 В последний раз я объяснял, что громфити и неверензи я definим в какой-то алжебрее цикле, но потом нужно реализовать коммолджию и это немного аноним. Например, если мою вариантура есть к и вы включите Клуткие комплексные номеры, а затем мы получаем, как беттихомолжа, беттихомолжа, which I did, I don't know if it's going to get b to x, yeah, c is analytic topology with rational coefficients. So, we get some finite dimensional rational vector space, but it depends on embedding, and there is an algebraic part of it, which will be, we can define, like, show a group of x up to co-chromological equivalence, to play by rational numbers, and it's the same as image in this or any other co-chromological series, in comparison, isomorphism, we get the same image, and the trouble is that this space could be too small, for example, we don't know whether it satisfies upon graduality, and so Grottendick designed something called standard contractions, essentially saying that you can see the Q-net component of diagonal class in H-algebraic of x-process, x is cycles, and also something about left-shifts decomposition, and if you believe in Hodge-conjectures, these things will be true, and then you can design the semi-simple tensor category of motifs, but without assuming conjectures many years ago, Evangere found essentially a tautological solution of this thing, again, for the case and characteristic of field is zero, you, in a sense, add by hand missing pieces in commulgia, missing algebraic classes, and you call this something like the motivated cycles, which is more than algebraic cycles, and in this case, we get canonical semi-simple abelian category, semi-simple tensor category of rational numbers, and if you consider some realisation function, like beta commulgia, you get some kind of motivic Galois group, which will be proriductive group, defined over rational numbers, and we checked on, let's say, beta commulgia, beta realisation, yeah, so group secretly depends on realisation, and fixed points, sorry, okay, you get this group, and this group, actually, there are several things, one can say about this material Galois group, it maps epimorphically to profine group, Galois group of q bar okay, it will be the action of h0 of, let's say, zero dimensional varieties, inspector of finite extensions, and it also maps epimorphically to JL1, or JM, its section on h2 of p1, and there is also another JL1, or JM embedded centrally, in the motivic, it's the torus responsible for the weight of pure host structures, and the composition here will be a kind of multiplication by 2 in raising to second power, yeah, so I get such a group, and then I define a smaller group, I just remove this motivic, it will be kernel of the sepimorphism, which is from the state module, this is a slightly smaller group, and it's also exon-comulgia, and you can see the invariance, it will be motivated cycles, and it will contain this algebraic part, and it's kind of like beta part, that is, like panker duality, and for this smaller group, you have a smaller thing, you get, instead of central JM, you get central Z mod 2, you get central embedding to the G, and still get map to profiled group, and what else I consider, what Hodge theory tells us, Hodge theory tells us, there is another action of, another JM or JL1 sitting in GC, in a group of complex point, namely, if you have lambda, it's x by lambda p minus q on HPQ, if you get a variety. So what is this map G and Q inside this G? Ah, because if you have a representation of this material group, we get some sort of pure motif, sum of pure motifs, and the compulsion on pieces corresponding to weights, and these weights will be action of some JL1 sitting in this group. Also that's the JL1? Yeah, JM or JM, okay. In Q, yeah. That's it, no, it's fine. I consider just commulgia, I don't forget about Duram commulgia, for example. Okay, yeah, so there's no periods, periods are not really well defined on these things, because you lose 2 pi i. But when you define this as a single arcohomology. Yes, yes. That's it. Yeah, sometimes like people consider like the vel restriction from C to R or JL1. Yes, yes, I don't do it, yeah, yeah, yeah. This is closely related to this one. Yeah, yeah. The other one is lambda dp lambda bar to the Q. Yeah, yeah, I have like action of, for usual pure motifs, I get action of 2 C stars, and with some real structure, because they are powers of P and powers of Q. But here if you get, you remember only difference P minus Q. Okay, yeah, so get this little thing, and now if you have X, it's smooth projective variety, then the zero gram of between invariants give a formal, maximal and low F bundle, over some base, over some base. Define over rational numbers. So you're taking gene variants in HP.x. Yes, yes, consider gene variant, yeah, it will be just algebraic cycles and a little bit more. I'm sure it raises motivated cycle story. Formal, maximal, log. F bundle, yeah, yeah, like last, let me explain, get some, yeah, it's a serious, yeah. And also in doubt with action of G, and by action of G, because all this algebraic, all this confliction of invariants are algebraic cycles, yeah. And now one can add, yeah, it's kind of like formal things, and you can add, consider bigger field, like to add rational numbers, some dummy variable, which means nothing, just convenience, take algebraic closure, completion, yeah, we get a couple of such algebraic closed complete field. And you get, by considering four kind of maps from spectrum of K to this guy, you get not formal things, but analytic, now without any log, maximal F bundle over some base, on some base, which is K analytic supermanifold, yeah. And central, Z2, central in evolution, GX minus 1 to the period on the supermanifold. I'm so sorry, but what is map from GL1 to GC? Ah, it's another, it's a space of complex map, because we consider commold with complex coefficients, you have hodge decomposition, and consider just grading only by P minus Q. You get grading by P minus Q, which is preserved by the algebraic cycles, and gives a cis-direction. So, you, okay, so you just go from formal thing to analytic, it's like an open unit. Yes, yes, an open punctured unit disk in Q-directions, yeah. And you view it as analytic either in the sense of Belkovitz. Yes, yes, exactly, yeah. And then what we have? We have this base, and still group GX here, we have fixed locus, and fixed locus is contained in purely even part, can you see my tool, which is even part. So, you get honest submanifold, not super submanifold, and it's smooth, connected, non-empty, I assume its axis is non-empty. This guy will also non-empty, even analytic in manifold, and whose dimension is exactly the rank of this gene variance. Okay, so you get this, so what is going to get some kind of like supermanifold, it gets kind of like fixed locus, yeah. And group acts kind of non-trivial outside. I define a spectral cover, essentially I can repeat what I said in the first lecture. Spectral cover, which sits on this, I think, multiplied by a fine line of my field K. It will be divisor, and kind of like fiber at any point, and let's take point of this field, not further point. It will be finite subset, in fact, this multiplicity, it will be spectrum of earlier field, and this earlier operator can act on kind of different spaces. You can act on all commode of X, you can take X on even part, you can act on commode of, you get smaller and smaller pieces. You can see the space of gene variance, these motivated cycles, or you can act on actual cycles. And spectrum, as a set, the spectrum is the same, as a scheme is different, but as a set is the same. And now consider spectral cover, and this is because it's commutative rings. This is commutative rings, or even the smaller one, and acting on some finite dimensional module. And consider elements of this ring, and consider its spectrum. But why the action on the full thing? Because if you have a non-trivial model of the ring and contains one dimensional submodule, then the spectrum of the model will be the same spectrum on the joint action of the finite dimensional ring. Now, if you have finite dimensional commutative ring, and you get finite dimensional module, which contains the copy of the ring, the module, and consider any element of the ring, the spectrum of action multiplication on the big space. Okay, so you somehow use the way, okay. Yeah, so you get the spectral cover, and the main definition, like atoms of X, will be a set of irreducible components of the spectral cover. It's just a finite set. Or one can describe in a different way. You have this BXG, and here there are kind of locus of kind of... Is there something discriminant? So BXG is again something like a poly disk or...? Yeah, yes, yes. Okay, in the... Yes, yes. And here has some divisor. The divisor is a closed divisor. And closed divisor consists of points where multiplicity of spectrum is less than a generic point. But the number of different eigenvalues... Different eigenvalues, yeah. Number of different eigenvalues drops. And the same thing one can describe the function. Outside, if you remove this projection to BXG, you can see the preimage of the complement to the discriminant. Then you get an unrefined cover, and you can see the pi zero of this. Yeah, okay. So you get some finite set as such to a variety. And then kind of final definition of what will be atoms for your field. Yeah, so what I do? I take this joint union of isomorphism classes of smooth projective varieties. I can see the atoms X and divide by maybe the automorphism group of this variety, which was natural construction of this automorphism