 Okay, so it's a pleasure to have the talk by the April and on the geometry of hypercalamide. Okay, so thank you very much for coming in for the introduction. So, yes, today I want to discuss the, well, in this two seminars, the first part is introduced the main reasons of the second doc. So, basically, it's to mention why are hybrid better manifolds, what are the case of the two dimensional case, yes, so surfaces or K to surfaces. The main properties of K to surfaces that can also be for a higher dimension of that are the hypercalamide. There are other properties that doesn't come very pretty fast. So, the point is to try to generalize and how to get it for also for this special but I did. And then, and this is particularly I don't want to stay for that period. I will not shoot here. So, it's plenty of examples so please do many of these many questions because maybe there are some stuff that I just to skip it, but you have some questions just tell me, okay. So, perfectly, I want to say that this is over the setting of the complex numbers. Okay, and my right is here are what are manifolds in this case. And they are coming for a with a complex. They are complex compact. Okay, smooth off dimension. Okay, so, what is that K to surfaces because the stories that like try to generalize some properties of in the setting of surfaces to higher dimension. So, we'll be under the second accomplished come by the smooth surfaces. Okay, such that admitting to properties for the first one that is a single connection means that the fundamental group is. And the second condition is that they exist. I'm not a lot more. Before. It's not later. So that it's not the generate particular generate this part of the composition as a killer manifold. Sorry. So let's call it. Let me go see. And then the. And it's a polymer. Basically simple. Another way to define with the service and actually comes from the classical classification of algebraic surfaces is to say that they are a they are a good irregularity, but it's zero. And the other part is relating to the canonical device. We say that the canonical device of the street. So services, examples of classical Katie, so Katie surfaces are like coming from, for example, the first one is the baby for my. So consider the quality. In three. Okay. And then, so if you consider this one. You can prove that actually, this is an accurate results. In general, if you consider smooth. In three. So consider another polynomial of the before in three. This is also a on a Katie surface. Okay, so they are the big example and how do you prove that it's a satisfied with these two conditions that they have all of these main settings. But now, what we want to prove it is that H one zero, for example, and this is something that we can prove it by considering, for example, a hypersurface is right. So if you consider the of degree D in PN, then, if you respect the map of these obsessions of these a consider the one or the one forms this a the sections in PN, the restriction of this section is actually a induce a section for the for the device or on the, on the survive. So this is something coming actually for if you consider an arbitrary hypersurface in PN, and then, and the restriction of this bus induce and some of this. So there are no sections there. PN, but you can put it there. You can compute the common of your PN, and there are no sections in H one zero so then there's no sections or particular for the hypersurface. So this is just the road speaking how to prove it that in general and also apply it for this particular profit. And also, in the second condition, that is to prove that the canonical divisor is trigger. So this is coming from the adjacent formula. Since even for the qualities, you can see, for example, that this one. So can be seen like a section. And the point is that since the court, they can be seen like that. The omega of this keyboard is taking the omega of P3 tensor, the omega. And this is, we know that is all minus three minus one. And this is a whole four. So that give you in the second. And you're sticking to us. And this give you that extremely. Okay. So this is a way to think that actually I wanted to present in this case of this, why, how to prove that it's going to take the surfaces, because it's a natural way to think that which kind of hypersurface is can give you these properties. These numbers vanish, and that you don't have, and that the trivial kind of that you get that you get kind of. Okay, so then you can also play with this degree and try to prove intersection completed. So this is the first sample. So the second one is something that's more and more classical age related to the humor service. So for this part. No. Yeah, I can have one. Sorry. Yeah. It's not different from one of the ones you know the other services which have one thing is simply connected one thing is a one. Yeah, right. Yeah, but if you think this, if you, you get the composition of each one. I mean, if you consider each one of the. Yeah. You get this. So this is a very nice organization of the fundamental group. So this means in particular, if you have zero here, you get zero here, then you get zero here, and it's equivalent to say. No. No. Yeah. Yeah. You have to show that. They are free portion. I mean, but it's, you know, yeah, whatever you're somehow seeing it. You know, certainly, you have services with. Each one is equal to zero, which are not simply connected. But you know, maybe, maybe K is truly than. I mean, that K, K, but all I think my idea is something I don't know. Okay. Mike, Mike, I don't know something like that because I can. But it doesn't matter. Okay. No, because for me, it's, it's an old nightmare. You were considering where you were going. Okay. Okay. I mean, you have to pay store. Okay. I mean, also, also in the, in the classical classification of the services with the online bundle, you can, there are ones that are not simply connected and are, for example, complex terms. So this is an example that I want to say, and then the, and how to relate it to. So this is the comer, the comer surface. And basically you started by something that have the video canonical divisor, but without this condition. Okay, so how to get a key to surfaces. So you start with complex course. Okay. And then it's always here. And the door of it comes with a natural right that is the element here and send them to mind. This is that evolution. Okay. And in the course, it takes 16 points. Okay. So there are 16 16 points. By this evolution. And this is, if you will see, so there is a construction, like I do start by there are your context or you consider the portion, which is given by the natural portion here. And this map, and this map is the portion back here, the 16 points in views singularity in the course. Okay, so this singularity are all of type a one. Okay. And when you resolve