 Okay. So let's start. Okay. So, yeah, yesterday I finished the lecture with, I mean, explaining that I saw variations of our structure as a special case of biojewel-like structure. And then I gave a general heuristic about functional transcendent statement in the context of biojewel-like structures. And then if the biojewel-like structures are different of a cubar, some atypical intersection statement in terms of cubar by algebraic points being sparse. Okay. And then I explained that there was a problem for variations of abstractions because we don't know how to characterize the bi-cubar algebraic points. So I want in some sense first to finish the lecture of yesterday by introducing a general set of conjectures in the context of variations of abstractions that generalize zilberping conjectures. And then after that, the plan is to come back to act genuine for that variations of abstractions that I stated yesterday. So the result of Baker and Zimmerman. So I will try to give a sketch of the proof. And then the last part will be about, on the contrary, a typical intersection statement, namely the theorem 4 that I stated in the first lecture. Okay. So a typical intersection conjectures for ZVHS. So this is a set of conjectures that I developed, I wrote like three or four years ago. No, I'm not repeating, Ravi, so don't smile. Okay. So you start as usual. So you have your polarizable ZVHS on S, which is a smooth quasi-projective. So you get an associated period map where G is a generic moment for take group for V. Then what you can do is yet you will define a numerical invariant associated to this setting that I call the hodger dimension for S and this variation. So what is it? Well, you take the dimension of the horizontal tangent bundle to S gamma gm. So we know that the period map has to be horizontal. So it maps the tangent space here to the horizontal tangent space there minus the dimension of the image. So more than speaking, this is the co-dimension of the image inside S gamma gm except that you take into account only what is relevant here for Hodge theory, namely the horizontal space. And then I would say that if I have a sub-vy in S irredistible, I would say that it is atypical for SV if the hodger dimension of Y and then you look at the restriction of V to Y. So of course this means that you have to reduce yourself to the smooth locus. But okay, let's just do it this way. It is strictly smaller than the hodger dimension of SVS. So this means that phi S phi of Y, so maybe I should call this one phi S, means that phi S of Y as excess intersection with the collection of special sobriety here with what I call the hodge locus of S gamma gm. Once the surface transversality is taken into account. Exactly. So I will call S atypical, so this is still part of the definition. The union of all atypical sobriety, so this is a subset of S n which a priori has no nice structure, countable union of algebraic stuff. And then I will say I will refine this definition saying that Y is optimal for SV. If not only you have this inequality, but you have this same inequality replacing S by any Y prime containing strictly S. So if the hodger dimension of Y v restricted to Y is smaller than strictly than the hodger dimension of Y prime v restricted to Y prime for all Y strictly contained in Y prime, so Y and Y prime are reducible, strictly contained, well containing S. So optimal in particular is atypical, but this is a stronger condition. So it's easy to check that if you take the atypical sobriety which are maximum, then they are optimal. So now what is the main conjecture? Conjecture, well there will be four statements and it's an easy exercise to check that in fact they are equivalent. Is that this collection S atypical is not Zarski's dance in S. C2 is that, sorry. C2 is that S atypical is in fact algebraic and strict, so R is where the notation is bad. So this is strict inside S and this is algebraic, closed algebraic subspace. So the finite union, so a priori you have this infinite union of things, but the claim is that every atypical is contained in a maximal one and there are only finite in many maximal ones. C3 is that, which is as I said, is that S atypical of V is a finite union of strict special sobriety which of course are atypical and in fact they're optimal, so C because they're maximum. So is that S contains only finite in many optimal sobriety. So these are just different ways of thinking of the same kind of problem either in terms of algebraic statement, so this guy has to be algebraic or it might be easier to prove by some counting that you have only finitely many, but what I claim is that who easy to check is that C1 is equivalent to C2 to C3 and so on. So let me maybe explain on special cases what those conjectures tell you. The subpar is, well, for S and V, I define this, I mean, I've been defining since the beginning, so these are irreducible components of the jokers and these will be maximal ones. So maybe you are a bit surprised if you think of the case of a Schimoir variety, but let me explain. So if S is a Schimoir variety, let me just write it, connected Schimoir variety, and you take for V the standard variation of structure on a Schimoir variety given by a representation of your Schimoir datum, then it's easy to check that this atypical locus is empty. Second case, suppose that S is a closed irreducible sub-variate of some Schimoir variety and now you look at the restriction of V to S. So Y inside S is atypical. This is the same thing as saying that the co-dimension