 Thank you very much. In mathematics, there are talks about theorems and talks about conjectures with evidences. I think you like one and two, but you don't like three so much. But I am afraid that my talk is concerning this third subject. I got old and my brain already collapsed. I lost the ability to study those two, one and two, but because my brain already collapsed, everything became the same in my brain. I can correct anything, any different things as the same thing. This is a really big advantage to become old. So, Vojta compares height functions. I will explain. So, here we have a kilometer, and then here we have... If it is a number building and if it is an algebraic variety, I am allowed to say this is a disk. So, if we assume that we are using it as an open object of an algebraic variety, and then in these normal functions, functions from C are explained to VC. And then this is an algebraic function to reverse such that if C is not contained in the boundary. This is about the number theory and this is about the neighborhood theory. And the subject here is to study the question, how open F has similarity or 4 or 0 on C? And that is how often FZ belongs to the boundary. And this is the neighborhood theory. Vojta compares the number theory and the neighborhood theory, compared to many structures comparing these two, and there are many, many deserts are known. L is a polarization? L. Oh, yeah, yeah, sorry, sorry. L is a diamond. L is a... A diamond on reverse. And then, yeah, we can show this on this. We can compare... What is R? TfLi. TfLi. This R, sorry. This R is non-negative diagram. So R moves. R moves, yeah. You guys are... FZ is not contained in the boundary, not contained in the boundary. Not contained in the boundary. Yeah, thank you very much. So we can compare the two. And then we also... Yeah, and FZ, FZ is a set of isomorphism classes, and that gives, over... These are so-called videos, I mean. People are... This is a... This is the structure with that same thing. Notation, morphism are called a CO-alert, horizontal map. And searching up with this condition... We respond to the variation of the hierarchical order that comes in right here. We go with that here. But here, this is a period. And everyone, and then it's like that. Education is defined. But this is not to complicate about some personal conversion. But here, we don't talk so much about this one. I don't explain so much. This is consisting of we report into all bits, all bits, in the theory of the variation of all the structures and all bits, with run-quits. This is, we add some points, all the important all bits, to exceed them. We don't discuss so much. And then, so the variation is that here, you have a variation of h. And then h, we respond to in the relation that it is h and this h, f is a period map of h. That is at each point of y, then if you restrict this h to that point y, then a small y, point of y, then you get the point of the h structure at each point. And so, then you get such, that is for the period map that is at each point. Why you have a usual h structure? This is the variation of h structure. So at each point you have h structure. And that is, but there are similarities. Because you have a log designation, but if you use its present, that becomes a true model. And then, so the lever-in-the-theory here, the TCC is an analog lever-in-the-theory. But so the subject here, subject here, subject here, subject here... How open are f's? This is the theory of this relation which is structure, so this is some nice subject here. And then I hope to define those high functions. Ah, yeah, that's right. And lambda, lambda define lambda. And so then those x here is not, this x is not usually not an algebraic variety. And so we cannot, so this subject is not contained in the usual numerical theory. We cannot directly apply the construction of high functions with a linear numerical theory to this subject. And so if I asked, so that's our first section, I define x here. Motives and structures, but today I hope to make the story simpler, I will consider only a pure case. Pure, pure, pure case. The pure motifs are pure, pure, pure structures. And so we fix the, so we, and this is our Ho Chi number. The condition is that h, w minus 1 is h i, and h i is h 0. h 0, q is a, is a, maybe h 0, I use this notation. h 0, q is a, q vector space of dimension h i, the sum of h i. And then, and then we fix h 0, q times h 0, q 2. q times 2 pi i to the minus w, pure, pure linear form. And, and symmetric w is given, and anti-symmetric, this is a filtration zone on h 0, c 0, h 0, q. And the filtration is, is, is over, see vector subspace. Such that, such that, this is a decreasing filtration, and, and depends, such that, visualization. Ho Chi structure, this is, should be a Ho Chi structure, such that, such that h 0, pure linear structure, plus minus is, is, is, is the