 Okay, there it is, very good. Okay, so all right, so we talked about the local period domain, the local Torelli theorem. Now, let's talk about the global period domain and the global Torelli theorem. All right, so see now, okay, there we go. Okay, so how do we make the global period domain? So when we look at H2 of XZ, right, that's a discrete kind of thing. So the idea here is that we can pretty much kind of identify all of the integral cohomology groups of all of these irreducible holomorphic symplatic manifolds over a given dimension, you know, so with given, so you have, so basically each time you can take the lattice structure of the second homology, right, and then you can also take the Boveal-Bogomol of quadratic form on it. In fact, usually when people talk about a lattice, they kind of implicitly mean that there is already a quadratic form on it, right, so it's not just a a Z-Module, it already has also a quadratic form on it, right, so when I say a lattice, I actually mean that I already have a quadratic form on it, right, so maybe I should write that, so definition. A lattice is the data of a free Z-Module, which I would call gamma of finite rank with an integral non-degenerate quadratic form, which we will denote q sub gamma, so this q is, interestingly, comes with the lattice, so the homology of a, the second, you know, h2 of xz is naturally such a thing, right, so it is a free Z-Module of finite rank and it has an integral non-degenerate quadratic form, which is the Boveal-Bogomol of form, so we can, we can sort of forget, we can look at, in some sense, we can look at all of the, all of the hyperkele manifolds that have the same lattice, right, and we can sort of map them all into the same period domain, so for a lattice we can define a period domain, so the next definition given a lattice gamma, q sub gamma, the period domain is, and we will call it q gamma, it's just like before, so q gamma is, by definition, the set of alphas such that q gamma of alpha is zero and q gamma of alpha plus alpha bar is positive, right, and this naturally sits inside q gamma bar, which is, by definition, the set of alphas, such that q gamma of alpha is zero and this guy lives inside the projectivization of the lattice tensored with c over c, right, so I make a complex vector space, I projectivize it, and then I can define a quadric in it using my quadratic form q sub gamma, right, and again I have a, I have a period domain in a similar way, right, and then we're going to kind of map all of the holomorphic synthetic forms with a given whose H2 can be sort of identified as a lattice with a given fixed lattice, we're going to map it into this period domain, and that will be our global, global Torelli map, so let me, so we're going to construct first a modular space of marked holomorphic synthetic manifolds, so let me start with a definition, a marking of some holomorphic manifold, sorry, this is what holomorphic manifold x is, a lattice isomorphism, or if you're like you can, you can, if you want to be a little bit more precise you can just call this a gamma q gamma marking, right, it's a lattice that isomorphism phi between H2 over xz qx tilde and gamma q gamma, okay, and this pair x phi is called a marked manifold, marked manifold, then two marked manifolds, x phi, x prime, phi prime are isomorphic if there exists a holomorphic map and actually a bi-holomorphic map such that phi prime is the composition of phi with f and we write with f up or sorry, if you have a map from f from x to x prime, it will induce, you know, it will induce a map in homology, right, from H2 of x prime z into H2 of xz, right, and then I can compose this with the marking of x, right, and then I get, I get, I get phi composed with half of our stuff, right, so we, and then we write, we write it like this, we write x phi is isomorphic to x prime phi prime, okay, so it's very natural, right, so you, you define the isomorphism between marked manifolds and then what's the marginalized space, so the marginalized space of marked irreducible holomorphic manifolds, holomorphic symplectic manifold, sorry, is the following, so I will call it m sub gamma, gamma for the lattice, it's by definition the set of all of these marked pairs, modular, the equivalence relation, which is the isomorphism that we defined in number three, all right, so we have a marginalized space of marked manifolds, right, and then, so what do we, so what do we do, so then we have, we have a, sorry, so what, so then we have the global period map, right, maybe, let me tell you another definition on the next stage, what's the global period map, it's we're going to call it p, and it goes from m gamma into q gamma, which we said is inside q gamma bar, which is inside p of gamma tensor with c over z, and what does it do, it takes the pair x phi and it sends it to, well if you have you, the marking gives you a map from h2 of xz into gamma, so you can take the image of sigma under that marking