 First, thank you very much for the very kind invitation and I'm glad to give a lecture for Luc and on the topic which I hope is not too far from one of his manifold interests and I'm also glad to be able to see him at Bure, so it's a pleasure and also many friends. So I gave a lecture on the same topic very recently but we made, I mean like a month ago or something, but we made a progress so I feel very well to repeat part of the previous lecture and then to improve a little bit the result we had. So the main motivation comes from the work of Simpson which among many things says the following. So if X is smooth, so first let me look, I mean on my computer it's 41 so I can go to 41 after 12. So if X is smooth projective over the field of complex numbers then and we fix the rank here, a natural number then he showed that we have modular spaces, one is called the modular space, Betty modular space, so it's a complex points here are isomorphic classes of semi-simple rank R, I'm sorry, complex local systems and one of the geometric picture of this modular space is that it is affine. And then there is a Durham modular space of the same rank so that's the same thing, the complex points are some direct sum of simple connections, the whole thing of rank R and between the two here there's a Riemann-Hilbert correspondence and the Riemann-Hilbert correspondence is a complex analytic isomorphism. Further he showed the existence of a so-called Dolbo modular space and the complex point of it are rank R hitching local systems which are semi-stable of degree zero and now the new thing in his work I mean among many things is that there is a real analytic isomorphism and consequently here there is a real analytic isomorphism and geometrically there is one further property of this Dolbo modular space which has been used by many people, not only on gobao chao, the so-called hitching vibration to an affine space, it has two main geometric pictures. The first thing is that this map here is a proper map and the second thing here is that this dimension here is just a dimension of the complex number from h0 of x and then the symmetric differential form which go from 0 to R, R being the rank. So this has immediate consequences here, those geometric facts here is that to say that if we look at isolated components in the three versions here they correspond because those are isolated points and they are called rigid objects and then Simpson conjectures that rigid objects are motivic. So let me not give a precise for lack of time let me not give a precise definition here of motivic but say roughly that this is a direct cement on the dense open of x of a Gauss-Mannin system coming from a smooth projective morphism and recently very very recently this has been recast in a more general context by Petrov who proved using the methods of Peter Schulzer that in fact this conjecture is a special case of the Fontaine method conjectures. So that's one consequence we can draw from the fundamental geometry here of Carlos Simpson and another one is that if n is 0 and more generally if you assume that on x there are no symmetric differential forms then I should put like this I'm sorry here then this morph is being proper and the Dolbo modular space is proper but proper means compact topologically consequently the affine space is compact and consequently it is zero dimensional. So all the m's here are zero dimensional in particular in the terminology of Simpsons they are rigid so they should be motivic but a while ago I thought that it should imply more than Simpson's conjecture not only they should be motivic but in fact they should have finite monodromy and this has been proved here there is a positive answer to the question here this has been proved by Bruno Bach, Klimger and Tataros in this has been published in 13 is that this vanishing condition here I'm going to call it vanishing for x so this means no higher symmetric differential forms the vanishing condition for x implies all complex local systems have finite monodromy so this is of course a enlightening fact and we can make here a remark here is that varieties with these properties there are many of them for example if we take x smooth projective in pn of dimension sufficiently large yeah sufficiently large then we have the vanishing condition this is one condition and this is one remark and another remark is that if we take a Shimura variety of rank at least two then the fact that then we have that the dimension of all the space is always zero so that means all local systems are rigid but yet there are local systems with infinite monodromy in fact there are many of those because it's a Shimura variety so infinite monodromy so to force finite monodromy it's not enough to assume that all local systems are rigid and now before going on let me give a hint let me give a sketch of the proof of Bruno Bach, Klimger and Tataros this is a pure application of positivity theory in hot theory so the first thing one has to realize is that it is again due to Carlos Simpson is that due to the fact that all points are isolated this implies that there are all complex variations of hot structures so that's Hick's theory which is telling us this so this is due to Simpson and furthermore this is this has been proved by Katsarkov and so using the associated periodic representation so in GLR of QP bar is that if a representation here of the topological fundamental group is unbounded viewed as a QP bar representation then this produces a differential forms symmetric differential forms but more generally and