 Very punctual. I should share now the screen now to make sure that it works. Okay, good. Yes, please. Just a couple of recommendations, please, interrupt me anytime you can raise your hand or if you feel a little bit shy. You can ask on the chat questions. Maybe I am sure that other people here are very knowledgeable about a billion varieties they can answer, in particular, Pavel Borovka, who will be leading the exercise session and he is already there, connected, so he can, he is also free. He will be answered if you write on the chat, if I cannot check immediately the chat. You can also do it live, so you can please ask me, interrupt me, and we will make a little pause after 45 minutes. Yes, I think I remember correctly, five minutes pause in the middle. Okay, let me tell you about the course. The course is prem varieties and premap. And my goal of this course is to show you not only the prem varieties and why they are useful, I wanted to motivate you the geometry of the premap, the geometry of the fibers of the premap. Okay, so roughly the plan of the course is today we are going to just talk about the billion varieties, which is, so today is generalities on a billion varieties. And tomorrow I hope we start to talk about prem varieties. So prem varieties are a special type of a billion varieties and introduce the premap and premap and sorry. And then I decided to go through mainly two papers where they compute the degree of the premap. So the degree is just an excuse for me to analyze the geometry, so how this construction comes about. So one case, we compute the degree, the degree of the premap from R6 to A5, so I will explain what the options are. The answer is 27. And the second paper, so this is the paper, essentially it's donagismid and also paper of donag, the fibers of the premap. It's a beautiful geometry. I hope you will also like it. And the second example is when we want to compute the degree of the premap, let me see, 43, I should homogenize my notations. So this is, this is not, this is another space code basis. Okay, so there are many, many parameters here, but essentially both the first case are about double covenants over course and the other are cyclic covenants, triple cyclic covenants over a genus four course. And this is the paper of father. Yes, and we will see some similarities and other constructions. And in the first case, the degree is, the degree is 27. And the second case is the degree is 60. And both degrees have a very geometrical meaning. Okay. And to end the course, I hope in the last lecture, I would like to do an overview of all the finite premaps. And I will state some open questions. Overview of finite premaps. So we have a list, a finite list of finite premaps. And all, all this happens in low, low general, so covers on low general. And yes, and we are interesting on the, on the geometry of these maps and more, more largely, I'm interested in in the common denominator of all these finite premaps. Okay. Let me start with a billion varieties. So. Yes, all the material is uploaded. There is a list of exercise already on in the, in the web page. So they are the exercise that the public will start to discuss in the first session. There are more exercises. We have more exercises prepared, but to start with this, I think is enough. And all these papers I mentioned here also upload. So you can yourself go and have a look at the papers and, and, and check if I'm not saying nonsense. Yeah. Okay. So try to end also all the material, all what, what I'm writing here now, it will be uploaded. I, we will save this. So do you want to take notes, take notes, we can also, you can also check after. So. I'll be on varieties. First, well, I will work all over complex numbers. Yes, the whole course is about complex numbers. It's enough. It's interesting enough for the complex numbers. And let me tell you what, what the first probably, you know, a complex torus is. So a complex torus. A well it's a quotient of, so a is a quotient of C vector space over the lattice. So we, this is a more of a complex vector space of dimension G. So what we are going to do is, we are going to take a look at the complex numbers. space over lattice. So the isomorphic complex vector space of dimension G and lambda is a full rank lattice inside full rank lattice. So that means lambda is isomorphic to integers to G twice. And this is a complex torus. So an abelian variety will be a complex torus together with a polarisation. So let me tell you what the polarisation is. A polarisation on A. Well, easily is an ampoule and an ampoule line bundle on A. And that's all. That's a polarisation. Okay. So an abelian variety is an ampoule, let me call it L is the line bundle. So an abelian variety is overseas, is a complex torus and meeting a polarisation. So a complex torus meeting a polarisation. So it's a couple A, L. L is an ampoule bundle. And so we call here a polarised abelian variety. And so what is the role of this polarisation? You