 Thank you, yes, so recently I have a, we posted a paper in Akai, 200, some same pages, about their, called Aedic line bundles, a quasi-projective variety, and the motivation of this article is trying to develop a theory of heights for quasi-projective variety, which can be used to study the varieties, or separate the modular space of a billion-variety dynamic system, so all their universal family. And the paper is pretty technical, it's long, even introduction is very long, so my lecture today is given introduction to my introduction. Okay, so I will start with the, since everything is technical, I will follow the history, so my, we'll talk about three parts, the first part will be the, maybe first part, I will talk about the heights and Arachael theory, and the second part, I will talk about my basic construction Aedic line bundles, the last part, I will talk about man, at least some theorems, you can come out from this thing called equal to equal distribution, so that's the three parts. Okay, so for the heights, I need a, I think as many people didn't really, probably not all of you study Arachael theory, start with the very beginning, the theory of heights was formally introduced by André Veyi in the 1920s, but André Veyi, he introduced the theory of heights, was introduced for study, so the height was introduced for his proof of Maudet Veyi theorem. I think this is, was a part of his PhD thesis, and the title of the paper is called Arithmetic of algebraic heights, so that's the, the thing, and then the theory was studied, this height was the very study, the theory was studied by Noscutt, I forget the time, probably 1940s or whatever, then the first non-trivial theorem that Noscutt proved is called Noscutt property, and he used this to prove that the pre-prepared point on PN is over, for example, over K is finite, so what does that mean that he started out, he have a morphism from PN to PN, then he can see all the pre-prepared points, x is at PN, this orbit is finite, fm of x, because fn of x, for some, m not cos n, and if you require this thing, it's some number filled, then this is finite, so actually you don't really need a field, you just need a, this can be replaced by a degree, with a boundary, so, so, so even at a very, very beginning, the theory of heights was very seriously studied by dynamic systems, not by, let it be by K2 or 7 man, it really went back to Noscutt, and then in the 1970s, 1960s, the Neon annotates, separately introduced, called the Neon Theta Heights, and so this is, it's a series of heights, so this was used to prove, then formulated by Burge, and soon it will die, and the big difference between the heights of Veyi and Neon Theta Heights is, so Veyi's height is not really honest height, it depends on, so there's a, to understand that there is something called a height machine, I forgot who introduced that, maybe there, the height machine, and is that, so you have, if you have X over number filled, there's a project variety, and my prop is good enough, it's a project variety over number filled, and then for each, then you have a pick of X, so this is, I saw more of them classes of line bundles on X, then this height machine says that you have a map from pick X to the functions of X over K bar to a real number, all functions are modular that bind to the function, and this actually is a group homomorphism, group embedding, I'm injecting homomorphism, injective, let me try to write that, injective homomorphism, okay, and with the property has some factorial property, and so I mean that if you have X, you have Y mapped X to the F, you have a line bundle here, then your pullback, the height of this guy is equal to pullback of your height, okay, so this is defined by HL compose, I'm sorry, compose with F, so this is first property, second property is that if X is Pn, error is the O1 bundle, then this is the height of error, the very height, so I have not yet defined the very height, for example, if you have a point inside Pn of Q, and okay, and with homogenous coordinate, X, you have homogenous coordinate, zero, n, and the AI integer with the GCD equals the one, then it's a height, called a very height, or naive height, whatever you want to do, it's a log of the maximum of the absolute values, so that's basically for a height and machine, height and machine, height and machine, question, yes, so the, you said injective homomorphism, yes, you mean, oh yeah, tensile, modular torsion, yes, this guy is a vector space, yes, modular torsion, okay, okay, modular torsion, and for die fountain analysis, the height and machinery machine is more or less enough, right, because the theorem we prove usually is about the bounded of heights, for study birds and so on and on, this is not enough, because the conjecture is about identity, so it's somehow, it's already the interesting question, so the nanotidal height is one height inside of the class, okay, so this is basically the story between, then in 1970s, it's a part of R-Kelop theory, it's a partial R-Kelop theory introduced an intersection theory, an arrhythmic surface, and this intersection theory gave a new height theory, the purpose of this theory is to prove, I mean, for proving modular conjecture, following the same line of the proof given by partial R-Kelop of a function field, and there, and this is fully developed by Fortunes in the 1980s, and then actually, and in the end of the 90s, the paper in 1991, but I believe, so the bottom line of the theory is that you say