say, I don't know, D-NIS and D-CDH or something like that? Yeah, so the variance that we have in the category is, okay, so if you allow a solution of singularities, then we have the cleanest result. And then any proper morphism, which induces an isomorphism on the complement of the boundary, will induce an isomorphism on the motives, okay? So... We've all revoked resolution of singularities. We've, okay, so if K has, let me write it as a resolution of singularities, then let's consider f from x to y proper, such that so x minus the boundary of x is isomorphic to y minus the boundary of y, then the motive of x would be isomorphic to the motive of y. Yeah, so that's exactly answering the question. Yeah, so that's what you... Of course, this means essentially how do you use a solution of singularities, right? So you dominate any such map with a tower of blow-ups along nice centers, okay? And the centers are all contained inside the boundary, okay? And then you can further sort of dominate this... This is actually a result of Niziole that you can, so refine log blow-ups, okay? And then, yeah, and then sort of boils down to the dividing invariance, which is built in the category. So this is true in log dm dNiz, okay? Okay, good. Yeah. Yeah, and so that's a pretty... You don't want to use... You cannot want to use De Jong because you don't want to invert P, you know? Well, I could invert P, but somehow I think it's boring. Yeah, okay. Same, you could try to use H topology, but it seems that result will be the same. It will destroy... Yeah, exactly. It will destroy the interesting P information. So, no, I mean, we are... Yeah, somehow we have to... No. Oh, another question that came to me. So you do have a homotopity structure on the log dm or not yet? Yeah, not yet. But so I know that Alberto Medici, so who was actually listening to the talk before, he's working on this. So, but yeah, we, of course, we expect to have a homotopity structure. So probably by working out sheaves with log transfer or something like that. Yeah, I mean... Okay, so we know that local objects are strictly... Okay, so we definitely don't have the analog of the Voskis theorem saying that if you have a sheave, which is a one invariant, then the homology is strictly one invariant. So you have to sort of work to begin with strictly p1 infinity invariant sheaves. Okay. And yeah, I mean, I suppose that this is... In log dm effective, this is probably the heart of the t structure, of the homotopity structure. But there's no obstruction for the Voskis theorem to be extended to that setting. You mean the Voskis theorem, you mean the fact that f... I mean, how much p1 infinity invariant would imply strictly... Strictly p1 infinity invariant. Things like that. I actually don't, yeah, I actually don't believe it. I actually don't believe it. Okay. Okay. So I think that close the series. Thank you so much. And I go, thanks again for, thanks you for the nice talk.