see, first notice that V is an abelian group with the sum. So the quotient A inherits the group of structure from V structure. So the plus here from V. So it's also an abelian group. So A is an abelian group and abelian group. First observation. Second observation is by definition of amplinus by definition of amplinus. So amplinus, a line bundle, ampoule on an abelian variety is by definition a line bundle such that some power of this line bundle is globally generated. And this globally generation allows to embed our abelian variety into a projective space. So we know it exists a positive integer K such that first of all, this is the tensor Lk is generated by global section. It is generated by global sections. But even more, there is a power of this line bundle such that the map, the map I would call it V, L, tensor K, defined by the sections of this line bundle. So you go A into P, A, zero of a L tensor K, do you want? So what is this map? You take a point in the abelian variety and you consider a basis of sections of this line bundle L tensor K and you evaluate the sections on K. And it turns out for K bigger enough, that is an embedding. Actually, it's an embedding by leftist theorem that this is for K at least three. It's a limit. So the section around my polarization allowed me to embed the abelian variety into a projective space. So in conclusion, in conclusion, this is where the thing we had to keep in mind is an abelian variety is an abelian group, which is also a projective variety. We have these two features. Okay. So an abelian, let me write it down. Color, because it's important. So an abelian variety is, it's an algebraic group actually. It's an abelian group together with a projective variety. And this is given by the polarization. Okay. So let me tell you a little bit more about the line bundles. So now I'm going to give you a couple of definitions that might sound a little bit fast. But my idea is to show you the different incarnation of this polarization. Because polarization, as I said, is a line bundle, but it has different incarnations. And in order to give you these different ways of thinking of polarization, I need to give you some definitions. First, okay. Let's denote by peak of A. So that would be the group of all holomorphic line bundles, holomorphic. Remember that this is, this is also an analytical variety of holomorphic line bundles on A. And this is a well-known fact that the holomorphic line bundles on A can be parameterized or are identified with the first homology group of A of the invertible. So the units of the A of the trivial bundle. Okay. This is a fact. Okay. So let's consider the following exact sequence. So we have a short exact sequence. Consider the short exact sequence in terms of shifts. So you have see the constant shift. It goes into, it's a sub-shift of the trivial shift over A, date exponential. And then you get into invertible shift functions. So invertible functions over E to zero. This is exponential. I think, I think I should write exponential 2pi. And then you put here the function. Okay. Well, this is an short exact sequence of shifts over A. And the associated long exact sequence. Well, we start with an H0 of each one. And then you have H1, H7, and then H1, A0A, H1, A, H2 of A7. This is the homology. And that's right. So the map, the connection map from here, here, is what we call, gives you the Cheran class, the first-chern class of the limelight. So the first-chern class of a limelight bundle here, if you believe me, this identification between peak A and this homology group, well, goes into the first-chern class of L, is an element in the homology group of A. Yeah. So this is the first-chern class, third-chern class. Okay. And actually, the polarization, what do we call the polarization? I say a limelight bundle, but actually it depends only on the class of the Cheran class. Yes. So, more generally, I believe that this has a very nice feature. So the homology groups are very well understood, actually have a very nice description. So we have this, for instance, the second homology group, this is a single homology of A with coefficients in set, is isomorphic to the second alternating group over the lattice with coefficients in set. Yes. So this is the set of alternating forms on omega. So you have then right like this. So you have alternating bilinear forms defined on the lattice with coefficients in set, non-degenerate. Well, in general, alternating bilinear forms. Okay. Actually, more generally, you do it for every homology group. More generally, you can identify, we have a map between N alternating N forms from H1 on omega, is isomorphic to the HN of, but this fact I wouldn't, I wouldn't, I would need it. It's just to know, it is just to tell you that this homology group of the lemuritis are very well understood. And this is introduced by the co-product right here. So you take one, one homology groups and the co-product and you end up in it. And they are all like