 So, thank you very much for the invitation to this conference. And I will talk about joint work with Andreas Langer. And it is about the deformation theory of hypercalar varieties from the point of view of this place. And I mean hypercalar varieties were considered recently by several people, for example Charles. And these are higher dimensional varieties, which are very close to case resurfaces and their properties. And studying this owes much to two articles of the line wrote down by Elusie in a lecture note, Syrophase algebraic in an old lecture note. And so let me recall what is the frame that consists of a ring, w and ideal j in this ring and r is a quotient and there are two maps. So, one map, sigma is an endomorphism of this ring. And w and sigma point is a map from j to r to w, which is sigma linear. It has a property that the image generates w as an, so the image sigma w generates the unit ideal in w. So, it generates w as an ideal. And moreover, if we consider sigma, okay, I mean I, I fix now a prime number p, which is bigger or equal to 3, but this p will, all the times become bigger during the talk. What is sigma? Sigma is any endomorphism. I mean, this is only the general framework, but I explained an example. So, these rings, w and r will be p-addict rings. Often p will be nilpotent even. And sigma modulo p will be the absolute Frobenius. Sigma x modulo p will be x to the power p, where x is an element in w. So, and we consider two types of frames here. One is we start with a p-addict ring and we take w, r, which is then also a p-addict ring. And here we have the ideal j is equal to the verschiebung of w, r. And then sigma will be the Frobenius on the width ring, on the ring of width vectors. And sigma point of vxy will be equal to psi, where x psi is an element of w, r. So, this is one example and this frame is called w, r. And also we consider s to r and the kernel is an ideal a. And we assume that this is endowed with a divided power structure. And then we have w, a. Excuse me, this is a p, d. And a is a p, d, ideal. So, and then we have w, a. And here we have the width polynomials, which are w, n, which are map. So, if we take the product over as a width polynomials, we get a map to the product of a, which is, of course, sometimes zero. But now, because we have divided powers, we are allowed to divide these in the sense of divided powers by p to the n. And then this here becomes an isomorphism of a billion groups and even of w, s modules. And where, of course, I had to say what w, s here means, but it means that on the n's component, w, s acts via the width polynomial. So, and so here we have. This product is indexed by n. So, this is a countable product, yes? Yes, it's a product that is indexed by n. The product is indexed by n. And here also for all n bigger than zero. This becomes an isomorphism. And here we have the set of all elements a, zero, zero, which, which we call the logarithmic Teichmüller of a. So, this is an ideal in w, s, which is isomorphic to a as in w, s module. It depends on the PD structure, of course. You know that this is stable. Usually this is not stable under addition. This a, zero, zero, zero. When you add two things like this, you get cost terms. I mean, this is not a width vector, of course. I mean, if I write this as a width vector. Yes, it's exactly the same. Very much of this product. Of course, if I write it as a width vector, this has infinitely many components, maybe. Yes. And, and then we, we, this, I call it a Teichmüller. It's equal, not content. Equal. Oh, yeah, yeah, yeah. Equal. So, and then, then I have here the map to, to R, which, which is a map w, 0 to s to R. Yeah. And the kernel of, of this map, j, turns out as a direct sum of a till plus ideal w, s. And, and this, this ideal, so this is an orthogonal decomposition. So if we multiply two elements here, it, it is zero. And this ideal has then also divided powers. Because on the width ring, we have here divided powers. And here we have the divided powers we already had. So this is also a frame. So the Frobenius SSB4 is a Frobenius on the ring of width vectors and a sigma point of a till the plus is equal to psi. So this sigma point just annihilates the ideal a. And, and this gives another frame, w, s, r. And so if, if we have, if we have a p-addict ring. A p-addict ring means the work. Okay. I mean, it means a ring which is complete and separated for the p-addict topology. And, and then we have x is a, x over r is a proper, proper smooth formal scheme. Say first for the p-addict topology, but maybe for some other topology. What? The model from x to r is abic, in order to analyze. Yes, of x. Yeah. Yeah, maybe not always. Yeah. For example, yeah. It's a little bit problematic. How? Very smooth. What do you mean by smooth? Okay. By, by