 Okay, so we are getting recorded. Alright, so yesterday. We started talking about. Sorry, it looks like I need to stop my video because again my computer's overheating sorry about that. Okay, so. Yeah yesterday we started talking about modular hypercalors. And so let me remember and the, let me remind you a little bit what we were doing. So we said we want to we want to study the modular spaces of these hypercalors that we've been studying right, and I gave you a couple of definitions. The definition of Taishmuller space. If we have a differentiable manifold X, then we, the Taishmuller space is by definition, the quotient of the set of all complex structures on X modular and equivalence relation down here. The definition of the equivalence relation is that two complex structures are equivalent, if and only if there exists a diffeomorphism pulling one back to the other. And such that there's an extra condition we want that diffeomorphism to be isotopic or homotopic to the identity. And then the, the marginalized space of complex structures was very, very, very similar. The only difference was that we changed the equivalence relation will remove the condition that the diffeomorphism be isotopic to the identity which is that there exists a diffeomorphism that pulls back one complex structure to the other. All right, so that's what we had. And then, we said, well, these two are related. That's a relatively simple relation relation between two. If we call diff of X the group of diffeomorphism of X and if zero of X it's connected component of the identity. And then the group of the group of the group of components of the diff of X is this quotient diff X by diff zero of X, which I called the G yesterday. And the relation between the the modular space of complex structures and the type of space of X is that complex X is types of X, modular, the action of G. There's an action of G on types and then you take the quotient you get comp. All right. So, we had this and what we said also was that, well, a priority I mean if we want to study to understand you know various complex structures on X, we should be interested in this comp of X. But the problem is that it usually doesn't look very nice for instance it's not house dwarf. It's not going to be singular, you know, sometimes badly, but Tash of X is actually much much nicer, which is why then we that's the one we we study most of the time in practice. So, in practice actually we usually work with small opens, open sets of this type of X. Okay. So these these small open sets will describe small deformations of complex structures on X. Okay, so now let's talk about deformations a little bit. So I'm going to first just define what the deformation is so first of all let's define what a family of complex manifolds is so that we just we're just all on the same page right. A family of complex manifolds for us you know in this in this series of lectures is a smooth proper morphism of complex spaces I'm not asking that baby, they they themselves you know this curly X and then the base s. I'm not asking them to be manifolds but I am asking the fibers to be manifolds. So, the morphism itself has to be smooth and proper so that that means that the fibers have to be compact complex manifolds. Alright so that's what we mean by a family. And then, now let me say what I mean by a deformation. That's, this is another, the continuation of the definition if you if you like, or I will just write that as it right as another definition. So a deformation of. Now I'm going to write it as a pair X, I X is the differentiable manifold, and then I is a complex structure on the on the differential man on the differentiable manifold right. And then X is a family. So curly X to us and I'm going to define to denote the projection map by PI of complex manifolds compact complex manifold sorry with the data of a point. S zero in S an isomorphism between the fiber, pi inverse of S zero, which we will denote let me let me give that a name so I will call that X zero. So this is by definition the fiber over little s zero of our map PI. So this is an isomorphism between this guy and X. Now what we mean here this is an isomorphism of complex manifolds, where I end out X with this complex structure. I. All right so this is a deformation. And then we, we will say that a deformation is universal for any deformation for any other deformation really right. To S prime and let me call this one PI prime. There exists a unique morphism fee from S prime to S such that. fee of S prime zero is S zero and X prime to S prime is the pullback of X to S by a fee. In other words, we have a commutative diagram. Like this. So, another commutative diagram sorry Cartesian diagram. So we have S here, and we put X on top of it that's PI. Here I will put fee S prime. And here I have X prime so I want this diagram to be a pullback diagram right. And of course because this was defined via a universal property. What we have is that the universal deformation is unique up to