 And thank you so much for inviting me. Well, I should say that I first got acquainted with you, wonderful mathematics when I was a student at the, in the 70s. But we met first in the 90s and since then Luki is a dear friend and amazingly kind and gentle person, deeply appreciating books, music and nature. And thank you, Luke, and many happy returns of the days. Now, this is a small work with Spencer and let me start with a construction due to him of some cycles, homologically equivalent to zero. And as usual, all we know that cycles homologically equivalent to zero are interesting object and construct them with a sort of arc. So we start with a family of algebraic varieties over a complex disk and initial pointers. Zero is here. Oops. And I assume that Y is smooth. The one that singular point of F have just one point, Y is zero. And it sits in the fiber over zero. And the point is quadratic. Well, and then let's consider a blow up of Y zero at this point. Let's call it X. And then we see that the exceptional divisor of the fiber of pi over small y zero is an undegenerate quadratic denoted by Q. Well, and I assume, now let's assume that the dimension of Y equals to N, sorry, to N. And then Q has dimension to N minus two. And so we can, it's even dimensional quadric. And so we have, we have it's two rulings. And definition, I will refer as block cycle B is the difference of two rulings of the two rulings of Q, but I wanted to view it as an element of the Chao group of X. Well, the co-dimension would be N. Well, again, we can see that I'm not lying on the quadric, but inside of who X, and you can ask if this cycle would be homologically equivalent to zero. And here is a small M. The next conditions are equivalent. So the first one is that B is homologically equivalent to zero. The second one would be this. Let's consider the local system of homology over the puncture disk and the monodrama action. So the first one is that B is homologically equivalent to zero, and here is a small M. And the monodrama action. Well, I can see the homology of degree two N minus one of I versus the rational coefficients. And it is not triple. Yep. We don't see the bottom line. You don't see the bottom line. Okay. So I will. Okay. So let me pass to the next white board. And right. Now you don't see anything. Okay. So. Unfortunately, I don't know how to scroll the thing. Oh, beyond she's at all. Okay. Okay. So the monodrama is not trivial. Okay. Well, okay. So anyway, well, so this condition often. And you check actually the second certainly and this happens pretty often. And for example, let's consider the case when M equals to one. And this means that we have a family of course. And with the generation with single quadratic points, a point and there should could be two possibilities. We have either one, one component. Or two irreducible component of the fiber. And in the first case, the local monodrama is. Non trivial. So the lemma, the conditions of the lemma hold. And then the second case, it's trivial. Maybe I should also say that if we get our family from a left sheet's pencil. On a given variety of them. The conditions of lemma will, will be mad if the degree of the left sheet's pencil is sufficiently high. Okay. Now, let's formulate. Let me try to formulate the theorem. Let's consider. Excuse me. Yes, yes, I assume that it's proper. Sorry that I did not tell it. Well, from now on, I will assume that the condition of the lemma are mad. Now consider our family. Variation of mixed hot structures that I can see that I mentioned in the lemma. You, and maybe I will twist it by 10 to make it to have a weight minus one. And then I do not buy the limited hot structure. In other words, the nearby cycles for this local system. Since the monodrome is non-travel and the point is quadratic, that's fairly easy to see how the hot structure, how it would look, would look like. It will have three weights, minus two, minus one, and zero. And the associated graded portions for the weight filtration. The zero is one will buy the morphic to Q of zero. The next one is actually intersection. Camology of degree two and minus one of the singular fiber. They stop. Oh, just a moment. Something bad is with pencil. I should call my daughter. Wait a moment, please. Yeah. Stop writing. Now wonderful. And the one weight minus two is Q one. Again, it's as a morphic to Q of one. Not exactly canonical as here, but at least you can, if you consider the action of the monodrome as a thing, then they open the bottom quotient are identified by the operator one minus monodrome. Or monodrome minus one. Okay. So the picture looks like this. And now let's also look at the hot structure. Well, and let's look at as a real hot structure. And then there is abstraction to do it splitting as a real hot structure. So when we have weights, which are next to each other, then the things splits as a hot structure, but there is could be a non-trivial abstraction of splitting of the whole thing. And this would be actually a real number. Yes. And our hot structure. This is a real number which is defined, could be defined as follows. Take a generator in the quotient Q of zero and say one. And then we can leave it to say one tilde, which lie in the zero term of the hot filtration. And then we can actually leave it so that its difference minus with the complex conjugate to itself. So the imaginary part of one tilde will actually lie in the minus two term of the hot filtration. There is a choice of such lifting. And this is a real number, which is imaginary part of the hot structure. And this is our abstraction. And well, I don't know the right name. I don't know which local index or something like this. And the theorem gives an interpretation of this abstraction to splitting in terms of, well, it relates to the block cycle. So let's