group X. Somehow we get quotient set. And more doubt by equivalence relation. Relation generated by three things. Yeah, first if consider atoms of the joint union, let's say of two varieties. And that I can identify with atoms of X1 and atoms of X2. Why so? Why these atoms for this joint union split? So you're taking isomorphism classes of smooth? Smooth projective varieties. Not necessarily connected. All smooth projective varieties. Yeah, it's all story about smooth projective varieties. Even those with different dimensions of components. Could be different dimensions here. And the story is folk. Why atoms of the joint union? Sorry, it's union of atoms here. The joint union goes to the joint union. The reasons are following. When you have this maximum left bundle, I told you you have earlier vector field and get also identity vector field for any maximum left bundle. This identity vector field, kind of earlier field to power zero, it's a vector field acting on any base of maximum left bundle. And what it does with eigenvalues is just shift eigenvalues. So this means that if consider generic point this eigenvalues kind of like freely move in K. And if you have union they can freely separate them from one to another. And that's isomorphism. Second is this blow up formula, blow up identification. Namely, you take this Z and X will be pure co-dimension R. Then produce two varieties X prime, which will blow up. And X double prime is the joint union R minus 1 times. And what explains this theorem of Irritania implies that certain domain here and certain domain here connected to non-empty domain here and here identified. And then start to identify some pieces of this cover or some domain. So you start to make some identifications. Gives some identification. Between atoms of X prime and atoms of X prime. Also last time I told you that there was some potential trouble with adic convergence. But I checked with Irritania, everything is completely okay. So it's the convergence of the... Yeah, yeah. In his formulation there was some kind of badly looking rings, some additional variables and some badly looking rings. But at the end of the day it doesn't matter. So it's some technical story, it's related to the formulation which I didn't even give it to you. So Jb is smooth here, right? Sorry? Jb is smooth here. Yeah, it's all smooth projective. No, no, Z also. Z is also smooth projective. Closed. Yeah, subset here. Okay. And then there's a sort of thing which is convenient to add. You'll see in a later way. If you have a vector bundle of again rank r greater than 2, then it can again produce two varieties. X prime will be projectivization of this bundle. And X double prime will be disjoint union of now of r copies. And there is a similar result for this story. Okay. And... Sorry? Okay, one question. But for this story and the application, can we take the field K to be C and G just to be the amount for the... Yes, yes, it's completely fine, yeah. You don't need at this point a final. Yeah, yeah. If you don't want to go to 100 definition, you can take gamma forte group. Does really matter, yeah. Yeah. And then, what we get? Yeah, but the man forte group is not the same as the even there. No, it's only simple. No, yeah, it will be kind of one story. It's universal story which maps to more specific stories if you go to man forte group or stuff like this. Yeah, yeah. Now, if you get variety, then you get kind of class of X will be... Sorry? In three, something different. If you have vector bundle on a variety to get another two varieties, namely projectivization and disjoint union, and have again kind of the same... Some identification between atom of X prime and atom of... Yeah, yeah, it's another previous result of retinas. It's like a year ago, he proved it, yeah. It's called quantum layer Hirsch theorem, something like this. Yeah, so it will be kind of finite sums of the space of finite linear combination of atoms of K with non-negative coefficients. So you get some kind of like some numbers multiplied by some atoms. Just look on fiber, generic point, you see this formal linear combination of these guys. And why do you call atoms atoms? I mean, what's the inclusion? Why are the elements in atoms X? Sorry? Why do you call the atoms atoms? Yeah, because if you kind of stay... It's not generic point, you get less spectrum, but this kind of wants to decompose to more elementary pieces. And eventually kind of this... These relations make a... I mean, it's the same kind of relations you see on like whatever the non-commutative motive associated to X. Yeah, yeah, I will go to it. Yeah, yeah, it's all... Yeah, we got... But are you gonna... Do you know if this association factors through? Yeah, I will explain in a minute. Yeah, just a cool story. Yeah, just before going on, I just want to say that the set of atoms of the field is filtered. Like it has increasing filtration. And these things, what are atoms less than n? It's something which can appear, atoms which appear in the composition of a class of X, where dimension X is at most n. And in fact, one don't have to think about all varieties of dimension n and choose one representative in each vibrational class. So by the way, if you have a non-algebraically closed non-field k, do you get the same atoms as for the algebraic closed? No, no, no, it's different notion, because you have different notion of invariance and have different notion of generic point, and the spectrum could be different. So this thing really depends on the k. So if you change the embedding in C, it will change, no? Yes, yes, yes, yes, you get a kind of fuzzy decomposition. You have more room to deform, you get more eigenvalues. If you change the embedding in C? No, it doesn't depend on embedding, you can see it's kind of... On embedding you can see that nothing depends, it's kind of purely in this event or category. It's this way, that is not... Okay, so it doesn't really depend on... It doesn't depend on embedding you can see, yeah. So if the base field we have complex number to start with? Yeah, but for interesting complicated... No, no, it's also interesting, yeah, but in principle there's barotian geometry over non-algebraically closed fields, so it's... That's a descent, it's... These things. And what's kind of like basic principle, how to apply it to barotian geometry, if you have two kind of like barotian equivalent varieties, or dimension N, it'll be connected, non-empty, and... Oh, no, this field K, for any field, yeah. I fix my field and work always with this field, yeah. Then the difference belongs to finite linear combination of atoms of my field, it belongs to filtration at most N minus 2. So, let me see. So the atoms... No, the class of X is... How does it go to the Freiburgian group on atoms? Ah, because if you have... You can see the generic point of this BXG. Okay, then you count with multiplicity... Ну, multiplicity, yeah. What are connected components? The connected components. So each connected component is counted... With multiplicity degree of covering, yeah. You represent it's kind of like... You remove this discriminant, you get an undramified cover, maybe not connected, and then each connected component covers certain number of times. Okay, but you have got this spectral cover, which is... In the spectral cover, you take it with reduced structure. With reduced structure, yeah, yeah. Yeah, you can see the pi zero, you can see the reduced, yeah. Okay, so it doesn't depend on this story there, so you just take the degree in the reduced... Okay, okay. Well. Yeah, okay, so you get these things. And why it's so? The reason is because all vibrational transformations come from blow-ups at centers of co-dimension two. And you see, you get appear in new terms, which come from co-dimension two or more. And can appear, disappear, so if you get vibrational equivalence, you get automatically these things. So that comes from this relation. Yes, yes, yes, yes. And of course the class of X disjoint here on X2, X1 disjoint here is the sum of the classes. Yes, yes, X explained. Ah, yes, yeah, definitely, yeah. The joint union goes to sum. Okay, making that. So again, my question about the non-algebraically closed K is it possible... So of course you have a Galois action on the single-fordalgebraic closure. So can one describe the situation of a non-algebraically closed K in terms of the Galois action on... No, not really. No, because you get kind of a strictly smooth variety and get some different number of eigenvalues, which are hard to control here. Once again, what is this BX, how to... BX is... Ah, no, you have this kind of like formal neighborhood of zero in cosmology and you can see the points. It takes this field, which I removed. Ah, no, this... I can see the kind of like formal path in cosmology. Okay, because the Galois will appear in G and so it is really arithmetic. Okay. So it's all cosmology. All cosmology, yeah. Yeah. Yeah. Yeah, so you get... These things. And again, there's also some kind of like easy fact. Ah, sorry, just before going on. And if you consider varieties of dimension N, which are not equivalent, I mean, suppose you consider in the same very independent model or previous varieties of previous... Well, in the motive. Do you get different atoms? No, no, no, no, no, no, no, no. For example, for rational... Yeah. Yeah, there are many things. For example, consider like flag varieties, it will be sum of points, and number of points will be total dimension of cosmology. If consider like projective space or flag variety. Then it's a symbol of x will be some dimension of commotion multiplied by class