 喔  bees tsí qóntsí dí orter thrust  благодар bōb jīsī to arri m'ţ bích b'ehez క౬ఠరప్ మాటెటిల్నా ఠండ్ ఓర్ళ్స్ రోం దిరదిన్ ... చ్ఆ next week so I first I have to maybe it's a good maybe it's useful to recall what is irregular autonomic demodule to give a definition okay so ex is a complex manifold and you have ex mod current demodule autonomic demodule on inside you have the category of regular autonomic demodule so I give a definition so called lambda the characteristic variety of m in t star x so a closed so we assume that m is autonomic okay so lambda is a closed complex analytic conic for the sister action sister conic and apriori if we are here it's coisotropic so lot of people say it's coisotropic by gabber serum but this serum is due to satau kawakashiva on later gabber gave a purely algebraic proof but the original proof is to satau kawakashiva so if we are here then it's lagrangian that's the definition okay so a module is autonomic if the characteristic variety is lagrangian so the question is what means regular autonomic so denoted by i lambda the ideal of the grad grow dx okay of functions vanishing f restricted to lambda equal zero defining ideal of the lagrangian manifold land then so definition is now m is regular autonomic if locally an x there exists a good filtration such that i lambda apply to the graded of m is zero or if you prefer if locally there exists a good filtration which is reduced okay so that's the first definition so it says serum that i i don't going i don't prove it here which is due to kashiwara i guess maybe the other people work on this that mod regular autonomic of dx is a sick sick abelian subcategory unstable by a lot of things by duality proper direct image duality for the module okay i don't give all definition now all result now this of course when we take proper map we need good filtration but okay we will come back later when we shall need it and if i have time i will construct not a functorial filtration but it's another okay so i don't insist on this because it's a it will be another course and it's considered as classical more or less maybe i should mention also here kawai because there is a paper kashiwara kawai which is called regular autonomic three anyway so we shall use these things so i recall a result that i mentioned last week about the properties i know first i i give you excuse me i give you definition again so you take d a normal crossing divisor so we choose coordinates x1 xn local coordinate and x such that d is given by x1 xr equal zero and we say that m has regular normal form along d if it is isomorphic or if it's a local coordinate system m is the z power lambda so let's call with lambda lambda equal lambda 1 lambda r in c minus no positive integer so what does it so is that clear j is x the coordinates are x becomes z so it means xi dz i minus lambda i for i equal 1 to r on dz j for g equal r plus 1 n okay you take this ideal on m is d over i lambda okay so regular normal form is very explicit and very easy to calculate so the game is to reduce regular autonomic demodule to regular normal form so there is a i don't know a lemma or a crm as you want consider a property pxm so it's a statement on x for all m regular autonomic so we assume that px is local maybe i i wrote it last week it means that maybe i write it pxm is true if and only if for any x for any covering then assume that it's a invariant by shift it's invariant by shift it's stable by triangle distinguish triangle if you have a distinguish triangle if you have pxm and pxm prime then you have pxm double prime so it's in the derive category excuse me you are right thank you in the derive category so this is some statement no this is you understand what it means of course because this one is six so it means the full subcategory so there is a condition which is useless here maybe i should not write it but it will be useful in the irregular case so if you have pxm plus m prime then it implies pxm in the regular case i think we don't use that but it's so natural and so now comes the two main properties in practice these property are absolutely obvious to check maybe i write here i want to keep this blackboard remanilbert for example i will apply this to prove remanilbert so i've not finished if f is a projective map then if the property pxm is true then pxpy of the direct image of m is true and here we assume that m also is good it is automatic but when you don't use it it is automatic it's known that regular autonomic module are good but we don't use it so we assume that the good it means you have a good filtration but not everywhere on each compact set each upon relatively compact set it's enough not globally semi globally on the last one the best if m has regular normal form then pxm holds so if you skip the first property which are more or less obvious the important property is stable by proper direct image and you have to check it in the normal form case so i give a a glance a sketch of proof so dilemma the conclusion is that it is all the way down yeah i forgot the conclusion yeah i forgot so what is the idea of the proof first assume that d is a normal crossing divisor and m is