the singularities, you consider the minimal resolution of this portion, you will get an assortment is that is okay to solve it. Okay, so let's say. This is the minimal resolution of these 16 single items of time. Okay, and this is the criminal associated to complete the diagram, because we like committed. And then the way that you can complete and actually if you the way to prove that this number is zero is to do any next move where. So here I have these 16 points, I can do blow up with these 16 points. Let's call this blow up there. Okay. So this involution here can be extended in an involution here in an upload where because just following the description of the blow up. And now when you consider the portion of these services with this new information that this is this give you a map to the humor of the service is not really an exercise at the point is that you can complete it. So you can also obtain the tumor, like the portion of these blow up in 16 points of these complex stores, and by the natural involution extension. Okay. So when I introduced this also this example because it's not what I could be, I will mention in the higher dimensional analysis of cases, how can we get it also hyper fundamental office, but in higher dimension. So here is that also you can think about, and this is on people work on this problem that also it possible to do these constructions for other orders for other autonomous the difference orders. Yes, the question is, yes, you can do it but you can ask that this map exists in the, in the case of the torus, and it's compatible with some map coming from the KP service. And this is that will be the second ingredient that I will stop in the in the second thought that is simply so. For example, and you can study the family of KP services coming like a resolution of complex stores by an automorphism of all the three in the, in the process. Okay. Okay, so this is the second one. And the third one, they want to say is the double tolerance of P2. And the third one that I want to say is the double tolerance of P2 so verify in some particular devices. So this device or will be a smooth 60. Okay, so this is kind of curve. Okay. And if you consider the following in this branch of this, this is your, your card that you will get one one this map. The surface here is an activity so. Okay, and this is another interesting example for, for this, for the services. You can also prove it that by the properties of this final degree of maps that the canonical divisor is previous. Okay. And it's basically using the fact that the degree is six. Okay. So this is the way how to prove that is also a key resource. So, I think that I was there examples that I wanted to mention. And now I've got my phone. Okay, so now what I want to do is try to generate life. These examples for higher dimension. Okay. So, let me see. This example that if I wanted to think how to generalize these facts for an authentic in dimension for at least, at least four. You can think that a, for example, let's take this one. One of my favorite. If you consider. We are thinking now. Okay, let's just define it first the day so white on a hyper color. Now, it's basically the sense seems that I impose this condition in this general setting, and actually the argument that I presented as a decomposition in the first degree of the homology. So you can actually ask this number is equal to zero, but when you just generalize the definition on hyper color medical, actually what you really asked is that the fundamental do this for them. Okay, so this is the same definition for hyper color manacles. Okay, and the fact that the canonical divisory Sweden actually is also in the in the in the general setting is requested by this by this property. In other words, hyper color manacles in the in dimension to correspond to Katie surface in the difference, equivalence definition that you have for a meeting for a piece of this. So, these ones. So when, when you want them to do that. So the first example that you have in mind is, try to do something like this kind of a game, but now consider product of Katie surface. So, let's, for example, let's ask the surface. Okay, and now I will consider the problem of this deeply surface. Okay, for instance, and then I will also get the portion here by an evolution. Okay, and this, when I was up when I get the involution here so you get the permutations right. So, what are the pixel office here, the pixel office is the dialogue. Sorry. So, this is a set where it's not defined. So it's not well defined. And then you consider the blow up in the product of this set of points, and then you get a new part. So, so there you are. Okay, and of course, you get the minimal resolution of this also. And then what actually you get here is a variety of dimension. Okay, which corresponds to what the professor that introduced like the human skin. On two points of s. Okay, so it's not really not immediately that this variety is that I'm a hyper killer manifold of dimension for this actually was proven by mobile. Okay, and then you can also do the same game, but consider now any product of this s. Okay, and then also you consider the action of this a symmetry group. You study again, the set of points where it's not where it's fixed by this map, you resolve it, and then you get a similar construction, but now in any call so the human skin of endpoints on a K2 service. You need to do any times. There is an, of course, there is a more a description of this variety in terms of source teams as a result of, I say, in the in his top, and also those correspondence are the same like this is the two medical models in some sense, and then yeah, this is an example. So what happened here, if you do, for example, instead to take a K2 surface, you consider an avidia surface. Okay, so that means that your tools is projected. Also, and if you do the same construction here, you can define the scheme of endpoints, all the tools, and what actually get is a hyper filament, also another type of hyper filament. So this one actually if you do the construction, if you consider him. And, oh, some appeal your surface. This give you is another example. I mean, yes. I mean, it's not simply connected or whatever. So you, you take that is something of that. Okay, or a fuller construction is something. You, I mean, this is not to have to kill a man. It's a complex service is an avidian variety. Yeah, it's not a fuller type, the generalize. So you have the map from me to the end. Yeah, yeah, yeah. Okay, yeah, you're right. Yeah. Okay. Yeah. Yeah, yeah, you're right. So this is something that generalize me the generalize of this in the car in the case when it is equal to one to get the tumor, exactly the tumor. So you have the tumor, the tumor surface, which is allocated surface. And then if you consider for any greater than one, then you consider, you