unique isomorphism. And we usually denoted. So that the top I will denote it the same way but the base I will call death of X right the basis is kind of special. So we will just call it death of X. So we have a Cornish's theorem. So because of this theorem of Cornish. These universal deformations are called Cornish families actually. So what does Cornish's theorem say it says that suppose that Xi is a compact complex manifold. So that's the top of X, Tx equal to zero. So here, you know up to now we were thinking of X mostly as a differentiable manifold and Tx was a differentiable tangent bundle here Tx is the holomorphic tangent bundle, tangent bundle. So H0 of Tx is zero what I mean is that there are no holomorphic vector fields on X. So the condition means H0 of X, the X equal to zero means that there does not exist the global holomorphic vector fields. So suppose that X satisfies this condition that it does not have any global holomorphic vector fields. Then, is there a question, then a universal deformation with with that complex structure I exists. And this is another nice property. It is universal for all of its fibers. So as I said then you know this this space then X going to death of X of sometimes people call it the Cornish the Cornish family of X. And there is another nice theorem which is going to be very relevant for us. I mean, it's the fact that, you know, deformations of K trivial Kelly manifolds are unobstructed. Okay, so I will explain what that means but let me first give you the title so on a trap are obstructed miss K trivial Kelly manifolds. So, so suppose. So we have this. So again suppose just like before that Xi is a compact complex manifold. And let's assume you know that we are in Korean in Korean issues situation which is that H0 of X Tx is is zero. And then so then let, then we have this universal deformation right. The Cornish family. So these are our assumptions. And what we're going to assume also that we were going to assume that this work we're taking a germ of the definition. Okay, so we assume that therefore X is very small. So it's just a very small open set. A very small open set around the special point little s zero whose fiber is our manifold X right. So, so this. So then what we say usually is that this extra depth of X is a germ of a deformation. So, so then another thing that can issue tells us is that for in such a case right for T in depth of X close to close to the point little s zero right. So here's my map pie again so pie inverse of a zero remember this is X with the complex structure by we have the following. We have that the tangent space at T to the space depth of X, can be identified with H1 of X, T, X, sorry, X sub T, and then T of X sub T. Okay, so again, so what am I doing here I have. I have my my deformation right X over depth of X. I'm taking a point here, right. Then I have the fiber, which is, you know, one of my deformations of big X and what I'm saying is that if I look at the tangent space at T to my to my base space depth of X, that can be canonically identified with H1 of X, T, with values in the whole more of a tangent bundle of X, T. Okay, so this is, this is a standard fact from deformation theory. And so then and then we have obstructions we have things that we call obstructions right so the obstructions, I'm not going to get too much into the obstructions but let me just give you the intuitive idea about the obstructions. The obstructions to deformations and these these obstructions exist to various orders so you can have obstructions that order. I guess that depends on your convention exactly whether you start at one or at two, but you know that the order kind of goes to infinity so so the obstructions give you local analytic equations. So the tangent space to depth of X in a neighborhood of zero in H1 of X, TX. Okay, so when I said that the tangent space to depth of X was H1 of X to TXT, what do you get from that you get that in particular, the tangent space at, at the point can be identified with H1 of X, TX, right. So we have that. And so in some sense so what we're doing here we're thinking of. So depth of X is an analytic is a complex analytic space right so its tangent space as a complex analytic space could be could be quite big right that depends on how singular depth of X is at the point s zero. So what we're saying is that this tangent space can be canonically identified with H1 of X, TX, and then we are sort of because we have a very small neighborhood, you know we have a very small depth of X, we can sort of think of depth of X as sitting inside its own tangent space right which we often can do with with complex manifolds right if you take a small or complex analytic spaces rather. If you take a small neighborhood of the point, you can always think of that as sitting inside inside its own tangent space. So if you think of this depth of X as sitting inside its own tangent spaces inside H1 of X, TX. Then you have these things called obstructions, which will give you