a regular autonomic and x and also which satisfy two things m is equal to its localization along d and there is no singularity apart from d or if you prefer m is a flat connection on x minus d then p of x m yeah so in this case in this case how can i prove the result okay i assume all conditions are satisfied so locally is there exist a filtration not a good filtration a filtration by d module m equal m0 contain m1 contain mj contain mj plus 1 equal 0 such that mj divided by mj plus 1 as regular normal form honestly i guess you are told me it's very classical but i believe him i've never seen the proof but for specialist of this question it's very classical so you reduce by this result the result is proved in this case so how to pass from this case to the general one using desingularization so i don't give the proof i just give an idea of the proof and i cheat okay so more or less you make an induction on the dimension of the support of m and you will construct a morphism so support of m containing z so you will make a desingularization such that w is smooth d is a normal crossing division and then what you do so assume so it's where i cheat because otherwise it's a little more complicated assume for simplicity that z equal x and call this f then you will use this some triangle like that you take the direct image the inverse image of m localized it's not the same d it's horrible on n so it's just an idea of the proof it's not a proof but z you choose it no? z is the support of m so i assume the support exactly it's contained contained it doesn't matter that you can choose x yeah but i do it by induction okay so support okay so for short i assume z equal x so you do something like that and here this one is case a when you take the direct image you apply one of the properties property e okay so you take the so this module will satisfy the property okay for this one and for this one the support of n is contained in s so by induction it will satisfy also the property and the property is stable by triangle your notation for the operations in d modules it's not mine i hate this notation no no because what i remember is that there is well i so that the usual with the each operation is two versions i mean there is like like for shifts a plus a plus a yeah but not for the module it's just a shift it doesn't change for the module there is only one inverse image up to a shift there are not two inverse image but for direct image you have and for direct we take direct image when it's proper so you have two notation but in practice they are the same we never take direct image when it's not proper okay so it's actually no for the module there is one direct image essentially and one inverse image okay so it's the idea of the proof it's not the proof okay so now i want to apply this big lemma to remanilbert ah no before excuse me before there is something rather boring to do which is to blow up real blow up so what i doing now will be used by masaki intensively and i think it's here again it's very classical it's like polar coordinates but with more variables okay so maybe some notation so we take this action positive number cross c star cross 1 to c star cross 1 so a z t gives a z a minus 1 t a is real positive okay so we denote by c see what tilde tot this is c times cross r divided by this action and we also need the most important one this one c times r positive divided by this action and also c tilde positive equal c times r positive divided by this action okay so the important one is this one okay and so now if x is c r cross c n minus r and you have d the divisor z1 that r equal 0 then you define x tilde tot as c tilde tot power r cross c n minus r x tilde and x tilde positive similar I'm obliged to erase this blackboard and we have a map pi which goes from c tilde tot to c to z t gives tz so this map we find a map we keep the same notation from x tilde to x on here you have x tilde positive which is isomorphic to x minus d here is the situation so if you take n equal 1 you will see that it's exactly polar coordinates this is x x tilde ok so x tilde tot is not intrinsically defined but if you have a normal crossing divisor x tilde is intrinsically defined and this one is only defined as a germ in the neighborhood of this manifold with boundary and now we put we put we put sheaves here maybe I skip some details essentially I will define o t x tilde as pi upper shriek o where is that x t so you take temporary homomorphic function with poles on d on this is inverse image and it will be it can be shown so of course this map pi tilde is proper ok we use it always of course we can prove it's not ok on a x tilde is alpha t x tilde so this is a usual it's a usual of homomorphic function with temperate growth the boundary of the manifold with boundary on d what is the notation a x tilde is a x tilde tensor with pi minus 1 o x pi minus 1 dx so we will do d module on this manifold with boundary x tilde so why is it so useful maybe I give the CRM without proof so it's an exercise it's more or less of use proposition or lemma if l has regular normal form along d so l is a dx module then when I maybe I should have defined it