if you have a variety, say X over a K, spec of K, you have a line bundle L, what can you do if you take an integral model, so this X is the integral model, over a spec, okay, and suppose you have a point, say a point of P here, the X-K bar, and then there's a P bar, that's also closure, you're getting something like this in here, right, and of course, this is not proper, and this is not good enough, so you need to add one more point, you add the places, V divided by infinity, for each infinity, you get X, V over C, so you get all unions here, so this whole package, it's kind of, it's called an arithmetical variety, then if you have a line bundle here, you get a line bundle, let's say L bar, this L bar is the most important part, it's of two parts, it's L over X, right, so this is a bundle, on this integral model, this is X, and then here, you add some metric, so this is a smooth metric, I mean, for example, if you assume, at least it continues, some nice metrics, so like when X, V is smooth, for example, and then their theories say if you have a, it's a line bundle, line bundle, it's a fancy way to talk about devices, so you will get, so you can define the heights, so you can define this bundle in the six with your P, right, because P is, this is like a curve, so you can, okay, so you can define this number, let me write it different way, so you can, you can define the degree, a arithmetic degree of your bundle under the P bar, and yeah, this essentially is a height, but sometimes we normalize, and it takes the P bar over okay, right, spec okay, so this is a degree of points, so this is a height, so this is an essential definition of the RLKL theory, and then you might ask, what's the advantage of this thing, okay, the advantage of this thing is that, so you not only can define the height of a point, you can also define the height of the sub-variety, so the picture, if you take a sample picture, you have x here over a spec of k, you have this model, x here, you have this, put this, some, some remaining manifold, xv of c over to the infinity, if you have a sub-variety inside the sub-variety, then you get a sub-variety here, right, you get sub-variety here, then you can define the height of your z to be, in the same way, the top of the c1 of bar, some powers and dimension of your z and z, yeah, this will be a real number, and then you normalize, the normalization, I mean, at the very beginning, there's no normalization, but in my paper anyway, I figure out that you should be better way to normalize by some numbers, you normalize divided by z here, and then you also divided by the geometric degree, so it's a c1 of L of k of dimension of z okay, so this L, of course, for this definition, suppose this guy's not zero, okay, so usually if L is ample, that's good, so not only, the advantage of RKL theory is that not only you can define the height of a point, you also can define a height of a sub-variety, but this idea was also discovered by Philippon and others, you know, using chow forms, but the RKL theory will give you a much more flexible way to understand, then, for example, from this thing, then you have other beautiful and arithmetic analog of employees, right, then you have something, another question, so the dimension of z here, is this the dimension of the k-scheme or the dimension of the integral model? Integral scheme, so nz is the point, and this dimension is one, yes, okay, so you get it, you get all this positivity, so you get some theorem, and I can record one theorem we're going to, it's called a noscar, and then here's some of your theorem, and you get some theorem called successive minima, all these theorems are related to the height of your bundle and the variety of generative fiber to the total height of your variety, so I give you a flavor of the theorem, it's called the volume, for example, I just give you one example, which we used, that I called a volume, so suppose a volume of the line bundle, and this will be defined to be the limit, m goes into infinity of the leading coefficient given by the polynomial, so usually it's m to the, let me see, yes, m to the dimension of your x under the m factorial of the number of the section of the H0 or error bar to the tensile m power, so this section is a kind of super norm, it's less than one, so this is the analog, this is really analog of the first and the column of the section, or the group of sections, but yeah, a reasonable sense, so there's a number here is analog of dimension, so that's the typical thing, and but this thing is quite important, so if L is m whole, then you can show that this guy is exactly the c1 over L to the dimension of x, so this is, you could keep this up, keep it some your formula, so it's a, it's a problem that the main two from our kind of theory to study the heights of variety, so because this thing is construction, it constructs on the sections and so this thing tells you that if there's a height of the positive, that what's going on, so I don't want to go into the detail of the study there, anyway the model constructor was approved by Barton's and just used the definition of heights, he didn't even use the power of Hebert's formula, he just defined his own height of a billion varieties and he used the line bundles, used the integral models, but he didn't really use other fencing from the field, and I give you an example, it's always good idea, you don't really need to do anything, so if we have a pn of z, this integral model, so here we have a bundle