same, is, is the same thing, but here, here we, we take here this 0 or minus of that is, this, this, this is replaced by, and g, g is a, a, so as you, as you group up, such that there are s and t, that is g, g-s-y, g-y, g-y. So then, as you have reactions, and then k. k is the quick, we fix, we fix k, this is the, So there is a polarization, and lambda is the k plus of q.317. So k can act here, and then we take the mojirou k plus. And this should be our order, and there should be some condition that there is some polarization. And also some on n, and some as morphism. This is a bit clear. I forgot to say that everywhere is a bit clear. And this is a bit clear realization. So I think that we have a morphism between this and this. Which sends this polarization to this h0 out of something. And also this pyramid should send also something. I'm a bit too imprecise. And I print which always shows a bit imprecise. But actually that is all these precisings, but it is written in a print which is not yet uploaded. So in the present time, there is a kind of a correct many mistakes. And the correct version, and also the correct version and part 2, will appear in a kind of soon, after 2 months or 2 weeks here. I'm reading this part now. So we don't think it will appear there. So to have a nice theory, we should have a good condition. Some need a condition on k, but I don't talk about this. But I don't talk about this so much here. If I have height functions, I explain about the simplest part 3. Then we have h structure. See what's a projective musical complex level. And then h is the variation of h structure with rock destination. The local model is important. You have a vector bundle, how you get a vector bundle? With filtration. So this can have a singular point, but still this vector bundle, which is three times h, the local system in the non-similar. And in the place of the singularity. Extend to the singular part point. Point with singularity over h and h, but by doing that. So you require the residues to be zero. The residues to be nidopotent? Ah, yeah. Local modem is actually nidopotent. Yeah, local modem is connected, but on the boundary point, you have logarithmic connection. And you require residues to be nidopotent. Residues to be nidopotent. Residues to be nidopotent. I'm sorry. I'm a poor type. Sorry. You don't care. You don't care. You don't care. You don't care. You don't care. You don't care. We have an extension of vector 1. Ah, this is something like that. The DREAMS, the DREAMS-Chanan-Karek Station. The DREAMS, the degree of the... And you can take that. The degree of this vector 1,000 on the DREAMS project is... Ah, okay. Then you multiply, put, by c, i, and then you take that. Yes. Never in the version and the number... In the version, I need some preparation. So that if h is a cogestructure, the trick is defined to be a tensor p inverse, the metric of this element here, a divided by d bar. Absolute value. So this is... And then b goes inside h. And because identified with r bar to r bar, complex conjugation, by complex conjugation, you have i and you have minus i. And this is... You have complex action, the complex conjugation from this to this element, to that extent to the determinant module. So you have b bar. B bar can be defined here. And it is also a basis. So you can take the ratio of two bases, and then this ratio is in c cross and c. You can take the solute value there. So this is a canonical metric. And then the height function for the... For the Never in the type is defined to be... For n bar, this one... Then this corresponds to some h here. h on... Same with the log resolution. The variation of this structure here. And then lambda r of the kappa... This is the... So this you have... This metric becomes... Can be extended on this. That metric can be extended to this family. You have a family of metrics on this. How you purchase structure. And then you can take the kappa chart form. kappa chart form is... And you take one or two. Because this... I and here you... We compare this. Here only you have... But this w minus i plays the role of the... This is the degree of... This is minus of that. And the authorization. This is twice the role. So you take... See what the product of i is. Thank you very much. This is CRC. Oh, something like that. Something like this. And this doesn't depend on the choice of the... You don't take log? Log? This doesn't depend on that. It's the choice of the... Longer bits, so it can be defined in the U-mode. And actually, this has some similarity at the point, this one is a 2-fold, but that has some small similarity at the point hr, which has a similarity, but still this integrable component can be defined. This is a, so that, this is a, this is a, when we are in hid, by this height function here, we fix h0q such that z hat h0k such, such that z lattice, z lattice, z structure exists. My, my