this I mean this xn, pi n, smooth for any n. Yeah. Yeah. Okay. But man, sometimes I also consider this here. So I have some other headache topology. Okay. But the morphism is abic. If you change topology down, you change it also up. Yeah. Yeah. Okay. Yeah. It's abic. Yeah. And then I say I have a, I have a lifting x tilde over w, r. And then there's the following properties. If I consider, so a lifting as a formal scheme. I consider the Dramco homology. Then I, I will assume that this here is locally free free. And, and so if I take this as a one spectral sequence ej, this should degenerate. And then, then if I consider, so, so assume r if r has no p torsion. But in general for the finiteness theorem for formal scheme, usually one needs the ring to be neaterium. Is it as here maybe? Yeah. Okay. The ring, you are right. The ring is nursery. So it will be always. Okay. And if it has no p torsion, then we have a, so that this homology. X over w, r. This is for, for n smaller than p. This is a display. Yeah. Let me, let me recall what this means. So first of all, I mean a display. This means that, that we have here for the display for, for the frame w, r. Yeah. And so first of all, a display consists display over f. Yeah. Or pre-display. It consists of a series of homomorph, of w module homomorphisms. And this here, this will be, then the, in the example, this will be the crystalline homology of x over w, r in some degree n. Yeah. And, and also it consists of maps from J pi i to pi i plus one. And, and of sigma linear maps f i from pi to pi zero. The f i is a, is a, is a series of sigma linear maps. So, so this here is linear. And this is sigma linear. And, and we have that f i plus one of alpha, say eta. Yeah. If eta is an element of J, x is the same thing as sigma point eta f i of x. So, is this equation? Slightly more general than the definition in your book or it's equivalent. Because in your book it's, there is just two things. Yeah. Okay. I mean, I, I also have, I, I defined this place for p-divisible groups. But, yeah, but now if we want to consider higher homology groups, we need, we need higher displays, exactly. And then, so there I could add more axioms, but this is not absolutely necessary because so that these data, they should arise from standard data. Yeah. So, so display. So, standard data means just we have modules, LD, these are W modules which, which are finitely generated projective. And, and we have a map phi. We have maps phi i from l i to the sum LD and this, this module will become p zero. So, we define it now as p zero and so, and, and, and if it takes a direct sum of this we denoted by phi it gives, so this is sigma linear. It gives a sigma linear endomorphism of p zero and this should be an outomorphism. So, for, for example, if this module are free then this, then this phi is maybe regarded as, as a block matrix in, in GLW. Yeah. So, more or less you can say a display as a block matrix in GLW. But you, but we have to, so then, then for example, we, we call p i plus one as j li plus one as j l zero and then we have here p i and, and here we have the maps yota, yota i and it is here. It is a multiplication by p. Yeah. And here we have li. So, here is always a multiplication by p but here is an inclusion and here is the identity and, and also we have g tensor and we have g tensor p i, j tensor and so on. And here we have the maps alpha to pi i plus one which is a j li. And, and now, now we, we have to distinguish. So, if we, if we are, for example, in the case, yeah, so this is first for the case wr, yeah. So, and then this map here is, if I, if I have v psi tensor v eta. The verschiebung of eta, yeah, times some element of x, it goes here to v c eta x, yeah. So, this map is called the, the Veryungung. And here, excuse me, here, yeah. Because you go from zero to one. L, L zero, yeah. So, and, but I don't multiply, but I, I multiply and I do it like this. Yeah. Yeah. So, this is, I usually call the Veryungung, which is a word from tensor. From tensor algebra in German, in Germany. So, if you, if you take a tensor, then you can take the trace with respect to a co-variant and a contra-variant index. This you call in Germany the Veryungung. Okay. And here you, you take this also and here you have j li, li plus one. And here you just have the inclusion, yeah. And, and then in the end you, you have the Frobenius. So, if you have here Pi, then you have zero, yeah. And here you have the map to P zero, yeah. But you don't take this map, but you take the map phi zero tilde, yeah. The phi zero tilde of V xi x is xi phi zero x and so on, yeah. G li minus one, phi i minus one tilde to P zero. And then you have here li. Here you take phi i and then you take P times phi i, two pi zero and so on, yeah. And, and then you get a, you get