in S of Y is strictly smaller than the co-dimension of the special closure of S in the special closure of Y in the special closure of S. So this YSP is the smallest special sub-variate of your Schimoir variety containing Y, and S special is the smallest Schimoir sub-variate containing S. So these are some examples where you can make those conditions kind of completely explicit, more or less. Then a particular case of this conjecture is a generalized under-out conjecture. So the lemma is the following. You try to understand when is a CM point atypical, and you have the general following criterion. Suppose that you have your variation of a structure of S, and suppose that Y is an irreducible sub-variate of S, which is special. So this is a preimage of by the period map of a special sub-variate in S gamma gm and of Schimoir type. So I already introduced this terminology. This means that this is the preimage of a memphortate quotient, which is in fact a Schimoir sub-variate. So here you have your period map phi from S to S gamma gm. So here you have some horizontality condition, but it might happen that you can embed horizontally, totally geodesically a Schimoir variety inside this thing. And then I will say that Y is special of Schimoir type if it is a preimage by this period map of exactly one such Schimoir sub-variate horizontally embedded inside my S gamma gm. And now with a second assumption, not only it is a Schimoir type, but it is very big in the sense that if you look at the restriction of phi to Y, so now it goes to Y, so you know that at the end you arrive in some S gamma gm, but you know that you factorize through a Schimoir variety embedded here. And I ask this map to be dominant. In other words, Y is really truly exceptional. You are inside S, and essentially you are a Schimoir variety embedded for hodge theoretical reason inside S. Okay, so these are two strong conditions. And then the claim is that such a Y, which is this kind of Schimoir variety, will always be atypical except if S gamma gm itself was already a Schimoir variety. So Y, let me write it this way, is not atypical for Sd if and only if already phi S satisfied exactly the same property gm of Schimoir type phi S dominant. So this is an easy exercise that the condition, those numerical conditions gives you. And the corollary is that a cm point is always atypical because this is a cm point is just a case where Y is a point. Okay, then obviously it will satisfy those conditions. And so it tells you in particular the corollary is that Y, any special point, any cm point, a cm point in Sv is atypical unless phi S is a Schimoir type and dominant. So this result that the cm points are not atypical in Schimoir varieties, in some sense it extends here. In fact, cm points are always atypical except if you are already in the Schimoir gaze. Okay, and so the main conjecture implies the following one. Suppose that the union of special sub varieties of Schimoir type with dominant period map is Zeissky-Densen S, then phi S from S to S gamma gm is of Schimoir type with dominant period map. So you see easily that this conjecture is equivalent to the same conjecture but only for cm point. So you have your variation of a structure. Suppose the set of cm points is dense in S, then in fact the period map is dominant to a Schimoir variety. Okay, and so you see that this conjecture is equivalent to the usual under the odds for Schimoir varieties which now has been proven for Schimoir varieties of a billion type plus the conjecture B, which is a purely group theoretical statement which is same assumption as in A prime and then the conclusion is that the target is a Schimoir variety. Then so I wrote all these as conjectures are very little evidence for this. Okay, well it's just that I think that it clarifies at least what should be looked at. So maybe this should be just some questions. So really this conjecture B is kind of very strong, right? I mean it tells you you have a variation of structures, you suppose that you have a Zeissky-Densen set of cm points, then in fact the target has to be a Schimoir variety. Your generic moment for the group has to be a Schimoir type and the associated period map takes value in a Schimoir variety associated to that group. Okay, so maybe I would give one example where this conjecture recovers a conjecture which is well known to people working in Calabasas. So example, suppose that X is a smooth Calabasas 3-fold, okay? So this means that X as trivial can be called bundle and it's simply connected. Then you look at the homology of X in degrees 3, so I may be busy. Cm points in conjecture A5. Yes. Cm points are well defined. So our just points was a memphortate group is a torus. Okay, yeah sorry, maybe I did not mention that. I mentioned cm points yesterday, but this is a general definition. Just memphortate group is a abelian. So it's automatically a torus in that case. Okay, so suppose you take the third common energy of, so this is a way 3, h structure. So this means that you have vz, well sorry I started with h, one of my notes that's right h. So you have hc that decomposes at h30 plus h21 plus h12 plus h03. And this guy is of dimension 1. And so by h symmetry is also this one. So the only interesting parameter is the dimension of h21. And then, so there is a first remark is that, okay, those things are way 3, but essentially they are understood by two weight one hot structures. Namely you can construct the veil, a hot structure of weight 1 associated to this. So this is a weight 1 hot structure