this. And that holds for any N at this one. Okay. Right. Let me, let me. Okay. This is the only thing. So as I said, my polarization will depend only on the chain class. Yes. So it's good to define the image of this map C1. We define NS from A, the image of the map C1 from H1. Associated to each line model is for chain class. Okay. This is called the Neronseveri group. Neronseveri group. So it is where all the possible polarizations of A happen. Leaves. Yeah. They, you pick one element here, they come. So an element here, that means that it exists a line bundle such that the first term class is there. Okay. So, so I'm going to use this identification here. This is very important to describe what is the image of the chain class C1. So the question is, which homology elements in H2 are actually chain classes of some line model? And this is also known as a theory of, let the, well, first of all, I define it over V is the C vector space. So let's define an alternating linear form. So the following, then the following two statements are equivalent or equivalent. First, X, it exists a line bundle L on A, such that the alternating form, yeah, E, that's right. Alternative E is the chain class of L. So it's represented by this linear by this linear form, the linear form if only if. Well, two things happen. So the lattice, it takes integer values on the lattice and that we knew. But moreover, is they have to respect the multiplication by, well, respect has to be somehow compatible with respect. So V, okay, and that's the whole for the ABB. You see, it's a very concrete way of seeing the chain class of polarization. You have a linear form, alternating linear form with these two properties. Moreover, there is a very simple exercise you can do it yourself, I think. There is one-to-one correspondence between the Hermitian forms, Hermitian form on V. That's right. And alternating forms, E of V times V, alternating such that respect is. This is just an easy identification. You send an Hermitian form and you send it to the image of this, sorry, the imaginary part of H. And vice versa, you take E, you send it to the Hermitian form defined as EIVW plus E times EV. So it's an easy exercise to check this. This is one-to-one. Okay. Good. Okay. Now, there's any question now? M, too fast, too slow? Maybe it's too slow for the people who don't really know, maybe too fast for them to know anything. But don't worry, I just, you don't have to absorb everything now. I just want to, I have a precise idea of why I'm telling you this. Okay. And so they have another one, incarnation of the polarisation. And for that, I need to define what is a dual complex torus. Yeah. I have a question, maybe I should have asked earlier. But in the proposition of the two equivalent statements, you are saying that there is a line bundle such that E is the C1 or CN? I see one C1. Yes. I have defined only one term plus is the image of this map. Yes, C1. That's right. Thank you. Yeah. Okay. The dual complex torus. So, okay. We have our complex torus. And yes, I should say that. I wanted to skip that part, but I need it. Excuse me, Angela. Yes. Yes. I wonder if you want to say that the ampleness of a line bundle gives you the positive definiteness of her mission form. That's true. That's true. Yes. Yes. So, actually, we know a little bit more, in the sense that if you want the line bundle to be ample, you just want its her mission form to be positive definiteness. Yes. True. Yes. There is a distinction. So, in principle, yes. So, what I have said until here is it holds for a general line bundle, but the airline bundle such that it could happen that this bilinear form is degenerated. So, the condition to be ample is equivalent to say that this bilinear form we wanted to be non-degenerate. Okay. So, you are right. So, I will consider only non-degenerate cases, and they will be positive definiteness. Yeah. So, no, no. Yes. Mm-hmm. Yeah. Thank you. But I will point it out precisely. Okay. Yes. There is another nice characterization of the line bundles on an abelian variety, which is the Appel-Humbert theorem. But I will skip that. I will just let you know that, should I say it? Let's say it. Sorry. Let me mention that. What is the Appel-Humbert theorem? So, the Appel-Humbert theorem gives you an isomorphism of, give you a some diagram like this. So, you have on one side the kernel of this chain class map is peak zero. So, corresponds to the line bundles with chain class zero of A, peak A. And the image is the Neyron Severi group of A. This is rejected. So, Appel-Humbert theorem tells you an identification of each part. So, this part will be isomorphic to some, okay, I will define something that depends on the lattice. Here we have equality. And here, with the homomorphisms from the lattice into the C1. So, C1 is the circle in the complex norm. So, C1 are the complex norm with complex number with norm one. Okay. That's right. So, essentially, so, you can identify a line bundle in peak