have to do something. And you have to do something for. So the cover services to take the head with him on two points. And you, you take the, you have the map from the other team of two points, which is in the group, the group up after two points take the image of zero under this map that's a comma right. So the point has to mention, they have a small one point is not. I mean, the human scheme of one point just a little point would be something else, but you take the image of zero under this map. I mean, so it's not precisely that but the comma construction is precisely, you know, taking the human scheme but taking the image of zero under the map of summing up the points on the search. Okay, no using the group structure. Okay, okay. So it's a. Okay. So the different way of driving the form of construction if you just see what, what, what you were written down what that does. You know, this just decides the means that you kind of take just took off the inverse image of zero ending up. Okay, you know, I think that, okay, I'm more in the in the title. But I would think that, okay, I was reading also provided the seminar that for any equal to one that actually justify the name of the generalized tumor type. So the title to one in the in that description is just correspond to boomer surface, and it's something that comes from a complex source and then give you a key resource. So then if you do for any larger than one, then you get in higher dimension other examples of hypercalamide. Okay, so that. Yeah. So, to a particular example that I just want to mention, like for the existing dimension six and 10. And are so, I mean, at least how they appeared in the in the in this story is relating to something that I want to introduce it in the last part of the result that is relating to mobilize space of ships. Okay. And the point is that when you start to study these mobilized spaces. In some cases, when you put the good hypothesis on the on the shoes of the topological invariance, you can get the game that the structure of the key resources, or but in higher dimension. And the point is that they are provide another kind of examples of hypercalves. But the point is that if that is this mobilized spaces, you can find two exceptional cases in dimension six and dimension 10. And their examples were introduced by already. So, and it's been noted by over the six. So the point I will mention just I would be I will, I will do some comments about how they comes in the in the construction of the mobilized spaces. And, okay, so what happened for that. The big question this topic is trying to find new examples of hypercalematic difference to the ones that I described. So this is something that I mean, for instance, you need to impose some in some condition setting to say that those that I presented here are the now. Basically, it's coming from the notion of what is the formation. Okay, so the formation type, we would say that, for example, they want to say many things of the. So what is the formation but it's in the classical formation setting, but if you know what is that good enough, just assume it like a definition. We say that it is a hypercalib. Oh, any type. If it is the formation equivalent. And also, if you put that in this setting of the deformation, you can also say like, is the type of a general generalized comer, if you can deform it as a comer, a generalized comer. And also odyssey and odyssey. So, but why are why they are very different, because this information you can also read it in terms of the how to structure, and that's the point of the of the second part of this first part. So introduce what is a. And then, and also their role of the second group of college. So, this is a model. Okay, but more, more than a model, it's a lot. So, the point that is that is a group of commodity is a lot of is that he comes with it comes with a millennia for which it's not degenerate it's given by the top problem. Okay. So, we can start it is a group of commodity. And actually, if we consider. Okay, so, yeah, in the case of the case of the office is they have it. And then when you get here, the this. The complex structure of this model. You know that this comes from another complex structure given by the fact that is killer. And actually, you know that this is can can be the composite by these pieces. What's important here is that when you consider so just for notation, this is something, this is a zero of each on omega forms. Okay. And the same, but to zero. Okay. So the point here is that we know this is the part that contains the, the olomorphic to force right so this is actually generated by this omega piece. So this is the part that contains the geometry of the of the hybrid. It basically is encoded. If you consider the set. If you consider the model, you can see that this intersects with H2. This is actually the peak of x, which is isomorphic to that also is what it's on a hybrid element. Here, we know that I mean, the role of this part in the next part of the talk is that continue formation of the many devices. Okay, and the cons give you how to precise the complex structure of this. So this is something that. So this is the mobile for that actually, if you think this is a homological group is almost unique to us so it is. So what you can do is that if you study latitude in an abstract way for these families that I mentioned, you can get precise format of how it decomposes as a lot. So, for instance, if you have the silver scheme. If you take the surface, you will get that this one is isomorphic to three copies of the new copies of this. You can find lattice, E8 plus a part coming from the exceptional device. If you consider kumer type is just I think. You can distinguish that those examples were different because when you computed this number nine percent in here I'm a little bit shit and because now you can count it. What is the right here and the numbers are different right, but the point is that when you, if you are constructing a new example, and if you find that these numbers are equal, then we be correspond it's possible that they correspond to the same type of information. Okay, so this one are different, like for instance, all g6 have dimension six. So you can think that, for example, if you consider n equal to three and n equal to three, if you compute the circumventing number of this one is different to the key various things on three points of the case resources or the. So, this is how already, in some sense, found new examples of hybrid elements and all again in the same for all of them. So, I have a couple of music, I wanted to say something about the third year. So the third year is that in case the surface state a way to take the surfaces are isomorphic or not in a nice way. So, for instance, let's say that it's and why here are the resources. So what I'm trying to say is that if