a pre-display, yeah. So, so then this is a pre-display and a display is a pre-display which is isomorphic to a standard one. So, this is this definition. So, we get Pi, Pi i, tilde i, alpha i, f i. So, this is a standard construction, yeah. This is our construction. And this play is a, this play is a pre-display which comes from the standard construction, which is isomorphic to, say, the star from the construction, star, yeah. Construction xi phi phi is well. So, if you have sheaves, it means locally it comes from or it? No, global, yeah. Yeah, I mean, in the applications we have always local rings and so on. Maybe you, you should make this definition more general, but now for this purpose it's sufficient to do it like this. And, okay, we could also give a definition without referring to the LIs, but this is more complicated. But for example, if we have, if we have, for example, if R is a perfect field K, yeah, then a display is the same thing as a module P0 with a Frobenius. So, this is from P0 to P0, which is, which is injective, yeah. And so, so an effective isocrystal. A display over K is the same thing as an effective isocrystal. Yeah, because you... You don't make things vertical, right? No, no, no. So it is effective, yeah, okay, okay. It is a Frobenius module, yeah. A module with a Frobenius. So you, you consider P0 and then you have here F to the minus one P0. And then by the elementary divisor theorem, this is a direct sum of L to LD such that this here is L0 direct sum P to the minus one, P to the minus D LD. Yeah, you can find such modules and then you make out of these modules a display. And this is an equivalence of categories, it is. So if you, if you are over a perfect field, there is no difference between these two notions. But so this is, this notion is made to, to study the crystalline cohomology relative to a ring which has nilpotent elements, yeah. So over an R-team ring, for example. And I mean we have also this place for WS over R. Then we have to, then we have here some other ideals, yeah. So where J i is the ideal A i plus E w of S. And, and we also have the, the Fajungong, which is a map from J times J to J, which is new of A tilde V xi tensor. B tilde plus E eta is equal to, is equal to A tilde times B tilde plus V xi eta. And we have also the map pi of A tilde plus V xi and it is equal to A tilde plus P V xi. And then you have to replace this P here by pi. Then you get what, what is called a WS over R display. Excuse me, W. Do you need the assumption of the generation of this? Yeah. Because if, if this is not the case, then, then the display structure should be on the complex and the derived category. And I was not there. I didn't make such a definition first. I have to, to see that this definition works in some reasonable sense. Okay. So we, we, we think that in general, if we have X over R, say, smooth and proper, and now say R. In our case, R will be a local R T in ring. And, and this X has suitable lifting properties. Yeah. So then, then we would like that to X be associated. We associate PS WS over R display S to R as before. Yeah. So, and this should be a crystal in this place. So, for this one has to say, if, if we have such a commutative diagram, P, N, R. But we will also use it sometimes for other things. So if we have such a diagram here of PD extensions, then, then we have a base change from WS over R dash displays into WS over R displays, which, which goes like, if you have here a standard data li as in the standard data goes to WS tensor WS dash li. Yeah. So here you have a display which is here, defined by a standard data li. The base change will be defined by this standard data. Yeah. Yeah. So it's associated to X for if you have a variety X over R, you have for any PD extension relative display. Okay. I mean, if you, if you have, if you have, for example, PS, yeah, which, which is here PIS, yeah, then, then the P0S will be the crystalline chromology of X over WS. So you choose a degree for the crystalline chromology. The sum of all degrees of some. Yeah. Okay. We fixed some degree N. Yeah. Excuse me. So this is here, HN Chris. It comes in with a display structure. And then. So here you don't assume any behavior like before on the locally free and degenerate. No. Yeah. Okay. You do. You need such a lifting. Yeah. In fact, you will need, you need any frame over R, which is P torsion free and where, where this X has a lifting in this frame to W with a spectral sequence degenerates. Yeah. Yeah. The display itself has some mystery nature. So it's a kind of feature also in the picture itself. Because you are fixing a frame, so frame is also a feature. Yeah. We will now see examples versus holds. Yeah. So the first example is a theorem of Eichelau. So this PS exists for it's a billion varieties. Maybe it