we will come to this later on in fact there is a more general result of Michel Grosjeanik and myself which says that this implies really integrity so integrity would be that this is bounded for each P or said equivalently that the representations have value because they are rigid they have value in GLR of Q bar and to say that it's unbounded for any piece this is saying that in fact it is conjugated in GLR of Q bar to representation with values in GLR of Z bar so we have integrity the result here is more general we will say a word on this and then the key point is a CRM of Kant's law we will come up again in the lecture later on which says that now if we look because it's integral we can look at the map which is induced by the local system to the period domain and then if the image has dimension which is more than zero then X the vanishing condition on H is also not true that means the image has symmetric differential forms and consequently X has symmetric differential forms so that's a sketch of the proof and as we see this proof is highly transcendental so the problem we can pose here as a general problem is now if we assume that X is smooth projective over K algebraically close but now of characteristic P what what geometric condition forces motivicity and because motivicity is a very difficult concept we can start with finiteness and of course we can we can pose this question for isocrystals over capital K which is a field of fractions of WK we can pose this question for LAD local systems of course L is not P here and we can pose a question for crystals over little k that means flat connections so so this is a general question and of course before I proceed if we are interested in in the eladic theories or not the crystalline theory is then of course we have a we have a general answer here if K bar is Fp bar and X is a curve because now the length correspondence is going to tell us that the local system is geometric if and only if it is arithmetic and arithmetic means that it descends to a finite field so that's a consequence of the length length program but even this consequence of the length length program affects us higher dimension we cannot answer this question once you want to assume that this is true but we cannot answer this question so I'm not going to talk on the eladic local systems here but let me already make a small corollary of the serum the analytic serum here which on which we discuss so we have here so first before I do this let me write here a general serum which is due to michael and myself from from the same period here is that if we assume if we just assume that all this so we are overseas here if we just assume that all this modular species here are zero dimensional then as I already briefly mentioned earlier on then we have integrity of all all local systems here and in fact the result is more general here but I recast it I apologize in the context of our lecture here and now if we go on the Durham side if enabla is rigid then we go to good model of x over c so let's say x over r where r is a ring of finite top over z as usual and then we can look here at the w point of x over r and then we can restrict e over w tensor capital k then the serum here is that it is an isocrystal and not only it is an isocrystal but with a Frobeno structure so as I said earlier on so we are really searching for conditions which should imply a multiplicity and that is a background of what I want to explain here so let us come back to the problem here which we post here and discuss just as a simple application of the serum of Bruno barb clinglor and totaro discuss our first application here so we have a first serum here which is due to michael and myself but now a more recent serum so let us assume that x leads to w and which is smooth projective let us assume that we have the vanishing condition and let us assume that yeah so the first consequence here is that all eladic local system m over x here are a finite monodromy so in fact I can if I want to sketch the proof here one is there is nothing in it I mean this is the Riemann-Hiebert correspondence plus the serum of the three here and if in addition k is fq here then we any convergent isocrystal for any convergent isocrystal then there is y over x a finite detail which so let's say h such that h over star of the isocrystal is trivial so for short here I'm going to say that it has finite monodromy the isocrystal has finite monodromy and y is too smooth it's not as I say it's a very easy consequence but let me nonetheless discuss it very rapidly so first the vanishing condition for x by best change so the vanishing condition I apologize implies the vanishing condition for x over capital k so this implies in particular that if we realize the isocrystals as connections for x over capital k then some if we look at the orbit under the Frobenius of x nabla over capital k then this orbit is finite because as we saw earlier on the moduli are zero dimensional and consequently this is pre-periodic and if it is pre-periodic this means that if we replace in nabla over capital k or if you like if you replace the isocrystal by a Frobenius pullback of an isocrystal it's going to be a Frobenius pullback together with a Frobenius structure on f n opposite of nabla over capital k and this implies by the theory of companions and in that direction this is due to Tomoyuki Abe and myself and also Kedlaya this implies that there is an eladic local system which underlies the situation but by one here this eladic local system has finite monodromy so this serves here too