local analytic analytic equations for depth of X in a neighborhood of zero in H1 of X, TX. Okay, so then we say that we say that the deformations are unobstructed, if the obstructions are zero. And what that means then is that a neighborhood, a small open set of depth X can be identified with a neighborhood of zero. In H1 of X, TX. So, um, so this would this would be a particularly nice situation right because this means that the base of your current issues deformation is just an oh you know and it's just an open ball right it's just enabled of zero in this vector space right so it's just an open ball so the community family is particularly nice looking right locally. So, this is so now it's a theorem of Tian Todorov and Bogomolov, which says that the deformations of K trivial manifolds X are unobstructed. But me what this means basically is that if your manifold is Calabria or if it's hypercaler, then it's, then it's unobstructed so in particular, if X is hypercaler, then it's deformations are unobstructed. Any questions so far anyone. Alright, so, and then we have a bunch of other nice results. I'm not, I'm sorry that I'm not being very careful with my attributions you know I'm citing results of other people in the lectures but I'll try to be more careful with it in the in the notes you know that will be published in the proceedings so I'll try to get also put in all the references and also. But it's a little bit difficult to do you know in the lectures anyway. Okay, so I was going to talk about that. Right so some some other nice nice nice results here. So other nice results. So if X is caler, then so is any small deformation. So this is again very useful for us right. This tells us that you know if if X is princess hypercaler not only are the deformations are obstructed but they're also caler. And in fact, and we have more so if a, again we have more that applies to us if X is caler and K trivial, meaning the canonical class the canonical bundle is trivial right. The deformations of X are also caler and K trivial. And then we have a little bit more actually even if X is holomorphic symplectic, meaning hypercaler basically. Actually, sorry, I take that back when I just when I say holomorphic symplectic. I suppose strictly speaking. That does not include. Did I include caler in the definition of holomorphic symplectic. Now I'm not sure if I did or not. I guess that's really a choice that one would make. I think maybe I didn't. And I don't think you'd need it strictly speaking we don't need to be strictly speaking assume that it is. Yeah, no I think maybe I did include. Okay, never mind, but let's just assume that it is caler for simplicity it makes things simpler. So, alright so if X is holomorphic symplectic, and that for us includes caler we put that here. Then small deformations X are also holomorphic symplectic. So basically if X is hypercaler also small deformations are also hypercaler. And furthermore you actually even have a little bit more so if X is irreducible holomorphic symplectic, then all fibers, so not just small deformations all fibers of any definition this is a stronger result of X are irreducible homomorphic symplectic. This is particularly nice for us right if we want to study deformations of holomorphic symplectic manifolds, then it tells us that if we take one deformation then all the fibers will also be irreducible holomorphic symplectic right so that's, that's kind of nice. And the key, the key to really understanding the deformation, the deformations of holomorphic symplectic manifolds is to actually is the period period domain right so the key to understanding the deformations is the period domain. Sorry, Professor, I have a question. Yes, please. But in the definition of irreducible holomorphic symplectic, do we include the caler. Yes, yes. But we said previously that small deformations of caler manifolds are caler so why couldn't this happen when our manifold is a, I mean, why all the fibers stay caler when X is irreducible holomorphic symplectic. Well, okay, that's because the irreducible homomorphic symplectic is a stronger condition so it will it will force. That's a priori only small deformations are going to be caler but but when you when you when you impose irreducible homomorphic symplectic you have a much stronger conditions and then it forces all the fibers to actually be caler irreducible homomorphic symplectic right. Okay, so, okay, okay. Yeah, so it is some. Okay, maybe it's a non trivial result. It's not. I will give you the references you know in the in the published version of the notes. Yeah. Okay, thank you. It is it is an entrepreneurial thing. Okay, okay, so the key it's so the key to understand the deformations is the period domain, and I'm. Yeah, sorry about that. I mean I guess I put. I'm not sure I can give you any more insight on that. Anyway, okay so. So how do we how do we use the period domain the thing is that what we're going to see, you know, in a few minutes is that we will construct this