 da je poživit. So, perhaps, will start a little bit slowly. So, if you remember, yesterday we had a description, we had a discussion about how the tautologica ring of MG looks like. And actually the description applies both to the chao one and to the one in chromology. kar je človek, kaj je... Ne bomo poživati, da se tko tega postajem vse pravi. Zdaj sem tko, da tko liniče, jih je vse način, in da je do vseh pravi, da je do vseh način, tako ford, da je no, 23, je tko učin, da je to, da je vseh, za slj巫te, there is a perfect pairing. Let me write for l-h, it actually works in both cases. Now we have some kind of a fake top degree, the one g minus 2, which is the last one for which the total logical group is different from zero. in to je tudi tudi v r.h. g-2, ali to je isomofičko q, kaj je genetil kapa g-2 klas. In to je prejfektiv. Tako, as we pravdaš, je tudi nekaj prejfektiv, da je tudi tudi g-24, tudi nekaj prejfektiv. Tako, je tudi nekaj prejfektiv, in so prišlične, da je to prišlo. Zelo na različnih, da sem je zelo prišličnj, ta vse sebe, če je to prišlične, je bilo prejfect, da je zelo prišličnje, je bilo način več vsebe, da so prišlični, da se prišličnje. But the point is that the new expectation is that perhaps this just the unifying construction that generates all possible relations in this thing. As what with a relation. If you remember we have a set of generators that services of the relation would simply be a polynomial in the cottage classes that vanishes. This unified construction arises from something ki se je vzelo vzelo, da je tudi tudi za 2.000. Tudi zbijaš formula, je tudi zbijaš formula, ki je tudi za tudi za tudi, kaj je začel, z vsem kaj je zelo, ki se zelo, kaj je zelo, ko je zelo, tako, ki je nezelo geometrič, je to zelo numerologija, kaj je zelo, da je srečno vzelo, je to, da se vzelo, this is the way to ob Tinder, and this is the same kind of formula but this is just for the ng, and by now it's known how it generalize this kind of relations actually to the totalogical ring of ng and b even, and then put the successor on that parallel plane there I think, at about 2010. And again if you wish I can write these family of relations for you butt there is nothing geometrical about this, it just a way to write some formal power series and then taking the coefficients of it, give the relations which we want to be satisfied. So what do you prefer? Would you like me to write the relations for you, or shall I simply state what the thoughtful ideas are. So anybody wants? No, no, because the single relation doesn't say anything, it's the first formal power series that comes from the information, so either you get all relations or you get none. The property. The property. So the idea is that they are expressed as coefficients in a formal power series in some dummy variables. First there is some kind of variable t, which keeps stock of the degree, and then one has a strange bunch of variables p1, p3, p4, p6, p7. You know, they are labeled by natural numbers, so by positive integers, which are not congruent to 2 modulo 3. And then, so this is a set of variables, then the coefficients have to relieve in the homologi ring, and in a certain range one has that all coefficients that power series vanish. In particular, if we fix the genus G and a fixed co-dimension, we get finitely many relations. So let me just give two remarks. So if these are all the relations that exist, so the relations we can describe in this way, and then all multiples of them, this would imply that certainly this pairing is not perfect starting for 24. So somehow this expectation substitutes the previous one, and I don't know whether it's nicer or less nice, because depending on the purpose they're still interested in. And secondly, since this expectation holds for both for Chao and for homology, of course, if this is true, we get, of course, that if the same conjectural description applies to the relations in both rings, of course, given they have the same generators, they are isomorphic. It may be, well, as we discussed yesterday, we have not proved that it is a counter example. So people have constructed tons and tons of relations, but if you remember, there were two degrees, if I remember correctly, ten and twelve, and in one degree the dimension was known, in the other degree we couldn't, people can't choose, perhaps it's the same, but as far as we know it may be one more. So one would need some additional relation that nobody knows, and nobody has seen in practice, and considering there are so many independent ways of constructing relations, that's why people expect actually this to be a counter example. If we wish to prove that this is a counter example, you simply need to study better the classes which are non-totological, because perhaps you can prove the independence of the 37 generators by intersecting all of them with sufficiently many classes which are not in the tautological subring, because it's a subring, so somehow if the intersection numbers with other classes in the subring are equal, the classes may be equal. But if you want to know whether they are numerically equivalent, you need to know whether they intersect in the same way, also with all other classes which are non-totological. So perhaps there is some nice geometric class that has this property, but so far nobody has appeared with that. No, the relation with the genus and the degree is that if you take the formal power series, there is a range starting from which the coefficients have to be zero, and that range depends on g. And then there is some kind of divisibility conditions for the things you need to take, and that also depends on the genus. So