if we remove if we draw the condition for of conversion see not for the isocrystals itself but for some Frobenius pullback of the isocrystal but now if we assume that the isocrystal is convergent the Frobenius pullback is an equivalence of category so this finishes up proof here so this is a it's an easy consequence but of course it raises immediately the same question can we just as a so to speak nearly small provocation without more evidence at this small theorem here we can ask whether the vanishing condition here over x but no liftability implies the same result as the same same consequence and here we don't know we cannot even start okay so this is this is one part so now I'm going to go to the third part here in the list of problems which are here so I discussed already this a little bit this a little bit but of course under a strong liftability assumption and now I'm going to turn my attention to crystals and so and this is the main part of the lecture so maybe I write here crystals over k here and I will assume so k equals k bar equals f p bar so I'm assuming that I have a finite field which underlies the situation here and of course and we want the recalls that we want to to find an analog over a finite field of the fact that the vanishing of the highest symmetric differential forms forces motivicity in a very very strong sense over the field of complex numbers usually when we are so we assume that x is smooth projective over k usually when we are over a finite field over f p bar then is to look at the global sections of differential forms and symmetric differential forms this is not always a very good idea because because that we have we know that we have pathologies in characteristic p but thanks to the work of deline and delusie then we know that if we assume that the variety lives to w2 so assume x lives to w2 then we can expect a better behaved properties for the shift of differential forms and as a consequence for crystals over little case that means for connections of rank at most p so those are going to be the assumption which we have here so those are really stemming from the work of luke and an extension of it so the serum here which we want to discuss a little bit is the following one so this is our assumption here so we have here we assume that x lives to w2 so x is smooth projective so we have those assumptions here we are going to consider only r at most p and now we test the condition which i had predicted in in characteristic zero in this situation so let us assume that we have this condition no higher symmetric differential forms and what does that imply then the result is that it implies fanatness so there is so given in a blah of frank i yes i i insist here frank at most p which is which has c i equal zero and which is so my simple but so my simple here is just for decoration because we can also treat the extensions here but just for the lecture let me assume so my simple then it has fun at monodromy in the sense and rank the rank is at at most p but we can prove it so far as i said it's work in progress we can prove it here for rank at least three at most three i apologize i should say rank at most two this is written rank at most three we have only nodes so i should be careful here but it seems that it works and then it gives of course a confidence in general even so we cannot we cannot we cannot prove it so let me write the theorem here is that if we fix in a blah then we will find here an a y over x which is finite et al so let's say h such that h opposed to r of in a blah is trivial so that's a theorem here so in the remaining time here let me explain how the proof works and it's only an indication here even though it's not completely difficult it's not it's a big technical so i will start with rank one which is not technical assumes that the rank is one so but the rank is one there is not much of theory behind but it gives so to speak it shows what one has to know at the more theoretical level in order to have a chance to argue similarly so uh when we have rank one here uh then we uh so then for a historical reason i'm not going to denote the connection by in a blah but by el na blah then we can look at the Frobenius orbit of el na blah so now those have called yields fq points um of let me write here picna blah uh picna blah of of x so now picna blah we have the chance that it's really fine but more generally we can we can say that because the field is c1 i so to speak here i predate a little bit uh uh in advance what's going to happen later on uh because uh fq is a c1 field then it is a case that if we have an fq point of the modular i mean due to hard on our c-man perhaps the fq point of the modular corresponds to an object so uh we have here it's a bijection so we look at this um at this orbit and because points are the same as objects over fq bar then this orbit is pre-periodic so now if it is pre-periodic it's going to tell us that again we will have here replacing n n na blah by some Frobenius pullback of n na blah then we have so there is n such that some Frobenius pullback of n na blah in fact is stabilized by uh by some Frobenius power so it has a Frobenius structure but uh if we regard here from this information the information which is which it implies only on the underlying line bundle so this implies that we have an equation of the type let me write n here fn opostar of l isomorphic to um fn m let's say opostar of l for n not m so this implies so let's say larger than m well i should say little n here i apologize i mean the little n which is here so this