there exists an isometry of lattices of H2X and H2Y, which is a hot isometry is isomorphic to what. So it's enough to consider. So, it's powerful, the fact that this cap product actually determines the complex structure of the both cases of this. And this is something that you can want and also for higher dimension. But the point is that in higher dimension is not immediately that if you construct a hot isometry of these two lattices, then you can get it that they both hypercalium and both are isomorphic. And this is the, like the difference on the on the hypercalium. So that far, there are so many versions of this. And actually, I mean, personally, we'll prove it by Berbicki in a, you know, more in a potential geometrical setting. And then the hybrid then stated more in terms of lattices, but the was a weak version, and then Markman basically formulate this question in terms of monotony operators. And the formations are are very, very in some close the idea of how to distinguish these families. But I don't want to present in the monotony operators because I need more time to introduce what is that and what I need. The idea is that thoroughly for hypercalers can be a summarize in say that, but we need to study the course. So, there's the alcohol cone. There is the net cone, and there is the mobile phone, a hypercalium. We're assuming that it's projected to get some stuff. And, but it's also true for, for, for non-projected so just mark the case. So the point here is that the, you can also statement in some sense, like this, but asking more conditions on this whole taste on it. And the point is that, and so let's say that if they exist. And this area, which is so. So all of these, these conditions, if we satisfy satisfy that the mobile phone, the phone back, not sorry, this one. And the mobile phone. Okay, that this is also coming from, so there exists here a map, which is the isomorphic such that the pullback correspond to this map to the isomorphism between the two cases of stuff. Now I want to say the same for hypercaler manuals, but I will impose some conditions of this map. So I need that be sense. So the mobile phone, and put it in the right way. No, I'm sorry. Okay, the mapping commodity is actually the mapping commodity correspond to the pullback in this I saw on this. So if I put this notation, I will ask that the math income or a, the pullback income of the sense, the mobile phone, why in the mobile phone office. If you ask that you're isometry satisfied, you're how they saw me satisfy this condition that send the mobile phone of white is more of x, you get that x and y are irrational. So that's the point. And that's the point that you just on this level, you can just distinguish when to a hypercaler or not, but how can I extend it. I'm going to ask the rational morph is to the property isomorphism asking the second condition. So, if he starts up the net enough to meet us. This is an isomorphism of x intersect. This is an isomorphism. Okay. This is called the geometry of the of the hypercalem in two levels by rational and isomorphism. Okay, and I think that this is also under the setting that I'm projecting so they are projects for a general statement. So, you need to introduce the notion of one of the groups and and parallel transport. So, I want to skip that version, but just to keep an idea, this way for this in this setting, and because I will use it also in the second part. So, the, so we can say about the napkin and the movable cone for the K3 surfaces. So this is, so this one actually, the ample tone is the interior of this network. And for K3 surfaces, this is an equivalent. And this is an equality. Okay, so that's why for K3 surfaces, these two cases are the same. Okay, so it's, it's, it's a, it's, it's a big motion. But I have just one minute for the first part. So maybe I can just in the second part introduce what is the mobilized space and how we would use it to study, why we study mobilized space of the stable chips in order to get all the examples of hypercalamide. So, yeah, so the example will be study mobilized space of stable chips on K3 surfaces. And we will see that when we fix some topological invariance, we get it that actually these mobilized spaces are examples of hypercalamide, but higher dimension. So they come, they still have the geometry of the K3 surface that you are studying, but now they are parameterizing other objects. So these objects actually are the stable chips that I will stop here. And you have some questions. I'm going to leave a comment for other examples of hypercalium. There are so many examples in there, so many papers for these examples, but in another presentation, like you consider that there exists a Lagrangian vibration that you put on some other conditions that you can get that that one is deformation equivalence of this one. Okay, for example, to see some audience. And, and yeah, they're also in just in this type of deformation that is K3 and 5. We have this commodity here, which is very, very useful in order to study a one, a hypercalium that lives in this family can be seen like a Hilbert skin of and points. Or is just birational to the Hilbert skin of nothing or totally different. So, we will study the next part. The ones that are in this deformation type. Yeah, yeah, it's an amorphous in terms of the lapses that respect the, the whole structure of both. So you send the pieces in the right way. The form. Send into the exactly to the other something in each to zero something in each one one. No, no, no, it's not possible you have to send the two zero part here in the two zero part here. On one homology. In age one for an age one. So the question is, the, the, the model for that before, how does it know, do the pairing with respect to the different types. So, if you have something of type to zero, multiply with something of type to zero, what, will it be zero or will be something else. I think. Yeah, that was the question. Okay. Okay. Okay. And you think I don't understand very well. Sorry, but it's not part of the, it's not part of the definition of. No. I mean, we can discuss. Okay, another question. Yeah, I guess that are also many nonexistence. Yeah, I was there. In 1994. But, but, but, but I mean, yeah, there are so many bounce. I mean, many. There are some bounce of the, of the hot numbers. First, of the possible hyper killer medical yes. There are only left. Very few. Yeah, yeah, exactly. So yeah, yeah, this is kind of sad. You start to, to think in something like that that you compute this number and then you see that it is less than this bound and then you are maybe have a good candidate. Perfect dimension eight, how many also we have to have another. That's a good question for that. Yeah, yeah. Good question. Yeah. Yeah, I think that you also precise the dimension, you can get more restrictions. Nice question. Okay, so