explain it, let's consider the situation when everything is defined over Q. Then everything, I mean the family came from a family of course over Q. And also let's assume that the block cycle is also defined over Q. Then we can consider the height. The block cycle is logically equivalent to zero because of our assumption in the lemma. And so we can consider its height, its height bearing with itself. And the theorem asserts that actually this height bearing and I will recall how you define it is actually equal to this abstraction to the splitting of the hot structure on the limited hot structure that we talked about. Let me formulate it more modestly that this will be not a question of the real numbers, but model combinations with rational coefficients of logarithms of primes. In a sense, you have to do it because if you multiply our function asked by some rational number, non-zero rational number, then certainly the zero fiber would not change and the height bearing would not change but this abstraction to the splitting will be multiplied by the logarithm. One should add to it the logarithm of the number will multiply our function. So you will get this. But in fact, there is some unfortunately conjecture on the level of our due to our ignorance at the moment. So model of some rationality conjecture there will be actually true equality of the numbers and again I will say a few words about this. Also, let me say that the theorem is actually very easy and the proof is essentially a recollection of all the definitions and then it comes by itself. Though the two numbers are seemingly different but still the proof is quite easy. So what I will do, I will explain actually a geometric version of this assertion and then say a few words how to change what should be changed in the argument to make it to make a very specific statement. Well, so I need to recall collection on I should say that I was playing with this a long, long ago in the middle of the 80s and somehow both at that time the two subjects the high-pairing and vanishing cycle somehow became disjoint in my head and this work is just an attempt to draw thread between two seemingly different points. Well, so in the geometric story what I will do is to replace so before we considered your familiar course over Q but now I will replace Q by the generic point of a smooth complex curve I see and let's see what will happen. So we have a smooth complex curve C and we will consider it's smooth compactification the smooth compactification C bar and below we will play all our shifts we will play with Q-shifts Q-hot sheets as were defined by Marie Heikassaita so all this everything I will do this will be about Q-hot sheets. Well, now suppose that we have on our C we have a local system of weight 1 weight minus 1, okay and then we can extend it as usually by an intermediate extension or which is by invariance of local monodrama well and now I need the following which is which comes very easily just playing a little bit with weights and the claim is that let's consider extensions the group X to 1 of extensions between Q of 0 on C bar and the middle extension and then we can restrict to X to 1 of Q of 0 C Well, so we consider extensions of Q-shifts, constant shifts with H on the compact curve and restricting it to extensions over C and the claim is is that the thing is actually that the map is an isomorphism Well, now this helps to define economical pairing on the right hand side for the reason that well we have one corridority for the middle extension to Q well and then certainly we have a map from from the corresponding well we can pass to this which are pure Hodge structures which pair into Q of 0 and then we can pass to well to Hodge cycles inside from Q of 0 to these spaces and then we will get an on degenerate pairing from this and this groups maps to maps to Hodge cycles inside of H1 of C bar with coefficients in H so we have pairing between the corresponding group for H and H star of 1 and so we have the pairing on the corresponding X groups okay so let me write it let me write it down okay now we can use the lemma and just invert the arrows so we have a map from the pairing between X1 of Q of 0 on C and H and the second group will be similar thing for H star of 1 but I prefer to rewrite just passing to dual extensions as X1 between H and Q of 1 and then we have the pairing to Q so again so we used here the lemma in order to pass from the extensions of Hodge local systems on curve to the corresponding extensions between the middle extensions on quantification where you do have the Pankara dialect and this I will call they the nice thing is that there is a way to compute there is a global high pairing but there is a way to compute it using local invariance some local invariance but in order to define this invariance we should have some extra data well the extra data would be a local system E with the following with the following properties so the following extra identification so the weights should be minus 2 minus 1 and 0 and also we should have the identifications of the associated graded quotients this should be Q of 0 this should be H and minus 2 quotient is Q of 1 so now maybe I should I should have said before that suppose that we have 2 elements of the corresponding X groups one is I will call alpha and the second I will call alpha V and my aim is to try to compute to compute the high pairing well now the assumption is that we have this identification such that the corresponding classes of extension of this will be alpha and of that will be alpha V now given such E we can define the local indices maybe let me denote in this manner X and this will be a rational number and they will be defined for every X at the infinity point of C so we consider the complement of C in C bar and then to every point given such an E