this this thing that we call the period domain and then small open sets of the period domain will be isomorphic to the deformation space of X. So, but the period domain is something that we understand a lot better. So that will tell you that will tell us quite a bit about definitions of X. And how do we make the period domain. A period domain, usually, you know you use the carmology of the variety to make a period domain for it right. So in the case of these hypercalamide faults. We will make the period domain by using the second carmology. And as I said, you know, yesterday. So, a lot of things about a hypercalamide fault are controlled by second homology now, unfortunately not exactly everything, but a lot of things. So, um, so this, this had so let me say that this has to do with, with the second homology, the second homology. H2 of X Z. And usually, you know when you take the carmology you need a polarization, right. So we need a polarization to to define the period domain. And that that is what people call the Boville-Bogomolo form. So now let me let me explain what the Boville-Bogomolo form is. So now I'm going to assume that X is is irreducible holomorphic symplectic. And for us this will be the same as irreducible hypercalam of dimension two n of complex dimension to choose a generator so choose a generator of H2 of X. So a holomorphic so a holomorphic two form a non-degenerative holomorphic two form on X, right. Such that when you take the integral of Sigma Sigma bar to the end on X you get one so this is just the you know just a normalization condition. You can't satisfy this you just multiply it by some by some complex number and it will satisfy this equation right so it's just just a normalize things. So how do you define, I'm going to just drop the definition of the Boville-Bogomolo form, the way that people came up with it. So this is something that you know for a K3 surface actually agrees with the intersection form on H2 of the K3 surface and the same for a complex torus of dimension two, it will agree with the intersection form on H2 of the complex torus of dimension two. So it's a generalization of the intersection form in dimension two. And it's also the way that people came up with it is that they computed some examples you know this of these Hilbert schemes of points. In fact, more specifically well maybe I'll say something about that later. Let me first give you the definition and then I'll explain more a little bit about where it came from so for alpha in H2 of XC right so we can perform we will define qx of alpha. So there are two ways of writing it down. This is one, this is the first way, integral over X of alpha squared sigma sigma bar to the n minus one plus one minus n times the integral of sigma to the n minus one sigma bar to the n times alpha and then integral over X of sigma n sigma bar to the n minus one times alpha one. Okay, and you can show that this is equal to if you write alpha equals lambda sigma plus beta plus mu sigma bar. Well now I'm doing the hard to become position right so I'm writing H2 of XC as H2 zero plus H11 plus H02 so here my beta is in H11 right and then the sigma of course is in H2 zero right this is this is H2 zero and sigma bar generates H02 then this qx actually has a nicer shape I mean you can write it as lambda mu plus n over two and then the integral of beta squared sigma sigma bar to the n minus one. Okay so it doesn't have such a terrible terrible shape right but it has a very nice property so Boville showed there exists a natural number dx in n such that if I take alpha to the 2n and I integrate that on X then this is equal to dx times qx of alpha to the power n. So in particular this is this gives you a little bit of insight about what this qx is if if I take the real positive root the real positive n through this integer dx then I can define a new quadratic form qx tilde by multiplying qx by rx and you can see that if you do this then you can absorb the rx into the nth power right this guy is then an nth root of the nth power cup product on H2 products and so then in particular this also shows what I was saying then in particular if you're if n is equal to one then you just have that if you have a surface right then what do you get you get that so rx will just be n minus one then integral over x of alpha squared is equal to qx sorry you get that this is dx qx of alpha squared right I think I mixed it up no no no no sorry sorry there's no square that that's just that's just the one right so it's going to be dx qx of alpha right okay so you see that then yeah so then dx is equal to rx in this case right and then your your qx tilde is the is just the intersection for right so qx tilde is just intersection and as I said the way that people came up with this right so people came up with this interestingly this was discovered via the example of the form of variety of lines of a cubic four fold so how does that work it's a kind of a nice picture so if you have a cubic four fold so if