if l is a dx module I set la by definition it's where is that it's dx a tilde tensor over pi minus 1 dx pi minus 1 l so when you have a d module here you associate a dx tilde module on this manifold with boundary on the result if l has regular normal form on d then local a on x tilde la is isomorphic to oa that is a so once you have make this desingularization not desingularization this blow up d module with regular normal forms become flat connection ok why is that true it's more or less abuse it is the reason is that z power lambda is locally invertible in the x tilde a in x or in a you see z power lambda if you work locally it's invertible function ok so maybe I skip some details to state the main result then I will come back to the details later but here really you don't need all this power it's too big for this so just to realize this you know what it is to be the ring you are on this complex one it allows you to put it you can adjust this one is not too big we need it you need it later but for this it's simply that z power lambda which is not invertible outside of zero is invertible on the blow up maybe I state the main CRM all these things needless to say are not due to me but to catch you are it's old result maybe it's a new formulation with OT but essentially it was due to catch you are at least the two first one so you take L which is regular autonomic then the first result is the temperate the RAM I don't remember my notation is isomorphic to the usual the RAM on the second one is the same for solution so these two statements are equivalent of course by duality because it's a statement for all regular autonomic demodule and you pass from the RAM to solution by duality so it's the same statement formulated differently on the other one so this one I did it with Kashiwara and it's stronger than Riemann-Hilbert as we shall see if you take OT derive as a demodule with L it's isomorphic to RM Sol of L with value in OT now this is the transfer product as a demodule of course everything is the arm it's not the transfer product over D it's the transfer product over O but with a structure of a demodule of course it's the arm so before to go further I want to show you that this is a deep result so why is it Riemann-Hilbert first? sorry but with this procedure do you have something like the function muhom for demodules or not? muhom for demodule you mean it's kind of macrolocalization? no no there is no no macrolocalization here where have you seen muhom? taking the blow up you cannot localize no I don't know what you mean but of course Kashiwara when he is in his study of autonomic demodule with Kawai made a macrolocal study using macro differential operators but here it does not appear to have a real allergic amines and see the functions of the pyromorphic take the blow up and produce what do you mean? okay so let me give some application of this result in particular to none of the morphic functions let me give some application of this result to show you that they are deep very strong results so the first application see you apply the function alpha x to the isomorphism 3 when you apply alpha it commutes to tensor product and alpha of ot is o so alpha of oxt tensor l is l so you find that l l is rm sol l oxt so this is Riemann Hilbert by Kashiwara in 84 or 80 even even in 80 okay Kashiwara this thing was denoted tm by Kashiwara this time so it means that you recover l from the knowledge of solution so we did not have oxt at that time exactly no you need oxt let me take an example assume that sol l assume that ml is hd z ox okay then sol l will be the constant shift up to some shift okay so rm sol l ox is hd z ox and if you put ot then you find hd algebraic comology that's why you need ot okay so we de-formulated originally originally Kashiwara defined a function that he called tm which was defined on the category of constable shifts and you prove it with this function tm okay but tm is a particular case of rm with value in ot okay because tm f ox but of course all the id's are in tm okay so let's let us give another application so as a particular case of this formula 3 we recover Riemann-Hilbert okay this is the Riemann-Hilbert correspondence so let's give another application of 2 r constable shifts and I apply rm f to this isomorphism 2 so I find that rm over dl rm f ot is isomorphic to rm over dm rm f ox so why is it a excuse me m on l why so on the hypothesis that l is regular autonomic so let's take a particular case take f take m real analytic manifold x a complexification of m let's take for f for f the dual over x of the constant shift on m then the left hand side is rm over d of l in the shift of distribution on its isomorphic to rm l in the shift of satos hyper function so you see this is a very strong result because so this result here contains all these regularity result that you can guess it proves that if m is autonomic the complex of distribution solution is the same as the complex of hyper function solution with the same techniques we can prove the same duality in real analytic also another example take f the constant shift on z then you find rm l algebraic homology supported by z there