should be not too difficult to prove this conjecture. But I'm not so far. So, and now, now say P is a two display. I have to say something else. So if we have an arting ring R, yeah. And then we have not only W R, but we have also the small width frame W hat R. Yeah. And the small width frame, it is like this. You have, you have here the ring W hat R. And you have a surrective map to W. So R is, say R is artingian. And R by M is a perfect field K, yeah. Then you have, this is here contained in W R. And here you have W hat M, W M. And these are the width vectors which have only finitely many non-zero components. So it turns out this is an ideal in W R. And then you can form this as a small width ring. And you can... Yeah, if you take the sum here of two, yeah. Yeah, okay, this holds, yeah. No, I'm not sure I understand. Okay, in any case this ring here exists, I don't want to discuss it now, yeah. Okay, it depends on the assumptions of R. It depends on the assumptions of R. Of course it's M, so M is nil-potent. M need to be a nil-potent ideal, yeah. If M is zero, this is the same ring as the ring of R, right? Yeah, of course, yeah. Then it is the same ring as W of K, yeah. So in fact this, this here has a canonical splitting. We have a canonical splitting and this ring is the semi-direct product of these two. And I mean we have also... I mean we can, if you want we can also consider this place over these rings. These are also periodic rings. You can also make this place over these rings from varieties. So what is... For example, we have a variety X, X zero over a perfect field K. Then we call, we say it's ordinary if F from H2. I consider in the moment X H2. It's a Frobenius. The Frobenius is an isomorphism, yeah. So if we have a deformation X over R, yeah. And... So this is a usual Frobenius of this. Yeah, okay, okay, this is true. But here I consider only this Vika condition, yeah. For example, for case resurfaces, this is equivalent, yeah, to this other notion. And then I take the two-display, yeah, P. And then I call it ordinary. If X zero is ordinary, then I mean this here is a two-display, yeah. So P comes from a standard data with only three modules, yeah. It's a two-display. And so it is given by a matrix, yeah, by a block matrix. So this V may be written as a block matrix, yeah. And ordinary, in our sense, means that this is invertible. And so I could... If I, yeah, okay, I have the following theorem. We have the following theorem. So P, an ordinary two-display, relative to WR. And the dual is also ordinary. I mean, I should say that this plays form a tensor category which has duals, which has all the operations of linear algebra. But we can also consider the inverse matrix. And then we can also write it in a block matrix. So the dual is ordinary means that this matrix is invertible. If I have S over R and a PD, then I have a base change, yeah. And so here I have P. And if P1 and P2 are a WSR displays lifting P, then there is a unique isomorphism between Pi1 and Pi2. Uniquely isomorph... So there is a canonically isomorphism. Yeah. So in this case, automatically if we have this display, it already is a crystal automatically. We need no... Why do you need this base change of displays? So the displays are from the WSR part. Yeah. Here I have a display WS over R. And here I have a display over R. And here is a base change. I mean... P1 and P2 are what? WS over R displays. Yeah. P is a WR display. I have a WR display and I have this base change function. I ask myself whether I have here relative displays lifting P. Of course I have this because this play is simply a matrix. Of course I can... Because this is nil-potent any... The kernel is nil-potent any invertible matrix lifts, of course, to invertible matrix here and gives me here a relative display. But these two are canonically isomorphic. And also... Yeah. Now I have not much time. Now I consider X hypercaler. And so if I... If I have, for example, X over R smooth, then hypercaler means that if I consider X of the tangent bundle X over R is 0 and H2 X of the tangent bundle X over R is 0. And also there is a form omega in X over R which does not vanish in any point. Now we are vanishing. So which defines now we are vanishing and H. So this is a symplectic form on this many fold. H2 H0 omega 2 is equal to R. The differential of omega is 0? The differential of omega. Oh yeah. The differential of omega is 0. I mean it is not in the definition but maybe it follows. And also it follows then that this TXR is isomorphic to omega XR which is given by this form omega. Omega 1. Not normal vanishing but everywhere non-degenerate you mean. Non-veh degenerate. And then if I have X0 over K is hypercaler, excuse me. Then to any such X this is again an artinian ring. Then the display also every so the