let's do the boss boss. I will not. Okay, so to the second part of the talk is relating to the problem that I was a study in the in my PhD, which is a relating to a study in some depth of action. Okay, sympathetic actions also actually in cases and then to a particular type of differential that I was focused in the, the K train a type. And then I would just give some motivation of this type of actions that just indicate to keep resurfacing they come from out of office and they are very, very interesting because it's another way to construct so many examples of K-3 surfaces, giving us a minimum solution of quotients of K-3 surfaces by simply out on the face. And basically that they are and say that on the last selected actions you would get again a K-3 surface, a new family, and then it is not simply, which is the, the, the not satisfied with our condition. Then it produced another examples and other families, which is relating to rational services for instance. That happens in the case of K-3 surfaces, but when you study sympathetic actions on hypercarous and higher dimension, you will also get that this condition that being being also obtained for by rational maps and the action will be a study in that in in comology, induced by this new sympathetic by rational maps. So, means in fact, the here it's that the map presents the period so presents the tool of multiple forms. So let's say that it's a map here. Okay. For instance, a very rational map, such that when you consider the action in comology and you compute what is the image of the period, sometimes it's called a period of these hybrid weather metaphors. It's the identity. Okay. This kind of actions can be changed in general, it's a some power of a handbook. But when it's implanted, it's properly one. Okay, when it's not simply if you get something different one here. Okay, and there are people and so many people that started this kind of actions. So, and what I want to finish the, the, the previous stuff with an interesting example, which is coming from a study model I space. So, how to, how to get a good mobilized space and how to impose the geometry of a hypercalia many for for this mobilized space, the coming from a from the fact that I voted this, this I fall. Okay, so conditions. So, stable here. So stable is according to the stability condition in terms of the healer polynomial. It's a free portion ship. And then you want that satisfy the condition is stable with respect to you consider a ship here, and then you consider a sub ships here, and then the stability conditions. And that means that for this street come, come. Yes, so she is the hillbill polynomial, which is something that depends on the polarize of the K3 surface is less than the 5050 less. You would say sometimes the same stable, you will put here. So, for the same escape. This is the case. And there is other pieces in terms of the hillbill polynomial, but there is a factor in this hillbill polynomial that is this low. And actually, you can ask conditions like it just slow stable, and there is good properties like that for instance for a ship. So stable, which is not stable, just a pieces of this hillbill polynomial implies stable implies semi stable device in use. So, if you want to study this kind of spaces, your parameter on a key to the surface. So, let's say that F is a key to the surface, a polarize key to the surface where it is the polarization of embedding in a projected space. Okay, this condition is something that depends on the polarization so big this class that you choose in your, in your case to solve this. And then, and if you want to parameter like this object stable ships, you need to see that these actually correspond, at least on a scheme and quasi projected of schemes and then if you put more restrictions on the variance of the chip, you will get that actually is a hybrid. So how to do that. So that, for instance, just keep in mind that this condition in place, this one is always true for this, this a precaution shifts. But to get another a implication is something that we can do it using some topological environments. And, and this is what I want to introduce, so if not just parameter these stable ships. If you consider a key to the surface without polarization, you can see a particular lattice, which is calling the mobile lattice of associated to us. For some of a zero of s interverse coefficients, plus each of us. Okay, and this is like a vector here, we call it a mucay vector. If you live here with some condition with some properties that I just want to skip it for a moment. But we will say that in the second component, the second component live in the Nelson very, you remember the Nelson very, or the people that where I summer, I saw more of it. They is it's considered H11 intersecting to this to last. Okay, this mucay vector. So, let's say that is the one, the two, the three. Okay. So, since these are lattice comes with the linear fall. And the linear fall is that if you have two elements, the problem is given minus be one is lost into minus. So this, and this, this, and this, this times, right, we are going to stop. The point is that this is the linear fall in the Nelson very so in the middle of this room here. Okay. And then, and this is something that is my neighbor. Okay, and this is also I'm bigger because the linear for our right to be to the universe numbers. So, if you consider there is a natural, a more peace that is called it the mucay and more peace. And if you consider achieve. You will send them these sheep to the rank of the sheeps and see one of the sheeps and key of these one minus. Okay. This is the color, the order. Okay, so this is a map. Okay, we find it by mucay. And then, and the point is that now I want to parameterize table shift objects on my key to surfaces that on this map, they have this number pieces. So this is a vector B. So I probably hear fix B. And then this model is space that I will do nothing like this is just the model life space. Okay, so this model life space is parameterized this stable cheese with respect to this polarization, but now, under this map, all of this imbalance are fixed by this B exit. Okay. This actually it's that nice model life space. This is not actually that exercising. And, but it's also, if you consider some, so that is not compact. So you can also compactify. Okay. I think that it goes. This is an open set in this compactification and they compactification can be written like. I think it's, it's chips that are saying a stable. Okay, it's probably I mean it's right it's technical, but the point is that you are more chips that now are saying a stable. So this is going to give you the compactification of this model life space, and now it's coming. Okay, so now, if I want, so at this level, we have something that is a good mobile life space a good candidate, but I don't have that