is also a comparison with solutions in the formal series the completion does it follow from this kind of thing yeah sure not directly but you can deduce you can deduce by duality for example that's the shortest all these comparison result you can also do like completion along the sub-variety which is at the point but along it is also it is like the old result of the homology now you can prove something like that rm over d l beta of o x isomorphic to rm over d l o Whitney so when you apply this to then you find things like that rm over d l o x restricted to z isomorphic to rm d m l o x formal completion that was your question ok so maybe I will enter the proof of the result at least the first two isomorphisms so ok so you see with this language of and sheaves ot or ow it contains lot of results this isomorphism is very strong contains lot of things so let me give a sketch of proof maybe I want to write down the rm a little bit because I will keep it ok all the results ok ok so so so so so we have seen that the two statements statements one and two are equivalent ok by duality ok so we just prove one so we have to check all properties that I erased of the lemma so the first one are of use there are only two properties which are not of use direct image and regular normal form ok so what about direct image I have to prove that if pxm is true then px direct image is true assuming f x to y is a projective and m is a good but if you remember the deram of the direct image is the direct image under this hypothesis of the of the deram of L we have seen that it was a consequence of temperate grower C1 at this level and the same without temperate is true this is not so easy again it's used grower but not temperate maybe not it's not easy but it's old ok so you mean the usual grower for coherent shift not for demodule but it's easy to pass from coherent shift to demodule yeah but we have a good filtration it's not written now but it was written in the statement of the lem we assume it's written good it's written we assume m is good ok so we see that the property is stable by direct image so finally we have to check the property for normal form ok in the case of regular normal form oh I cannot you won't really it says that if you have f on x to y if f is coherent on x f is proper on support of f then the direct image of f x t is now we have to do again the proof it does not unfortunately ok so ok so what what I have to do now ok I resist I have to prove I want to prove so now we assume that l has regular normal form along d and we want to prove that l that omega t x tensor dx l so I recall we have x tilde to x pi x tilde positive x minus d and this is an isomorphism ok and there is some so there is some technical part that I so this is the the RAM comparate of l so I skip a technical point which is not difficult but it's too many things I skip that r pi star pi upper streak of the RAM temperate of l is isomorphic to the RAM temperate of l ok it's some technical lemma so what do I want to prove so I want to prove that this thing so the question is this thing is isomorphic to alpha of the same thing is a shift I want to prove that maybe here this is alpha of the RAM temperate of l so I want to prove that this object is isomorphic to alpha I never write yotta of course it has no meaning to say it's alpha it's because I don't write yotta ok but alpha commutes with direct property with this function ok so it's enough to prove that the inverse image of the RAM temperate of l is usual shift is a usual shift it's not a n shift it's equal to alpha but this we can calculate so the inverse image of the RAM temperate of l is isomorphic to maybe of course it's difficult to follow like that there are a lot of technical things hidden so roughly speaking you will find omega t x tilde tensor over d a l a and now we use the fact that l has regular normal form so this is isomorphic to o a so finally this is isomorphic as the inverse image of omega x t tensor over dx o x locally and this is so it's a shift so it's what I've done I said that after the blow up my demodule is nothing but the flat connection and then it's put ok so maybe stop and I will prove the third part after and I will give an application an example mainly what I will do after is to treat an example of irregular demodule to see what happens ok so we have a short break so now it's not it will not be in the notes what I will say now but it's one of my ideas it's very long it's how to undo a demodule autonomic demodule with good filtration so I take this last formula you have l and I want to undo l with a good filtration or with a filtration let's say but functorial so an idea is to undo oxt with a filtration so what does it mean to have a filtration and this strange object of temperate autonomic function with temperate growth so oxt is the dolbo complex of temperate distribution on distribution have a very classical filtration by some left spaces ok so I don't know so now I take a real manifold so is it possible so the problem is that on the subalytic site you have the pre-shift of subalephe spaces s is negative don't ask me the definition of subalephe space but we know