phi, right, phi is the map, and then you can take its point in the projective space p of gamma tensor c, okay, so then we have a version of the global Torelli theorem, right, so you two, the map p is generically injective on each connected component of m gamma, so let me, let me explain a little bit about, you know, how this, how this will work, so normally I mean for those of you who are a little bit familiar with Torelli maps, right, this is not exactly what people usually mean by global Torelli maps, by global Torelli, right, so this is not a usual, this is weaker than a usual global Torelli theorem, so let me give you some examples for instance, so in the strongest global Torelli theorem that we have is for complex tori, so two complex tori are isomorphic if and only if their first homologies are isomorphic, so this is basically the strongest global Torelli theorem that you can have, so you're basically saying that your complex torus is fully determined by its first homology, and you're not even talking about a quadratic form here, so you don't need, you don't even need a quadratic form, all right, so this is, this is the strongest you can have, the next strongest global Torelli that you have is for, is for curves, so what do you have for curves or Riemann surfaces if you like, so two Riemann surfaces are isomorphic if and only if their first homologies are isomorphic, sorry, sorry, sorry, I should have said isomorphic as hutch structures here, right, you do want the hutch structure, so for Riemann surfaces you need a little bit more, so their first homologies are isomorphic as hutch structures, you also want the intersection forms to be, to be, to correspond to each other, and under the given asomorphism the intersection forms on the two curves coincide, and we will say for short we will say that the two, the first homologies are isometric, the third example which is slightly less strong, I will explain how this is less strong, for 2k3 surfaces isomorphic if and only if their second homologies, and actually sorry, instead of what I mean here is hutch isometric, because that indicates also that you preserve the hutch structure, their second homologies are hutch isometric, okay, so these are the best Torelli theorems we have, so we see, you see that in this case, you know, with the with the global Torelli for, for the Hapikela manifolds, that's not what we're saying, we're just saying, so the Mach p, what does it do in some sense, so if you're, so you send, you have a, you have an isometry between the second homology of x and this fixed lattice, right, and the image, when you take the image of the symplectic form phi, right, sorry, sigma, when you take the image of the symplectic form sigma, right, sigma generates, so this is on a side if you like, so note that sigma generates h20, right, inside h2 of xc, and if we have the quadratic form, so you know that qx tilde of sigma is zero, and also this is something that I didn't say, but this is true usually in general, if you, if I look at sigma part, the part for, if I look at the orthogonal complement for my quadratic form, right, then this is actually h20 direct sum h11, so if you know, if you give yourself the line that sigma stands inside h20 of xc, then you, if you have the quadratic form, then you also have its perp, which means you also have h20 direct sum h11, and then, well, you know, sigma bar generates h02, right, so you pretty much have everything, so, so this means that, you know, this is the class of sigma, right, determines the h structure on h2 of xc, so it determines the h decomposition, right, it determines the h20, it determines the h20 and, you know, the h11, and it also determines the h02, so if you have your, if you're saying that, if my global trailing map period, if my global period map had been injected, then I would have a trailing theorem, right, I would know that if I, if I know that there's only, if two manifolds map to the same point of the period domain, that means that they have the same sigma, and they have the same quadratic form, that means then they have the same intersection form, they have the same quadratic form, and then they also have the same h structure, right, but that's not what I'm saying, I'm only saying it's generically injected, right, so for, for irreducible polymorphic symplectic manifolds, global trailing actually fails, and there are examples, the first example is, is due to the bar in 1984, so he produced non-isomorphic, so there exists non-isomorphic, however bimeromorphic compact hypercalamanifolds with hodge isometric second comologies, they were also, these guys were also not algebraic, and so, okay, so people saw this and they thought, well, these guys are bimeromorphic, now if you look at k3 surfaces, if two k3 surfaces are bimeromorphic, in fact you can show that they are, in fact isomorphic, so people, you know, people thought, well, maybe this isn't too bad, if, if these are the only counter examples, then