 Okay, so, so we talked about the surjectivity of the period map. Okay, so the last thing that I wanted to talk about during these lectures is slightly different topic. I wanted to talk a little bit about hyper holomorphic bundles. Okay, and sorry. So, and it's related to the, to the twister families. So, and actually, oh, no, no, before we do that actually I need I wanted to say something else about the twister families which kind of, which is which is rather nice so. You know, we introduced this, the twister family right we, we took. We chose, we fixed a hypercalametric right if each time you fix a hypercalametric you have three. What you have an S2 of complex structures that are Kailer with respect to that hypercalametric, and then you can construct your twister space right which is a one parameter family of Kailer man of compact. And this is called parametrized by P one, and this P one is the P one of the complex of the killer complex structures, and the killer form for each of those killer complex structures is what you have here, this omega sublander. Right. Okay, so we have this. Now we can each time you have one of these you can produce many more, in fact. So, and that is. So this is the use that uses the Columbia theorem. Okay, so let me explain what the Columbia theorem is. So this is a club is conjecture, and it was proved by yeah. And what does it say. Yeah, so we start now with a complex manifold. We choose a Kailer metric on it with Kailer form omega, and remember that when we have something like this. And then we have the Chivita connection you know for the metric G, and the levy Chivita connection we have the curvature of the levy Chivita connection and then the Richie curvature which was more or less the trace, obtained from the regular curvature by taking a trace right. And then that guy was symmetric, but then you can in the same way that we go from the symmetric tensor G to the anti symmetric tensor omega, we can go from the symmetric tensor, the Richie to the Richie form which is now an anti symmetric tensor again so and that's what we call the Richie form. And usually we denote that row right now this guy is is a one one form right for the complex structure I. And all right so this is our data. Now we we we choose one more thing so suppose that we have another one one form, of course you know it's with respect to the same fixed complex structure with co homology class, equal to the co homology class of row and if you recall this is equal to two pi C one of the canonical class. Then, what's the conclusion. There exists a unique telemetric. G prime. So again it's killer with respect to the same complex structure with California. Let me say that differently. Who's caliform omega prime satisfies the co homology class of omega prime is equal to the co homology class of omega and row prime and such that row prime is the Richie form for G prime. So what are you saying here you start with a fixed compact complex killer manifold right, and you choose someone one form. The co homology class is equal to the co homology class of your Richie form of your fixed Richie form right, then you're saying basically for any form so what does it say it says for any one one form which represents the class of the Richie form. That's a unique telemetric whose killer form has got the same co homology class as your original killer form, and whose Richie form is a row prime. So, so that's, that's the statement. And now it has a nice it has a it has a nice consequence for the manifolds that we are interested in right for remember that hyper killer manifolds are what we call Richie flat right the Richie forms are actually zero. So if you like the C one of km is zero right so corollary suppose. Now I'm going to put the G in here is is compact killer with C one of km equal to zero. In each killer class. And I will explain what I mean by killer class in a second in each killer class on M. There exists a unique Richie flat metric. By which he flat we mean that the Richie form is zero for that metric. Okay, so what is what is a killer class a killer class is the class is the co homology class right. A killer class, if you like this is a definition is the co homology class a 11 form which is killer for with respect to some metric. It doesn't we don't know which metric it just it's just killer with respect to some metric. Okay, so. So these are the killer classes so what what we're saying here for in the in the Richie flat case and where C one of km is zero which is the case for hyper killer guys. So every killer class contains a unique Richie flat. Calumetric, right. All right, so now what does it mean for us for the, for the twister families, so we're going to go back to our twister families and apply this right so. Okay, so given me say I need to make sure I get this right. So, so given given our, our, our, our family, this family that we had of X lambdas right where land was a I plus bi plus CJ right in as to which we identify with P one right for each lambda X lambda. Well we have that C one of K X is zero right. So for any. Oh, I forgot part of the theorem sorry. Let me first do that. Okay, so this was just a parenthesis if you like so that's what we're going to do. What is this if you like so let's go back to the result that I was stating so the corollary says okay for each time you take a killer class, there exists a unique. Richie flat calumetric in that in that killer class right. So, um, furthermore, the Richie flat killer metrics, what did I call it I called it Emma on M for a smooth family of dimension. Each one one of M isomorphic to the killer calm. Okay, so now we're going to apply this to to our X lambda. So my ender is you know AI as BJ class CK right, and this belongs to us to which we identified with P one. So if we apply this to that what do we have we have that basically for every killer class for every killer class alpha. Remember this belongs to each one one of them. There exists a unique. Now that it says. Yeah, a unique hypercalametric G. Let me call that. G lambda care for lambda such that the class of the killer form. Omega G lambda is equal to alpha. Okay, so. So what do you have so you started with one telemetric and for but for each killer class for each time you pick a complex structure which is killer with respect to that to your original fixed telemetric. You can produce another hypercalametric so geometrically so so so then if you have a new