 Thank you very much for the invitation. When Luke was around 60, he often told me that he became too old and he could not work well anymore, but after a time passed, he became more active and more younger. And I hope that he will become more and more active and younger and younger. Luke is like the father of all mathematicians and that is a great thing. I respect him also like my father and I am thankful to Luke very much. Today, Luke is the founder of the logarithmic geometry and I talk about the log geometry and logarithmic varieties. So I use the document camera, I hope it works. I am not sure. I hope I myself cannot see so well, but I hope you can see this. Okay, I see now. This is a space with sheath of rings and log scheme is space and sheath of rings and sheath of monoliths. And this scheme was started by the great teacher, Lutandik of Luke, and the log scheme was started by the great student of Lutandik, Luke, and Conten. And the theory of log schemes, log geometry was started. Luke was very much excited and probably he was happy that he is making a new world of algebraic geometry like his teacher did. And so we discussed Luke and I shared many dreams on logarithmic geometry. And I think the logarithmic variety was already in the dream from the early days of geometry. And I am happy that today I can talk about logarithmic-carberian varieties after 30 years. So here now there is a nice book, a nice textbook on logarithmic geometry, which was written by Sir August. In the first line of this book it is written that logarithmic geometry was developed to deal with two fundamental and related problems in algebraic geometry. One is compactification, one is degeneration. And concerning how compactification is important and how degeneration is important, there are two famous, there are famous golden sayings and famous observations are introduced to these. And for the compactification, a golden saying is by Algero Vistolo, working with non-complete original spaces like keeping a change in a pocket with holes. This golden saying is often quoted by Abramovich in his talks. And for degeneration, the king observation by Patec Saito is that in the story of beauty and beast, beast, ugly beast became a nice man by the love of the girl. And in the degenerate object, ugly objects in algebraic geometry become a nice object by the power of the log. And so the power of the love of the girl and the power of the log are similar. So then here is the mysterious coincidence of the letters. And so by these important observations and golden saying, we can see the importance of the compactification. Who is the first one, Algero? So then these things are, in the case of modular space, these two things are connected in this way. So the modular space is a set of space of nice objects in usual geometry, but it is not compact, often not compact. But then we can compactify the modular space by thinking about nice objects in log geometry. These nice objects in log geometry can be ugly in the usual geometry, but they become nice objects in log geometry. And today I hope to talk about the modular space, to talk about the modular space of Arbelian varieties with TEL, and then Laun has a nice theory of horizontal compactification, charismatic horizontal compactification. And then today I give the new formulation of joint work with Kajiwara and Nakayama, in which the horizontal compactification is understood as the space of log Arbelian varieties with PEL structure. And then log Arbelian, usually the degenerate Arbelian varieties have no group structure and rather ugly, but it is nice here in the log geometry and they have group structure. And here the PEL is the polarization is the endomorphism ring and L is the level structure. So the endomorphism ring is fixed or something like that. But the endomorphism ring in this is a ring because the ring, this endomorphism preserves the structure of the Arbelian variety. And then we can have a ring because two endomorphisms can be added because the log Arbelian variety can have a group structure. So without a group structure then it is very hard to consider the endomorphism of the degenerate Arbelian variety. But we can now have such a nice object in the logarithmic world. So it becomes very, very natural and nice. So the formulation becomes very, very transparent and nice before you use log Arbelian varieties. And at present our work is only the interpretation of the work of Brown, but I hope that in the future the log Arbelian varieties will have good applications because the theory becomes so transparent. And then in the case there is no endomorphism then the theory was made by Farting and Chai in 1990. And also I hope to say that Fujiwara had also some nice work on the case of PEL type structure for Arbelian varieties and for the other part of the work. But unfortunately this work is unpublished. It was done around 1990 but unpublished. Then I just introduced the rough idea of the log Arbelian variety. So log Arbelian variety over FAS log scheme is a contravariant factor from the category of FAS log schemes over S to the category of Arbelian groups. And so it is naturally the structure of the group structure. And if K is a completely discrete variation field and AK is Arbelian variety over K with semi-stable reduction, then AK extends to log Arbelian variety over OK. Here OK has a standard is endowed with a standard log structure and AK extends to a log Arbelian