of a point. Also this important fact. If canonical class of x is f, numerical effective, Тогда это класс X, это только один атом, который я делал не по атомам в зависимости от X, с мультиплицей T1, с плицей аген-вельдии, и причина в следующем. Элья раперейтера, в том числе гамма, который обладает roughly 2h0 plus h2, roughly of my space, X on commolder of X, preserving filtration, it can only increase the degree of commolder, and consider part which preserves the degree of commolder on associated graded, you get multiplication by identity multiplied by constant, and constant will be coefficients of 1 and h0 in my class gamma by unit axiom. If it's 0, we get an important operation, class is 0, otherwise we get shift by constant, so we get just only one again values, and this thing follows from the dimension of virtual dimension model space of curves, because first-chain class plays a role. So we got such things, and now what will be invariants of atoms, the most naive invariant will be isomorphism class of nonzero representation of group, my group, and because I extend my field, unfortunately I have to change scalars to the larger field, of an dimension representation, maybe superrepresentation, where minus 1x by parity is before, and such that representation will be alpha, and such that the alpha admits a structure, I changed scalars from q, I don't know how to do, extend scalars from q to this bigger field, such that it admits a structure, it's kind of super of unital commutative associative superalgebra, and I extend what is wide saw, because we have atoms then in the neighborhood of generic point, we get decomposition of my F bundle in the product, and then on all tangent spaces we get structure at any point of this commutative associative algebra, and why it's kind of the same everywhere, if you get concrete variety, then this huge group acts through some finite dimensional quotient, it will be not proriductive group, but through some actual finite dimensional reductive group, it has kind of like contrablament different representations, and definitely they stay the same in each connected component. So admits, but the structure is not... It's not given, yeah, the structure is varying with the point, yeah. Okay, and the group respects... A group respect, yeah, and G invariant. In particular, you see that this representation always has a trivial subrepresentation, because it has a unit in the algebra, so it's not a reusable representation at all. And on the other side, you take the standard homology or the primitive homology? I mean, when you have these whole structures... No, no, no, I forget about this primitive... It plays no role for me, yeah. But then the man 48 group, if you don't put the polarization, it's not reductive. No, no, no, no, consider polarizable hot structures. For polarizable, it's reductive, yeah. So commutative means graded commutative, I take it, right? Yes. Yeah, super, yeah. And yeah, and then one can draw some consequences. That's already on this very rough level. It's superrepresentation, but is it... Was the superstructure recovered by the central... Yes, yes, yes, exactly, yeah. Yeah, but commutative algebra in supersense, yeah. So you should remember it. So it's anti... It's just Z-moto graded. Z-moto graded, yeah. And now one can immediately get some discovery. Suppose you have two Calabia varieties, Baryational Equivalent and Calabia... Of certain dimension N. Then... Then commulger of X1 is isomorphic commulger of X2 is GUK. Formally, I have to make this extension module, yeah. But in fact, one can forget about this thing here. Yes, yes, okay. Let me tell you what's going on. In fact... Kind of many years ago, following idea of butterf, I proposed something called motivic integration, which implies this commulger of six equivalent, even in stronger sense. S, G, motivic modules together with weights. And the proof was kind of like magic proofs using this motivic integration, with no identification between two spaces. Then about maybe five years... So this identification is what? Isomorphism or GQ modules? It's isomorphism of GQ modules, yeah. Okay. Maybe I'll first say the story and then give the proof of this result. And then about five years ago, Mark McLean, in case embedded complex numbers, produced a completely different proof using another proof, using some symplectic topology, very involved, actually, proof. And here I have still another proof of kind of a bit different nature. And let's me explain the proof. Yeah, so we get... If you have a rational equivalent, first I get this X1 minus X2, yeah. It's different of two atoms, because each guy has just one atom. So we have to prove. I claim this atoms cannot come in co-dimension... Cannot come from co-dimension two. So you're giving a proof of this isomorphism? Sorry? So you're giving a proof that those two are isomorphic GQ modules? Yes, yes, yeah. I claim it's actually zero, eventually it's zero. There is a fault. These things cannot come from... Both atoms cannot come from varieties of co-dimension N minus two or less. And the reasons are following, because you have to, like, Hn0 of X1, or X2, is one. And this in P minus Q decredition you get difference P minus Q equal to one, equal to N. Part is non-trivial. And because for smaller dimensional varieties you cannot have difference P minus Q, at most N minus two, they cannot come from co-dimension two. On the other side it's barotian equivalent, it's linear combination of things. So it means that this guy is equal to zero. And then it means, if you use this invariant, you see that this representation is the same. Relation between GQ and what you said GMO TV? GMO TV gets slightly more. It keeps tracks of weights. In fact, maybe in the second part of the lecture I've explained how one can even recover these things with some extra work, as well. So I didn't understand the... Why... So you have different... What is alpha X1 and alpha X2? Yeah. No, I claim that these things, if a barotian equivalent should be linear combination of atoms of a chemical variety co-dimension two. But it's impossible. So it means that... But you didn't understand why... Because you make blow-ups and blow-downs, you base this blow-up conjecture, you get things coming from center, which is co-dimension two or more. But why they cannot... I don't understand the law. You have to use the calavi of this, the dimension of H, and also the triviality of the canonical class. Yes, yes. How do you use it? I use it because it means that for my... Atoms, when I get this representation, then in kind of like Hodge part only by P minus Q, I have an retrieval part with P minus Q equal to N. I have H and zero in my variety. Yeah, but you consider... Yeah, you get C star acting on commode of X. It's complex coefficients on HPQ, it acts with weight P minus Q. And there will be property of this representation that in Hodge realization you get P minus Q component is equal to N. And this cannot appear from N minus 2 dimensional varieties and appears for this atoms, because it's the whole commode of X, one commode of X2 here. But when you take the difference, this thing goes away, so how? So it means that it should coincide. So it means that atoms coincide. Because it's free abelian group, it's either generation of the six, not coming from a co-dimension 2 or should come from dimension 2, should vanish. You said that the atoms correspond to isomorphism classes of non-zero representation. To atoms correspond to isomorphism class of representation, yeah. And the way to X is written as a union effect. If X is... if kx is left, then the class of X... It's exactly all commode of X, yeah. The one single atom. Yes, yes. And representation is a representation of commode. And why is this? Because it's completely obvious. If you take some of atoms, all my local systems, like tangent space was split in a direct sum over in the composition series of some subvarietes, then tangent space to maximum bundle was all commode of my variety and only one atom. Why there is only one atom with kx in it? Because multiplication, action of earlier recta-field by algebraic classes increased degree in co-homology. Yeah, that's... Because you get... operator is only one again value at any point. Only one eigenvalue... Yeah, because you get a kind of diagonal term and there will be coefficients of h0 some number and the rest will be strictly upper triangle in the... Why... where do you use kx is nothing here? You're saying that the separator preserves this filtration. It's come from calculation of dimensions of modulus space of curves and going back to definition of ground return invariance. Ah, okay, so this is the part where... Yeah, yes. And in the Calabi house we know that kx is nothing, and then you have this. Yeah, then you get this. So you use only that h0 of kx is... Yeah, one can make weaker properties. It's canonical class on non-negatives and gamma of x1 kx1. It's a variational variance, it's not trivial. You can make it slightly more general result, which I don't know how to deduce using material integration. Yeah, so it really shows that it's a kind of different idea. Are there some examples of these atoms and points for g1b? Yeah, yes, yeah, I will eventually go to examples. Yeah, there are stupid examples coming from a variety of general type, or Calabi house, yeah. But usually you start like projector space, you get decomposed to points. Can you give the result of this statement, like from the 90s, I think, about the use of motivic integration to prove... Yeah, to prove this guy, yeah. What is the reference? I think Lazar wrote it. Deneff and Lazar wrote