it exists so we have this but this pre-shift is not a shift so it's difficult to undo the shift of distribution on the subalytic site with a filtration because they are not shifts usually subalephe space is done on a manifold on a manifold that you use but when you have things with subalephe boundary what do you say exactly about the boundary ok so let me explain what exists so the subalephe site is not is not enough on with Guillermo we construct the linear subalephe site so what is a linear subalephe site the open set are the same but the covering are different u1, u2 is a linear we call it linear covering of u1 union u2 if there exist a constant c such as the distance of x to m minus u1 union u2 is less or equal c distance x m minus u1 plus distance x m minus u2 that's why linear because with Lajasevich inequality there was some constant n here but we assume n equal 1 ok so there are very few coverings but with this covering you can define the subalephe shifts subalephe site is it let us say u is relative to compact in in rl and subanalytic so do you take the one for rl we support in u in some sense or the closure of certain ok as I said at the beginning and ask me what is the exact definition of subalephe I just tell you what exist so there exist a shift which is constructed by Lobo which is a shift on the linear subanalytic site on ok on which let me one second to finish my sentence and you will it will become clear ok and you have direct image let's call it raw again so I work here unbounded unbounded from below and so this is it's partly my work with Guillermo and partly the work of Lobo at this stage it's my work with Guillermo so by the brown corm there exist a left adjunct to the direct image ok so you can define the subalephe shifts on the subanalytics site as the raw upper shriek of the shift constructed by Lobo yeah so they are object of direct category ok but the good the good new the theorem if you is subanalytic with lip sheets boundary then our gamma u of the subalephe shifts is in degree 0 and is equal to the usual subalephe space I change my notation what do you say no that I have not defined ok as negative real negative ok anyway there is something in mathematics which is called the usual subalephe spaces ok which is horrible except but when you it's not horrible but what we prove is if you is lip sheets then it's a good definition if you is not lip sheets it's not the good definition if you is lip sheets boundary by a lip sheets invertible function it's a half space by a lip sheets change of coordinates it's a half space so it shows that the subalephe spaces are naturally object of direct category they are not spaces they are ok ok and once you have subalephe this thing is to be subalephe or not yeah yeah yeah absolutely we have written the theory for subalephe maybe you can generalize no but the question is whether if there is a by lip sheets or we are more present whether the subalephe one or we assume the subalephe I don't remember this point maybe you ask the graph a five to be subalephe I don't remember anyway this object of direct category is concentrated in degrees zero and it's the classical one so finally you see that you have a subalephe filtration on db so if you look at this maybe I need I somewhere it's an object of demodule of filth so of course it needs much of development but we can define the direct category of filtered demodule it's not a abelian of course but it works on an object of demodule on distribution are well defined dbt I forgot the t so by taking dolbo we see that oxt is well defined in the the graph category of d and dx module and once you have a filtration on ot by this formula you have a filtration in some sense on d so of course you can ask me what can you say about this filtration is it in degrees zero I guess very rarely is it good in some sense but here of course we cannot say anything so it is apparent sorry at this point you take quasi abelian categories no I have not defined this but it's easy to define the graph category we schniders who have written some paper on this subject ah this filtration with the maps you mean what's not in fidx filter filtered and object of demodule no I'm not precise here it's just to show you that maybe it's interesting maybe it's not and so this doesn't give for regular thing it doesn't give the canonical good filtration no I don't know if there is canonical good filtration but this one is functorial maybe but it's in the d half category but I remember vaguely that there was for regular there was a way to yeah but it constructed at ends by you know like this is that distributions you get a canonical distribution distribution yeah but it's another it's another thing no it's another thing and maybe I don't know maybe this filtration works also in the irregular case you replace t by e but it's anyway it's too when we want to prove something we need the result of analysis that we don't know so we have not been very far the result of this thing I remember the L2 lattice L2 lattice no but there are a lot of filtration and demodules each mathematician has constructed this filtration so I give mine okay it's a parenthesis