variety over OK uniquely. And this semi-stable reduction, how to treat semi-stable reduction nicely was the starting point, the motivation of the creation of the log geometry in the usual Arbelian variety in the case of usual Arbelian geometry. AK extends to a robust scheme with semi-stable reduction but it doesn't have the group structure. So the group structure cannot extend to such schemes but the Arbelian, log Arbelian variety has a group structure. And for example, if AK is a Tatum elliptical curve with Q invariant Q, then the A is if we restrict the Functor A to the FS log schemes over OK module MK to the N, so over this Arltinian quotient of the OK, then this restriction can be written in a very simple way. It is a quotient of GM log Q divided by the power of Q. Here Q is a Q invariant. And this presentation is the presentation of the Tatum elliptical curve, an analytic presentation of the Tatum elliptical curve in the analytic geometry over the field K. But that can be treated in an algebraic way, not an analytic way, an algebraic way in this log geometry. GM log Q is a part of the log such that Q to the A divides T and T divides Q to the B in the log structure for some AB. And then we divide this part of the MT group by the power of QZ. And this is very similar to the presentation of the Tatum elliptical curve as a multiplicative group by the powers of Q in the analytic geometry over QK. And here the Q is the element of MK minus 0 and it is regarded as a section of the log structure M of the spec of K for the standard log structure. And then for log structure, the log structure is the log structure C for monoid of the log structure is regarded as the extension of the unit group. So we have a bigger unit group and this is a monoid, but this monoid is embedded in the group associated with the monoid. And you have also a map, a multiplicative map from MT to OT. And then here for these schemes over OK Mojiro MK, Q is nilpotent. And if n is 0, then Q is 0 here. But you can, in this case, but Q is put in the log structure and then Q becomes invertible. That is, we are making a nilpotent element invertible here. And in the case n is 1, we are making 0 invertible here. So we can make 0 invertible in the log geometry. And by such thing, then we can have this group structure of this degenerate of the Arbelian variety. So a log geometry is such a theory to make 0 invertible. And now I put another comment that a log Arbelian variety is a log algebraic space in the second sense. The usual algebraic space is something which is covered by a usual scheme by a termorphism. The morphism here is a relatively etal. X is regarded as a functor, but it's relatively it is representable. This morphism is represented by a termorphism. And U is also a representable and you have such surjection. That is algebraic space. The algebraic space is something which is covered by a representable object by a termorphism. And the log algebraic space in the second sense is a similarly a functor covered by a representable object by log etal morphisms. So etal morphism is replaced by log etal morphism. A log Arbelian variety is such a object. And the log algebraic space in the first sense is such a functor x covered by a representable object by the classical etal morphism. And this is also equal to the algebraic space with the usual algebraic space with an FES log structure. So there are two things. First sense and second sense. And log Arbelian variety is a log algebraic space in the second sense. Now I don't explain more about log Arbelian variety if I explain too much then I will lose time. And then now we go to the problem of Mojira with PGL structure, polyporealization and endomorphism and level structure. So then in such theory, we fix a semi-simple Arbelian algebra over Q and with such an operator, it's two four locations identity and it changes the order of the multiplication. And then we fix also. So when you say log etal, you mean the kumer log etal or the more general log etal? This is general log etal. Yeah, yeah. I think so, yeah. Sorry, I forgot which one was better, but I think that all log etals morphisms used in my memory. So I think we have also another question from Bruno Padotto. So Bruno, do you need us? Sorry. We have a second question from Bruno Padotto. So Bruno, do you have a question? It was a mistake, I apologize. Sorry, sorry. So then in the Mojira, so I don't explain all things because there are too many things to be put, just I explain the main part and the satisfying certain conditions including this size skew symmetric non-degenerate by linear form. And then another, G is the group of similitudes for this pair, V and Psi. This is the algebraic group over Q, yeah, algebraic group over Q, reductive algebraic group over Q. And then by some, I don't explain some condition, but under some condition then we have a similar data for this algebraic group and then we have associated reflex fields. And then we can start the Mojira. But this is the usual things in the theory of PPL structure. I hope this is something we'll know. Then following the usual story, we fix the structure of V, which is the order in V subring V, which is stable under this star. And VZ is the integral version of V, VZ stable Z lattice in V, satisfying this condition. And then we fix a set of prime numbers here. From now on, I follow the notation of run in his work. And this square is a set of prime numbers without