some. I gave a talk but it never wrote anything and Lazar put it in written. There's also maybe Buraki seminar by Layanka about it. Yeah, and yeah, so get this application kind of in any dimension. And if you want to prove this varieties are not rational, give them axes greater than 1, greater than 2, then if axes are rational... So x is just in Calabi house? No, no, no, it's a general statement. It's for any smooth project, connected smooth project variety. Then axes rational implies that class of x belongs to linear combination of atoms of degree at most than minus 2, because projector space actually linear combination of points, as I explained to you. There's no hypothesis? Do you need the dimensions of these global sections of the canonical model to be the same for x1 and x2? No, if they are rational, it's the same. Ah, yes, thank you. If it's a rational... Yeah, so get this criterion of rationality, or better, non-rationality, and I just briefly remind you, like for my first lecture. Yeah, but you use it for projective space, points, it was one of calculations of... Yeah, yeah. So the projective bundle thing, when you write p of... You can project bundle of a point? Yeah, but the projective bundle thing is for the projective completion of a vector bundle. Yes, yes, yeah. This means it's not the general p of E. You don't have it for general p of E, the space of lines or hyperplanes. I don't know if you... I see a projectivization of vector bundle. Ah, okay. For this you have a formula, but not for a more general project. Oh, no, no, no. It's another story. I will maybe return to this sort of last lecture. What's going on? Yeah, so there's some story. I just want to say some little words. I can repeat from my first lecture. There was application to this very concrete case. If x is generic cubic fourfold in p5, so degree 3 in dimension is 4, then it's not rational. And how one prove it? Yeah, one can calculate this quantum multiplication, this specific point, and essentially like zero, you don't put any corrections, like zero point, more or less, of a commulger space. And then one get kind of... When you get spectrum of earlier operator, you get zero and maybe some cubic roots of three roots of one. At all roots of one, you get atom corresponding to a point. But what happens in zero? It's a pretty fine space. It will be some 24-dimensional commulger space. And you can see the p minus q decomposition. You said that the spectrum of the earlier depends on the point. In general, you say it is... So the spectrum... Ah, spectrum, it's some specific point on this base. Yeah, like giving time calculators quantum multiplication. And then one can see that you get four eigenvalues. You get zero and up to some multiples, three cubic roots of one. Oh, the cubic roots of one. Yeah, yeah, it's very typical picture. You get zero and roots of one here. This is for zero or for a generic point? No, it's point zero, it's not generic point. So there are exactly four... Four eigenvalues, yeah. I don't know. But it does matter. I think it's gain four. But what happens? You get representation. You got maybe not atomic representation, because maybe it's still not very generic point. So I get here some representation. V. So it's a 24-dimensional representation and dimension of rank of fix thing. It's exactly for this generic thing. It will be two. It will be only two algebraic classes in this story. Because commolder of this hyper-surface decomposes something coming from projector space and primitive commolder, and they assume in primitive commolder there's no algebraic classes. And what you can tell, calculate part of primitive algebraic classes. Only two and... Which algebra you mean? Yeah, primitive commolder, again some algebraic classes. And this is generic. It means exclude this situation. Generic means it's no... no motivated cycle, say. No. V corresponds to... V is what? V is the... No. V corresponds to... V is some piece of commolder of X. So V is not the whole commolder. No, no. All commolder of X will get X of three algebraic classes. So it will be 27-dimensional. So it will be... Numbers of dimensions will be the following. Dimensions of representations will be like this. Yeah, it's all roots of one will be. Yes, exactly. Yeah, it's some kind of very specific calculation. Yeah, so you get the situation. And a claim that's already shows that the things cannot be rational. Yeah, in principle, I don't know whether it's... Because there are two algebraic classes. There is still possibility that if you deform it can split further to some smaller things. It can split further, but... Maybe y' but still what we get. So the things we get dimension of v' and p-q equal to 2 part will be at least one because you can give the part with this h20 class. And rank of fix locus will be at most two. And then it gives... Then the things come from varieties of dimension to most two. From points, curves it's clear because from points of curves you cannot have this p-minus equal to number at all. So it can come only for surfaces. But for surfaces we need not all surfaces up to variational classifications. There are good variational classifications. There are some various types. Something like, you know, 12 different types of surfaces. And you have surfaces with... You only have this section of canonical bundle trivial section of canonical bundle could be either case re-surface, a billion variety or surface of general type. But surface of general type will have maybe some ADE singularities in minimal model. Then we can resolve these ADE singularities, get smooth surfaces. And for all the surfaces, so all surfaces with gamma with h20 or non-equal to zero have variational models for which canonical class is non-negative. After blowing up this ADE singularity get not general type surface, but slightly borderline surface. But canonical class is non-negative. And then it simplifies the number of invariances at least 3. So we get this kind of tiny contradiction. So we get 2 and here we get 3. So we get surface, we get class in h0, some ample class and class in h4. So we get at least 3 different library classes. So it's some kind of... You mean you have h0 because you have h0, h4 and polarization in h2? Yes, yes, yeah. So it's kind of like... So that will prove that they're generic? And yeah. And then one can try to play the same game with some other varities and I think it could be interesting for knowledge break closed fields and maybe for three-dimensional manifolds, four-dimensional manifolds. It's dangerous to go to high dimension because we don't really know classification of varities in version classification high. Yeah, eventually maybe in three dimensions close to completion so we can use this information go to five dimensional. It's not mathematics, which I like. Yeah, so it's one can... It's really using many... So you will need to do Kodami-san too? Yeah, if you know kind of complete understanding of all very useful things up to convention too, it can grow some conclusions. Yeah. Yeah, but it's very unpleasant direction. Some people like... What did you use exactly for the classification? Sorry, generic because in this otherwise if it's not generic it could be three. So you also use a previous remark the KX is nef then then you have a... Yeah, it's only one atom. Use the structure of general type surfaces that there is a minimal model which is... Exactly, yeah. You can resolve it, you get... Yeah, yeah. Again, one check, because it's AD singularity then you get... The pullback of the one from downstairs Yeah, it will be nef. It will be still nef. Yeah, because it's ample downstairs it will be nef upstairs. Yeah, yeah. Yeah, just maybe before we finish. Now I just hold what Dustin was asking about motivic measures. There was something some another story in this motivic integration there was a use something called K0 of varieties of my field K which can be described in two ways. You can see the constructable sets up to cut and paste. In fact contains Z you said that X is equal to Z plus X minus Z. Yeah, I can see the nef kind of constructable cut and paste group by constructable sets and then those are theorem by Wittner that this things has kind of really nice representation by generators and relations. It's kind of Z-module generated by maybe symbols, I put kind of like M, it will be motivic measure for these things. By classes X is smooth projective variety and you're modelled by two relations, kind of like if you take disjoint union it will be some. And one relation and second relation is the following. It's actually very similar to what we have here if you get again some sort of variety another variety you can see the class of blow up and remove exceptional divisor exceptional divisor which lies over these things exceptional divisor sexual projectivization of normal bundle. So if you remove exceptional divisor it's kind of satisferically gets the same as X minus Z. So this uses resolution of singularity. Yeah, it's a story yeah. And concrete zero. So we get just two relations just in this. MM, motivic measure. I just not to distinguish with my symbol which has no MM. Yeah. And then you immediately see from this relations we almost we come by the things we get a map from k zero variety to additive group generated by atoms of my field k by X MM goes to X. Yeah. This bit in the relations will be satisfied. Good. But in fact here you see that this thing actually is a ring. It's a ring. You can see the X1 MM X2 MM will be multiply it's a product. Yeah, it's a tomatically ring. But as the model it's kind of horrible. People found some torsion. And this thing is much nice