A. Peak A is equivalent to give a couple of H. So, H is an element here, which is an Hermitian form with these characteristics. And the element Psi, which are called characters, semi-characters. Psi is a semi-character for H. So, a semi-character for H is a map from the lattice into C1 such that with the property the sum of two elements in the lattice times exponential from Pi, take the imaginary part from H, lambda more. Okay. So, for all lambda more in the lattice. So, in particular, so, the elements here look like, like, correspond only to the characters. Yes. So, as H is 0, C is 0, and Psi is the character. So, this part you don't have it when it's 0. So, it's just a homomorphism between the lattice and C1 with the C1 is the circle. Okay. So, again, this is very nice from a Billion Wright's complex story, actually, for complex story. You can describe any line bundle as a couple of the Merron Severi group as an element of the narrative plus a character. Okay. So, this already suggests that the big A, or actually, so you see, big 0 is already a torus. So, this suggests, oh, it's clear, that's big 0, is going to be the quotient of this group. It's a group, modulo, the Merron Severi. It's also a torus, and that will be our dual torus. So, let's construct the isomorphism. So, you have less of the omega, the anti-alumorphic maps between V and C. So, no anti-alumorphics, anti-linear. C, anti-linear forms, from V to C, the whole of L. Okay. This is a vector space, a C vector space. And we, together with a linear form, a linear form, we have a canonically a linear form on here, times V to R, which actually just defined, just take the imaginary part of this anti-linear form of L, V, just eat with L, just eat V. And this is non-generator. And we can define and lattice via this, using this linear form by taking all the anti-linear forms L in omega bar such that it has restricted to the lattice, it takes integer values. And this will be the dual lattice, so dual with respect to this linear form, dual lattice to of omega. So, the quotient, we have a, we have naturals, we have a quotient, we define a hat as the quotient of these forms, model of these lattice. Okay. So, here's the proposition. And the map from this omega bar to the morphisms from the lattice into the C1. So, this is, again, this is the circle and complex numbers, induces an isomorphism between the a hat, the dual complex torus and pixel. And for that, and for that, we are using the apple theorem, apple humber term that I mentioned here. Okay. So, let's see. Yeah. So, the proof essentially is, so what is this, this map to take L and the exponential of psi p to p. So, since this, this, the linear form is non-generated, then this map is subjective, is subjective. Okay. And the kernel happens to be the kernel of, of, of this map. I didn't give a name. Alpha is precisely the dual lattice, or where do you take integer values? If you take the integer values, this exponent is one. That's right. And this is proof. So, hmm. Can I ask a quick question? Yes. And this short exact sequence you wrote from pick zero to pick to the new answer. Yes. Is this somehow split? Because you wrote pick zero is quotient of pk, pk and new answer. Well, you can, you can think, you can think it as a quotient of these two. Split will, when you say split, I would, I would be thinking of a map from here to here. Right. Right. I mean, since you quotiented by answer very, the answer very must be somehow inside pk, right? Not necessarily. You can still define the quotient. Oh, no. Wait a minute. Sorry. This is, thank you for the observation because I realized my mistake. What I mean is on this, on this one. Yes. Okay. Yeah. Okay. That makes sense. Yes. Yes. This is a tourist. Yes. Because you're right. Yeah. But actually, the, yeah, that fact that you don't need, I don't need. What I need is this. Okay. So what this shows, what this shows is that, yes, that is often, we will, in the future, I will identify both things. Yeah. So for the identified, so pick zero. So you can, you can think of, of the line bundles of trend class zero over A as, as also complex, a complex torus given by, by, by the dual, dual torus. This is an isomorphism. So you can identify them. Okay. So, okay. Here is my next information. So given this line bundle, a polarization. Oops. I didn't want to change the color. For, for a couple AL, polarize a million variety PL. We can always define a map between A and A and the dual by the following, by the following is the stake. So you have to see, you identify this with pixel. So you take a point in A, and every time you have a point in A, you define a translation to the point. So X, you have a translation. This is a good thing of a, a Nabilian group. So you can, you can translate with A, X. And every time you have an automorphism of every variety, you can, you can define the pullback of a line bundle A. So you have a line bundle here. Yeah. So you can pick the pullback of this line. It's a new