 Okay, thank you. Okay, so. All right, so let's go back to where we stopped. Let me actually also go back a little bit to touch me the space right I, I spoke about touch mirror space right. And so let me, let me remind you what it was. Let's say types of x right this was the set of all complex structures on x module the equivalent selection that I called till the zero, which meant that which was saying that two complex structures were equivalent if one was the pullback of the other via a morphism which was isotopic to the identity. Right. Okay. And then we had complex, right, which was the set of all complex structures, just modular differing morphisms so we don't ask anymore that they are isotopic mister to the identity and we, we said that this was. This was the quotient of tish by the action of the group of components of the group of the few morphisms of x right. And in between here, we have our module I have marked. And then we have our scalar manifolds and gamma. Okay, so. So these these these two, this M gamma, what can happen is that sometimes the touch mirror space has got actually infinitely many components, it could have finally many components but it could have infinitely many components. But the M gamma usually I mean has finally many components and then the comp. Often people hope that the complex really has that only one component but that's not necessarily the case of course we don't we don't we don't really know necessarily. Anyway, so this this. And the other thing that. So, so M gamma is the quotient of tish you know by some discrete group action right which is a sub, you know, which is some group of G. And basically, you know, basically speaking you, you know and gamma is a modular space of marked. Complex structures right, and so the main difference between the touch mirror space and then gamma is the basically the automorphisms of the lattice right so you could, you could have some. You know the manifold which will act trivially on the homology lattice right so then those guys would give you the same marking, but they would not necessarily give you the same point of touch mirror space right so. Okay. So that's how it works, and we have the period map, we define the period map. And then we have this modular space of marked complex structures right. Again module sum equivalence relation. Let me write that one like this. It's just to work just to differentiate between that and the other one. Um, so then. Okay, and then we have the period map right so the period map we define the print map from and gamma. But we could compose it with the map to from types. And then we do the period domain which I think I called that Q. Q gamma, right, this was a quadric right. It was a real open sub manifold of the other complex quadric inside the projectivization of our lattice cancer would see. Okay. And this was a period map, the gamma right. And what we what we also saw was that so this and gamma is not necessarily house door but it is a complex manifold so this map here is is again a local isomorphism. So, both touch with our space and in gamma are complex manifolds but neither of them is house door so in gamma is a non house door complex month. Q gamma on the other hand is house door and simply connected. So what, um, so what what do people do so did the, you know, where which is clever idea what was it he said well okay this guy is not house door. What I'm going to do is I'm going to make it house door. Okay, so we he you know he introduced what he called you know the. I don't know the house door if you're like I don't know. I don't remember exactly what what they call these things but so you can you can sort of you can make a new. A new version of and gamma which I will call and gamma s s for separated right. The period map will factor through this right this guy is here is the period map and m gamma s is now house door. And my m gamma s is basically you know m gamma modular. So you identify the points. The infinitely near points of m gamma and by that what do I mean so I mean that so a point P is infinitely near to Q. If every neighborhood of P contains Q and vice versa. So, um, so you can you can construct a new space which is house door and the period map factors through it because Q gamma is house door right. So then you have a new period map from this house door visualization right and and so now what do we have this m gamma s now is a house door complex manifold. And we also know that the period map we we saw this before this PS is a localized morphism right the differentials are injected locally. So then, so we would basically almost get a Tarelli if we could show that the period map is surjective so if we can show. That's the theorem if you like, that's the main part of the result of the of the proof of Tarelli is that PS is surjective from any component from any connected component and gamma s to to Q gamma. So if I restricted to any connected component I get. I get a surjective map now combined with the fact that that PS is a local isomorphism, you get as a corollary is that PS induces. In fact an isomorphism between components. So maybe I should say isomorphisms plural induces isomorphisms between components and gamma s and Q gamma. Okay. So that's modular surjectivity. And this uses of course that that Q gamma is simply connected right and then that the map is et al the PS is et al and Q gamma is something connected if you have something that's surjective. It's going to be a nice morphism on every connected component right. So now there was, there was a question last time about you know what exactly is the difference between M gamma s and N gamma so. So in other words, when do two, when do two hypercalia manifolds give you intermediate points of the marginalized place of marked hypercalia manifolds right. And so there's a almost a complete answer to that. Let me see. Okay, so. So this is mostly due to high brash