former student Koshikawa proved that the, the height doesn't depend so much of the choice, so much on the choice of the such integrable structure, yeah, yeah. That is the one which is named about the height of the changes. And then, and then, but here hrm, hrm is, but here hrm is going to be a, here the, here the, here, you, you define it to be a determinant of a gruelite entramment. This is a, entram, entram, entram, and you take a, this is a, element, the, the, activity and then, and medium, then, it corresponds to a, an, embedding of f into c. But the realisation of this is that you have a metric on L, which is written by the P.R.G. P.R.G. P.R.G. P.R.G. P.R.G. P.R.G. P.R.G. P.R.G. You have some integral structure on this thing, because you have to satisfy. In the definition of the lambda, k-level structure, lambda satisfies the sigma line should satisfy such conditions. This is the Galois action representing. Then by this, then, this is how the artist is Galois table in a meta. And so, then, actually, so sorry. In this theory, they don't need any Galois action to define their real integral structure, so sorry. So this is not necessary, but anyway, I forgot to ask you this. Yes, I have a question for Paris. Can I ask? Yes, okay, so I want to ask this elementary question on your definitions, because you take the sum of the whole i. So you have the, in both the analytic and the, and the, and the motivic things. So let's say, take the analytic first. So you have i and w minus i. So because you have an odd number, c of i is equal to c w minus i, and the line. H i is the H i. So H i, H i is, c i is any, any other, H i was H w minus i. And so c i is taken, I'm trapped. So c i are any. Ah, okay, excuse me. So another question, so another question, so another question is that you, when you're using, integral p, and the code theory, do you assume some good reduction or? No, no, no. I, I, I, I, the, the good reduction is not actually any reduction is okay. Yeah, yeah, but the reduction is okay. And they have a nice result in the case of good, good, good reduction. But, but I think that, that, I think that comparison with geometric, geometric theory, has a, they have, in the case of, good reduction. They have, they have, good interpretation of the integral structure by, by using crystal element, as, along with this, I, integral is the factor, but may also extend History to, this. Send these, send them over, this one over here. Sorry. I'm more high, than them, obviously. I'm, I'mMMMM, more, than them all. So, j, I, I'm sorry. This Neb hot, these two, j, j, j, j, j, j, j, j, j, j, j, j, j, j, j, j, j, j, j, j, j, j, j, j, j, j, del, j. Cessna vicis. Cessna vicis. Extended to the sumo of the work. Cessna vicis. Cessna vicis. Catch to teach. Thank you very much. Thank you very much. He was leading us this semester. So that is the interest factor is here. Divine, divine by their theory. Without any interpretation. First, the good luck showcase their show. So there are different structures in each area. Integral rank, homologing, and generalization are the same as the original term. So there are different structures in each area. And now, then we have this variation. And then we have many, many questions. So I just introduced some few questions. But if you open the papers of Boiter, then he has many, many conjectures in comparison to the other theory. So then we can transport their problems to this station. So we have actually many, many, many new problems. And so I can complete inviolatly many conjectures without any impedances. But I like to have conjectures without any impedances very much. So, but today I only introduced some of them. So the first one is that we can, yeah, each one I should address. So for example, the equations are, for example, all C and all H are 0 x 3 C, such that H lambda m is F, Q are all F and all n are all X. What are the questions? And for all, you have tf lambda r. So that is lambda is, if you consider the same lambda, then you have 3 C, 3 types of functions for the same lambda. And then we ask that they have some common properties. And maybe here may be not the same, but here there is a meta-linear theory concerning not only C, but concerning the lambda as a covering of C, a complex plane. And then if you put such generalizations, then maybe you have equivalence. But that's a question, a question, a question. It is a question. And then, by also by Green, Griffith and, and Peters, let the lambda equal star means the, means the, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah. Peters proved that the fact is that the H star, H is always a negative. And the H star, H is 0. So that means that H is constant. It's that special number. And that, if it's constant, it can be proved by using the variable. Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah. Actually, this height function, each of the stars defined by Griffith's longer, yeah, yeah, so, so, so, I am generalizing his definition to know what you're doing, height function. And then, then under this can be also, this follows also from the other side. Yeah, I can find a support, I can formulate the support of Boita Conjecture, but preparations to do that, so here I present some simple form of Boita Conjecture, not the exact analog of Boita Conjecture. Yeah, that is, so, so, first of all, this star, there is some B such that some C, maybe I could do it. So, you have a picture on C, then you have H as part of that. So, this is for maybe one, this is for one, H as similarity. There is a, there is a some C such that, such that, such that, such that, if has similarity, if Z is outside, if has similarity, what's the similarity? What is the role of this C? Oh, a C, you know, we have plastic here. Ah, plastic, yeah, yeah. And so, from Boita Conjecture, not exactly, but to perfectly, the, the Boita Conjecture, you know, we have, I mean, the analog of, preparation, so I, I just need to do some, ah. So, the C here may depend on F? Yes, yes, yes, and so, yeah, ah, yes, yes. Ah, yes, I, I think, yes. It is the paper Boita. And so, I feel, show that these are not true, then please, please, please make the papers and on the, on the, this type of conjecture. So, I, I just, happy to, to, to bring such, such as, to his, ah, to his, to his, to his. So, yeah, that is the, in here, ah, duty of all the person. So, then the, supported by, the conjecture was, ah, because, ah, because the, because the, here, you fix the, ah, good reduction, ah, if we have this good reduction, ah, outside, outside this, then, this part appears, we are only, we, we, in this can contribute to Sati Emra. So, so then, ah, this part is bounded. And then, by, by this, so, ah, this Sati Emra should be bounded. So, so, so, these two are, are in place. So, again, maybe I, maybe I don't remember exactly, but you just spoke about a statement that you called one, Kapishel one, and there was an notation N of V. What was N of V? V in the, in, in one. N of V. N of V. N of V. Ah, N of V. Oh, down. Ah, down. Yeah. What's the projection? Pencil. Yeah, so those points, See, Brother, that one. No, no, one, one, one, one. Kapiso one is the projection of a theorem. Kapiso one is the conjunction of theorem. Kapiso. Ah, conjecture, conjecture, all those are conjectures. Yeah, yeah, yeah. Conjecture all. Yes, conjecture all, so that I, ah, I have to finish the story, but there are also issues supported by the fact that this can be proved by using the Samohochi theory. So, the theme of the New Crest-Kamiemarine theory is that property correspond to a property that is a vvvvvvvvvvvvvvvvvvvv. So, such a thing is the conjecture of an angle, that is, this is a so-called hyperbolicity. And the hyperbolic variety has only, finally, many rational points, is the conjecture. This is the compagion of this conjecture. And so, I have never been, I have never seen you up such a conjecture. So then, if you set an arrow with two then, this is a path. Thank you very much. So, first ask a question from Paris. There is no question in Paris. Okay, so then I'll have a question. Sorry, sorry. He has a height of a better variety. And the first step is to interpret this height as a height of a point in a modernized space. In some cases of your height, or is it? We don't have the interpretation to set as a usually, we don't have the height. This height, we cannot, we have a modernized space of motifs. So we cannot prove the finiteness. If this is the origin of variety and this is a different point, then we have a finiteness. In the case of partings, then he had, in his case, the variety is understood as motifs. And then, the modernized space of motifs, the variety is realized as a variety. So this finiteness follows from the theory of height of a modernized space of motifs. But in this motifs version, we don't have such a good understanding of the modernized space of motifs. So we have no, at present, I have no way to prove the finiteness. But my dream is that if we, by using long-range correspondences, the motifs should correspond to the automatic forms. So finiteness may be, may be proved by studying automatic forms, but by using motifs by automatic forms. But I have, at present, I have no good idea to that direction. Thank you. So that's all for today. Okay, then a question from Beijing. There are no questions from Beijing. Okay. Then I have some questions from Tokyo. So you formatted the concept for this pure domain. So if you have some periodic analysis for using a periodic height of domain. I wanted to define periodic heights for motifs, general motifs, but I have no good idea. Yeah, so a periodic version should exist. Okay, so that's the finite places. Yeah. So there should be a theory of periodic heights. Yes, yes. Any other questions? Okay, thank you very much.