o1, so this is a bundle of homogenous coordinates, so in particular the section of pnz o1 is nothing, it's just the summation of z, some si, so si is homogenous coordinate, and of course each si doesn't have any meaning, but a quotient qsi is really a rational function, and then you can add one point here, you get a pnc, you get a c point here, so I can have o1 put a metric at infinity, and so in other words, I mean then this si and infinity will be a function from a pn of c to real number, right, because, and this actually can be defined rather easily, it's defined to be the ratio, if we add a point of p here over summation, I'm sorry, that's just a maximum of s0 of p sn of p, okay, each of these quantities is not well defined, because you have homogenous coordinate, you rescale it, you get a different one, but the top and the bottom seven are scaling, if you use this quite a naive way to define the height, then you will get the very height, of a p is equal to this rk of height of o1 with this bar here, let's give a name called this guy, o1 of bar, okay. Can I ask, yes, about the volume formula, is there supposed to be a log when you're given a log of the number of sections, or? Oh yes, you're right, okay, just thinking to make it more like- Yeah, this is, yeah, this is a log, you're right, I mean, it's pretty clear of a function field of a finite field, right, the number is not a, we're not going to get a dimension, you get a, yes, a log, yeah, so the rk of theory is roughly speaking to connect your points, height of points to the heights of variety, and in this case, your height of o1 bar of pn is actually zero, actually the other theorem, you can show that if you have x inside pn, h o1 of x is zero if and only if this x is the closure of subgroup of, let me kind of say, the species of right, let me try to give a name, so species of right in that, it defined by, okay, so it's x is a subgroup, or maybe component of subgroup, of gn to some l, inside gn, inside pn, pl, this is inside pn, okay, so you take some hyperplane, you get zeros, then you get this thing, so otherwise it's a positive, right, so this is, so we're talking about rk of theory, so this is something about rk of theory, and it's not really even used in the proof of moderate conjecture, I mean the full power is not, so in the 90s, then I introduce the idyllic line bundles, and the purpose is to understand Neuronthedra heights, and something called a boggle model of conjecture, so I'm going to write down, the idea is very similar to rk of theory, so it still starts with with x over a spec of k, for example, that I believe in the right spec of k, then I start with the integral model, I guess integral model, but in my integral model I don't require over spec of k, so it's a sum of parts, so u here, so u is a part of spec of k, right, so it's open, so then I need to add, so let the s be the complement, spec of k, take a u away, so including a kemenon place, non-kemenon place, then for each v, I need to glue all of them together, so I needed to glue all this v inside s, let me write the answer here, s here, and over here I have, I needed to think about all the points, v inside s, xv over the cv, so I need to consider this thing, so once I have a line bundle here, let me write l here, I have a fancy line bundle here, l u here, I needed to add a lot of metrics, so this is vatic metric, so there, well this idea is quite trivial, the difficult part of the theory is that you needed to add some metric here, so that you still can do, so you need to add a, you add a metrics, so then you get the error bar, error u, and the difficult part is compare ways, the original idea is that, the original idea, if your variety is smooth, you add a smooth metric, right, then you do intersection theory, but in a periodic word we don't have such a thing called smooth metric, so the smoothness is difficult, so then, yeah, but still I can, v inside s, so we add a, so we add a, this thing more systematically, and to get so-called integrable metrics, so I, the reason I call it integrable, because I don't know how to define a smooth, integrable is, in my feeling, is the worst singularity, so that it still have an intersection theory, right, so I can't define integration all these things in there, and I will not going to repeat that, because in the second part of my lecture, I'm going to redo all this thing, so the bottom line, so we get the, so we get, we can prove everything, we can extend everything, everything from the, the, the, the traditional Gillesile theory, Gillesile, Arakello, whatever, to this, this part, that the order put, one order put is some kind of thing called equidivision theorem, so let's have an equidivision theorem, the theorem is approved by the Yulimov, Shapiro, Yulimov, and myself, and we're not going to, you know, do the same, because I'm going to talk later on, but I just give you one example, one example is the following, say that A of K be a revealing variety, open number field, and we let L, let the H, Neren, it's Nerenthedrheids, Nerenthedrheids, associate to Ampolline bundle, and let, we fix the place, let K, C be a fixed place, a fixed Kimberden place, so, so, so then, and then, okay, so there's a same setup, so then let it X1, XN, and be a sequence of points, and A, right, and we, we have two assumptions, first assumption is that this XN is small, that means the height of XN goes to zero, second part is XN is generic, so I mean that no proper sobriety of, of A containing a