then I can define a local X so let me let me define well in order to define it I consider let's consider the middle extension now of E to the whole of C bar well maybe even let's do the shift so to make a Y1 so to make our things perverse and let's look how this middle extension would look like well now it's easy to see that so E shifted by 1 now has weight 0 and so the whole thing could have sorry it will E shifted by 1 will have weights 0 plus and minus 1 and the middle extension would look like this so you have the top associated quotient is Q of 0 now the 0 the middle weight quotient the thing looks as follows well so first it is a pure pure shift of weight 0 so it splits and it splits into pieces since it's restriction to C equals equals E then the thing would be just a direct sum of some delta which is supported at C bar skyscraper supported at infinity at C bar minus C and the middle extension of E and then we have the bottom part this is suddenly it's usual so now E of H it was E oh no no no it's H I'm sorry yes you are perfectly right it's H yes yeah well now we can define the number well we know that so we have success of extension so we can consider classes of X1 so we know that classes of Q of 0 by means of H is alpha and here we have class which is alpha E but now we have extension also of Q of 0 by delta by the skyscraper shift and let's call it beta and here we will have beta V well and we can compose them beta and beta V and then we will get a number so for delta is skyscraper at C bar minus C so it's the same as if we do everything at every point and so we get a collection of numbers because well it's composition of 2X1 so it's X2 with support in a point at infinity and this is just Q so and by definition this local index of E at a point at infinity it is just well let's take number with minus sign it is a composition of beta V beta which is again is viewed as X2 with support in X of Q of 0 and Q of 1 and this is a rational number so for every point at infinity we have local index it is essentially immediate that the sum of this well that the sum of this local index equals the high pairing of alpha and alpha E well it's immediate because well here if you just look at this successive extension of W you see that the composition of 2X1 this one which has 2 components alpha and beta and another which have component alpha V and beta V is 0 as element in X2 and this is exactly alpha alpha V and plus beta beta V and so so it's essentially an open assertion so but it is very useful for for the following reason well first the first question is how to find well E this E first the abstraction to the existence of E lies in lies in well if you just look at it it lies in X2 with Q of 0 on C and Q of 1 on C and this is the same as Picard group of C and so shrinking C you can you can find E and also I did not say but I should have said that this high pairing itself it does not depend on if we shrink C we shrink well and locally and different E well it's if we have such sort of a double extension that certainly we have just an extension of Q of 0 by Q of 1 we can consider the bar sum of it with E and we get another E and it's easier to see that this makes all E's a torsor over X1 between Q of 0 and Q of 1 and this is this is just invertible functions on C and well so well if there is one E this will be a torsor so essentially so we have just a torsor with respect to invertible functions standard Q on our Q so this is the data of E and it's easier to see how local the global certainly the global high pairing does not depend on the choice of E and depends on E on E of the local indices certainly if I would just add will translate E by means of of an invertible function F here then the corresponding pairings well this will be the sum of certainly I don't remember the sum but let's take the clues the divisor of F at X so that is the picture well now let me formulate let me explain what the story has to do with algebraic cycles and then I will explain how the theorem comes the geometric version of the theorem comes well suppose that we have let E at high pairing of algebraic cycles suppose that we have let E at be genetic point of C and we have a smooth project variety X E at over E at well and suppose that we have two elements of the Chao groups alpha Chao group of co-dimension I of X E and we have alpha V which is in Chao J of X E and I want that I and J I will be chosen so that I plus J equals well that they would not intersect but just just just a bit so it will be dimension over E plus one so and then and then suppose also that alpha and alpha V are homologically equivalent to 0 on the generic geometric fiber well in this situation can well the previous construction they help to define the pairing of their high pairing and this will be a rational number and in order to define it we should just apply the previous construction to to define we should do the following first spread our X E to project a family over C say P and then let's also spread our cycles and let's define H as the direct image of Q of I minus one of Q of I on X and this will be a local will be a local system and we will have the classes of classes of alpha and alpha V exactly define the elements because alpha and alpha V are homologically equivalent to 0 on every fiber they define the corresponding extension classes we have which will lie in the next one 0 and H same for alpha V and we can just just apply the previous construction so this will be the global high-paying but we also can define local high-paying and well the only procedures that I knew before was that we should pick representatives of alpha and alpha V as true two algebraic cycles and spread it so that the corresponding well the supports of the corresponding cycles over C would not would not intersect and this is possible after