yes, there is a dependency on the genus, if you look at the formulas it's a little bit hidden, but it's there. So somehow the case of genus g without maripoints is the most exciting one, because it's the most intuitive one. I mean we know many things about geometry or smooth curves, so theoretically we can exploit any kind of geometrical construction to explore the tautological thing, because in general geometric constructions have some kind of interpretation and they depend on the curve and some kind of choice of data over it, some additional structure, I don't know where divisor, mark points, whatever, so you can interpret them as something that depends on some modulate space, which has an actual vibration over mg, and then we can do things there using the geometry and push everything forward to the tautological thing. To mg, hoping that everything ends up in the tautological thing, it needs to be proved, and then using this to discover relations. But anyway, these kind of problems we discussed, they generalize to mark points and to the compactification, giving something which I will call for you, faber type conjectures. This is just the way in which the properties we have seen or we have discussed as conjectures for mg generalize to the partial compactifications. If you remember, we have three of them, we have curves with rational tase, in this case we have components of genus G, and all other components need to have genus zero, sorry that I didn't know the mark points for you, but of course we need them to stabilize it. This lies inside curves of compact type, I mean this guy here is interesting to study it separately, only if G is at least true because otherwise it would coincide with the next one. And then there is the whole compactification, somehow mg and rational tase is the natural generalization of mg in this kind of situation, because if we want to have rational tase, well we can add to other rational tase we need to have at least two mark points. So mg n, so mg n t is the same thing as mg n, if n is zero or one, so somehow we should obtain again what we obtained before by taking rational tase and n equal to zero. So if you remember the first reason where we were expecting there to be some kind of perfect pairing, which we should, well even it's not perfect, some kind of pairing between tautological classes that should encode at least most information, at least the information that's coming from tautological classes themselves is that there was this vanishing phenomenon, the tautological thing vanishes starting from a certain degree and in the last degree, which is non-trivial, it's one dimensional. Here actually constructing a pairing may be easier because if you look at mvargy and well here certainly we have a pairing because we can take two classes and just take their cut product, for instance in mvargy, the product in the charting, so we would like to be able to simply restrict this pairing to these two spaces and the idea is that we take the pairing of the lift of two classes and we multiply them by classes vanishes on the boundary. So what are the classes vanishing on the boundary, for instance, on the complement of the locus with rational tase? So if you remember, the, well let's start here, curves of compact tags were just the complement of a divisor because we wanted to remove our curves with non-separating nodes, so we are taking away the divisor parameterizing irreducible nodal curves and all the degenerations. So what is vanishing there? Well the idea is that we have our hodge bundle that depends on how many independent differential are on the curve and if you take a curve which is irreducible with a node then its normalization has genus g minus one, at least if we have just that single node and this implies that the hodge bundle degenerates, it does not have full rank and so if we take its top 10 class which is lambda g, it has to vanish on the locus, on the divisor delta zero. Just the locus of all irreducible curves to a node and then we take its closure. Recall, this is exactly what we have to remove if we want to obtain the open subset of curves of compact type. So the idea is that if we take two classes here, we can lift them there and if we multiply them by lambda g, the degree of the intersection does not depend on the choice of lifting we took. Then on the same principle one can prove that the product of lambda g and lambda g minus one vanishes on the complement of the space of curves with rational types. So why is that? Let me assume as I said, just this two otherwise sub case of what we had previously. The idea is that we have something which is not of this form, it will lie, it will be contained in one of the boundary divisors in which one component, in which both components have degrees smaller than g, because this is always, such kind of curve is always contained only in the divisor that has a component of genus g and one of genus zero and then we add the mark points to stabilize the thing. So let's assume we take any of the divisor, so we just take two genera that add up to g and then we want them both to be positive, so this will imply that both are smaller than g. So if we take the hodj band and we restrict to this stratum, how does it behave? Well, I mean, here we are just taking the product of two curves, so if we want to take a differential form, well, we can, it will be a product of differential forms on the two components of the curves. So this is actually the direct sum of hg1 and hg2. I will put a prime here to say, well, they live of course on different spaces. So if we want to compute lambda g, then lambda g is just the sum, oh, wait, sorry, it is just the