immediate subsequence, okay, so, so that's the thing we have, then, then the Galois orbits, okay, let's give a name called OXN on, on A v of C is, R, equidistributed, so, so I mean that if you, if you have any function from L of C, A v of C to a real number continues, then you can see that this is a finite norm here of O of N XN, you take the cardinality of this O of XN, this will convergence the integral of F, D mu on A v of C, the D mu is the, the, the probability of harm measure, and the harm measure, and V of C with the total volume one, so that's the order put of the theorem, and even for abelian variety, this is kind of nontradial, you know, it's a, for example, you can replace this XN by torsion points, even replace torsion points, I really don't know how to prove without this fencing theory of an additive line numbers, okay, so this is, okay, so finally, finally this, this exhibition theorem is approved by, used by, we use, then we have a boggle model of conjecture, approved by Udomar myself, and for subvariety, we have a variety of A, if X is not a translation of B plus point of P, so this abelian subvariety is the torsion points, then there is this epsilon bigger than zero, such as that if X inside X, part of X less than epsilon is not the recipients, okay, so that's the funny thing, so our color theory eventually used to prove something in the boggle model conjecture, and the additive line numbers was introduced mainly for this purpose, for understanding nontradial heights, maybe ask me where did they use the abelian, the reason is that it did the picture here, so if it is abelian variety, then we have a narrow model, but narrow model itself is not proper, right, that you can have a toradi computation, but if you want to study the question about the boggle model of conjecture, then you need to make a best change, right, because you study the points algebraically crux, so that's really complicated if you have a trace of toradi computation and this thing together, so my idea is that I take a U so that abelian variety has a good reduction, when U does not have a good reduction, I completely forget about the integral model, so I just introduce, it's very similar to acumbitum place, so I want to study this as an identification to see what's going on. Okay, good, so this is the first part of most of my time, so I go to the second part and the third part, actually it's much easier, so part two is abelian bundle over quasi-projected writing, so the idea is the same, so it starts with X over there, let's do the simple thing, it was spec K, it's a quasi-projected variety, okay, the quasi-projected variety, then I want to find a dedicated line bundle, so I need to do compatibility, I'm compatible by two steps, so I have a line bundle here, the first step, I pick up X U over a U, the U is still a spec of okay, right, so this guy is flat, and I don't really require a projective, it's flat, and okay, quasi-projective, but this is far from compatible, so now I'm compatible it, so now I'm compatible, it's kind of tricky, what I'm going to do, I need it to add, so I have this picture here, this is X over U, this is U here, I need to add a lot of things, I need to add something like this, because this is not compatible, also I need to add something like this, right, because I need to add more places, so this is S here, I also need to add this thing here, and this thing is kind of easy because only finally many of them, and this guy is kind of complicated because we don't, if we get a quasi-projected variety, there's no unique way you get a compatibility, so our idea, okay, so there's a whole thing together, okay, there's a whole thing together, right, we get a, so there's a whole thing together, let's say it's X, and I have a line bundle, if I start with line bundle here, I have error over U, so I add a whole thing here, I get an error bar, and then, so I get one X error bar, right, so this is the Hermitian line bundle in RKLAB theory over a project model, and then I have many obstacles, but each of them, for each of them, we have a, for each of them, we have a height theory, U of K bar, so the question is how to define limit, and let me just illustrate it by one simple example, so for example, my X is A2, okay, A2 over Q, right, A2 over Q, so let's say, so I get a really nice model, I can get A over Z, expect Z, so I have infinity, I get A2 over C, so there's a C square, I can add this in, I can add a very standard way, I can add a P1, P2, also add a P1, cross P1, or I can blow up them, so in fact, the way you do these cuts is much more than you saw, so you get a kind of an infinite tree over the infinite place, so the paper, the paper with Xing Yi is that we carefully, we carefully defined a topology on all these models of Erva, the set, and then define the limits, okay, and there we end up a theory of integrable matrix line bundle, ethylic line bundle, because it's really ethylic, and so we get, to make it more precise, we need to do this notion called XAN, so this is called the Bercurve space of X, and if you never see the thing that's actually, it's not that hard to understand, let me try to give a slight definition, so if you have, say X is covered by affines, not the union, so it's union, right, so this, the spec AI we know is just the set of prime ideas of AI, and the XAN is the spec AI of AM, the spec AI, AM is the funny thing, is a set of semi-norms on AI, the semi-norm is just the AI, you map to a real number, non-negative, such as