for okay so maybe before to come back to the subject I want to make another parenthesis more interesting maybe I want to discuss an explicit example of irregular demodule okay so until now we have spoken of regular demodule what happens with irregular so it's an example that I think which is illuminating for what Keshawar will do next week we will look at the simplest example so dimension is 1 so it's something I did with Keshawar in 0 3 but I think it contains a lot of information that will be generalized after by Danilo and Keshawar so X is C I know in algebraic geometry A1 C but I call it C for short U so the coordinate is Z U is C minus 0 J is embedding and my demodule is associated to this equation simplest one so M is D over DXP or if you prefer it's D exponential which sign 1 over Z so I want to try to understand this shift first I will try to understand H0 of this that is kernel OT is the complex OT OT and first I will try to calculate the kernel so OT is a concentrate in degree 0 and it's it's a sub and shift of OX OT in degree 0 because H0 of X is one OT is a complicated it's an object ok so let's call maybe I will call Sol 0 I don't know so H0 of Sol T of X is a sub shift of our sub and shift of H H0 Sol X M so when this is not 0 on an open set V yeah maybe I should say first that H0 of Sol the classical shift of solution is a constant shift on U easy calculation this one is not 0 only if V is contained in U V is subanalytic by definition and also exponential 1 over Z this function restricted to V is temperate this is more or less tautological ok temperate solution of this equation the solution of this equation exponential of 1 over Z with a constant so it has to be temperate so on this function restricted to V is temperate if and only if particular 1 over Z is bonded on V otherwise it will not be temperate on this is bonded if and only if V is contained if and only if there exist epsilon positive so that V is contained some U epsilon where U epsilon is a set of Z in C so that I prefer to make a picture U epsilon is C minus B epsilon B epsilon is a closure close ball centered at epsilon on radius epsilon so I make a picture here is U epsilon so with this remarks we find that maybe I skip some detail we find that so T of D exponential 1 over Z do you mean bonded to both or bonded bonded bonded to both real part otherwise no no bonded not a bond you want this to be temperate so the real part should be bonded no but you say bonded means absolute value is bonded no no it's original it's real but it's less no not absolutely so finally I skip some details we find that this is inductive limit in the sub-biotic topology of the constant shift on U epsilon so I recall U epsilon so we have a explicit description of the complex of solution of this irregular D module so of course if we take alpha this is the classical things but it's clear that with temperate we have much more information so what is what is good and what is not good in this result so with this this is much more precise that the classical that this result but it's not so good the bad things bad news is that if you take D exponential 1 over Z on D exponential 2 over Z they have same temperate solution so temperate and homomorphic functions are not enough or C maybe are not precise enough to distinguish this kind of D module ok but good if you take D 1 over Z D exponential 1 over Z square for example then the solution are different so it's a tool to distinguish a lot of D modules but unfortunately not all D modules ok so I have made two parenthesis maybe too much on paraft I can come back to my proof ok so ok so it's the end of the parenthesis I think this example with cashier you will see the more general example exponential D exponential phi where phi is meromorphic and he will calculate the solution chief of this ok it's more difficult but it's very similar to this calculation ok so maybe I come back ah no ok no there is something else I want to do I think I will not give the proof of the set I think you are tied me too so I will give an application another application of this result ok the proof is the same line you show both terms are stable by direct image a new test in the regular case in the normal case regular normal case no I want to here it's I almost forgot there is something important that I want to do I want to apply this result to integral transform so in the regular case so what integral transform means a correspondence ok in the complex analytic case so you have a manifold and morphism of manifolds so they are all complex manifolds and I want to compare I want to see what happens to the Durham complex of a an x when I go to y by inverse and direct image I introduce so m will be a D module on x l a D module on s then I define the composition as a D module so you take the inverse image the direct image if I have a so if l is a shift or n shift on f is a n shift on x maybe I take g on y I define a psi psi l of g as no f excuse me as a direct image of r i on l inverse image of f or maybe I define also l compose with g ok so there is a theorem which now it's just a corollary of all what I have said so theorem is the following