product of the top ten classes on the summands and if we take lambda g minus one, we get something which is very similar, so one of the two factors decreases by one and the other one. And I am putting the prime just to know on which component we are. It does not have a special meaning, but they are not living on the same space and we should keep track of this. So if we work with the restriction of the product of lambda g, lambda g minus one, restricted to some state, then what we are taking is the product of the two top ten classes and then here we have the expression for the previous one. So let's see, in this first summand we have lambda g2 twice so we have the second power of lambda 2g prime and that is multiplied by lambda g1 lambda g1 minus one. And then we get something similar because we get lambda g1 square and then it's multiplied by lambda prime g2 lambda prime g2 minus one. Now if one looks at the way in which things are defined one can prove that one, if one takes the top ten class of the Hodge bundle on mg and take it square, this always vanishes. So the point is this vanishes. I should write it with colored chalk. I might be able to do this. This is equal to zero. This is equal to zero. Because on any mg prime we have the square of the top lambda class is equal to zero. And so in this way we get the vanishing of the boundary. So this ensures that we know at which kind of intersection numbers we can look because we get some natural pairings. So I want to do to write all three cases at once. So I will put a star in the place of the decoration should be put and then I say, well, I'm writing for that h, but the same pairing exists in chow, the two discussions are completely parallel. Some things are easier to prove in chronology. So we can take any total logical class and then there is some kind of top degree in so we take the complementary degree with respect to the top degree and then we can get a pairing with values in q. So if we take a degree d class and degree n minus d class then what we can do is to take the integral of mg and bar. So if we are on mg and bar we simply integrate the curve product of A and B of alpha and beta, but then of course it would not be well defined if we want to work for instance curves of compact type. So in the case of compact type we have to multiply further by lambda g this is only for the case of compact type erasional case and if we want to also take erasional case, let me add a new color, we need to multiply by lambda g minus 1 and because of the fact, so here I'm using the same notation but the idea is that we are lifting classes to so we extended an arbitrary way to mg and bar but then the fact that the lift was arbitrary is accounted for because you multiply by class that anyway vanishes in the boundary and of course this top degree which we consider formally is given by the dimension of the space minus the degree of these additional classes we need it to have a well defined pairing so we have a top degree which is what is called to be what we say is the degree of the circle of the pairing so that the last non-trivial so of course if we are just taking stable curves this is going to be the dimension of the modular space and then as I said if we take curves of compact type then we can decrease by g because we are copying with lambda g and for curves relational case we decrease further by g minus 1 and I'm interested in this only for g at least I don't want to hear this unless there is so if the g is equal to 1 then these two guys are equal if the g is equal to 0 everything is smooth curves yeah, yeah, yes it's again the restriction so it's actually the same kind of idea one can realize it I mean yes that discussion would give the pairing with lambda g lambda g minus 1 it's some kind of dual way to look at it in the discussion yesterday we instead of taking the cut product with these two classes we said well but we have a nice generator the kappa class and we use that to realize this isomorphism but if we do it here in this way we get something equivalent so for the period discussion you can already guess this guess it, this is a good candidate to have a perfect pairing but of course we know that already in the most intuitive case the evidence seems to go in a different direction so we don't expect that anymore but anyway that are also as I said the kind of structure which was conjectured for mg by father well gives rise to gave rise to conjectures also in this case and two of them are known to hold so now they are properties and not conjectures the first is the vanishing and soccer property actually I would prefer to state this in čal because it's actually less trivial in that case so the idea is that we take the homology of the compactification of any of these partial compactifications in the top degree this is indeed isomorphic to g so in the two cases so in homology for the compactification this would be trivial because it simply means that that the fundamental class of mg and bar has the expected dimension or something like this and if we take a degree which is larger then what we find vanishes so this explains why it's enough to take values in q here because indeed we are taking as I said we can identify this pairing we are taking with an appropriate identification of the generator in this isomorphism and this has several proofs so the case of rational taste is what we is actually the same the proof is the same one as we discussed yesterday yes, thank you and then if one takes it in čal and one takes stable curves this is actually the fact that the top tautological group in čal is of dimension one is actually an retriever thing has been proved by Graben