at least as the module has no torsion. And the claim so the things actually goes to some quotient. And you mod by relations that class of p1 is equal twice the class of p point. Or in terms of affine space affine line is equal to point. Yeah, so model by some ideal it's so it's again a ring. It's a quotient ring twice a point. Yeah. So this is like modding out by L minus one. Yes, exactly. quotient ring. And this quotient ring actually goes to something else. One can see the group generated by isomorphism classes of smooth proper categories dj categories mod out by relations coming from semi orthogonal positions. And conjecture will be map from here to here but it's from atoms you can get a category. That I'll explain on the last lecture. This atoms actually gives a class of isomorphism class of category but it's very hypothetical. And just what I want to say it's a ring in fact here is a product structure with structure constants are non-negative integers. It's again by pure thought. I don't know what is going on here. What is going on? You take two varieties and take product of the things where we decompose the eigenvalues. It's some part it can be embedded to this guy and there is some notion of tensor product for this maximal of bundles. Very funny. You kind of take two maximal of bundles to make some tensor product will be something of dimensions will be product of dimensions. So it's pretty natural procedure but one can kind of go to some even get splitting on this part you get some kind of generic eigenvalues we get some special points then move a little bit further on and the six will decompose further. This is like the corresponding some trivial components or something. You take independently some dichomology of X across the unit. Yes, you can see the kind of kind of class is kind of split coming from here times identity plus identity here. Then this operator failure product will be tensor product of operators and and then because we have tensor product operators we get product of sets and the product of sets will be embedded to spectrum here but then move a little bit further on and things start to split further on. So you get some kind of funny non-negative structure constants by pure sort. Yeah, yeah. Okay, maybe I'll now make a little break for five minutes. Maybe I just just too little remark. There are things which I should mention but before there is some story about duality just for F bundles. If we have F bundle some H with some metamorphic connection or some base multiplied by spectrum or some field some variable U. Then one can speak about things with self-duality when you had a pairing between... so you get these things you consider dual bundle and you identify with the same things coming from antipodal evolution in U-variable and such that the pairing will be symmetric and non-degenerate. So it's It means it's undegenerate. One can speak about symmetric pairings and what really what falls from the story if you get this structure if you have on when I consider decomposition theorem have this duality it's persist on on factors in decomposition theory so it means that on all atoms you get the structure because it's yeah, this structure is natural for gromovetron invariance just use Poincare pairing and enhance it means that on atoms you also have pairing and in particular when you get this representation of isomorphism class So what about preservation of the irreducible component under this yeah, if you have F bundle with the pairing then maybe at some point whatever it splits locally in the product of F bundles of smaller bases they also have inherited this pairing as well the automatically inherited non-degenerate pairing yeah, yeah you have your kind of a bundle and your point is decomposed in the product of some bundles of smaller bases and they also inherited the pairing yeah so it means that if you have the alpha representations of G whatever this K Q representation it has this symmetry of hodge you get symmetry of hodge spectrum there was I recall there was a group sitting in GC acting by P minus Q on HPQ and then this number is P minus Q the hodge spectrum will be this number, this will be symmetric respect to zero around zero so you have this hodge spectrum this pairing is another tool to deal with the things and this story with cubic for fault original proof which I use parity of pairing instead of this other stuff here so there are different tools here one can use pairing as well and there are again unrelated remark GAL group of Q bar or Q external space of atoms for any field preserving product reserving multiplication yeah, because what really goes with atoms we can see the eigenvalues and can see like like points with rational coefficients essentially solve equations algebraic equations with rational coefficients and there will be some GAL or action I don't know I don't have any example which is non-trivial but it's something which exists for free if I get a base change to this field yes, yes, but it doesn't really matter because one can analyze it I really don't go to the details and maybe some last remark it looks like abstract story we don't can do any calculation but we can try to think about it and make conclusions and for example one can do some develop a very easy theory if we get singular variety like singular like normal and what usually people do can do resolution singularity variety and contain some singular part then one can do the following one can see the all possible blow ups resolutions so we get some X prime maps to X which will be proper map and X prime will be smooth and it will be one to one over X minus singularity we can see the all possible resolution of singularities and now what you do you do the following in this situation we can see the growth of it in invariance of X prime using only curves X minus X minus X what do you have X one to one over X minus singular locus on smooth spot isomorphism over that yeah it's resolution singularity yeah it's called yeah if you consider only curves subject projection of curve is a point in X it's only vertical curve it's kind of closed rotate also gives maximum left bundle a point of consider generic point of B X prime with this modification invariant under this my group G represent by algebraic classes and said that coefficients of kind of like class and commolder of identity element and H zero is zero a deform only by class of degree 2 or more then the theorem is that what you have sitting over eigenvalue zero doesn't depend on resolution singularity and so to get some kind of like contribution of eigenvalue zero eigenvalue zero of Euler-Recto field is universal doesn't depend on resolution singularity it's kind of interesting object so you get some some sequence point credibility it looks like intersection homology some classes coincide with intersection homology but it depends and it will be kind of like common part of all commolder of all resolution of singularities and intersection homology it's only no there's no product structure you get quantum product but it depends on point on the fragment you don't have product structure you don't even get degrading but you still get some sequence point credibility and here X is projecting yeah, yeah it's another kind of worms here one can do things not projective one can define this quantum multiplication for these things it will be well defined one variety doesn't matter so it will be some shifted object but I will not go to it yeah so now I'll just try to briefly see if you've worked about very funny theory of weights or maybe still a little remark another potential invariant of atoms which is the following I recall what was the atom you get this all BX and there is fixed locus there is a fixed locus and we make this product decomposition and we have atom for atom you get the following you get a germ of analytic germ of maximal F bundle end out with over certain space depending on atom maybe super manifold depending on and end out with G action and germ at point belonging to fixed locus and now we get something very specific for this atom on this fixed locus the other field will get only one eigenvalue everywhere only one eigenvalue G and these things are defined up to analytic continuation what do I mean by analytic continuation you have two analytic germs imagine there is a big some larger analytic F bundle with connected fixed locus and you consider germs at one point and another point like in complex analysis you can make up to analytic continuation so it looks pretty inaccessible notion but what here goes on I have some algebraistic conjecture that it's not so wild I think namely let's consider such guys and now what we do we consider for some large integer n we consider n germs so you consider germ at some other point not zero but the germ of a germ at a point of fixed locus ok but then you can move it just along the fixed locus connected point of fixed locus yeah now what we consider you consider n jets this maximum of bundles instead of at points of a truncated form power series ring of this G-equivariant F bundle and you consider n jets at points belonging to this B alpha G up to D-pherophism invariant because you don't have canonical coordinates here D-pherophism is engaged transformations so you get some finite dimensional variety consider this set and consider the risky closure yeah it's a so you get some finite dimensionality family in some algebraic stack you take the risky closure you want D-pherophism and D-pherophism D-pherophism because bundles it's a base and it's a germ of manifold you have to consider the form of D-pherophism of the base commuting with G-action also