bad primes. Bad primes contain some primes for which this integral version of Psi, Psi for this integral structure is not perfect pairings at P, at the prime. Such primes are bad primes. Maybe we also remove some bad primes. So then now the notation is that this localization of Z around this square is the localization of Z by inverting all prime numbers which are outside the square. And we have the localization of this integer ring of the stiff reflex field by this square, which is just a cancer product. And Z hat square means the product of all ZL for L, prime numbers L, which do not belong to this square. So that is, this is not the for L for in delta. This is the usual notation for people in the study of Shimura varieties. So if you put such notation primes in upper place, then we are removing the searching. And if you put lower place the prime set of prime numbers, then we are keeping all these things like this. We fix the open compact subgroup H in this group because following this integral structure, we can define the Z hat square points of G. And we assume that it is neat. I don't explain the meaning of neat here. By this neat condition, we have nice modular space. And so then are Mojirae punctures. We have a Mojirae puncture. So I think to, is it to 20 minutes, 20? I forgot the end of the time of this talk. It is until 20 or, it's okay, 20 or, yeah. So now we consider the Mojirae puncture. We consider three Mojirae punctures. And the first one is just Mojirae puncture for Arbelian varieties. And these second and third are punctures for Mojirae punctures, Mojirae punctures for log Arbelian varieties. And they are all punctures from the FES category of FES log schemes over this localization of the integer ring of the reflex field and to the category of sets. And these are FES and less F bar S. So here we first define F1, F bar, and F bar, sigma is, comes later. And this sigma will become the toroidal compactification. But then I have to explain sigma, cone decomposition. This sigma is cone decomposition. But F1 and F bar can be defined first in a very, very simple way. That is, we can connect, we can use the same presentation of the definition of F1 and F bar just connecting by less. Because our logarithmic Arbelian varieties have group structure. And then we can define the endomorphism and the level structure in the simple way. So, then Arbelian, so the FES and F bar S, less F bar S is such A and U and A and P. U times the endomorphism, A times the level structure, P is the polarization. All those things can be defined nicely by using the group structure. And then, so the A is the, here A is the Arbelian scheme over S and less logarithmic Arbelian varieties over S. And U times the endomorphism from BZ to end A. Here we are using the group structure of A in the quarter, because endomorphism means homomorphism for the group structure. So we are using the such group structure for this. And then A times the level structure. So this is the isomorphism between this standard object tensor. Here this is the square part, removing the square part of the state module. So the state module with the upper square is the L-addict state module for L, not in the square. And it is the inverse limit of the kernel of the multiplication by N only, where N ranges over all integers feature prime to square. So here we are using the group structure and the multiplication by N on the log Arbelian variety. And so just like the case of Arbelian variety. And then the level structure is the isomorphism module H. So we H is acting here. And so then H is acting here. Then if two isomorphisms are connected by H, then we regard them as the same level structure. And this is, and then this is the busy isomorphism on the pro etal side. This is the busy isomorphism on the pro etal, this pro log etal side, or S etal, or S log etal. And finally, P is the polarization. It is a homomorphism from A to the x-shaped of A and gm log. This is also defined by using the group structure. And so this polarization, I don't give the precise definition of the polarization. It is such homomorphism, satisfying some condition. And then these three things, A and A and P should satisfy some compatibility. So this already defines the definition of Fs is well known. That is the Arbelian variety with PL structure. And here the log Arbelian function for the log Arbelian varieties with PL structure is defined in the exactly the same style. So the definition is very simple. So then actually one of the main result is that, if F bar is, I will explain this sigma part of F bar, but if we don't think of sigma part, then F bar is, F bar sigma is coincided with, will coincide with the toroidal compactification of run. But F bar is bigger as a factor and it is a log algebraic space in the second sense. This is the fact, this is the log algebraic space in the second sense is one main theorem around here. So then the remaining part, I still have a big time, but I'm almost finishing my story because the definitions are so simple. And then, but still for this sigma part, I need some complicated story. So then this part will have time, need time, I think. So, but roughly speaking, this sigma part of F bar is element of F bar such that all element point of S, the local monodromy of A at S belongs to sigma. And I hope to explain this more precisely. But I am afraid that my talk finishes soon, because the story of sigma part is not so simple. Yeah, so