maybe, you know, still saying that, okay, global trailing, if two, if two polymorphic symplectic manifolds are hodge isometric, then they're, in fact, bimeromorphic, that's not a, that's not a bad thing, that's still a pretty strong theorem, and that's nice, but that was, that's not true either, there were other counter examples, much more recently, this is Namikawa, and I'm not saying these are the only counter examples, I think there are others in the literature, this is not an exhaustive list, but these are the ones that kind of jumped at me, so there are, he produced an example of non-virational, so projective, four-dimensional hypercaramide manifolds, so it's like the worst possible thing you could get, they're, the smallest possible dimension that they could be, they're projective, and they're not birational, with hodge isometric second homologies, all right, so, so there is no hope for, for a real global trailing, the best thing you can hope for is basically what, what Warbicki did, but there is, it's still, it's still, you know, this is the other open question that I wanted to mention, so question, so is it possible, is there a good characterization why by good, we have no idea what good actually means, right, of holomorphic symplectic manifolds, irreducible ones that are hodge isometric, but not isomorphic, so can you, so what kind of, so people were hoping that maybe they could say okay if they're hodge isometric then, then at least they're birational, but that's not the case, you know, that they're bimeromorphic, but that's not the case, they're counter-examples for that, so is there, so what else can you say, you know, if two things are hodge isometric, can you say anything geometric about them, right, how do you, how do you link them together geometrically and not just via their second homologies, okay, so that's, as you can imagine that is a hard question, so no idea what the answer would be. I have a question, so along this line, so if you, what is it about generosity that makes it injective in verbatis serum, is there anything particular of which? No, the proof doesn't tell you anything about that, I mean the proof just, just looks at the map globally and sort of shows that it has to be generically injected, but it doesn't tell you anything about, you know, where it's going to be generically injected, you know, and why exactly or anything like that, so yeah, the proof is just a clever trick basically, so that's the, one of the problems with clever tricks, you don't get much insight from them, but yeah, okay, thank you, yeah, sorry, yeah, this just not, yeah, let me think a little bit, yeah, basically, yeah, that's it, I mean the proof is just a clever trick, it uses simply connectedness of the period domain and that's basically it, so you don't get anything from it really, so yeah, I could say something about the proof if people are interested, maybe tomorrow, I mean, I don't know, I mean, tomorrow's the last lecture, so I'm willing to let people kind of direct me a little bit into what they want me to do, I could talk a little bit about the proof of the global Torelli or I could, actually one thing that I will definitely do is talk about twister lines and twister spaces, and so I was thinking either to talk about the proof of global Torelli or maybe talk about hyperhologomorphic sheaves or I don't think I can really do both, so yeah, and I could say just a few words maybe about Lagrangian structures, yeah, anyway, does anybody want to express any preference or okay, I can also say something about all of it, you know, just keep it, try to make it fit into the time that that is left. All right, so let's see, how much time have I gotten right now, 10 minutes, 10 minutes, okay, all right, thanks. Okay, so let me say in the 10 minutes that are left, then let me say something about this M gamma, this marginalized space of marked, so let me remind you what M gamma was and gamma is, so M gamma is the marginalized space of marked irreducible holomorphic synthetic manifolds, right, we defined it as the quotient of the set of all pairs, you know, all marked pairs of holomorphic synthetic manifolds on the marking while modular isomorphism, right, okay, so one of the things that are a little bit surprising is that M gamma, so we can, so just a little bit, a few words about M gamma, let's say, and the toralium and the period map, okay, so one can use, so you can show that, so we defined the local period map and then we defined the global period map, of course, the two are the two, you know, the local period maps glued together to give you the global period map, so let me explain a little bit about that, so, and what that gives you in the end that is that, is tells you something about M gamma, so we can use the local period maps to show that M gamma is a smooth non-hostal complex analytic space, right, so how does, how would that work, so if you, if you give yourself given an irreducible holomorphic symplectic