differentialization of your vector bundle risky closure in what again sorry I consider space of all possible n jets kind of like describe finitely many Taylor coefficients of the story consider things and consider dimension the conjecture is bounded by constant independent on n yeah so so there are kind of like differential equations and what you at the end of the day you get some maybe finite dimensional algebraic variety with some algebraic fallation and we move along leaves of algebraic fallation so the n-jet of a particular j-quivand oh no no if consider this for each atom you just move point and consider choose some kind of local coordinates at any point and consider Taylor coefficients to realize bundle somehow consider Taylor coefficients up to order n bundle and you look at it different points different points and different realizations yeah okay yeah so starting from given bundle okay yeah for given analytic guy one can give this kind of finite jets and claim that satisfies some differential equation which kind of mirror symmetry seems to indicate this is the case at least one can analyze for Calabria varieties what is going on and get actual some finite dimensional varieties but since the risky closure will have a lot of irreducible components sorry is the risky closure will have values irreducible components no no only one because it's connected analytic variety it's only one it's only one connected component here so the just key closure is stabilizing that's what you are saying so it is one irreducible component yeah one irreducible component here yeah so it's something which we can in principle calculate and distinguish two atoms to see that the risky closure are the same or not the same but it's kind of purely theoretical yeah okay so now I can say the following yeah it does about oh this maximum of bundles where they come from yeah I just say this ground between the variance blah blah blah but there is kind of like universal source of F bundles and the maximum of bundles you start some a very general story start with a smooth proper z2 graded dj category of my field and here what does it mean it means that this category should be perfect complex of some dj again it will be z2 graded dj algebra and this notional smooth and proper it means that a is a perfect bimodule and proper means that a is a final dimensional commodule of my field yeah so this well known notion and maybe I'll just go to some kind of basic example for such category suppose you get smooth algebraic variety of my field key here maybe k to t is 0 and you get a map from x to a1 for y to a1 now sorry you said that it should be a by module yes bimodule has direct sum of finite extension of copies of a cross a opposite I will not go to infinite dimensional commodule yeah yeah it's smooth this guy it was a translation if you consider category of perfect complexes on schemes of let's say separated schemes they can be written as perfect models of some modules and this is exactly correspond to properties of schemes proper scheme no proper correspond to proper smooth correspond to smooth and where is this the construction of the digital correspond to a scheme yeah it's kind of like bundle Wunderberg's theorem one can do it yeah it's um is it way yeah but what examples for such guys yeah yeah it's kind of like smooth proper large break varieties but there is a little bit more you can add potential here you can see the function such that critical locus is proper and f restricted to critical locus sorry critical locus is proper okay yeah and f restricted to critical locus is zero maybe some multiplicities it says theoretically you have only one critical value which is zero variety itself could be not compact and f is proper the function f is proper no critical locus is proper yeah then you get a category which can be described in different way it will be z to graded category and people prove that it's a smooth and proper you allow f to be identically zero yes allow yeah yeah it's a category it has two descriptions it will be very brief there are some called matrix factorizations roughly it's very rough it's not really precise things you can see the like two vector bundles on EI like vector bundles on x on y and I get two maps so the delta square is equal to identity multiply by f and such sink form some DG category or equivalent description if f is not identity equal to zero you can describe as some which is called matrix factorization singularity category it's called db sink except so you can see the category of perfect db of coherent shifts f is a subscheme and mod out by full subcategory of perfect complexes and it turns out to be two periodic things and people proved that it's belong to this class yeah so you get this so it's include usual it's very dramatic even f is equal to zero db you are db some oh sorry why db somewhat db sink of this critical locus okay so that's that's a quotient of category of and people proved that this is equivalent to something in this complex over some A yeah yeah but you assume that just one critical value come on critical value yeah critical value usually you have more so yeah the story is a phonic this first definition no here I just kept zero and here I get this if I add constant to a function for example critical value not zero then both categories will be trivial but why in the second case you get triangulated category but in the third two per your a well the shift by two is yes yes yes one can treat it you can say that you shift by interchanging no no you don't shift because our homes in obvious way will be z2 graded complex and it has some usually you have functions which have more than one critical value no here I see that critical value is zero only so you don't know what to do with that no no if it's not zero for each critical value can shift function by hand define some category but have two critical values no I don't want to consider this yeah no in this case one for each critical value shift function by constant so zero became critical values and define for each critical value you can define a category they don't speak with each other at all yeah and then in this situation you get the following just for general category get various homology theory get hawkshield homology which is kind of derived product by models from A to itself then we get negative cyclic homology hc- yeah so it will be actual space of k negative cycle will be samsikov series in one variable and we get periodic cyclic homology it's overall run series in view it's just basic in this example what does this homology will be hawkshield homology will be hypercomology of y with coefficients and shift of forms with differential multiplication by df and negative cyclic homology will be hypercomology of y shift of forms now take series in u and consider df plus u times derame differential and periodic cyclic homology we get the same but you took run series you get this 3 you have hypercomology and you talk about hawkshield homology hawkshield homology, yeah it's algebra because dual to variety no no no this example it's hawkshield homology of algebra yeah so gradings are yes yes yes z2 graded spaces yeah I can see yeah the whole thing is only z2 graded yeah and then one half always inequality is that dimension of hawkshield homology over k is greater than equals than dimension of over periodic cyclic homology over this thing by some spectral sequences and total dimension yeah yeah actually this thing is finite dimensional it follows from this smooth and proper property it's in place because it's perfect module with finite dimensional get automatically finite dimensional thing here yeah so it's so things are in this case really finite dimensional and so one can make various conjectures that dimensions are the same it's called the generation of hawkshield homology and it holds for things coming from also holds in this example in this geometric example and in general it's open so people prove it for z2 graded algebra it's proof doesn't work one can make this conjecture and then what follows from this story already from this conjecture I claim that you get F bundle over point namely in this case this implies that negative cyclic homology is free module of finite rank it's actually supermodel because even in our part so you get super vector bundle and the claim it has canonical connection which has second order pole I don't really have a proof that this has second order pole where this comes from this second order where this story comes from in general we have not just one algebra but consider like family of algebras then we get something called getzer gaussman connection and in this z2 graded case we get canonical deformation for example getzer gaussman you said that you consider if you get family of dge algebras then I get something called getzer gaussman connection it will be connection on periodic cyclic homology along parameter space but not in u direction along parameter space yes yes and where comes connection in u direction in z2 graded case there is a natural action of jl1 or jm on space of algebras you for example multiply both differential and multiplication by constant like differential multiplication multiply by constant so for each algebra for one parameter family it actually corresponds to in algebra algebra I didn't say what is algebra in the examples correspond to your scale function start to rescale functions for example every such object has canonical one parameter deformation which actually equivalent to rescaling variable u so it automatically gets a connection variable u so it gets some mysterious connection in this variable u and if you if you