they, they, they are hybrid. So, important some conditions on this topological environment so this is the first class. This is something in terms of the first and the second class, the other characteristic, and then I will fix this in variance and impose some conditions on the square of this element with this linear form, such that we can provide examples of a patient attack. So this is the result for many people. So, first, hi. Yes. Okay, so how do I write in a nice way. No, it's okay. So, so this is a. So what they say that if you can see for example you have to impose a condition of the other of the ample bundle. So, this is a generic. This is a condition that basically say that it's not living in is not a wall in the home of your catering service. So, you can also ask that the is pretty deep. It means that you cannot characterize for some other a vector, okay. Yeah, other mocha vector. So it's not possible like to take a value for case and liver. Okay. So, you are also asked that the square is greater or equal to minus two, in order to to get that it's not empty this model a space. And I think that's enough to say that this model a space is a hybrid. meaningful. Oh, KB. And actually, the dimension here to the dimension. And can be obtained. Okay. So, this interesting example is relating to the second part, because when I told the condition of, let's consider, and a hyper killer of Katie and a type at meeting this symplectic action, coming from a rational map. Okay, enough to say that is this introduction is not really good. Okay, that's a that the, the hyper killer meaningful have to be a mobile a space that as I described in that way. Okay, so that's the first result on this on this paper. So, so the first statement. So there are a lot. It's a hyper killer. At meeting. Exactly. By right now. Not. Not really yet. This is a model. With a mobilized space. When I take this office with this, with capitalization. Okay. So, we will see that as a consequence of this theorem. And this particular hyper killer meaningful cannot be general of the cannot be general. So that means that the, the bigger number cannot be one. So, actually, have to be at least two, but in the particular case case. Then, the bigger number of these, these, these two. Then, the order of this symplectic by rational map. This is possible. So, it's just a meeting in relations. Okay. And then, and the time result that we get it. It was more relating to another questions. So it was more relating to try to find or a or how to prove the system of this symplectic by rational maps for the candidates. So the point is, of the third result is now thinking that under this hypothesis we get it that it's a mobilized space. So how can impose conditions on this mobilized space in order to obtain simply active by rational maps. Okay, so maps that respect the period and Oh, I'm sorry, I forget to say an important hypothesis is that it is a symplectic by rational map such that the action in comology. When you consider the action in the portion. It's not really where it is a definition, the discriminant group of the of the of the slides. Okay. What is condition, because I want to do something that is property by rational. There is also an outcome of this, and the outcome of the satisfy that when you're speaking to this discriminant group, the actions video. Okay, so I impose that is, there is something that you cannot understand. Okay. So, and if you're entry, I have to write. Yeah, like, I received. Yes. Yeah, I hate this office. It's be like that. This is a pretty big record. Consider also that are different as you. And if you fix. This this one, the reflection on this class is inducing. A very rough. Not really a lecture. Hey. So now I'm going to say something. If I don't leave. Some conditions are satisfied. So here just I will say some comments. So the conditions are that is another question. It's a particular. I will introduce. Only if the satisfying the following condition so divided to see a maximum and the greater common divisor of part and as a sequence of one or two. And also different. Okay. Not one. Okay. He has to be different to this one. I will explain just in a moment. And if I have time, I will give some example. There is to say that these conditions are used. So what is the meaning of this theory. We are trying to find a nice career to to to prove the existence of these types of simplicity by Russian maps on modern spaces. But I really think there are two say that if you look the rank that is equal to two, you don't have to wait that the order. So you have just involutions. So what are the good candidates for involutions reflections marks, but reflections are max in the commodity with coefficients in Q so rational coefficients. So it's not always a map. Okay, so how did you get that your map with coefficients in Q can be actually a map with coefficients in that. So the point is asking some of these conditions, but it's something that comes from the from from results of my man that in this monodromy operators on reflections for some particular class in commodity. So this is the mysterious of these conditions and this and so in particular this one. And the other part is like a, if you are that you look at a vector is different to all of these options, then you will get a good candidate on a hypercalid of KDFI, which is a mobilized place. And with an evolution. Given by the reflection. Okay, so that's all the idea to prevent the system to say, good candidates that are in these conditions. Okay, and, and the point to how to prove these results. So, okay, it is more a corollary. Why, because you need to also study the combs for this particular case, but let's say a couple of words of this theory, one and two. So the first part is something that comes from the classical facts to study these simple actions, these infected actions you study in commodity. Okay, and if you call the conditions of this, that is, this action in the discriminant group is not really because there are two options or it's really are not for tradition for classical results. You can get it that and this a hybrid is birational to a mobilized place. Okay. But the fact that how can I study all birational models of these a mobilized space is using a second technique relating to the technique of the therapy, and it's the bridge and stability conditions. So, then here, the model, I'm a right to the we are answering to a birational model of each a first. So, first we get it a birational model. And then we can say that is isomorphic to the mobilized place using some facts, so strong facts on the stability a conditions. Okay, so that they are some of this. So why they are interesting for so why people is that is implicated by our helmets. For instance, if you're taking the old you then. Okay, so I have to say all the time that was this a exceptional example in dimension 