and Wackel only slightly later than this and then the case of course of compact type has also been stated by Graben and Wackel but there is also some previous work of Faber and Padeli Padle and by now there are sort of unifying these and then also the intersection number conjecture generalizes yes please so you yeah I guess I don't no longer have this what I wanted to say is that when we take curves with rational tails if we have if we have no mark points that can't be the curves have to be irreducible because we have no mark points but then if we have no mark points we don't have anything so if we look at curves with rational tails and we have at most one mark point then we are simply looking as smooth curves so I simply don't understand which part of the blackboard you are no I mean here is the one for rational tails and it should be compatible with what I discussed yesterday yes there are conjectural less conjectural generalizations of this kind of constructions to MGM without any decoration so just smooth curves and a higher number and of mark points and they are compatible with the one with rational tails but of course if you don't if you don't stabilize at least not even partially and you have several mark points the theory will need I mean the complexity will increase much more in N than it does here here somehow it looks like all generators we are taking at the same and there is just one so I guess that there may be some circle issues in that case but I'm not discussing that kind of of conjecture so the only the only structure we are discussing for smooth curves is up to one mark point and in this case we are thinking about rational tails as a special case of rational tails what happens with smooth curves with more than two mark points so it's not something I wish to discuss the spirit is similar but there are some technical there is some kind of technical evolution of the way to state things which is which makes them more complicated it's a lesser symmetric theory Are there any further questions? Ah, yeah it's a I don't know it's something that comes from the theory of graded rings when you have when you have a pairing the pairing has a soccer which is the the graded party which you take the values and morally if you have a pair which is what we have the last class which is non-trivial and you want that it should be one-dimensional so if you are in this situation you can so if you are taking a pairing of graded rings and it takes value in the top class in the highest class which is non-trivial you call that a soccer so the soccer refers to this space and the ring which has a perfect pairing in this sense is something which is called the Goldenstein ring yes this is coming from singularity theory as a concept there is the intersection number conjecture which is actually well I'm not going to write it but it's actually even easier in this cases because if we consider all possible values of g and n at once then we can use the natural maps to deduce intersection numbers to one k to the other so somehow this allows one to restrict to knowing all powers of the psi one class g one bar to know everything else so anyway this is also known and this allows to compute all intersection numbers between total logical classes this is the determinant uniquely all intersection numbers between total logical classes and this is now proved by work of many authors depending on the case as I said yesterday perhaps the case was the hardest one so there are by now several proofs some coming from mathematical physics other pretty geometric and I would say the earliest one was around 2001 most recent one I could talk was around 2011 and then so this is what is known so we are exactly in the same situations we were before and as I said with intersection numbers the fact that we are in those spaces because we are allowed to have mark points is even making the formula looking more compact but then so this is what is known and then there is the Gornstein property which I am calling a Gornstein conjecture even though in this form it has never been conjectured by anybody but it had been previously expected in the past so don't go to any mathematicians because they have conjectured this and this is wrong because actually nobody did so the pairing we defined before is a perfect pairing morally in all three cases so we just give the most general false conjecture we can get and note that if this is true in the charting then it is automatical true in chromology so if one wants to contract a counter example it is enough to construct in chromology even wants to prove the property since the thing would be stronger so this is indeed the case if the genus we take is efficiently small for instance in genus zero when everything is known so for instance in the case of of mg and bar this goes back to the work of Kill who described the child groups of mg and bar that they are the same as the chromology so in this case everything we discussed works without any trouble and then for g equals 1 and if we look at stable curves well if you remember in this case I already stated that the tautological ring is the same thing as the even part of the chromology and this statement had long been claimed by Gessler with the only issue that he shared he is prepared only with his closest friends so a prove was not general available and then came Peterson in 2014 who actually proved this and if you look at this and if you are interested in the fact whether the pairing is perfect or not where the pairing in this case would coincide with Poincare duality so the stated that the pairing is perfect is equivalent to Poincare duality for m1 and bar and given that