hypercalametric, then you can construct a new twister family. So, so then we can construct the twister family for this G lambda. Okay, so so basically so what's the picture. You have your original twister family he won, right. I'm picturing this I guess really this this is happening in the period domain, but each time I pick some complex structure here, then I can produce another one. Right, this is the new one. This is this is this was for the original G. Right this guy here is lambda. And then this one is for G lambda. Right so you get lots of lots of these guys in the in the in the in the period domain like this. Anyway, I just thought this was kind of a nice picture to look at and it's useful for. So this is very useful for moving around you know in the period domain right so this is how you kind of connect things together. Um, okay, so now I'm going to talk a little bit about hyperhalomorphic sheets. When how much time do I have guys, when should I stop would say in 20 minutes, right, 20 minutes. Okay, alright, I will stop in 20 minutes. Okay, so just a few words. I'm going to talk about hyperhalomorphic she's actually not she's bundles. Okay, so um, what is a hyperhalomorphic bundle. So, um, let me let me maybe just drop the definition and I will kind of explain what it means afterwards. So let's, um, given a her mission vector bundle. We say that, but let me call it give it a name that the bundle be on X with connection with her mission connection data. We say the data is hyperhalomorphic. It is compatible with all the complex structures, lambda in the S two of complex structures that we have on our month on our hypercalamin fold. X. All right. So, um, and this is. Okay, so what do I mean by all this. So what's a Hermitian vector bundle where it's a it's a C infinity vector bundle with a Hermitian metric. And actually it's a C infinity complex vector bundle right so be C infinity complex vector bundle is Hermitian, if it has a Hermitian metric, which I will denote like this. So by Hermitian metric it's the usual thing you have a metric on each of the fibers of B, which is Hermitian, you know for the complex structure on that on that on that fiber each fiber is a complex vector space right so we can talk about Hermitian metrics on them. And theta is a connection on B right and we asked the connection to be Hermitian, which means basically that the connection applied to the metric is zero so the metric is again is a tensor and you can you can apply the connection to it and you can you can ask that it's zero. So, so this is similar to what we were talking about you know with with a with a Riemannian metric we talked about the connection being. What did we call it. The metric the metric was flat with respect to the connection right so that the connection applied to the metric have to be zero so it's the same thing here except that you're in the permission setting instead of being in the Riemannian setting. So this is, so we want this guy to be a Hermitian connection. And we have the curvature similarly curvature of theta. And it belongs to the endomorphism bundle of B, and served with wedge two. PM do as usual, right. And we say that. So, I said that B is a sin theta vector bundle. But B is not necessarily a does not necessarily have a complex structure right so if we have, if we have a complex structure, if we are given a complex structure on B, we say that data and I are compatible. If the current, the curvature form theta is a one one form with respect to I. So that's the compatibility. Okay, so now let's go back to our definition of the of the hyperhormorfic bundle. So we say it's what is it it's a Hermitian vector bundle with a Hermitian connection, and it's hyperhormorfic if it is compatible with all of the complex structures lambda in s to so what does that mean it means that you can. First of all, you can lift all of these conflicts, you know, there's a lift up your bundle B has a lift of these complex structures, you take a complex structure on your manifold you can lift it to your bundle and then in, you can lift it in such a way. There exists a lift right which is compatible with your permission connection. So I mean this is, you know I know it's a lot to digest and we don't have a lot of time, but I kind of think of these guys as. So if you like you can. You can sort of you can go to the twister space right so so what's the intuitive thinking here. So you can you can go to the twister space. So what did they remember that as a differentiable manifold this is just x times P one. Okay. And you can do the same thing so you can put the bundle here you can put B times P one as a C infinity manifold as a C infinity bundle. And what this kind of tells you is that this be times P one has got similarly to x times P one this guy has got a structure of a complex vector bundle which is holomorphic. Actually it's a holomorphic. It's a, sorry. It's a comp it's a. Right, it's a complex vector bundle which is holomorphic would be on each of the fibers. So, right. And that's why we call it in a hyper holomorphic so you have a you have a family of C infinity vector bundles a priori it's just once is a big C infinity vector on twister space, but then on each fiber it is a holomorphic vector bundle. And there is a very nice result of verbits key, which kind of tells you that if you start with some C infinity vector bundle. And that guy is hyper holomorphic, if and only if it's first to turn classes remain of the correct remain of hot stuff so. So what does verbits key say, you also need stability but so let's say given a vector bundle be on x. I'm going to zero. By that I mean the degree of the determinant. So, then if be has a hyper holomorphic connection. No, that's not the wrong one. No, no, no, no, sorry about that. I want to give you a different. Okay, given a vector bundle on X. With complex structure I, where X is hyper carer. If C one and C two of the R of types, you know, one one to two respectively so meaning these are hard classes. So with respect to all complex structures with respect to all complex structures lander on on X, then be is hyper holomorphic. Okay, so why, why do we care about this. So what is he telling you he's telling you okay you take your bundle. You put it on the sister space and it turns out that it has this complex structure. And with respect to each complex structure on each fiber. If you look at the first two turn classes, these