implies that series uh by looking at the length also of rank one which in this case is just a kumar cover of degree p to the power n minus one so the important information that the degree is point two p in this case so there is here a finite etal cover which kills which trivializes the underlying coherent object here such that let's say h here h opostar of fn opostar of l is o but now on this on this pullback here we also have our connection and here we have the trivial connection but now because the condition here vanishing here is stable by finite etal cover this implies so now we have the trivial shift and we have two connections on it then but we have no differential forms on the y so there is only a single connection so this implies here if you iterate going from fn to fn minus one and etc you will find that h opostar of l itself is trivial and then by the same argument h opostar of the connection is trivial okay so this is a baby case so now we address the rank two case and i will explain the method i cannot explain everything in the proof but i will explain the method so based on the work of delinie lucy which has been generalized by uh uh augus and vologotsky and those two statements here i mean first this one is a local statement so to speak on x we don't have to assume that x is proper or projective that's the same for the work of augus vologotsky which is really based on the work of delinie lucy putting coefficients which have a nilpotent curvature and then so to speak reducing to the case of delinie lucy on the graded on the associated graded object um so to speak i mean it's of course very close what i said and so now there's a question for rick so uh dimension smaller than p or dimension plus rank smaller than p what is the assumption there for me i can you repeat the question uh what are the assumptions on on x dimension plus two smaller than p or something like that and i think i just did rank one so i apologize thank you very much so uh i will always assume that the dimension is smaller than p thank you very much but we don't assume that i think in uh augus vologotsky you need a little more than that you need also is a combination of the rank and the familiar sum of the rank and the dimension of x so little smaller even smaller than p well i i we're okay i should check at any rate is the condition the answer dimension is the one which comes from no i think dimension smaller than p is enough but i can check later but thank you very much as i should have said that earlier on and check what is the correct uh correct condition uh thank you so um and then then comes uh now uh coming from from this work here but now it's really a global statement on x that's the notion which is quite fruitful which has been introduced by lang shang and swoo and uh in a way we can say that even so carles simpson has zero interest in characteristic p and ultimately it is based on an id due to simpson and let me explain but i'm doing the rank two case so uh that's where we are here so we start with inna blah here and uh and now we really do uh geometry so when i say global uh i even mean projective because the notion of stability so my stability is absolutely crucial in what i'm going to explain and without these there is no uh there is no notion in the theory is not done locally here so let me let us start with inna blah which has a the home object is a semi-stable and with vanishing chain classes so uh now uh what may happen is that as the home object it is semi-stable but the underlying coherent objects that means e is not semi-stable so because we are in rank two then if it's not semi-stables it means that uh i have a sub here of rank two and the quotient uh of rank one i apologize and the quotient of rank uh one and uh such that this one is positive and this one is negative so in the sense of mom fault so uh what we can do here is go to the graded associated to this filtration so of course the graded associated to this filtration produces for us an expander but since the situation here uh here is simple if i really took the destabilizing sub here then this heat spandal happens to be semi-stable as well and of course because the chain classes do not change that is an easy task here then it has also c i equals zero now comes the work of delin illusie plus augas voligotsky then we can apply and the work of lang chance four then we can apply the cartier inverse operator to this chic spandal so we find e1 nabla one so uh maybe uh in order to to write uh in a compact way here let me call it v theta and here e1 nabla one and then we can go on but uh now because the main thing here is that we keep this no this property of semi-stability we are moving in fq points of the same moduli so i mean of course we have two modulites we have the hicks moduli on one side we have the draw moduli on another side so this uh uh this is called by uh len short swore the hicks arm flow so using that our field is finite and using the small remark which is written nowhere in the work uh again that's the field is c one then we have a pre periodicity so um so let me uh so that is that is the situation and uh now let us uh see what is so it has pre periodicity so it has a period which i denote for example by f because this is their notation and uh let me assume here in the first case let me so let me replace we have pre periodicity so that means i'm going to replace i'm sorry i'm going to replace e nabla here in the flow by uh en nabla n which is the point where we have the loop where we have periodicity so this one is going to be uh