is given by h10. So you have to pair some of those spaces and you choose the right one. So h03 plus h21. And then because the polarization is alternating plus minus plus minus, then you see that this guy is in fact a polarized hot structure. So you can out of it construct an abelian variety, which is the veil Jacobian, which is hc modulo h10 veil and hz. So this is an abelian variety. So although you are in weight 3, nevertheless you can get this information in weight 1. So what is the problem of this veil Jacobian is that they were forgotten as soon they were invented because they do not vary holomorphically in families. So they are abelian varieties, but they are not, they don't have a nice variational interpretation. Okay, so there is a second weight 1 hot structure, which is a usual Griffith Jacobian that you can associate to this. So let's write it h1g, g for Griffith. So now the h10g is the natural one. So it's just h30 plus h21. So you are just, so this is still symmetric. This one was also symmetric. But now this one is not polarized because you are taking two consequent spaces. So not polarized. So you get also a Jacobian, but this is just a torus. So the Griffith Jacobian constructed exactly by the same recipe, replacing h1w by h1g. So this is just a complex torus. Now there is a nice result of Boccia that tells you that understanding the memphotic group of this weight 3 hot structure is essentially understanding the memphotic group of this 2 weight 1. And so in fact what he proves is quite nice. He proves that this weight 3 hot structure is cm. So the memphotic group is a torus, if and only if both h1w and h1g are cm. And moreover there a memphotic torus commute. And the memphotic tori. So you get two memphotic tori that are both embedded inside gl of hq and they have to commute there. Here there are subgroups of this general linear group. The commodities are living in the common, yes? Okay. So now look at the variational story underlying this. So it's well known. I mean you look at the deformation space of x. So you know that this is a smooth by 10th order of this is a smooth quasi-projective variety. So you get the corresponding, now I will call it vz with fiber. So on s and with fiber at x equal to hx. Okay. So you have your variational structure of weight 3. And okay. So what is the assumption in Calabria? The canonical boundary is trivial. So canonical boundary trivial, pi 1 trivial. So simply connected and smooth here. Okay. So there are examples where s contains infinitely many cm points. So people have been able to construct such examples. So the conjecture a prime says in that case that the irreducible subvarities of s containing zayskiden set of cm points are of Shimura type with dominant period map. In other words, in those modular space of Calabria you should be able to embed Shimura varieties with dominant period map. And in all known examples this is true and basically you embed bulk oceans. Okay. And this conjecture was done a long time ago by Gukov and Vafa for different reasons that I don't understand having to do with conformal field theory. But at least this is an indication that maybe this conjecture, this very general conjecture is not complete bullshit. But notice that my conjecture predicts more. It predicts also that there are only finitely many such things. Such Shimura varieties, maximal embedding there. Okay. So this is all what I wanted to say about this. Okay. So let's go back in some sense to yesterday and let's try to indicate some ideas in the proof of action URL. So of course I will not give all the details. And I have to say that in preparing the lecture I have to complain that the paper is very sketchy at many places but okay. So I think everything works but okay. So yesterday I give the general statement of the action URL heuristic for bilge bike structures. So now let me repeat the special case for a variation of our structures. So we are as above we have V over S. So we get our period map S gamma gm. Well in fact here I should introduce some modification but it's not very important. At some point it would be important to prove is that I do not really look at the period map associated to the mom forte group but I can replace the mom forte group by the algebraic monogrammy group. So I really look at the factorization through the smallest weekly special sobriety of my period domain containing the image. It will come at some point it's just that I want the monogrammy to be dense, Zeissky dense in G. Or you can assume you can start with a family where you are Zeissky dense in the mom forte group but anyway at some point there will be a reduction where you need this. Okay so what is the picture? So let me draw yeah. So you have d over d mod gamma let me call pi this projection. Then you have S phi and then you can consider in the analytic category you can take the fiber product of this. So you have a Cartesian square in the analytic category. So I will call w. So please remember the letter because it would be important. So this is w and I look at it as being inside s cross d and this guy I would think of it as being inside s cross d hat which is an algebraic variety. Okay it's just a product of my algebraic variety by this flag variety. Okay so what is the theorem? So this is actually a new one so this is due to Baker and Tumaman and this is the actual new one for variation of abstraction. So this is here. It tells you that if you take v inside s cross d hat so you start here an algebraic sub variety iridescible. Okay then you can