that the X, Y equals X and Y, and X plus Y less equal than X plus Y, but we, we do not require X equals zero implies X zero, so we do not require this part, if we do not require this part, then you feel you have a norm like that, and you see the current of this norm of V is actually a prime idea, right, so you, so this identification is really quite close related to the X, conversely your X, so we have, and we have the funny thing, we have X, XAN map to X, and we have a retraction of that, this one is, if you have a point XAN goes to the kernel of XAN, from this part is given by, if you have a spec of A for example, if you have prime P here, then you get A over P, then you can define the fraction field, right, I call it K, the residue field P here, so the norm you define it by A goes to K over P, goes to their one or zero, so they're the trivial norm, so their A, the norm of A is one, if A is not zero, zero otherwise, so you get a trivial norm, so you get it, so the identification is certainly kind of a closure of original space, so then, so our, so our identity line bundles of XAN, so this is, it's a matrix line bundle on XAN, and we define a subset inside, called a pick of X, called integrable, and inside here we have X here, it called the mod, mod is, this is really come from, from models, and this from this to here is defined the topology carefully, okay, yes, so that's the, to define this, the topology is complicated, that's the crucial part in our paper, so the consequence is that we still have intersection theory, theory, and we have, we also have the, the theory of volumes, and the one crucial theorem, which is really, we have, we have here for the sum, and one thing it's very interesting, there is some called the CODOS use inequality, and then we're not going to describe that there, so we have two consequences, that's the, goes to my second part, two consequences, two, two applications, one of them is pretty relevant to work of Philip, Ziang, and Dimitrov, is that if you have X over spec K, the quasi-projective, and L of Y is, is integrable a deligline bundle, and such as that L over this X here, L of K, and this is also integrable, you know, this is not a typical bundle, so this degree and X is, so for this guy is, say, is an F, this guy is a positive here, so this, in their terminology called non-degenerate, and they're, okay, actually you can do much offensive that, that you can map to some base, F here, then, then for any, for any C bigger than zero, there is an epsilon bigger than zero, and there's an open set in X here, supported here is a line bundles M here, for example, and such as that, the height, height of X of L is greater than epsilon height of F of X with M minus C, so this is a pretty much very general thing, but this is really a simple consequence of Sue's inequality, which was originally approved by in Xingyi's PhD thesis, okay, and then in, in Philip, Tian and their paper, this one was introduced by a form there, so this is one consequence, well, it's just a reformulation, it's simple consequence of Xingyi's volume inequality, the second application is, this is the first application, another application is equity division, so they, it's the same, so suppose you have X over spec, okay, spec K is a quasi-projective, suppose your L on X is always the same, same assumption, so L over X is the, is the F, okay, and L over X is the big, okay, this is kind of a funny thing there, so X is quasi-projective, so maybe you guys find it funny, so why quasi-projective, so my line bond is really a line bond on this variety, a really line bond on certain computation, actually there's a nice name on this, it's called a Riemann-Kirbler space, but I don't want to use that, if you have X M, a sequence of points, and X with the same property, which has a small n-generic, then for any place V over K, the galore of X, of X N is equity disputed, equity disputed with respect to something called a Schambler-Lewa measure, so this is another thing there, so you have nice applications, you know, to dynamic system, and then the typical situation in dynamic system is, or a billion variety, if you have X over the S, this is a dynamic system, relative dynamic system, you have a line bundle there, if you have all the things here that define over K, okay, so if you have a pullback of a line bundle L to the Q, Q is bigger than zero, so this is a relative dynamic system, a family dynamic system, a family, so you can apply such a theorem, in this situation you will naturally get an L bar as the integral line bundle, and is this a different F? There's a different F, of course, that's called a pi here, so you have a map, and a more F there, so in this way you can study the situation, and of course, in general, usually this is always quasi-projective, never be projective, so you can apply our theory to study the same, and yeah, I mean, some work can be done by many other, many people before us, you know, but a week, I think Laura DeMarco and their group work out something, yeah, the recently they, last week also used a contribution to study a uniform bond of rational points and curves, which have been generalized by Sian or whatever, so this kind of thing, so okay, that's end of my lecture, so sorry I didn't really have time to really define what's the identity line bundle there, so I say, if you read their introduction, so the whole point is how to define a topology, but to define a topology I need another 20 minutes, so it's too complicated, thank you.