periodic of period f and and then i i i i replace the notation by enabla and then i will make a remark how to go from e nabla n to enabla so let us look at this object here which i just denoted like this so we have two options here either the peak of a church here is non-zero or it is zero and and of course you know if at if you go back to complex geometry and if we want to show that our vanishing condition on the on the symmetric eventual forms implies multiplicity is implies finiteness is then in particular the underlying connection is going to be unitary which is to say via the fundamental work of simpson that the associated x bundle is going to have zero x field so of course the concerns here is going to show even though uh the theory is uh the x the drum the x the um x arm correspondence works differently intuitively the first thing we want to show is that the underlying x field is zero so um let us assume that it's not the case and that the peak of a church is not uh is not zero so then i simplify the situation here by assuming that the period is just one and then we end up if we entangle here's the definition we angle with the cross of the following shape we have here e is an extension of m by m by l i apologize as before where l is positive and m is negative and at the same time the peak of a church which is need potent is going to um uh to uh tell us that uh we have here if the period is one i apologize here is l so we have a cross like this and at the same time uh at the same time okay let us continue so um then uh uh we have two options here so how did i prepare the yeah we have two options here the uh so remember that we have our kodara spencer class it goes from l to omega one tensor m so i'm going to analyze this kodara spencer class and the method we use and i think this is really in the spirit of what should be done is to show that the vanishing assumption is going to force so the vanishing assumption should force the kodara spencer class to be zero so uh that is the goal here and we have this diagram here so um first this is if so we have two options whenever we have a cross like this so we have two options so maybe i write bullet points here the first option here is that uh this cross here yields a factorization like this but if we have a factorization like this if you look and you use the fact that the quotient m and the quotient else are torsion free then in fact you also have a factorization if you read the cross in the other sense you have a factorization in the other sense so this implies that maybe it is stupid because i i i some of got confused about the setup you we work now with in the theorem about convergent isocrystals or no no no we we are looking at crystals in characteristic p here so we just have connections only in characteristic p but then when you use for various pullback in some previous argument there is this is not no no i'm sorry if i i'm sorry if it wasn't clear so let me repeat the rank one case so here as i said the isocrystal over capital k in characteristic zero was done here but the assumption we were forced to have an assumption of lifting of the variety to characteristic zero and then okay then i say now we start a new chapter which is crystals over little k and in addition to assume okay and then i gave a model of what i want to perform here in rank one and this is where the Frobenius pullback was was was showing up because i was just taking the Frobenius pullback of the connection this is allowed so i take Frobenius pullback of the connection it's a new connection with of course the peak of it just going to be zero if i take the Frobenius pullback but this is allowed and then i was arguing exclusive no okay then i understand i understand can i continue okay thank you so i say that whenever we have a cross like this then we have two options i say the l factors through f of a star of m and this is the option here one and the other option is going to say that the composite map from l to e to Frobenius pullback of l is going to be non-zero and because this is a rank one is going to be injective so those are the two options for the first option i can observe that the moment i assume that i have such a factorizations and in fact if i if i turn the cross i have a the factorization in the other sense because the co kernels here are locally free are torsion free torsion free is enough as a as an assumption and consequently i obtain that l is isomorphic to f of a star of m but m is isomorphic to f of a star of l so at the end we obtain that l is isomorphic to f square of a star of l and similarly for m so i can again take this length torsion which in fact is in the rank one case here is a kumar cover of the grid p square minus one so let us write it which trivializes l and m so let's say y to x finite detail it's even more beautiful than finite detail but that's the only information we will retain here from the situation which trivializes l and m and now by by the previous argument on which i discuss then in fact the connections here so h o per star and h o per star of f l are going to be trivial connections but and but they are sub connections so that means my diagram is split here and that kills the coder spencer class but i was assuming that psi is zero is non-zero i apologize so my coder spencer class is not zero so this is a contradiction so now the second case so that handles this case the second case is this one but if i have so now i go up in the diagram here so we are looking at this case here and then i say the second case is that this diagram yields