get family of algebras over some base then you get automatically f bundle sorry once again you multiply differential in the differential graded algebra a and what you do is multiply oh both differential and multiplication multiply by the same constant sorry no in general if you have infinity algebras there are such high product equations are homogeneous for infinity algebras just rescale and multiplication yeah yeah so you get this formula of algebra you get f bundles and but it's not maximal of bundles and the story is a falling if algebra is calabia algebra even on odd and you have degeneration and conjecture one then something remarkable happens then you get maximal of bundle so what do you buy calabia algebra it's a bit long story but essentially maybe one need very little maybe one need this effect it's this dual to the subject a is by model is a shifted by some even an odd number something like this I think maybe a kind of a is by module because it's perfect you can do it's equal to a shifted by even an odd number just I think it's very big property it's sufficient and then if you have conjecture one then this implies the following things says you can see the co homological whole should come complex as by module if a is by module because it's perfect don't make a dual by module over opposite ring which is the same you interchange pieces and then it says it's as a morphic to a in some unspecified way then it implies the following derive sense you take the dual dual yeah make resolution derive sense yeah sure equal to a again in this category of by modules yeah and the dual is again a by module the dual is a by module of some by module is perfect by module is perfect by module I take our home yeah or home yeah yeah I take our home from a to a cross a opposite opposite okay and then use the a opposite opposite is equal to it's like a goldenstein in some sense goldenstein yeah it's like a variable variety is equal to a what do you write in brackets a home from a to a times the opposite equal to a and then in brackets shift even or odd yeah shift is up to shift calabio could be even an odd yeah and then you are doing z more to graded z more to graded yeah it's all makes yeah in geometric situation a calabio means that calabio implies that x y has volume element algebraic volume element okay and what I want to say you get you get this yeah if you have algebraic you get something called co-homological-horshift complex it's some dj algebra responsible for deformations and we only have the generation of co-homological-horshift and calabio property one can show that this guy is homologically abelian so it's as dj algebra is a morphological algebra with zero differential and dimension of co-homological is equal to dimension of homology and so we get kind of like more carton space of this dj algebra will be formal germ and all this germ we get maximal f bundle so like the solution of morphological carton equations is dj algebra omega equal to omega yes, yes, yes, yes this is it's represented by some algebras form power series in several variables, even in not variables it will get some functor on art and rings what proves the smoothness because it's follows from the generation of horshifter in calabio properties so this what do you have in morphological time does it have a connection? no, it has dj algebra you can try to solve equation like this up to gauge transformation close to zero yeah so calabio makes this duality thing yeah, yeah this is calabio and then you get maximal f bundle and also this padding this horshifter commode you get padding plus padding yeah, so that's a very general class of origin of this maximal f bundle so it comes from such story yeah, I explain you one dramatic situation here this horshifter commode will be this algebra namely you can have some model consider like over complex numbers you can see the sections of y kind of infinity sense you can see the d bar forms multiply by poly vector fields yeah, what is this modulus space in dramatic case let's assume my varieties over complex numbers now I can see the sections of d bar forms multiply by poly vector fields you hear it right calabio c c a a a is there yeah, it's a cogemological horshifter complex ah, conform a this is after quasi isomorphism after quasi isomorphic, yeah SDG algebra isomorphic to this guy this kind of very concrete object okay, so here it's an analytical description analytic description, yeah this dimensional guys like so taxinfinity forms of type only in d bar direction as well as in poly vector fields only in calabio direction it's arg of poly vector fields but here on this complex we get literally bracket and differential is it to graded are you take p, the same p in both sides? no no no, different that's what my question was it's all different so don't get too different arbitrary, yeah eventually at some point it will be equal, yeah the least traction you get scot and bracket here cup product here in differential it's differential here and commutator is function yes yeah maybe I'll go why it's maximal why it's maximal because the dimension of the base is equal to rank of the fiber yeah commutator and commutator have the same size yeah, yeah, so you see that it's come from this variety with these functions called Landau-Ginsburg model and now go what I call between invariance there is this notion of fukai category whatever it is and this will be the formation of fukai category and again strictly speaking there are some technical questions we don't really know fukai categories kind of correspond to smooth proper things but it behaves like this in many cases proven but so where do you use the dimension what because the dimension of the base will be the dimension of Hohschild-Kommologe Hohschild-Kommologe is something like x in category of b-modules to itself of course yes and here you get tensor product and they have the same dimension because of the Scalabio property and this is used to know that you have an f-bundle you get maximal of bundle but your claim about the relation of fukai categories was it if you have a smooth projective variety then it should have something like fukai category and then the formation series of this fukai category gives you this f-bundle which comes from gonvite invariance so you plug in c to be the fukai category yes exactly this f-bundle is the same as the other one in fact the story is very mysterious for example one of questions suppose have a variety defined of a number filter can embed in complex numbers in two different ways you get kind of the same gonvite invariance but fukai category has nothing to do with each other so it's really interesting story it gets completely two different categories with the same deformation series and another kind of question this gonvite invariance are defined also for a variety of a positive characteristic you only need common reserve with zero characteristic and then I have really no idea what could be fukai category for a variety you cannot embed in complex numbers at all what is fukai category for a complex variety when you take a symphlectic form I took some ample class and I take usual symphlectic manifold yeah so you get maybe just still maybe 10-15 minutes little is the degeneration conjecture known for these fukai category? no this conjecture is known for fukai categories for fun varieties and it's used it's actually used algebraic result by Petrov and Volagotsky and Ventrop it's not really symphlectic proof it is known for when you get fun of a variety then people proved and maybe with some section of anti-canonical bundle which is smooth then people proved Zeidel and Prameleana recently proved this degeneration result for fukai category but now I want to add some conjecture too to the same kind of general story even without the Scalabia for smooth proper proper a whatever yeah ok when characteristic is equal to zero the conjecture is that actually there are two parts first part is if you consider periodic cycle of homology with this connection which I describe you only informally it has regular singularity so it means that HP HP is kind of like KU model extends to KU model so that connection has pulse of order one it's not the extension of h negative which is some other extension has pulse of order one and part B assuming K again values of coefficients at U-1 of this connection for this extension belong to irrational numbers that's algebraically quasi-unimportant monodromy again it was proved for fukai category for this fun variety actually I believe it's true for any dj category but it's again very open and then this connection has pulse of order at most one but it's another extension and then the claim in this situation one gets certain spectrum so in usually you say smooth and proper so in many cases you have Z graded Z graded I fixed points of the earlier field ok but you can treat it for for algebraic varieties you get Z graded yes yes for algebraic varieties for algebraic varieties you automatically here get regular singularity you get integer numbers for free if it's Z graded category it's really automatic if you have Z graded you know A and B A and B are unknown ok for Z graded and the case of Y with a function is Z graded no it's not too graded but then it is also known yeah it's also true it's also true and I want to say something about the second values and one can define certain spectrum which belongs kind of like Q maybe Q and with multiplicities and responsible for even an odd pattern homology namely what you do it's it's really based on the things with regular singularity so what we have we have a bundle and with some connection with regular singularity