manifold, we can choose a marking free from h2 of xz to gamma, and then we have the Cornishy space, right, so the Cornishy family x to depth of x, we saw that this is locally isomorphic via the local period map to the period domain, so we can use, we can use this isomorphism, which you remember the local period domain, actually, sorry, let me, let me, let me explain this a little bit better, so you have your marking, right, so via the marking what you can do, you can write down, so the marking gives you, oops, the marking gives an isomorphism between the local, the local period domain, right, which we called qx, and the global period domain q gamma, right, so you can just, you, you identify h2 of xz with gamma, the quadratic forms coincide, so then the quadrics map into each other and positivity is again the same because the two quadratic forms agree with each other, right, so you get an isomorphism between the local period domain and the global period domain, right, and then what you have is that the, the Cornishy family x over depth of x is locally isomorphic to the period domain q gamma via the, the, you know, via the local terrarium, all right, so, so basically, so this is, so this is what we've got, right, so qx is isomorphic to qx is isomorphic to q gamma, right, you map this into p h2 of xz, and this one was, we can identify it with p of gamma tensor c over z, right, these guys are isomorphic, so then if you take, and in here, right, inside this qx, you have an open ball, so a small neighborhood in the deformation space of x, right, so, so then, so then what you can do is that these open balls, so the open balls above, give you a covering, so cover the, the, the marginalized space of marked holomorphic synthetic manifolds and gamma, right, and the analytic structures on the intersections of two different, on the intersections of different open balls coincide because the Cornishy family, this was the nice property of the Cornishy family that it is universal for all of its fibers, right, so if I take the Cornishy family of x0 and then xt, they are the same Cornishy family, so these open balls inside the marginalized space of markings, when they overlap, they have the same complex analytic structure, okay, so then, so then you see that then these balls, they will cover m gamma and this shows you that m gamma is a union of balls, so then you can put this complex, this, this smooth complex analytic structure on n gamma, yeah, so okay, so that's what I wanted to say today and tomorrow I think I will start by talking about the toaster spaces and toaster lines and maybe I'll say a few words about Verbitsky's proof of global Torelli and then I will probably say a little bit about Lagrangian structures and hyperholomorphic sheets, so I will kind of wrap up, you know, by making us some kind of a summary of the other, of some of the other important things that people do with holomorphic syntactic manifolds, are there any questions? Can I ask a question? Sure, so in the counter example found by Namikawa, so it should be that the two hyperkeler manifolds lie into different connected components of the modular space, is it right? No, no, no, no actually they don't necessarily, the thing is that this, the modular space is not Hausdorff, so that's what's going on, so it's the non-Hausdorff points actually. But if two points are, if I will remember, if two points are inseparable, they should give two bimeromorphic hyperkeler manifolds, that's why I asked the- Oh, is that true? I mean, I'm not sure about that, but okay, you're saying the non-Hausdorff points always come from bimeromorphic manifolds? Yeah, because if I remember, if I will remember, I mean from the paper of Markman, said that the fiber, if I fix a connected component of the modular space and the fiber of a point in the period domain, I should give, I should obtain a set of points which are pairwise inseparable, and so this should be pairwise bimeromorphic. Okay, can you say that again, if you take the fiber of what? I fix a connected component of the modular space. I choose an structure, I'm in a point in the period domain, I pick the fiber of this point in the connected component I fixed, I should obtain points, a set of points, I mean, which are pairwise inseparable. Yes, that's right, but that doesn't mean they're bimeromorphic, does it? It should be the next point in the survey of Markman, which says that- Uh-huh, I see. Okay, it looks like a lot of people are agreeing with you, so okay. Yeah, all right, let me, okay, yeah, thanks, sure. So that's why I thought that, I mean, I don't know the examples, so- Oh, I see, I see. So that's why you think the example, they belong to different components. Uh-huh, right, there's the example, right, okay, let me think about all this and we'll discuss it tomorrow. Okay, yeah. Thank you. Other questions? Looks like there are no other questions. Maybe we can stop recording them. Okay, in case there are more questions once.