right. So first of all, there's the following result. So, if, if I have two marked complex structures. So x, the, and, let's say, x prime. The prime are two infinitely been two infinitely near points of. And gamma, then x and y are by more morphic. And in fact, you can also say something a little bit more you can also say something about the period point right. And the period point of. x phi, which is also equal to the period point of x prime, the prime is contained in a particular type of hyperplane, which is the, the period domain q gamma intersected with the park of some element of the lattice. And the lattice is a freeze, a free, finally generated a billion group right. So, so we we take here the park with respect to the quadratic form right on the lattice. And then we intersect, we intersect that herb with q gamma so we get, you know, we get a hyperplane inside to gamma and you're saying that the, that these all of these non separable points belong to hyperplanes of this in the period domain. Right. Okay, so this is one direction. So, again, do the hybrids in the other direction, which tells you that which says the following. Sorry. There's a question, yeah, go ahead please. In the previous statement of what is why. What is why. Do I have a why. Oh, no, I meant, I, it's expansion. Okay. Yeah. Sorry about that. It's not why. Is that good. Sorry. Now it's all good. Okay, thank you. All right. Okay, so, um, so let's take us one moment. So the other, so the proposition which was which kind of goes in the other direction, right. So, um, we have the following, which is again due to progress. If, if again, if now x and x prime are compact hypercalium manifolds that are birational right so let or bimeromorphic so let x and x prime be compact hypercalium and bimeromorphic. So again, basically it tells you that they give you non separate points of the, of the module I've marked manifolds so so that that's the result precisely right. So, there exists families of complex manifolds. So x to s. And, x prime to us. Actually, let me, instead of calling it as mega should call it these for this, that would be better. x to be and expand to be over the same complex disk be such that number one. Now that the, you know, it's just a complex this so it has an origin right so x zero which is the fiber over zero is isomorphic to x, x prime zero is isomorphic to x prime. So these are the central fibers, and there exists a birational map a bimeromorphic map if you like, from x prime to x, which is of course which of course commutes with the two projections to the disk, which is an isomorphism over the minus zero. And, I didn't give the name, but a map between x and x one. Okay. Let me just rewrite that a little bit like I have a bimeromorphism that I will call here. Let me call this one little app, right. And then I will call this other one. Okay, so I have, I have a bimeromorphism between x and x prime right, then I'm saying that that bimeromorphism is basically induced from a bimeromorphism between two families over some this on such that. Well, that's it. So, which is a nice woman over over this minus zero and induces X zero. I'm just saying that big app restricted to X zero is equal to. So, so it's it's so what is it telling you then that it's pretty much telling you that what you're doing when you pass from, when you pass from M gamma to M gamma s, you are identifying pretty much the bimeromorphic. Okay. You know, the thing is that this is this. The thing that's complicated here is this group of components so we don't know. We don't know how many components this M gamma s has so you were asking about this Namika example right. The Namika example is just, you know, is just cool is just with kumars so you know you just you know we introduced the kumar the generalized kumars for complex. Tori of dimension two right so this is what you get here is that he what he what he produces here is that he takes. He takes a complex torus and then he takes just its second kumar which is a four fold, and then he takes the second kumar of the dual torus. And then he shows that they are hodge isometric so they have the same h2 and that with the same hodge structure and the same quadratic form, but they but you know but but but they don't necessarily have to be birational to each other. They don't necessarily have to be bimeromorphic. So it's a very simple example. It's a little bit strange because, you know, you take a torus and it's do normally, you know, a lot of the time, I mean they're just kind of belong to the same component of the marginalized space of complex Torah. So, so it is, you know, understanding these. These components is not is not so easy. Let me also just quote another result of progress for this which is kind of nice. Well, it is nice. He has another nicer nice results. So, it's a finite nest result right so. It's fine and this would be now interesting right so you can say okay you know that the bimeromorphic guys go to the same period point, but how many non bimeromorphic guys will go to the same period point and this is, this is again this this kind of tells you a finite limit. Okay, so that that's nice anyway. So if we fix. Given given a, let me see how should I say this. Given a period point, and let me call it, I don't know little x in, in q gamma. So this is knowing already that the period map is subjective okay so I'm going to use the fact that. So, let's say we already know, assume we already know that the period map is subjective. So the activity of the period map. Okay, so assume you already know that then then what do we get we get the set of the set of hyper killer complex structures with image, I should say marks I guess with image. No, actually that doesn't really follow here. Sorry, just one second, let me think for a second here. Does the marking change anything. Okay, let me let me restrate this in a different way. So that I'm