you take m a good not good, quasi good is enough D module you take a regular autonomic kernel you take as a shift roman l as a solution of l or inductive limit filtrant inductive limit of good the quasi coherent that is the homology shape ok the homology shape it's a sixth sub category ok on the hypothesis we assume that g the projection is proper on f minus 1 support of m intersect support of n some properness then the conclusion if you take the Durham complex temperate Durham complex of m when you make its image by psi then it's isomorphic to the temperate Durham complex of the image of m by l no l is regular autonomic the kernel so unfortunately it does not apply to the Laplace transform which is not a regular autonomic kernel but it applies to many situations so first I say maybe I skip the detail that all ingredients of the proof have been given already because we have to prove that essentially the Durham temperate also a remark this result is false without temperate it can be make true with other hypothesis but with this hypothesis it's false if you remove temperate and what is we have seen what does it says essentially the proof it says that Durham temperate commutes with first inverted image then commutes with a tensor product with l demodule tensor product over d so this was not written down it is in the notes but this last result is almost a consequence of this one I don't give the details but it follows easily from this formula so this formula 3 is a generalized Riemann Hilbert it's much stronger than Riemann Hilbert and it commutes with direct image and this we have seen many times temperate grower things like that so once you know that Durham commutes with all these things then you have this formula so maybe I want to translate it maybe excuse me maybe I write again everything here cl of m is isomorphic to Durham maybe I forgot a shift that's yeah I forgot to put a shift Durham of m composed with l and there is a shift here dx minus ds that I forgot before to erase I correct dx minus ds ok now it's true I can erase ok so ok and we assume that g is proper on f minus 1 support of m intersect support l db ok so to conclude let's translate this so I have erased it but if you remember psi on phi are adjunct when you have quasi good so it's a filtered limit of good the support is it the support the support of a shift is the support of the shift no you take the closure of the I don't know ok so if I I translate this so now I take g another unchief then as a particular case of this formula I find I see x l composed with g with value in omega x t tensor dx m I write everything dx minus ds is isomorphic to our i cy g omega y t tensor dy of m composed with m ok so why are such formulas interesting because in the literature you meet very often particular case of results like that ok here is the formula with temperate there exists a formula without temperate with other hypothesis but let's just give an example for example if you take projective duality the incidence relation then you find formula which says that r m l composed with g with o p n c some line bundle maybe with some shift is isomorphic to r m g o p star n c with a these are line bundles ok there is a shift that I don't remember so you have this adjunction formula where g is any shift and you pass from one shift on the projective space g is a shift on the dual space so you can particularize take open set also and it contains a lot of information what is the k and k star here ok is a integer it's a line bundle and I don't remember the formula k star is deduced from k by some minus k plus n or something like that so I want these formulas are classical maybe they are in Bryninski already but I just to emphasize the interest of these general formulas adjunction formula and I think this kind of formula with r m here are there are many formulas in the literature like that and I wrote some with Dan Yolo long ago on Kashiwara tenizaki but I think it's the first time that we write them like that without taking applying to reconstruction so I think it's a nice formula and it's still better in the regular case because then you can apply to Laplace transform but it's much more difficult and it's what Kashiwara will do next week so maybe maybe it's ten minutes before my time but I think I finish now so what is the L here the L is given by by the incidence relation you have a z containing p n cross the incidence relation on L you have some choice h1 z there are many choice or you can take the constant shift or something like that that's the radon transform in this case of course you can take other flag correspondence for example the penrose penrose construction of the wave equation is something like that which you take p3 m4 you start with a line bundle and here you find the wave equation m4 is a Minkowski for space compactified compactified space it's also it's another subject I don't want to enter it m4 is a flag manifold f12 of c4 ok so this will not be in the notes no and the incidence relation will be in the notes no this is this is the incidence relation excuse me so this is f134 and this is f12 this is classical but it's another subject ok thank you