consider an iridescible analytic component of v intersected with w. So w is what yesterday I denoted more or less by a diagonal. So if co-dimension of v intersected with w is strictly smaller so I'm making intersection. So I look at the intersection of v with w. So if it is a generic then I know what is the co-dimension of this section. This should be the sum of the co-dimension. So suppose I'm not generic I have an excess intersection so this co-dimension is smaller than the co-dimension in s cross d of v plus the co-dimension of well here you can take it here it doesn't change anything. Then there is the theorem is that there is a good geometric reason for this and that is that then the projection of v intersected with w. So to offer I'm making the assumption that I'm taking one iridescible component okay yeah this guy is not necessarily connect iridescible of course so I take an iridescible component even if I do not write it because it will be a pain all the time okay it's really one I fix an iridescible component and argue with that iridescible component okay then the projection of that iridescible component in s is contained in a weekly special strict sub-variate of s right so maybe in this situation this is more understandable than the statement the very precise statement that I gave yesterday but this is equivalent okay no no the general conjecture it was general conjecture is just a heuristic so I'm but when you apply this heuristic here you an equivalent form of the conjecture that I stated yesterday is this one and this was proven by Becker and Timmerman. So I made the conjecture in the same paper where I did this conjecture of atypical intersection and then they proved it almost yeah very fast no I mean I stated an equivalent form this is equivalent so I claim that this is equivalent to the actual annual that I stated yesterday for the special case of biodebrate structure corresponding to my paired map okay okay so the goal is to give some idea of the proof so of course I cannot give all the ideas so so basically there is a tame topology part and this is the one I want to explain and then there is essentially a purely negative curvature of part which in fact to a large extent is similar to what we did for Shimouar varieties with in one FF even if this is more tricky technically speaking but the ideas are basically the same so let's try to give an idea of the proof so the proof is a complicated induction so you start so you denote your datum will be this and your v inside close irreducible inside s cross d hat okay so this is your datum and associated to this datum you define a numerical invariant that I would call the type and the induction will be on the type so what is its type it's a triple is first dimension of s then there is a dimension of v minus the dimension of the intersection so again if I want it to be extremely precise which are not in the paper I should choose fix and one irreducible component of this and then the last one is minus the dimension of the intersection okay and now those types are ordered by lexicographic ordering so with lexicographic order then now we say that v is bad at some point p in the intersection v with w if ah let me give a name to this inequality because otherwise it could be a pain we call it star to be original ah if star is satisfied at p and now we proceed by contradiction so we suppose that we have a counter example and now we take the counter example with the smallest possible type okay so suppose that we are in the situation where we have some v to s and where we have a v0 in irreducible inside s cross d hat which is bad in the sense that it satisfies this inequality but not satisfying the conclusion of the theorem and with minimal type among all the possible counter examples minimal type in other words as soon as I have a type which is smaller than the theorem is true okay and then I want to obtain a contradiction out of this so the way to do that is try to deform v so we proceed by deformation so I will call m inside the Hilbert scheme of s cross d hat d chat and once more I should compactify s so that it really makes sense and of course they don't care about such details in the paper but that's okay I hope then I define m as being the connected component of the Hilbert scheme containing v0 the class of v0 so now what is ah there will be a big diagram that we'll try to keep so you have your m and here you have s cross d hat and here you have the universal family so this is the universal family right so this is the set set theoretically this is the set of couples p in v intersected with w but okay let's in s cross d such that p is in v okay and such that let me just write this and let me write the class here so this is the pointy so the class of v is the point representing v in the Hilbert scheme and I'm just taking the tautological family there okay so now here I have s cross d so I restrict myself from the complete flag to this open orbit so I can pull back this family and and now from this I can consider my intersection w which was my fiber product of a d mod gamma so which was my my closed subspace analytic subspace here and so I get the family okay so now I want to define bad locus in family so define b being the set of couples p v in s well in this family in fact over w such that you are bad at that point p so dimension at p of v intersected with w is at least n0 where n0 ah sorry I forgot to say what is n0 n0 is the dimension of the bad intersection where n0 is dimension of v0 intersected with w where v0 was your