m1 and bar is a smooth so if you look at the course model this case has only locally quotient singularities well these of course holes so the Gornstein probably holds and is implied by Poincare duality so this gives a minimal indication for the genus at which the thing may fail so for m1 and the curves of compact times table class but in this case the Gornstein property is known by work of taracol and actually if you go to up to genus 2 but you stick to rational type in a different paper more recent taracol proved that the property the Gornstein property holds so these are the good cases the ones in low genus in which the property is known to all as we know rational type and no mark points there is a lot of evidence and yeah certainly looking at genus 2 is not and no mark points is not enough to get any kind of counter example because indeed that was the first case in which the intersection theory of g was studied so these are the known cases the positive ones so that in which the the Gornstein property holds so up to here we have there the Gornstein things and then we start with the counter examples so somehow the one which is harder to break morally is the one that has been settled first because if you take if we take stable curves of genus 2 with 20 mark points well as long as we have no totological classes sorry as long as all classes are totological we can sort of say well but more or less we are working with punctuality so we need to go in a range in which there are non-trivial non-totological classes and in this this happens for the first time in m bar to 20 as was actually closed slightly later than our result because the original result was slightly weaker so let me put just an n here and see not going together we done Peterson and actually the refinement is exactly 20 relies on later work by done so the idea is morally you expect you need the existence of non-totological classes to disrupt the symmetry of the pairing which is coming from punctuality and then if you are sufficient if you can control sufficiently how many new non-totological classes you get you actually find it in the first case in which non-totological classes exist exist these also these are the fact that the pairing was perfect when working just on the whole even common as I said if you work with curves with rational days I can explain to you later why in genus 2 everything is still working perfectly so you may wonder whether proving or disproving this Gorinstein property for curves of compact type is easier or more difficult than what it is for stable curves well easier or difficult just the property of the proof and not of the statement sorry I can't tell anything about this but I can tell you about how many points you need so the idea is that if you take logical ring of n to n compact type this is not Gorinstein so the pairing is not perfect and you only need eight points to get this so the three cases I stated to you as a conjecture only the one of rational days is open and you know that so the only one conjecture which is open for the case of rational days as we discussed yesterday is g 24 n 0 but then there is evidence that one could also take g equal to 19 and equal to 1 and it's perfectly plausible that we already know where to look for the counter so why why does the Gorinstein property holds for curves of genus 2 with rational times well this is a general feature when studying homology the reason to study is that we construct the space of curves with rational days in such a way in which it's fiber over mg so in this case over m2 so by definition if we forget all rational days and all my points we get the vibration the basic idea similar to the one we had here in genus one for stable curves we said well actually the homology the total logical ring is the same thing as well the even part of the homology of the smooth variety or the rational smooth variety in this case it was actually the same space so if we have one credibility here then we have the Gorinstein property and this vibration gives a similar identification for stable homology but actually sorry for total logical homology but actually in a different way so the idea is actually a projective vibration with smooth fibers so if the point is nonsingular then we get the configuration space of n distinct points let me denote it by f and c where c is the genus two curve and this is just the nth power of c from which we remove all the diagonals so all the n tapers in which we have pi equal to pj with i different from j and the fiber is some kind of compactification of this which is actually known as the Fulton-McPherson compactification if we want we can describe this is some kind of blow up of c to the power n by solving the fact that over the diagonal as we have something different so anyway there is a morphism from here a forgetful morphism from here to c to the power n sorry I think I wrote the things correctly but simply I wrote them in inverted order so what we are saying is this is a projective vibration the fibers are smooth and are these configuration spaces f and n c which are some kind of compactification of the space of configuration so let me if can you read it here so what I would say this is the space of configurations of n points and this is the compactification of that and this is what is the fiber so morally we need to parameterize all cases all configurations of n points on a smooth two curve and when at least two points coincide they are the same we build a tree of rational curves in which we can let them move so every time we have a map and we want to compute the homology of the domain well we have a morphism we can look at the layer aspect sequence