are hard classes. So why why is this nice it's because. Then, then what it tells you is that these, these, these classes, which are priority are only hot classes are actually classes of analytic cycles. So this, you know, this has to do with the hot conjecture right this is this is, you know, this is a big thing and. So, so it's a very nice result it has already had you know applications you know in various nice theorems you know, for instance, you can use it to prove that any, any rational isometry between K three surfaces was came from a map between the K threes themselves. So this is, you know, it was it was it was a nice result so it's, it's, and anyway, so it has already had nice applications. So it is, you know, it's just something that I wanted to mention before, before stopping here. Maybe maybe I will stop here and see if people have questions actually before you know, I mean I can't really launch into anything. There's much more, I could, I could say a few more things I suppose. Are there any questions. How do these hyper holomorphic sheets look like over the K three surfaces. Um, in what way, sorry. Thank you as an explicit description and this. No, well that's the problem right I mean. I don't have explicit descriptions. I think Yoshioka has got some explicit examples of these guys over for maybe more for a billion surfaces than K threes maybe. And I think, I think Markman did it for K threes, or maybe it was again for a billion surfaces. I don't quite remember, but yeah there are some examples of these of these explicit examples, but they're they're kind of they're very complicated you know you use some. You use some four year mucaya transforms and stuff like that so it's, it's, yeah, I could I could show you I could send you the reference. Yeah. Writing down explicit examples is not easy. But yeah I think Yoshioka has got some. Do I guess people also study their modular spaces. If what is, I mean I guess I don't even know an example so it's too much to ask. What do people do people study their module I. Yeah, well there's a notion of stability right maybe I can say something about that. A little bit so. So let me let me maybe usually I mean usually you need that you need a stability condition so. So to look at module I right to construct module I have these. We need a stability condition. So what is so how do we how do we define that so you you first fix a caliform omega. And if I have a you can define this for just any coherent sheaves. Maybe I'll just. Yeah, I can do it for her and she so f. She then you can define the degree first of all, what's it going to be it's going to be one over the volume of your manifold times the integral over the manifold of the first chair class. The caliform to the power n minus one so n is the dimension of complex dimension of x and the volume by definition is the integral of the caliform to the power and. Okay, so this is the degree and then you can define you have a usual definition of slope, right so slope of F is going to be the degree divided by the rank. By rank of course we mean the generic rank. And so how what definition of stability do you take usually F is called stable. And of course it with it's with respect to this killer class omega right if for all. Subsheets F prime of F of such that rank of F prime strictly rest. That the rank of F we have that the slope of F prime is strictly less than the slope of F and semi stable of course you replace strict inequality with a large inequality. Okay so you have. You have these, you have these stability conditions and then. Then there's a there's a famous theorem of will and I can yell, which tells you in fact that if be is indecomposable, an indecomposable bundle on a compact manifold compact camera manifold. Then B is stable, if and only if B has what we call a Yang-Mills metric, and a Yang-Mills metric is a special type of permission metric. So let me maybe I can actually give you that definition as well so a commission metric is Yang-Mills, if, if it's curvature form is a multiple of the identity. Okay now but this, this, sorry this might not make sense. It's not the curvature form which is the multiple of the identity it's. The curvature form right I mean the curvature form is a one one form so it doesn't make sense to talk about the one one form being the identity. So you have to. Yeah, you have to kind of use the use the use the California to to. You have to take the adjoint of cupping with the caliform and apply that to the curvature form and then you can say that that guy is a multiple of the identity, but I think that I'm out of time and I'm going to stop here. There will be, I will send the. Handwritten notes to the, to the ICTP people and to post online but there will of course be a proceedings volume right there will be an expanded version of types notes with more references and more details upcoming. Thank you very much everyone. Thank you. This has been really fantastic and worthy closure for a very beautiful summer school. And so my thanks go from myself and other and Valentina to and Lota to all the lecturer to the two main lecturers to the exercise session leaders who have done absolutely great great job. Thank you so much. And to everybody who gave a talk. And of course, thanks to the participants for managing through this difficult time. We agreed that we are not going to get so the gather will be open, but I think we say goodbye here. Gather is open you can go anytime it stays open indefinitely. But we would like to take this occasion to thank the speakers again, you gave us a fantastic experience despite all the limits of the pandemic. Thank you so much. Thank you. Thanks for the invitation it was, it was fun. Thanks a lot. Thank you. And hope to see all of you soon in person. Yeah, yeah. Yes, if people are willing to get vaccinated. Yeah. Well, it willing and able and able. Yeah, that's right. That's right. Most of the world still is not able. Yeah, that that that is the problem that are unwilling and the many more are unable and and of course there's all the minors or kids. If we don't get everybody is not going to get. Yeah, but you know it's better than we thought I remember you remember it said that there would be the first vaccine would be in September 2021 so we are had a schedule. Yeah, that's good. Yeah, thank you very much everyone. Thank you. Bye bye. Thank you very much. Bye everyone. Thank you. Bye. Bye. Thank you. Bye bye.