original guy okay so this is a subset of vw and so you can think of it that as being the set of how to say at least equally bad points right you have some bad points bad points corresponding to the v0 intersected with w but you are also now you try to deform v0 into v and you look just at the locus where the bad the bad condition is still satisfied at dimension at least n0 and the claim is that this is a closed analytic subspace because this dimension is a person who continues so okay so we have this and now the main step using hominimality is to prove that this set is very big in other words you have a natural projection so you are in vw and here over here you have your s right so the claim is that in fact b to s is subjective so starting with a single counter example then necessarily if you take it of a minimal dimension then this set of bad points has to subject onto the base so this is the first proposition is that b to s is subjective and this is where we use hominimality so the proof is that we use hominimal chow so in this proof what is nice is that you have the two criteria of algebraization taking your role hominimal chow here and later there will be pilawelki so let's try to make the picture of this proof so you start you have your w then you have your d so i'm sure my picture will be disaster but and here you have d mod gamma okay so what is the idea of the construction well you have this universal family here and what you use now you want to use hominimality so basically if you remember the proof of the definability of the p of map what we did is we constructed abstractly something replacing a definable fundamental domain for the universal cover and the way it was done is we were covering s by poly disks and finitely many and such that this f become the union of Ziegels sets for those poly disks so some power r i cross delta dimension s minus r i a finite union i is equal one to n let's say okay and so of course the universal cover a map to w and so f also maps to w and you add that map okay okay so what i can do is i can pull back this family to this fundamental set here so really if you want to think geometrically think that this thing is living in the universal cover of course it's not true this is a disjoint union i've just taken finitely many charts okay but you can you can think just that you manage to construct a definable fundamental set in the universal cover this is really what you are doing so here i can call pull back this vw to vf here what do you write that sigma h is the power so remember as a way we define the we look at the definability of the pure map was to compactify this with cross normal crossing divisor take finitely many charts around those devices and then take universe in the universal cover depending on the monomy either you take a Ziegels set or you take the full disk okay the dimension minus just the total dimension is dimension of s these are charts right so sorry this is dimension s minus r i okay so now what is the idea is that basically you are just enrich enriching this picture to this universal family in other words the remark is that the group gamma acts on x cross d cross m by the usual operation so you do nothing on on s cross d cross m by gamma acts on my point p and my y tv by gamma p so the action here does nothing on s but now you have the action on d okay so trivial action so p is in s cross d the action is trivial and s but here you have the canonical action on d and of course because v is in the dual there is also an action of the complex point so it also acts on all right and by definition preserves vw which is inside this thing so this means that in the analytic context everything here is complex analytic so you can take the quotient so now this is a c analytic space and basically we are playing the same thing as the same game as for the diffanability of period map but at the level of that family basically so the remark is that as v is proper over s cross d at then you know that this new w which is just a pullback is proper over w and so when you take this quotient the map new s to s is proper okay so in this diagram up to now you don't really need this part i'm just saying that here i still have a gamma action and that this pass to the quotient here to give me some new s to s so i'm back down okay so this is the first ingredient is that this map is a proper complex analytic okay so that's good so now i notice that i can also in the same way as i realize s as a quotient of this by an etal equivalence relation i can realize this guy as a quotient of that guy by etal definable equivalence relation so but v s is also v f modulo this equivalence relation given by gamma but this one now and v f so induced by gamma but now what is the advantage of this presentation is that you see that it has a canonical structure of definable space so a new f in f cross m as a canonical definable structure because this is a restriction of an algebraic sub body of s cross d check times m to f cross m so by definition it has a canonical definable structure and for this this relation gamma this equivalence relation associated to gamma maybe i will not put the gamma because this is unreadable this is an etal definable relation right so i'm just replaying the game that i played before i write s as a quotient of f by this etal definable relation and i say that everything comes into family so v f is also v s is also the quotient of v f by this okay and so this means that v s has a canonical definable structure coming from that quotient all right so