an injection like this so it's going to yield an injection like this but that is telling us so this case here i apologize or what here it's going to tell us that the shift l tensor m inverse in fact to support p minus one and in fact the power p minus one of each single shift so either l or either m inverse is going to have a section but the coder spencer class being non-zero this gives a non trivial section of the p minus first symmetric differential form and again this is a contradiction so this is wonderful so that means we perform the first part of our program this implies that psi is zero and if psi is zero then i'm sorry psi is zero then i as per as per you are i didn't say yeah okay if psi is zero then we obtain so now you look at again i was making a simplified assumptions here then we obtain that in fact for benus pullback of enabla is enabla and again we argue as we did before we look at the underlying coherent information of this equation this yields now really a length tensor i mean a non commutative length tensor under the group gl2 of ffp here because we assume period one here so we have x over y which is what time is it it's okay which is finite at all and it trivializes oh the underlying vector bundle and now we argue again as before that because we do not have differential forms higher differential forms in fact one differential forms on the y this implies here that the pullback of enabla is trivial so and then yeah so that that gives you that maybe i'm a little bit ahead of time but this was a standard time that gives you the style of the proof here so in the higher rank so of course this is the case here there are two things i don't want to discuss the first one is how to go from because this way we kill we kill one object in the hickster arm flow and then we have to go to the other objects but i don't want to enter in the hickster arm flow so precisely so this is fine and the second thing is this is more serious is what can we say in higher range so this program here could be done at least f equals one here a case in rank three yields to interesting considerations of possible maps between coherent shifts and not any coherent shifts which are which are torsion free this is very important and not any coherent shifts and any maps but is the is the option here in fact i i asked at some point lasas failed coming from a completely different side of algebraic geometry how to produce sections of coherent shifts and it seems to be a kind of a new view on how to produce sections so how to produce sections of higher rank shifts via connections so it's a big vague here as a general phrase but okay so that's the style of the proof and i thank you for your attention i'm done thank you any questions so i'd like to come back to this restrictive hypothesis dimension smaller than p or perhaps dimension correspondence well actually what could be said without that so now we have this momentous result of trimsfeld says that assuming only lifting to w2 then you get z-mod peak rating on the derangue complex yes the derangue category and even more than that an action of the kernel of Frobenius with the bit scheme of the on the cp but just just did that so it means for example that truncations of length p-1 from 0 to p-1 from p to p plus p-1 etc are become are decomposed so of course okay the old thing but but i don't think anyone has investigated what can be obtained using this much stronger structure on derangue complex yeah some kind of a monster stronger because you have this grading and also you have some important operator on each part of the grading so grading is such that each i corresponds to with i so this is a wonderful thing and then you have also some kind of an important operator on each of the of the parts so i don't know how it fits with the now you have a pair of an x and l e and you have the connection your connection and so on so i don't know but probably something should be expected yeah you're right and in fact i mean in in a in a primitive version before i knew the i mean before dream i listened to to lecture by a vadim vologotsky recently where he was quoting our work with michael our previous work i mean the one maybe i can i should go back here uh don't panic uh um yeah so this uh no this is a general serum here that uh if enabla is rigid i'm sorry you don't read the the color here you cannot see the color but i just moved the pen here so if enabla is uh is rigid then then if we look at the w point of the base here when when we come from from complex geometry then it's going to be from a size of crystal so then uh vadim imi's lecture in moscow was pretending that our proof here was in the spear i mean there's one part uh which is the same as what you are you are saying from dreamfield he was trying to explain um a little bit the idea of dreamfield so that i cannot say because i haven't thought what i can say is that uh so far here we were quite glad that uh it works here but we can make it work only in very low rank here i mean uh the only we cannot go anything beyond three here so far uh and rank three is not completely written i mean period one is written but higher period there's some work to be done so uh i don't know uh yeah yeah as long as i don't understand here more this is a topic of future future investigations but uh see um i mean vadim was pretending something like this i mean there were pieces i couldn't recognize but not as the whole uh the whole part of the lecture content concerning this year other questions don't let's thank our speaker again yeah thank you very much