line. So I take the pullback of, of the polarization under this translation, tensor with L inverse. Yes. So under this identification, you see the trend class, the trend class of the line bundle doesn't change on the translation. So L, the XL has the same trend class as L. This is inverse. And the, and the trend class of the tensor is the sum of the trend classes and the trend class of L inverse is minus the trend class of the other. So this, this is a, a trend class C. Okay. It's an element here. Then it's not so difficult, but this is a nice, a nice theorem in a billion varieties that this is a non-momorphism group. CL is a homomorphism of groups. Thanks to the square theorem or theorem of square, square. So the theorem of square say exactly that. That is a non-morphism of groups. Just to write it out. They know square. This is not included in the list of exercises, but now I think it's a nice exercise too. Okay. In fact, in fact, this, this CL is an isogenic, isogenic. So I profit to define what isogenic is. So isogenic between two, two, a billion varieties or two complex, complex story is a map, but it's a subjective morphism, a subjective morphism between, between a billion varieties. I'm writing you between a million varieties of the same dimension. That is a nice question. So a morphism, a morphism of a variety. So in particular, it's a morphism of groups. This is always an homomorphism groups of groups. And, and since it's a homomorphism group, the kernel, I have the same dimension, the kernel is finite group. Okay. Actually, it's a nabillion finite group. More generally, more generally, this is just, I don't need it, but I just mentioned more generally. This, this but in a hat, it gives, it gives us a contravariant contour. So hat is a contravariant contour between in the category of a billion varieties of the same dimension. So, so you have actually, it's defined the complex story. You don't need, you don't need the polarization, but yeah, you don't need the polarization. So for instance, you have a map between two complex story. So that is, that is for complex story without the polarization that induces an eighth hat from X2, X1. In a very natural way, because the hat I defined it via this omega bar, since I defined the vector space as a nomomorphism, you can imagine a natural way of defining the f hat also. But I'm not going to go through, just going to mention that this is a contravariant contour. And, and it has a relation with the, in the case we have a polarization, you have a nice relation. Okay. Well, okay. Coming back to fiel, this is another important definition. We define KL, the kernel of fiel. So as I said, it's a finite a billion group, a billion subgroup, subgroup of fiel. And it, it can be also seen as, as, as a quotient of some subgroup KL modulo del lattice. So where KL is, sorry, fiel, okay. fiel, fiel is all the vectors in, in the, in V such that the image of the Hermitian form given by the polarization. It has integer values. Okay. That's right. So here is a remark is that this mafia L depends only, only on the first-gen class L. Yeah. On L. So if, yeah. So summarizing. So all the story what I say summarizing is, is to tell you these incarnations of the polarization. So a polarization, carnations of a polarization L on A, you have a first trend class of a line bundle, ample line bundle, bundle on A. That is one. Second. And, okay, and I have come to remark a non-degenerated, degenerated alternating form in cross V2C. Yeah. Did I define, wait a minute. In principle, I said over here, right? Such that it takes integer values on the lattice. Right. And I know the degenerated Hermitian form, Hermitian form. Oh, I think it's time for, for the polls. Okay. Finish this and then we make a small C with, with imaginary part of H, taking integer values on the lattice. And, and so as I said, equivalent to say an isogenic, you can give the isogenic to VL from A to L, just zero. And, or, and something I, I, I didn't mention until now, a vile divisor, a vile divisor. Because, you know, every line bundle, ample line bundle has an effective divisor, is, is CNA, such that the points in A, such as when you translate this divisor with this point, is linear equivalent to theta. Linear equivalent. Okay. And this, that in a one that is, this set is finite. So the line bundle is, is defined by the divisor theta. And this is the theta divisor. Theta divisor that is in the awesome picture. Theta divisor. So this last, this last, this last condition reflects the fact that I want this, that this VL is an isogenic. So when this linear equivalent, these points here want to be in the kernel of VL. Okay. And I want a finite kernel so that VL is essentially, it's an isogenic. Right. I think I'm not missing anything. So let me stop here for five minutes. And I will come back with some precisely examples of a billion brightest. So the billion brightest I am interested in. Okay. Okay, you can. I don't want, I will leave it here.