certain that it is correct. So the set of hyper killer complex structures on our differentiable manifold X. With a fixed arch structure, H two of X Z is so then is so non empty the non empty part as I said that's the subjectivity of the period map right now this is the interesting part consists of a finite number of bimeromorphic equivalence classes. Okay, so what you're saying here is that. If you identify all the bimeromorphic guys then you have only finally many hyper killer complex structures with a fixed heart structure on the H two right so it is it is a pretty strong result. Of course unfortunately there's no effective balance on this finite number. Obviously, any kind of bound would be would be very interesting right. Alright, so. Okay, so we have this. Are there any questions before I move on. I would like to do is say a few words now about the subjectivity of this period map or there any questions before I move on to that. For this proposition. So do we need to fix the deformation type like we, we fix the connect component of the motorized space. Always the finance like says the also the finance for the deformation types. Yes, you are fixing you are fixing the, the differentiable structure on M right. So you're throughout all of this, we had we have a fixed see infinity manifold compact. For all of this yeah we are we are fixing that yeah. So, can I ask a question. Right. So, so we are not anymore on the modular space of mark the paper killer manifolds right, as we are. No, for this one no we're not know. So the marking I mean understanding how the marking. That's a different thing that has to do with the lattice right so how many different markings can you have I mean then you're looking at the automorphisms of the lattice. Okay, okay. That's a separate question. And the can be expressed in terms of types Mueller space, the statement, I mean. Oh this statement here. Yes. In terms of the space. I mean, well, well, you see the cashmere space could have many many components I mean there are examples of. So, okay, so the cashmere space sits above everything right so already these fibers here can be infinite. Okay, okay, okay. Yeah, there are examples where it is infinite actually. Okay, okay. Thanks. Okay, any other questions. Okay, very good. So, so now let's see so now, as I said I'm going to say something about the subjectivity of the period map. This is where biscuits clever tip clever trick right. So what does it. It uses what what we call what people normally call twister lines, which actually knowing how cause very much more aptly he calls them twister conics, which I think it is my makes a lot more sense. These are actually impact really conics. So, so there are two ways of thinking about these. And let me let me start with the first way so if you let's go back to the to the period map right so we have this lattice gamma with with its quadratic form q gamma right and then we had this big q gamma which was the period domain right sits inside the actual quadratic which sits inside the project realization of gamma. And so we'll see what this was a period domain. Now, for a for a hyper cannon manifold, the signature of q gamma, after I cancer it with our you know after I extend the scalars to our is of the shape you know three. I write it as B two, but B two here is the rank of gamma. And I write it as B two because usually it's going to be the second betting number for the hypocritical right. Right. So this is the signature. What this means is that then you have. If you go into, you know, because the signature is is three B two minus three, after you cancer with are you can have three dimensional real positive planes right so. Each time. I'm sorry. Yeah, that's what we do. So, we can choose each time we choose over three dimensional real plane inside you know gamma tensor are positive for q gamma. So this is a twisted comic. So the way you do it is you just let me give a name to this positive plane let me call it P PR, if you like, and then I'll take. Now I don't want to call it. We call it something else. I don't know. Yeah, maybe I'll just call it PR. Let me call it projectivize it so they call it. Okay, let me call it FR then what I can what I'm going to do I'm going to take FR I'm going to answer it with see you know then this will be inside gamma and so it will see then I'm going to projectivize it so I'm going to say to say that P is the projectivization of this tensor with C. So this now is is just a P to write inside P of gamma. And so it will see. And now you can intersect it. So P intersected with Q gamma this is by definition. This is a twister coin. Okay. All right, and you can show it's not it's not that difficult. I'm sure that Q gamma is what we call twister path connected. What that means is that any two points of Q gamma can be joined by a path by a sequence by a connected sequence of course, of twister connects. And this is this is you know this is very elementary, you can find the proof in holy breath for Vaki seminar, seminar talk. So, so now you have this and, but then the, the, the trick is that you can you can produce these twister conics in a different way, and that's that has to do with twister families so that's the next thing that I would like to you explain twister spaces. So let's go back again so let's say that suppose that X is hypercalor. So G is now my hypercalometric right then there then then we know that we know that there is an S two of the Taylor complex structures on X right. We already saw that there exists. Okay, Taylor complex structures killer with respect to G of course. And then this gives us a whole S two of Taylor complex structures right so for any ABC in S two in the, in the unit sphere in our three. Right. Lambda equal to AI. Okay, so now we're going to define the, oh, and, and, and we recall also the caliform so recall that the caliform associated to land up with let's