10. And can be constructed by a mobilized space of ships, right, but with a not primitive mocha vector. Okay, so then if you have a right this table shifts. I think it's on a kitchen surface or on a, yeah. Okay, on a kitchen surface, the other case what you see is on a window. And you try to parameterize this family, but you don't have immediately an statement like the big one with the how many people do actually what you get it is a single or mobilized space. And you have to resolve the similarities that they produce the new example. Okay, so that for example, this mobilized space. And just admit by rationale simply. So there are no automorphists that are simply in order. So this is actually our recently resolved by people like Jonathan and grassy. And so they were studied this particular example, they are easy. She was also studied for the case of all the six, and then you'll get something similar for effective action so, at least for the known examples. So it's not necessarily for the big family of Katie and a type, but in the case of the office. It was something very deep study by Giovanni mongardi and hasn't, and collaborators where they study precise actions of these simple. In the case of by rational maps, this is the statement that generalize and finish this classification. So that's the phone in cooler. So, and I think that the, that are partial results in that on the number five. And yes, so. How many minutes I thought. So, let's say a little bit of the day for for the period one right. So, so how to study this is probably so, as I say the first step is study the action. In quality of these maps. Okay, so what is the other start in each exact or by rational maps, but this is something that you can do it. It's so big lattice. We'd have a good property, and it's a. You can enter in. And I'm not laughing lattice. That is, I saw nothing. Then look at lattice. So, so, I would like to do one of the space, but I don't have one of it. So I need to find a candidate a good candidate and ask me questions. So it is corresponding to the, the mucca vector I was looking for. So this is something that this is this idea comes from Markman theory and the end. The point is using this and then. So, the, the previous theory, the picture that I wrote here comes with an extra and an extra tool, which is an isometry. So, the summit is calling them okay so it's because really this mucca vector to the commodity of my hyper teller. Okay. So, when you get it's a multi space. This is the good. Yes. So, in fact, let us that is clear here. It's a. It's a zero. So I put a condition that is good at the minus two. Okay. When it's minus two you get the demodulated just a point. If the square it is an isotropic elements of the square is zero, you will get a game, and I can't restore this is probably not as important to the first one. It's possible. And then, and if this greater than, than zero, then you will get four points, this for whatever you want. And just that in the case when it's zero, then you this map is actually divided by. Okay, that's, for instance, I will ask this. And this is a particular. This is explicit. Okay, this is explicit. It also comes from in terms of the categories. I don't want to introduce it here because we don't need it just I need that there is this. Okay, and let me know to bite them. Okay, the input map. So this data make me see the commodity of this one. In this particular class. Okay. Just for instance I will be not to buy lambda. And then it comes to basically see a completely this part with a Mocha event or my possible Mocha event. So you can find a V in this abstract lattice. So it's not a. It's a. Okay. This is very upset and this is the stuff that is very powerful for the theory of lattice, because this, when you compute some properties of this Mocha lattice you get in a model. Okay, and you need more than it is very nice in order to get pretty different bad things and don't put some torsion. So, now I would like to extend this action in this bigger lattice. Basically, I need just to see what happened in this new vector. It's not a claim, but it's part of it's a, it's a proposition. So under the hypothesis that the action in this portion is not really an extension of G on V. And I still did not by the product of G is equal to minus. So there are no many options. Okay. So this is an extension. So the first, this is that basically extends the in in my class B. Okay. I can see this, like, I mean, sorry, I can see this, this new vector B, then in the part of the community is the central is the ones that I fix it. So this new action using tools. Sorry. You can prove that the coin body and lattice, which is taking the, the body and lattice. So I want to compliment to the body and lattice with respect to the Mokai obstacle lattice. Okay. Signature. One something else, which is the ground. This is the, this is the dislike is just considered the action in H2. Okay. So I consider the action in the community, I consider the body and classes, and then I consider the overall compliment to the body. Okay, it's called the point by now that the negative part of the lattice is something that comes just for the action on the community. And it is new class, this positive class is actually this. So, this one is for the, for the, and then this one is just the rank of the coin body and lattice on my commodity. Okay, so why this is useful, because now the theory say how to classify, in some sense, these lattices and actually that was the idea for the classification on husband and collaborators. And they get like a 279 and so possible a coin by a lattices and that's come from a particular at home. But they didn't for the after would be so. And here's working out so just for by rational. And the point is that you pass from this new em and then you want to classify it as well. Also, the step to is how to identify at the level of a lattice theory, you can impose conditions. In this abstract lattice. So, again, so this is lambda, this is a mocha lattice which is abstract because essentially, you can split for pieces like the copies of you to copies of the eight. And actually, for copies of you and to copies of that comes from the commodity of your safety surfaces plus the terms of a zero and a four. This give you another you. This is isomorphic always I've been to this lattice, which is really modular. And then you can as a lattice induce on a whole to structure. And then you consider the one one part of this whole to structure. And if you find some copies of you, if you find copies of you are one copy of you. And in this primitive. A lattice. This is equivalent to say that your hyper killer medical. Under this and that is actually by rational to a modular space of shifts on a key to so. We're so key to so. Okay. Actually, I want to say something more with details. We actually founded copies of you