two display X exists here. And so we have the following theorem. So if I have X0 over K hypercaler then and X is a deformation. Then I can so X is a deformation of X0. Then I can associate to it a display PX a two display which is self-dual. And if X is ordinary then this is a bijection between the deformations between deformations of X0 and deformations of P0 to a self-dual two display. So this is a bijection. In the ordinary case so if we of course we have in the ordinary case also another description of the moduli of the deformation space in terms of the generalized Brauer group. So there is an old theorem of Neigard. So this display is an extension of the display of the extended Brauer group and you can describe it as an extension class in the category of two displays. And I mean we have also other deformation theorems but which are not so which I cannot describe which are in the moment for lack of time but they hold for any X0. I mean we should remark that if we have WS to WS over R we have this base change then a display here is the same thing as a display here plus a lifting of the Hatch filtration. So this display has a Hatch filtration which comes from the PI's and so a relative display if you want to extend it to a display over S it means the same thing as lifting the Hatch filtration. So thank you. Any questions? If your base rank R is periodically complete and into your domain is there a kind of way to associate our representation of the fraction of R to this play and higher display in the case? Yeah okay I don't know. I mean I know this for one display of course. I mean even if you say if you have here a complete local ring and you have over this one display and so this is a complete local ring with a Galois with field of fractions K and then we consider Galois K bar over K so then to any one display you can associate. Yeah okay but you can describe this directly from the display without going. Yeah this I don't know. The problem is to construct a trade module for a higher display. I mean you don't know how to define a trade module for higher displays. Yeah okay but a more natural question would be directly. So here you can directly you can this to a one display. You can directly consider this Galois representation. I mean by sort I cannot describe it in a moment but it is yeah you have to go to a bigger ring. Question about the deformation here you consider infinitesimal or formal deformation. Well they are insured by the condition that is 0 and it is 2 or 0. Yeah so then you have to. But my question is in the case of case resurface is of course the formal deformation. Then also line bundles lift. Yeah okay you can also have a deformation theory with line bundles I mean. So what is I don't remember DH1 is 0 in this case? And then you can also imitate somehow the proof of case resurface. Yeah we imitate a lot the linear we see. By using shown class. Yeah yeah. And also since you are in the ordinary case you get some kind of group structure on the module. Yeah I mean. Because you mentioned that the power group approach of the Niagara which is nice. Yeah we should get this but in the moment I cannot. You are canonical lifting first. Yeah but in some sense you have here also canonical coordinates because this Px is more or less. Even group structure I think. Yeah. Of course binozid exists. Yeah. For case resurfaces. Yeah. But we haven't generalized. So I forgot a lot. Yeah. Also I mean the whole thing I didn't mention that it depends on the Bogomolov-Bovil form. The Bogomolov-Bovil form. But I had no time to explain it. So what is the Bogomolov-Bovil form? It's a guess. Okay I mean if you have. You have here the second the Ramko homology group. You have to explain it because otherwise it's unclear what self-cruel means. Of x. Of x over r. So and then you have here a quadratic form to r. And so in the case of a case resurface you take the crop product. Yeah but here it's any dimension. Yeah so we have any dimension but for hypercalia you have the Bogomolov-Bovil form which is given by some integral. And we have generalized the Bogomolov-Bovil form for any arithmetic base. And this Bogomolov-Bovil form makes this place self-dual. Yeah. Depends on the polarization or not? Yeah okay you can look at it. I mean it is not a polarization because it comes not from a line bundle. For example if you have the universal deformation you cannot lift a line bundle to the whole universal deformation. Yeah I mean you can also lift it to a sub-variety of some dimension. Yeah so you can of course take the form of this divisor but it exists only over a sub-variety. But the Bogomolov-Bovil form exists everywhere. Yeah and so it is given by a formula but I cannot write it down. You can look the formula. No so if those are questions let's thank our speakers.