the local part is the local monodromy. So this part is the part of the, we are discussing the local monodromy type of local monodromy. And then the local monodromy of the Aberian variety, a log Aberian variety is as follows. The usual Aberian variety doesn't have local monodromy. This local monodromy appears only for log Aberian varieties. And then for point of S, small s of the point, small s, we consider the separable closure S bar of the point S. So with the inverse image of the log structure of S. And so S bar is now, we consider the log Aberian variety over S bar to think about the local monodromy of A over S. And then we consider the log Aberian variety over our spec of separably closed feed. This has some FS log structure. Then the Tate module, Tate module, this square part of the Tate module has a filtration, weight filtration, W0, W-1, W-2, and W-3, 0. Such that, and then this quotient of W0 by W-1, as group W0, has some integral structure. And group W-2 has also integral structure. And then group W is the tensor product of this integral structure times this square, and group W-1, W-2 is similar. And then such things then are for element of the pi-1 log, log fundamental group of S bar, which acts on the Tate module. Then we have for element of the log fundamental group, E-1 square is 0. And then the action of the E-1 factors in this way, it goes to group 0 by the canonical projection. And then it goes to group W-2. And then group W-2 is embedded in Tate module. And so the map goes like this, E-1 goes like this. And then furthermore, this pi-1 log S bar is the completion of this integral structure. And it is if S bar is characteristic of the field of S is of characteristic 0, it is a profite completion of this. And if the characteristic of S is P, then this is the completion of this for the non-P part, in the case of characteristic. And so this is a dense subspace of this subgroup of this, which has this finitely generated Arbelian group. And if it is in this, then this action E-1 keeps respect to this structure. Such thing happens in the Arbelian variety. And in the case, K is a complete discrete variation field and A-K as before. So A-K is the Arbelian variety over K of semi-stable reduction and A is a unique extension log Arbelian variety over O-K. And if S is a closed point and S bar is a separable closure of the closed point. Then this pi-1 log fundamental group, pi-1 log S bar is the galore group of the maximal extension of K over the maximal undamaged extension of K. And so then such story is very well known for the Arbelian variety of semi-stable reduction over complete discrete variation field. And then this state module is can be taken over for A-K and for A are essentially the same, they have the same state module. And then such story is well known for the Arbelian variety with semi-stable reduction. And then, for example, if A is the Q-tate curve, then we are considering the story of the special fiber of the log Arbelian variety. And then the state module is the same as the state module of the unique fiber. And if A is the Q-tate curve, that is, this special fiber is written in this way. And then the group zero is the power of this Qz, Qz, which is as a move to Z. And the group minus 2 is the search, also is as a move to Z. These integral structures are understood in this way. And I am just, maybe I am omitting the, I am sorry, something like this is actually necessary here. But I am omitting this thing to make things description simpler. We have to fix some, no, I am omitting that. And then for this story of Sigma, we fix Sigma, the family of condi-compositions. This is such a thing as follows, exist. This is written in the paper, work of Rang. Existence is written in the work of Rang, but it is reduced from the old work of Manford and other people on the toroidal compactification. And they consider condi-compositions over complex number field. They consider toroidal compactifications over complex number field and they consider the condi-compositions. But in the work of Rang, because he did not update module, he did not have the group structure of the designate of again variety, so the combination becomes a little complicated, more complicated. Here, the combination becomes simpler, essentially the same as the Rang did. Here, these are families of condi-compositions and W ranges over binary generated BZ module. And he is the surjective homomorphism of this standard one to W after tensor Z hat square. This is the surjective BZ module homomorphism such that I don't explain this part so well. The kernel, the annihilator of the kernel for this site kills each set of such conditions should be put. And then for each such pair of W and G, we should have the condi-composition of the space of positive semi-definite symmetric binary linear forms W tensor, W tensor, R, with rational kernels such positive semi-definite symmetric binary form has the notion of kernel as the subspace of W tensor R and the conditions that it is rational. And we have to put more conditions here that all cones should be rational and a little more condition, but I don't explain the details. And the one main property which we require is that this condi-composition for W and G H, H in the open compact subgroup H is the same of the condi-compositions for W and G. And for surjective homomorphism from W to another W prime, then the condi-composition given for W should be related to the