we get fiber equal to zero what I want to say if you have bundle this connection and Z with regular singularity I consider this period exactly homology it's a vector space with connection with regular singularity this assumes that this you get this rational numbers then from these six you can produce immediately a devalued valuation on periodic cycle homology it's a vector space of finite dimension on a normal series and I claim this connection with regular singularity gives you valuation and let's we explain you informally how it goes let's imagine that my field is complex numbers and because of regular sorry? valuation on the vector space on the vector finite dimension vector space relative to the how it looks like I have what I do here I have a bundle on formal disk with formal puncture disk with regular singularity suppose I have complex numbers I can extend to actual disk just to simplify life I get bundle on actual disk and now choose array and along array I consider flat section along the array and now I can expand my actual section in basis of this flat sections and when I expand basis of flat sections I get some series with coefficients maybe if it's variable again U I can see the things and divide with log U I get sections coming with some piezo series in particular they grow and I get some power of U so automatically for each section I get maximal order of growth and it gives me some valuation by environmental numbers because I have an important polynomial I completely ignore this log U part so you can just take the operator U d over du I get values of this and then do the same thing one can do it completely algebraically so you get valuation or kind of norm but extension of bundle of HP to H negative this extension gives another valuation if you have a section consider what is order of growth C-valued valuation so what goes on you have kind of one finite dimensional vector space with two variations to norms and that is notional like in complex numbers you have two remission norms then you get singular values you can simultaneously diagonalize and you get some spectrum which belongs to rational numbers okay we have a basis which is autogonal for both yeah yes true for noncommittal case if you have two norms there is some kind of singular values or logarithm of singular values and you get certain spectrum of rational numbers yeah so it will be the certain spectrum associated to the story what I want to say this spectrum is symmetric so you can ask what I compare with what I have singular values 1 or 2 respect to 1 then you get the same spectrum because my bundles in this situation have duality and for dual we get opposite numbers so you get symmetric spectrum and this symmetric spectrum it's really very interesting okay yeah so it will be kind of another invariant of atoms what spectrum we get a generic point so right now so next time I will continue to talk about the spectrum and because I've considered a generic point of this BXG yeah so in fact it's a model of things BXG you get this only one eigenvalue you can see the locals when eigenvalue is 0 so it corresponds to one critical value function 0 then what can happen it could be regular singularity or irregular if it's irregular it will be kind of new beast it will be invariant labeled it's irregular singularity we don't know what to do but if it's regular then you get the spectrum a generic point you get some bunch of numbers and what does this number say it's really very funny numbers so what goes on we can see the complete intersection one can analyze this this quantum invariance and so complete this section of degrees d1, dk in projective space and assumes funnel we can see funnel it means d1 plus dk is less than n suppose it's funnel then the picture is the following it looks like this you get something like n minus n plus 1 minus you get bunch of you calculate some point and believe it will be generic point you get some roots of 1 of this order you get kind of like one dimensional spaces and you get kind of like huge big piece of homology corresponding to 0 eigenvalue and there will be some piece of homology kind of central part piece of homology of my variety and it will be this complete intersection this homology will be some of restrictions of homology of projective space like one class in each degree and direct sum with primitive homology in the middle degree which generally will be completely transcendental and this guy correspond to 0 but this some part of this guy also correspond to 0 and what will be this spectrum spectrum acting on this guys and spectrum looks will will be the following gadget you start with numbers 1 over di d minus 1 over di it takes a union with multiplicities you get collection of nq kind of multiplicities of some numbers now I apply the following transformation lambda goes to n plus 1 minus sum over di lambda minus dimension of x over 2 you get some other numbers called lambda and you order them when you order you see that's kind of really funny things it's kind of like sorry? it's all rational numbers and now I apply just each number replaced by some linear apply a fine function in one variable so you get these numbers you order them and now we do the following or maybe you order like L0, L1 you order them so it means that all the charismatic progression interlaces and now you start to split them so now these numbers will be strictly increasing you start to lambda 2 plus 2, 2, yeah you order these guys and add index in the order, yeah no, we get rational numbers we get rational numbers and this will be a spectrum you order, so you have 2 sequences of numbers where you put tilde no, tilde, no you apply these things you get lambda tilde, yeah and then dot 2 tilde from that yeah, from 2 tilde I just start to separate them then this guy will be rational numbers and there are some kind of interesting rational numbers I don't know if you take 3-dimensional cubic you get something like 5 over 6 plus minus 5 over 6 and these things will be this spectrum is going to be this one yeah, spectrum will be these numbers, yeah yeah, so multiplicity and what together oh, lambda it will be this spectrum this spectrum in the sense of sorry? for this connection, yeah with multiplicities with multiplicity, ah, no, but after you do it, it will be without multiplicities before they can coincide but after you enumerate them and index, they will no longer coincide and the spectrum this means that the spectrum is as possible multiplicities comparing the two norms as possible multiplicities yes, now, if compared to norms it will be spectrum without multiplicities in this case ah, ok yeah, on algebraic part in what goes on it's kind of really funny picture maybe I just for this central part like big part one can introduce an actual of course, here will be something like p minus q over 2 and here it will be central 0 and here will be p plus q over 2 roughly ah, this will be spectrum which here it will be action of of c star sitting in my Gallo group c responsible for p minus q direction and what I get here here I get integer numbers integer or half integer numbers maybe divide by 2 it will be transcendental homology of the piece and here I get some kind of like strange rational numbers corresponding to part of algebraic story so you get hodge diagram but with rational numbers on one direction integer half integer in another direction and what is nice here if you look for this for this intersection I look many many examples what goes on you get it will be like algebraic part of your variety and it will be transcendental part it's some part of algebraic which do not separate for the story and what is going on here we get like largest number here we start to draw a diamond inside by 45 degrees it contains this point coming from transcendental cycles on x and they should stay kind of like integer or half integer arithmetic numbers they should stay as much as possible to fit this to fit this diamond so you get things like sort of dimension of your original variety and it really fits almost to the very end so this number also semi-continuous and gives this abstraction to rationality and I told this which I can examples it really fits very well and if it's bigger than dimension minus 2 then you should get non-rationality result of all varieties but it's ok now I think I'll stop and continue more systematically about the spectrum dimensions next time here so once again you have two sequences panel with the eye spectrum is for these two norms is this lambda double tilde so the first one you simply don't use yeah it's kind of like you start with something very simple just make a little bit rescalic and then make this make six just maybe one little remark before I go on if you have variety with canonical class equal to zero then this spectrum coming for two norms it's exactly filtration by dimension the degree in homology yeah so it's you get usual weight filtration this follows from where one can analyze these connections make some conjugation concluded almost immediately again for the suppotrangle part reasons yeah yeah and that's that completes the proof for this result for 2 kalabiaus barychic yield kalabiaus have isomorphic homology not only as usual as usual motivical group not the smaller non commutative part which I used so it proves the result yeah it's another proof of this old result where you really need the kalabiaus the weaker here it's a bit more general yeah it could be canonical class non negative and it has at least one section okay slightly more general yeah okay yeah so I think it's so if you take let's say modelize space like a model sorry okay yeah