let's say t is a cubic four fold then so so t here will be inside p5 right it's a cubic hyper surface of dimension four and then I can I can call f the variety of lines in in t then people showed that this guy is irreducible homomorphic synthetic and it's not just Kailer it's actually projected right so we have this and then what you have is that the and people showed that you have an isomorphism between and I don't know which way is the best way of writing it I don't I don't think it really matters you have an average of the isomorphism which I would call AJ the fourth homology of t and the second homology of f and then the way that people got this book will go more form what what T is a four fold so this middle homology this has an intersection form and then via this average of the isomorphism you get the book will go more form here so this is basically how this thing was discovered so once they once they got this then it wasn't so hard to actually produce this in all dimensions you know just kind of look at the structure of H2 of a homomorphic synthetic manifold and then of an irreducible homomorphic manifold and then write down this you know this quadratic form basically you know with with the formula that we had it wasn't it wasn't too hard to figure out from there so kind of a nice nice nice picture okay so now there are some some properties of the quadratic form I will talk about the properties of the qx still there right this is after I rescale this so that it becomes an end through the end power cop product right um do I have to stop oh no I still have until 1055 I mean until yeah oh it's 1055 for me but yeah okay seven minutes right yes okay so some properties of this quadratic form this this quadratic form you can prove it it's not that difficult it's not divisible it is not degenerate and over the reals it has signature three and B2-3 if I if you restrict it to H2 of XR it is integer value on H2 of XZ right um I should write that down as well so it's integer value I should I should have said that first right sorry on H2 of XZ and you have a little bit more qx of sigma remember sigma is a generator of H2 of omega 2x right qx of sigma is 0 and sorry qx tilde of sigma is 0 actually for this it doesn't matter whether I write qx or qx tilde because I'm just talking about them being 0 right or neither for this one it's being positive okay and for t close to 0 in the core issue space of X you actually have a little bit more you have that qx of sigma t is also 0 and qx of sigma t plus sigma t bar is also positive what I'm doing here if I'm very close remember the depth of X is basically a small complex ball and if you're on this small complex ball you can trivialize you know the H2 right you can basically identify H2 of all the fibers with the central fiber X with H2 of X and then you can you can you know you can evaluate your quadratic form on H2 of any of the nearby fibers of the family right so that's what I'm doing here so I can I can apply the quadratic form to elements of H2 of xt for any fiber xt of my family alright so you have this so now this allows us to define the local period domain so definition the local period domain alright so what is this so I will first define qx as the set of alphas in H2 of X such that qx of alpha is 0 and qx of alpha plus alpha bar is positive and this guy is contained in qx bar which is by definition the set of alphas such that qx alpha is 0 and this one is inside the projectivization of H2 of XC okay so this is this is the local period domain this qx isn't is a real analytic open subset of a quadric this which is qx bar a complex quadric inside this projective space ph2 of XC right so that's our local period domain and so because of this property that we had up here right that qx of t is 0 and qx of sigma t is 0 and qx of sigma t plus sigma t bar is positive we can define the local what we call the local period map right and where does it go it's um it's it I will denote it px and it goes from depth of X into this period domain qx what does it do well it will take a point t and it will send it to the class of sigma t where sigma t is a generator of H2 of omega 2 of XT right so this is this is our local period map okay and then you have some nice properties so the local period map is just sending you a defamation space into the period domain which is a real analytic open subset of a complex quadric right okay and we have some properties of this local period map so what do we know that this period map px is holomorphic because sigma t varies holomorphically with t because it's a section it's a section of H2 of omega omega XT right omega 2 of XT right um and we have local Torelli which says that px is a local isomorphism and what that really means is basically that the differential of px at 0 is an isomorphism because then you can always just restrict the domain the domain of px to get to get an actual isomorphism all right and I will stop now for the five minute break thank you should I stop the share screen