condi-composition for W in some simple way. And then the factor F bar sigma S is such a yota, sorry I forgot, this is in F bar S. And then the conditions that for all S, point S for the special fiber for the restriction of A to S bar, the separable closure of S. And there exists for each point there is a sigma and then here the map induced by yota, so the closure, the level structure is the isomorphism from this to to to to R non-T, T, T, R, E to T square E with this, this is the level structure. And then we have a map to group W0 and it is a group W0, Z times Z, Z hat. So then we have a map G for this integral object and then now we are using this group as W here. So then we have a condi-composition associated on the space of semi-definite varying in forms on this space. And then the condition is that for each S, point S, we have some element of cone on the in this sigma, such that for all A in this subspace of, so we have now monoid inside the integral structure of the log fundamental group. And then we have the A' from group, for A we have A' from group W0 to group minus 2. And then the condition is that then we have by this A' associated to A-action of A-1 then it preserves the Z structure. And then we have a map from group W times group W0, Z, Z to group W0, Z times group W minus 2, Z by this E' and then we have a pairing to Z by this polarization. Yeah, polarization gives such a map and then this belongs to this, the condition is that this map is a Pyrenean form on this space and this condition is that this map for all A, this map should belong to sigma. So that is we have a map from this monoid to this sigma in such a way. Yeah, yeah. So this is the definition of the sigma part and the main theorem is that this f bar, so we have the f and f bar sigma and f bar and f is the usual monoid factor for Pyrenean varieties. So this f is actually the modular space of the Pyrenean varieties is PEL type and this is a classical object and this f bar sigma is the log algebraic space. So f bar is the log algebraic space in the second sense and f bar sigma is the log algebraic space in the first sense, it is an algebraic space with an efface log structure and this algebraic space is actually the toroidal compactification of land and like land deed, sometime we can have also not only large space but projective modular space, representable toroidal compactification. And such thing can be proved but it is already proved, it is proved by land and such thing is, but the main good thing about the proof is, the main thing is that the definition becomes so simple at least for f bar, the definition is perfectly the same as the case of usual Aperian varieties. And the sigma part is a little bit complicated but we can still use the group structure and such, so the group structure makes all things simpler and the proof of this main theorem is also becomes easy. And using the Arting criterion or log Arting criterion or and for example the case, so from the partings try to line the without endomorphism ring to the case with endomorphism ring was a hard process and that was a very, very hard process. But this becomes very simple because this is why just we can prove the space that the home space from between log Aperian varieties A and A prime is representable over the base and so then by using this is proved by some Arting criterion. And then this is relatively, so if we take as the base the modular space of without endomorphism as the base and A and A prime, the universal objects over it then such thing can give the represent such theorem for the, because if we use that this is the algebra space then this case becomes algebraic space, this becomes if we know that this is algebraic space, then we can deduce that this toroidal PEL case is also algebraic space because it is relatively the home space, this one is a relatively the home space over the, over the, over the case B equals Q, because we are, because we are considering endomorphism ring here. So just this is relatively home space over this and the home space is representable so we can deduce the representability of this from this. So from this type to run is becomes simpler. Yeah, yeah, so, so I had to present my this method gives only a new interpretation of the theory of runner, but I hope that in the future that this theory of log Aperian varieties will be useful in the theory of similar varieties and some other. Other stories. Yeah, I, I have still several minutes but I finish the, the stories so I finish here. Yeah, yeah. Thank you very much. Thank you very much. So we will take a few questions. So if there are any questions in the. Okay, so maybe Sophie. Yes, can you see the, can you see the stratification of the toroidal classification in a simple way. Oh, thank you. Maybe it is. Oh, yeah, that's just, just how, how, how is this, this W. Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah. That is like, yes, so that is, so if you consider the local Arachnian case, then the log Aperian varieties is like, like this. This is, this is generalized to some G log Y, Y, Y, Y are for Y is to the end and such is a semi-Aperian, Aperian scheme and so over Arachnian base we have always or in the case of the, yeah, yeah, yeah, yeah, so the type of the log Aperian varieties is like, is, is, log Aperian varieties over such log point is warlog Arachnian, Arachnian case, this is, is classified by using the, the dimension of this Aperian part, Aperian scheme part and, and this torah's part. And this rank Y and such rank, this is the isomeric and T is the isomeric to GM to R and these are Aperian, Aperian scheme and, and so, such a thing appears if we consider semi-stabilization over complete risk variation. For example, such R and such dimension of this P becomes the, the, should give us the stratification of the Proidal Compactification, I think, yeah, yeah, yeah. Thank you. So, Luke has a question and you have two questions from the chat, so maybe your question is, we finish with your question. So we take two questions from the chat, so how do we, so I have a question about, in which category do you take your text in your, for your slides when you define your F, I think F, sigma, you take some text, yeah. Coralization. This is a category of sheaves on, of the sheaves of Aperian groups on the, on the, on the, on FES, if, if we consider log Aperian variety over S and, and this sheave means a third policy in the classical sense, yeah, yeah, yeah, yeah. This is another question that, okay, I don't understand what I am reading. So do you expect to generalize from log Aperian varieties with the best sutures to a notion of log Aperian varieties with odd cycles and give a reformulated modulite function of the Ridal Compactification of the general structure varieties in the sense of, by the principle. Yeah, there are similar varieties of coach type of Aperian type of some general, general, more general similar varieties than the similar varieties for PA type and, and your question is about it, the possibility of the generalization to, to more general similar varieties. Yeah, actually it's not my question, it's a question in the chat. Yeah, you can, you can answer the question you think too, yeah. Yeah, yeah, yeah, yeah. So, yeah, we have not considered such a problem. And, and so, so there are some interesting things which should be considered by using log Aperian varieties. Maybe a level structures for P not invertible for for any level structure for N not invertible or, or, or, yeah, but this is already difficult for Aperian varieties, but, but yeah, yeah, yeah. In the such more general similar varieties should be considered related to Aperian varieties, but, but not included in the similar varieties of PA type. So maybe it is nice to think about them. Yeah, yeah, yeah, yeah. We have not yet considered such thing. Thank you. So, Luke has a question. Yes, some time ago, I think it's Nakayama Chikara who asked me a following question. You're over a CDVR and you have a commutative flat group scheme locally of an anti over this company DVR and suppose that on the on the special fiber, it is similar billion. Then it is it's nearly a billion everywhere. The question and the answer I do so at the time it seems that you didn't know the answer. And then I asked it repeatedly found some some arguments for that. So, showing roughly that if you have something really important in the general fiber and then taking some some closure then you get something also important in the special fiber. And I would say this has something to do with your work and the criteria and your the criteria you mentioned at the end. So I hope Nakayama knows this well. I hope in these days Nakayama is much cleverer than me. But is it in the art criterion in this work or not? Is it in different direction? Is this my useful for what is my useful or what you discussed or not? Is it different thing? I am sorry I cannot catch what you explained so well. So, so maybe we I will ask Nakayama. The same everything is different organization so that was it. It's an open property somehow. I could not follow this thing so what you are saying so well. I hope that you explain this later slowly because I have a second question. Maybe a much older question. So, now you have these logarithmic schemes. So, the group structure. So, is there a way to better understand or understand in a simpler way? Gotentic monotony pairing in SGS-7-9. In the SGS-7-9 it's confronted prime by prime and I think there are so of course you only have the neo-modern. You don't have this. I think it should come out on what you. About that conjecture, my student Suzuki-Solo, he is a good expert but I am not the good expert. I am sorry that Suzuki is much better than me. I have a third question. So, very long ago you wrote a beautiful preprint and published and maybe you are finished and maybe incomplete. So, long due to the NACRA. So, here a long due to the NACRA for these longer billion schemes. No, no, no. That should be. I hope to complete it soon. Because, yeah, the total points of a billion varieties, log-a billion varieties are so important. And I hope that log-crystalline theory of log-a billion varieties are important and the total points are important. And then such things should be considered well. Yeah, yeah, so I hope to complete that paper soon. Yeah, thank you very much. Okay, so I think after you have a question, so after you can ask your question, if you unmute your microphone. What was my question? Yeah, so the question was, what's the nature of the embedding of this F-bar, sigma and F-bar? Is it the closed immersion and open immersion? That is like a log-smooth, or some blowing up. So it is inclusion, but if we consider it as a scheme, in the category of schemes, then it is like a blowing up. Okay, thank you very much. So I think we have no other questions, so let's thank the speaker again. Thank you.