shrinking C certainly and then we can define our local system let me just write down just write down its fibers we consider first the local system with fibers let A and B with the closures A and support alpha and B and support of V and again they do not intersect and then we can we can consider the relative group for every fiber we can consider the relative group of B and therefore the local system on our curve and we can use well the thing is not exactly extension by Q0 by Q1 but here will be a linear of 0 power A and here will be Q1 power B and then we should use alpha and alpha V to pull back and push forward to get exactly E in the shape we had before and then we will have local indices this is a nice procedure the only problem of it for example is that really in order to define I think we should have cycles with non-intersecting support and for example computing self-intersection is sort of a problem but now let me explain how you geometric version of the theorem well so we should do just what we did in case of theorem but replacing a rational spectrum of rational numbers which was very base by the generic point of C so basically this means the following that we consider we have our curve and we consider it inside of surface T and these are zeros of a function F well and then we have a family YT over T and this is a project so YT to T the project and I assume that YT is smooth and what I want is and its dimension is 10 plus 1 and what I assume is that all singular point that the singularity of the projection of y over C but actually has a single quadratic singularity so that's our picture well and here we can do just repeat everything that we did in the very beginning we can let then we have Y this singular thing that leaves over C we have a section I which assigns the point of C the only singular point in the fiber and then we can consider the blow up of YC at that point and that will be X well and again the fiber over of X over I is a non-degenerate quadratic so we have yes I want to understand your setup C is a complex curve yes yes what is T? T is the surface T is the surface yes and it lies over well it doesn't lie over okay and C is given by equation F equals to 0 and they are complete no I don't well the projection from Y to T is projected and YT but about T I don't care and I'm interested in it only near C and actually of C I also interested in the generic point of C so the base is not complete at all okay now again I blow I blow up YC at this at the singular point and so I get something X over YC X now and the exceptional divisor is Q will be a family of non-degenerate quadrics over C and I can do the very same thing I can I can define the block cycle I consider the two rulings of the quadrics I consider their difference and then I will get the block cycle P and I will consider it as an element of the middle Chao group of X turns RQ and again let's assume that the condition of the very first lemma holds so that if I consider the homology of non-singular fibers near C then the local monodrome will be non-trivial what were what were the degrees N-1 is non-trivial again then B will be homologically equivalent to zero and we can in this situation as I explained before we can consider the self-heat-paying just a moment so I should just a moment okay well so we can ask what would be the corresponding self-paying and the analog of the theorem is this so we should consider again the local family the local system of homology and so we consider local system E and this will be the vanishings the nearby cycles they will lie over C the local system over C and this will be applied to a local system which is defined on the complement so it will be local system with fibers H2N-1 Y QFN if you will look how it looks like again you have three weights and the successive quotients this will be Q of zero on C and then the next one minus one it will be the intersection of Y0 with k-efficient intersection and again there will be Q1 and this is exactly the data we played of E on a curve we played with and with this we can associate the local indices and we can compute if you wish there are some and the thing is that the corresponding global the claim is that the sum of this local indices is exactly is exactly the self-pairing of BB and the reason for this is just to check it we should just check that the two extensions are corresponded exactly to B and this is maybe let me formulate well I don't have I don't have time but maybe let me just say that what happens in just a few words what happens in the arithmetic situation everything could be done exactly as in geometry except that well for Q we have for spectrum of Z we should consider all points there will be non-archimede points and there will be archimede points and the number we consider is actually the local the analog of EX at non-archimede points but at archimede points we have similar numbers but to define them we have to use a larger chronology the current level of our knowledge as far as understanding is sufficient to know that there are rational numbers but the moment we know that there are rational numbers well we will have both the height-pairing is defined and again the height-pairing is defined also only model of this conjecture of rationality and it will be automatically equal to the corresponding corresponding sound that comes from from the nearby cycles which structure well, that's it so so any questions from Luke very naive question in the local situation so the local pair ability dimension one and one and quadratic singularity and here I take a curve similar to the reduction then in this case is there a relation between this height-pairing this local thing local indices I think and the characteristics theorem and gotendic model in pairing well, in the sense they are different well both the half-layer