which associates to that for vibrations the form of the layer aspect sequence is particularly easy and for projective vibrations we know that the layer aspect sequence degenerates at e2 so this means to the tree this kind of vibration gives the composition of the homology of m2 and rt in terms of information that comes purely from m2 and if you want to know what this sentence means what this means is that if you want to know the homology of m2 and rational case in fx degree k then this will be the sum for p plus q equal to k of the homology in degree p of m2 the problem is where do we take values here because we don't we have to take values in a local system that's determined where the kth homology of the fiber so we need to take higher direct images of the local system on m2 and rational case so what are as I said when we write this well, if we take this over a curve this is actually natural isomorphic to the homology in degree q of the fiber which is this fault of the person so local systems on m2 vanish in degree zero unless they have a non-trivial sum and by this I'm stating this by saying this means to me that if we take p equal to zero we get the part of the homology of m2 which is easier to describe because the homology of m2 with constant coefficients concentrated in degree zero so if we take p equal to zero here this is actually isomorphic to taking the homology in degree q of this thing and take the part in variant under the action of the symplectic group so if you remember the homology the only interesting part of the homology of the curve is just the h1 which is given and on that the symplectic group acts by the standard representation this induces in action of course also on the powers of this and it is also induces in action on the homology of discompactification so we have in action of the homology sorry in action of the symplectic group considered as a group scheme on the homology of the fiber so somehow in the special case in which we take p equal to zero here we have the same thing as something which is natural isomorphic to the homology of a smooth variety and then we are taking the invariant part under the group action which will respect on the fiber and if you remember I mean the space is just parameterizing configurations of n points so it's something of dimension n small n is also the top degree in the pairing in the specific case of genus 2 because if you remember the top degree was always given by g minus 2 plus n for curves integration methods so what I was saying is when we study the Ganeshtan property the degree the fictive top degree we are taking is the same as the dimension of the fiber of this fiberation and as I said well there is a part of this direct sum the one in which we take p equal to zero on which puankade duality holds naturally well of course there are some non trivial identifications here if you really want to use then the ring structure but we are just checking that better numbers are symmetric then this is enough to say well if we take the part of this direct sum that comes from p equal to zero then it corresponds to some kind of vector subspace familiar vector subspaces on which the better number are symmetric and then what one can prove is that this the part of this direct sum that comes from p equal to zero is also isomorphic to the tautological part oh yeah yes that's what I wanted to state I don't like to write q because I think that the correct thing is to think of sp as a group scheme so that you can adapt to the situation but yes that's a and g was equal to two so yeah thank you I had not realized it's always a vibration for higher genus but there are two things that maybe offer the first thing is that if the genus is higher then then then the degree of the pairing is not exactly equal to small n so it will not be equal to the dimension of the fiber this might be disturbing and the other issue if we make genus large that is of course what I haven't written yet but it is that if you take the part of this that comes from p equal to zero that's something we see inside the homology sorry kormology and we can identify it with tautological so in this the composition the k is p equal to zero gives the tautological part of the kormology but I'll say I can even fit the statement here so what we can do if we take k even otherwise we'll have to scale everything but then if k is all this part will give zero because of general facts about curves of genus 2 yes it's independent of c because if you take all possible choices of c they are from a tautological point of view they are all equivalent because in simplecti group it does not matter which curve you take the point is that this identification is up to homotopy but everything is homotopy invariant so that's the reason why it does not depend on c as soon as you allow the curve to degenerate you don't have this kind of nice description anymore of course so as I was saying even then we get of course a tautological part of the tautological homology the point is that we can identify this with the case in which p is equal to zero so if p is equal to zero then q is equal to k so what we get here so somehow one can realize this inclusion inside the information coming from the respect of sequence how canonical this is there are of course some technicalities which I am not discussing but the basic idea is exactly this and once we have done this we can define tautological homology with something for which we know that one can have a duality hold so we are doing the same thing in a different way I must say, tautical probability in the showering and the proof that is more complicated than this so what is the punchline of