call that omega sub lambda, and you know we define omega by its action on a pair of vector fields right so those are my two dots my two vector field, and the way that it acts on vector fields is by taking G of lambda acting on the first vector field and then putting in the second vector field right so. So this gives us. The caliform right for each of these complex structures, and we have a family of these X lambdas right of compact killer manifolds. And so with this, we can define the twister space right so definition, the twister space. Let me call that, I don't know what am I going to call it. Yeah, I'm not sure, because I've used so many, all of these things well okay let me just call it curly X like I always do it's not exactly the same as the previous definitions that we had but. So X, it maps to P one of x g so the twister space is specifically associated to our given my particular manifold right is. So as a differentiable manifold is just a product. So X times P one, but. So as a same thing the manifold, and but it has a particular complex structure happened there. Okay, there we go with the complex structure. Actually it's really it's really an almost complex structure right meaning that it's only defined on the tangent space with the almost complex structure. So once I'm going to define this on the tangent spaces so the tangent space to the product is the direct sum of the tangent spaces to the two factors. And I will say what linear operator I have here as my almost complex structure. So it sends a pair of vectors to, of course, on one factor on X we want the complex structure lambda. That's what that one's going to be. And then on the other side we're just going to put the complex structure of P one. So it's the most natural thing that you could come up with. And this is integrable, meaning it comes from a real from an actual complex structure by a result of hitching. Linsprong, and rosec. So. All right so you have this, you have this fact this this product structure, you put on every fiber you so you see that the complex structure of the fibers is moving right so what you have here then is just. It's just a you know you have your X over P one right, the complex structure and for each each time you take a point of this P one which we think of. Also as S two right, so each time you take a lambda here, the fiber over lambda. It has complex structure. Lambda. Okay, so it's a very nice simple notion. And then you can you can map this way over the period map because P one is simply connected you can just you can just put a marking. Because P one is simply connected you can trivialize the the H two is along the fibers right you can identify H two of all the fibers with the H two on one fixed fiber right so it has a you know you can just define markings right. You can choose markings, consistently on all the fibers right. And we can choose consistent markings, the fibers to get the period map P from P one to the period domain Q gamma, which sends lambda to the point. Sigma X lambda. Sigma is a generator of H two zero right so where Sigma X lambda is a generator H two zero. Sigma X lambda which is equal to age zero or omega two of excellent. Alright. Okay so you get a you get a period map and then you can you look at it you can look at the image of the P one under the period map, and then you have a lemma, which I'm not sure who this is due to. I suppose I mean you know once you knew that this was a holomorphic family, you know, then probably, I mean I spoke hitching car hit the instrument rosec could have done this right or maybe somebody else did it before. I'm not sure. But the lemma says that the image, or the image P of P one is a twisted chronic. So this is now basically what gives you the surjectivity of the period map because. So the corollary is corollary of this, and the two surpass connectivity is that the period map from M gamma to Q gamma is surjective on any connected component of M gamma. Okay, and you see here. I'm saying on any connected component because if you look at your family, you can take any marking or of any fiber and extend that to a marking to markings of the whole family right so you can. So this this this just this work perfectly fine that's where you get you get your surjectivity. So what you do is that you know that there are. You know the map the period map is a local isomorphism right so you just take an interior point of Q gamma, which is in the image of the period map right and then. You have a point which is anywhere else in Q gamma you just connect that point to a point in the in your interior ball right via a chain of twister conics. And then you know that all of those twister conics are images of twister spaces, why the period map right so then that will give you the surjectivity. Did I. Remember you long sorry well yes but you can take the break now maybe or. Oh yeah yeah yeah I will I will take the break now yes was there a question or no. Okay, so so we will stop now for a. So I missed just one, like the lines of the previous slide. Previous slide this one. Yes, to get to exactly that. Okay, I can I can make the slides available if people wish. Maybe I should have actually done it sooner I'm sorry about that. I could have. Maybe I will. Can I email the slides to someone you could, if you wish you could post them on the, on the website. Yes, of course you can. Well you can email them to me if you want but also you can just email them to any of the secretaries, and they will post them so. Okay, I'm not sure who that would be that the usual address let me just give it to you the. Okay, thanks. This is the official email address of the school so then. Okay, all right, thank you. Yeah, or if you prefer just email them to me and then I'll. Yeah, I can that that's fine I can. You can send it to that data to SMR three six point all right. Okay so we'll stop now for five minutes until 11 o'clock.