shots and gotendic monotrabia they are somehow inherently inside of the story for example this leomated algebraic cycles the classes of algebraic cycles homologically equivalent to zero can be extended to x1 for completed curve with intermediate extension of psychology itself it uses you really play with weights here so you should have gotendic monadronic theorem in the definition of this local cycles you also use it and so you use it's just inside but in fact gotendic you eventually arise a quadratic form so you take all the and it was not so easy to find the characteristics so it's quite easily easy so I wonder whether this more geometric approach could give a simpler definition of gotendic monadronic bearing and perhaps cool that it is in fact a perfect bearing maybe to prove integrality well I guess in case of relative dimension one you could prove you could prove integrality these last things that the local indices are actually not play naladic numbers but are rational but in half dimension I just have no idea any other questions so at the end I didn't follow the role of T it doesn't appear so what is the role of T I forgot maybe I was tired so T is the surface in which T is embedded with some kind of thickening so what is the role of T T the role of small T the parameter no no the surface the local monodrome the local monodrome the local monodrome of the family local monodrome or what the local T was also I think denoted the local monodrome of the family on the acting on the homology and that is for some T-1 identified the top weight quotient with a bulletin tape quotient okay so can you formulate the problem in naladic homology and is the result that you mentioned in the beginning does it depend on this rationality you said that the theorem is that something well modulo combination of log of parts is rational coefficients or it depends on some the result depends on some conjecture or not the theorem that I formulated in the beginning no it is not conjectural if you do it modulo modulo this pure linear combinations of logarithms of primes but you could if you know the conjectural rationality then you could you could get actual quality also the high pairing itself it also it is what is the conjectural thing that you okay so let me let me try to say it so we have maybe you can write again okay so let me let me try to do it problem is it tells that it creates all possible whiteboards okay so let me first delete just a moment okay so great now I can well so we have well we have if you wish M i t f over q M i t f e just a moment which is extension of say I don't know q of q of zero and then something called weight minus one say h I don't know and q of one well in the sense of or of nori maybe maybe the sense of nori then you can define for every for every p you can you can try to find abstraction to extending to the whole zp as an extension of q of zero by middle extension of h by q of one the problem is that you don't you don't have you don't have the language well nori motifs they work if you work over field of characteristics zero something like this but you can ask the corresponding eladic question then then they take middle extension and basically repeat all the words and then they get then they get a number and this number will be eladic and this you will get for a given p you will get for all l you will get its eladic numbers but you want it to be actually a rational number independent of l and then you can you can take this rational number with multiplied by log of p and then sum up everything and this will be an archimedeon component which comes from coach theory and then this will be something which depends only on two extensions q of zero by h and h by q of one by q of one by h you don't need well dependence of e disappears on this and this will be what does a global high but it's defined only in case only if you have if you can solve this this problem of rationality otherwise you can with what you considered in the theorem? Yes, with vanishing cycles. Yes, and it's very simple. So you get E, which has exactly as the shape as I wrote, which comes from vanishing cycles. And it's very easy to identify the successive quotients, the two successive quotients that they actually equal to B. And the reason for this, if you wish, is that, well, you have middle extension there. And in middle extension, if you consider on the whole family, there will come middle extension from the complement of singular point of the hyper surface to the whole hyper surface. And this will be primitive cycles on the corresponding quadric. And this is essentially as a plot cycle. So it's this part is simple. But you didn't define, well, I remember that you had a paper in the 80s, I think, but I don't, in any case, you didn't, in your talk, you wrote B, angular bracket B, you didn't say what it is. What is the high pairing of the cycle B with itself is? Yes. What happened in this work was that it was conditional definition of the high pairing model or this conjecture of rationality. So in the theorem, the high pairing of B and B is defined assuming some conjecture, yes? Yes. Yes, itself it is defined as, if you don't assume, it is defined only as element of the quotient of R, model of Q linear combination of logarithms of primes. And this you can do anyway by another method. Yes. OK. OK, I'm sorry for what I'm saying. I have a question from the audience. So Bruno Kahn is asking, is he the realisation of one motif in general? Is there a one motif? No, I think not. Well, it is for a family of curves. But for higher dimension, you will have just something of three consecutive weights. And the middle is not of type, well, 1, 0, 0, 1. But it could be, well, so it's not one motif in the sense of the length, period de Hodge-Trois. So there was a question. So let's thank the speaker again.