this so I was saying m to innovation and this is the part that we can control so luckily for genus 2 these boundary devices which are not which are in compact type not a rational type so it is just something about the case in which we have two components both of genus 1 so this allows one to do some kind of genus reduction so the idea is that if we know what happens with rational type then we try somehow to add the information about the boundary to it and that is how we have found his counter example already for 8 points somehow adding these things coming from the case in which we have two components of genus 1 this wraps completely the symmetry of the pairing which we had here well of course also passing from compact type to rational type to compact type is also moving the target because of course also the degree in which we are taking the pairing becomes different still if any is sufficiently small it is going to work so it is not what needs some work to find out what is happening so I don't know whether there are any questions about this but I still have at least 20 minutes left provided my yeah no my watch is not particularly correct but I do have almost 20 minutes left and I wanted to discuss briefly the algebraic the algebraic invariance on cohomology of course of the modulate space of curves even if not writing explicitly so the idea is that if we look at rational cohomology of course all cohomology groups are q vector spaces and this also holds for the totalogical groups they are of course also q vector spaces but there is not so much structure so it could be useful to add any more structure which is possible to get as complete as possible view on the properties and if we walk over see the natural thing is to look at the hodge theory sorry the hodge structures on the spaces so if you remember every time we can recall that cohomology groups of complex varieties so let's say those are projective complex varieties always carry mixed hodge structures so in particular if we are working with smooth projective varieties or smooth compact varieties what we will have will be pure hodge structures so if we take a cohomology group it will have an invariant called the weight and it will carry a structure which is given some kind of the composition so we are just working on q vector spaces so we are working with rational hodge structure and as we saw yesterday in the third talk the idea is that we can complexify our vector space so that we have a complex conjugation there and then what we want to have is it the composition of vpq of v into vpqs where the sum of is equal to the weight w we fixed at the beginning and there is some kind of symmetry between them in the sense that if we invert the two indices the complex conjugate of vqp is vpq now if you don't like analysis in a particular way but you prefer geometry there is just one piece of information which is very valuable here but in geometric meaning that's the weight somehow the composition is purely analytic tool so if we take a smooth projective variety let's say then this carries a hodge structure of weight equal to the degree so for instance since we are taking rational coefficients it does not work whether instead of being smooth it has a rational it has a local portion singularity like the coarse modulus space of mg and bar but in particular this will also work with mg and bar itself so this is a nice nice case but this is a case which is one of the cases we are interested so we are working on mg and bar this is simply anxious that there are some particular structure coming from hodge theory that we can add to this and equivalently if we prefer to work with filthations rather with the compositions where we can instead take the hodge filtration which is a decreasing decreasing filtration associated to this to this decomposition we want something decreasing then we need to take all cases in which the first index is at most this is still called b in the example and then because of this symmetry the characterizing property is that if we take fp and f in the complementary degree w-p complex conjugate the direct sum should give the whole space because of the fact that we had this property and this decomposition so this is all good but as I said this works well if we are working with something compact and smooth but in general the spaces we work with may not be any of those things and in particular if we are working with a modular space of curves and we are taking the open part of smooth curves where we will have something which is not compact so does some kind of some part of this structure survives and the answer is yes it's only that the choice of the weight is no longer unique for the degree in which you are taking homology and this means that we need a second filtration to take, to keep crack it could be it should be easy to check because I want to have a decreasing filtration did I put it wrong? I mean you are looking at the blackboard so your brain should work better than mine because you have some kind of distance so never mind so a mixed hot structure in a cube vector space is given by two filetations freezing filtration which is called the white filtration usually it starts at zero but sometimes they may be negatively graded and then a decreasing hot filtration such that if we take the graded pieces of the white filtration the hot filtration induces a pure hot structure and so somehow the geometric interesting information is coming from the white filtration and somehow identifying the pure structure is more of an analytic property so how the strong invariance from the geometric point of view is known in this white filtration so the idea is that if we