a twisted copies of you. And I think that it's good to have a mobilized stable she a mobilized space of the stable she's on a key to surfaces, you get mobilized space of the stable shift on a twisted case this office with twisted is just an option in the, in the cheese. Okay, but the, but the condition is the same stability as the hyper polynomial. Just to be careful. And this is a, this is a fact that actually in the non twisted case was something proved by adding them. So in the case it will be analyzed by colleagues. So we use this fact. We use this new embedding, and we actually compute that, for instance, this is something that you have to believe it but the coin variant part of a simple action. You can always live in this a one one part on the whole structure of this localize. So, it's enough to prove that this twisted copies of you can be also find it in the coin variant. And the point here is that the new involves you start to see what happened to this number here. And you can find that actually this is greater or equal time for, and this is enough to, to get this, this possible embeddings of the hyper value plan. So this is a couple of ideas of the code, but it's more technical than any questions. And I don't know if I, if I have time. Just two minutes. Now, just, just to say something about the other resource and where they come from some natural way. So they are here to the irrational model. So you can see this theory that a biomecrate that say at the level of stability conditions. So your mobile space now can be seen as a mobile space is dependent now for a new parameter, which is an stability conditions. The way that you start this by rational models of this mobile space can be a can be a replacement or can be translate in a decomposition of the of this stability management. The point is that when you are up to the rational model, you can find another modular space, which is isomorphic to the, to the first one. So that's the end, what do you get is that this hypercalibre of K3F type is by rational to a sum of the space, which is isomorphic to another by rational space on a K3 surface, but not depending on kind of stability condition. So, then when you're right here, you will do the good computations in the good chamber and then you will forget about the stability conditions you are right to be classical once the geese care mobilized space. And then it's very natural to think here, what happened for this for these conditions. Here's the statement. So the idea is by your in my cream post a map, which relate that the composition the world crossing on the staff manifold, and the decomposition in shambles and work of the cone of your model a space. Once again playing our goal. And now this reflection maps are coming from the composition in my stability manifold, which corresponds to that vertical wall. So there is a company there that is very special because it's vertical, then other one can be from a circle. Okay, this one, the vertical, you get it but this map, and that give you a class in the code. Okay, this class, this class have the play of. And on the sun assumptions of this e, as I mentioned before, this map is something in common of your queue, but on the sun assumptions, you will get that this is actually a, and it's okay. So that give you one part, and the point that it's implicated and three election on the discriminator comes from imposing these numerical conditions. So that give you many restrictions of your more correct. I mean, because I find, I mean, at this level, you lost a lot of information of the geometry of S, even this stuff comes from totally theorems and existence and this, but they don't give you what is the precise. And later, that part, if you wanted that if an involution and blah, blah, blah, then you get precise environments, and this environment. Those ones. Examples of this. They are in detail after you, if you have another different to the management box is denied. But, but basically, yes, comes from taking for instance, okay vectors in this way. Okay, and then you play with the, we are equal to one equal to two, and then, in particular, for the rank. Do you get some involutions, some geometrical involutions with the full back corresponds to that reflection mark on a particular class. And I will end this. I mean, the composition of the coin is in terms of by rational maps. Okay. So, yeah. Okay, if I understand what you're trying to do. Well, you, first, you have to do you have to prove that the wall that you were studying correspond to a divisorial wall or a wall wall. You have to put some conditions on them on the technical comes for a lot of that have this class. Okay, and this result by a McPhee say that if you started this particular lattice that contains this a candidate as mobile vector, you have to make sure the difference possible was corresponding to a floppy, sorry, fake, and blah, blah, blah. So that once give you what is the rational and why they are rational. So they give you all contractions or so they definitely be rational so I don't know that. Yeah, I just wanted to know how in the end. Okay, okay. No, no, I mean, yeah, this is so that results very much. So you study. Okay, so let me stop. Thanks. I like that question. Yeah, I stated that in words. So yeah, and you have a map here. For instance, it's a component particular component in the stability manifold. And then here is. You get here. And it's a stability points that you associated here a elements in the code, right. So you are considered here at this composition so you maybe have something here. And then composition of this manifold. Okay, that this give you some the composition some the phone. Okay, so the point here is that if you're right, for example, let's say that this is the point corresponding to my mobile space. Okay, my brother. So the first statement that they say is that you can arrive to another. Let's say here, for instance, but you can arrive here to that true mobile space, but maybe on a different, a, with a different vector. Okay. And on the same case resources or maybe another case resources which is the right equivalent to the previous one, just doing some crossing on these walls. Okay. The point is that this was my family day. The one that is very rational to each. And then I get my true model a space in the other company. How did I get it just maybe by some operations on this word crossing. Okay, so that's the result. Probably speaking of the. But all of these by rational model from a mobile space are again mobilized pieces. Okay. So, yes, yes, you, you have to arrive to the good one. And even when you arrive to the good one then. All of this is a study in this. This is a good evaluation of this element here and here that give you the right. Yeah, the, the right answer. But that comes from the from the future. Another question. No. No. Step here. Okay.