take the filtration induced by f on the graded pieces of the white filtration so being an increasing filtration we can portion by the previous one gives a pure hot structure of weight m on it so the idea is that well let us assume that w0 is the first non-trivial thing so that the white filtration is non negatively graded as it would be for the homology of a smooth projective variety that one could have many possibilities then this means that the first wm for which which is non-trivial is giving something intrinsic inside the homology and the next one then classes of so classes of weight 1 will be so w1 will be an extension of w0 by classes of hodge weight 1 so somehow we start the lowest weight and we add new classes by taking some kind of extension so if there is a mixed hodge structure we have something which is an extension of hodge structures and we put the hodge weights in an increasing order so by the linear theories of mixed hodge structures also a projective variety that we may generalize this to the case in which so for the instance in which x is ngn or is coas modulate space since the homology is the same in the homology with two coefficients carries a natural functorial mixed hodge structures and then in general if x is smooth then the weights can only lie between the degree so if x is smooth but possibly not compact then the minimal weight the one which is giving the intrinsic part is equal to the degree in which we are working but potentially we can get classes of weight up to 2k so how this is giving if x is smooth the natural part for the weight filtration is equal to the degree if and if x is also compact this is everything we have and the other hand if x is compact but possibly singular then we have the decayed homology has weights between 0 and k so if x is singular somehow the natural invariance are looking at the classes of hodge weight 0 and if they are there they are not only in interpretation in terms of the desingualization of the space and the natural part the one we expect to have the one which has weight equal to the degree is actually the top weight we can have and because of the fact that these structures are functorial if we consider these structures for the modular space of curves that has a natural action by the symmetric group then of course they will respect the two filtration so it will respect the mix I was hoping that since there is not so much time left what was upcoming was which hodge theoretic invariance are we truly going to work with for MGM but instead what was coming in my nose was the example of PN so the idea is that the only non-trivial homology of PN is the one which we have in even degree between 0 and 2n if you remember this is one dimensional and the notation for this pure hodge structure which will have weight 2k is q bracket minus k this is called so we can use this actually as a definition of q minus k and this is the hodge structure so what is the basic idea here if the the easiest way to construct a hodge structure in one dimensional rational vector space is to choose everything as trivial as possible concentrated in h00 and this would give q here for PN of course we have non-trivial homology also in a degree which is larger than 0 and this produces some kind of twist of the structure in which we are taking and now all homology think of it as an alternative definition is just all homology is concentrated in the central part of the decomposition and then there is of course some additional structure that specifying how to embed this rational vector space into its compactification because there are also some final invariants there so this is the characterization plus q minus k it corresponds to the compact in which all homology is concentrated in this balanced part so the other parts of the composition are of course trivial so this is the nicest type of hodge structures they are the ones we expect which we find in all the spaces that have the composition into cells which are isomorphic to CN for some value this is of course also true for PN which is the union of a point C, C2 and so on and in general the hodge structures are the kind of hodge structures that we find on the algebraic part of homology so if we take the fundamental class of a sub-variety then it will also always have a hodge structure of tate we can think of this as of something generated by the fundamental class of some T n minus k considered inside PN and in general fundamental classes so algebraic classes in general and of course there is the hodge conjecture that is morally accepting the converse to this somehow that this is also a way in which you can recognize the fact that the algebraic classes exist but we will not need this and by the way everything we said before but for this property it is slightly twisted not only for homology of varieties but it also holds with homology with complex support and in general if we want to pass from homology to homology with complex support at least if x is smooth so that we can use point card duality then we have to take real dimension of x so we need to take 20 degrees so that we can use point card duality we take the dual space dualizing also the vibrations and then we take the we need to correct the weight by twisting with a certain hodge structure of type so anyway everything we did with homology also works with homology with complex support only the bounds on the weight will be different if x is smooth because then there will be smaller so the aim of this of course is to define to you which kind of hodge theoretic invariance are useful when working with moderate space of course but I gather this will happen tomorrow so that was it for now