now i do the same thing for b okay so my b here is inside v w and i i claim that everything passes in fact this b is stabilized by those actions so everything goes back also to this thing so likewise v f which i defined the set of guys p v in v f such that the dimension at p of v intersected with f is larger than n zero is complex analytic in in this v f and definable because the dimension is a definable function and so it tells me that when i take the quotient i obtain exactly the same way my b s inside v s and this guy is c analytic and definable and well proper over s because new s was proper over s so and of course this guy is still still equal to my b mod gamma so i'm just saying that the big action gamma on v w stabilizes b which is obvious on the definition so i get the same picture so now what do i get i get that i have b s of s proper so by rematch time i get that the image z is a complex analytic subspace in s n everything here should be analytic sorry everything here is in the analytic category so this z in s n has to be complex analytic this is the image of so let's call i don't know p this projection so this is p of b n s it has to be a complex analytic subspace by rematch time because the map is proper and now i use the definability so as b s as p is definable by construction this implies also that z is definable in some minimal structure in occurrence next and so out of a minimal ciao i get that z inside s is algebraic okay but then i claim that this implies that z is equal to s so z is the image of b i've proven this is algebraic but then i claim that because of the minimality assumption on s this guy then the claim is that then z has to be equal to s because otherwise and this is claimed very fast in the paper so i tried to check it and i think this is okay but you still have to make some computation if you restrict the variation of structure to z and then you take your v0 base change to z then this would be bad bad of smaller type and still contradicting the theorem in other words you have to check that you keep the inequality of dimension it's clearly of smaller type because then the dimension of z is smaller than the dimension of s and the first order is the dimension of the support of the variation in your lexicographic order so so this is where you use basically oh minimal ciao then well i'm very late but that's okay then what is the second part of the argument using oh minimality well in fact basically there uh you do not use oh minimality but you use the power of uh delin semi simplicity theorem uh to prove the following statement okay so now we have proven that there are a lot of bad points for this counter example and so the proposition two is the following so now i have my universal family over m so i add my new w uh of w but now i can also base change to w to new w and inside new w i have b which was closed and so i have new b so you know for family now over b right i know this is uh yeah so uh you have to think that this new b is the family of equally bad varieties okay uh then uh so this was just a definition then the claim is that for a generic fiber here the stabilizer uh in g of z of this fiber is finite so then the claim is that stabilizer v is finite for a very general so outside the countable union of algebraic uh sub varieties uh well complex analytic here but we'll see that we can go to uh bs of for a very general fiber v of new b this is the claim so what is the proof of this and this is where the induction is kind of nice using uh delin semi simplicity and that all this business is very complicated in the mixed case because you don't have splitting uh so uh so we have defined this bs which was as a quotient of b by gamma and uh we have proven uh that uh uh um this thing was proper over s and so this is still in the analytic category so uh pi one of bs of course acts naturally so notice of course that there might be many fixed points here so I don't know and I don't know anything about the singularities of that thing it's really bad but what I know is that the pi one acts naturally uh on b uh via well of course pi one of bs is mapped to pi one of s right this is over s and uh uh to uh gamma which is your representation of monodromy so uh now the remark is that uh this image uh as finite index right because this map is proper uh subjective which time you have a complex analytic morphine which is proper subjective then the image of the pi one is finite index in in the target so uh it tells you that the image gamma b of pi one bs in uh g and this is where I use the replacement of the mem forte group by the monodromy it has to be uh q is uh zarski is q zarskidens okay this is where I use the remark that I made at the beginning that you have to argue with the monodromy group rather than the mem forte group then uh now uh this implies now the action of g on d hat is algebraic so this implies that for the very general fiber uh v of newbie um um this image uh is uh fixed group yamavi the stabilizer stabilizer so I'm just looking you you take all the points uh in b and you look at the stabilizer of the fiber and then I'm saying that because uh it has to be uh an algebraic subgroup uh of g so uh you know that outside a countable union of analytic subvarities uh you have to be uh constant so the stabilizer of uh so I will call it yeah the stabilizer of v in gamma is constant is a fixed group uh gamma v okay and uh notice now what is the important point is that uh if v is in this locus of very general points so if v is very general in that sense that it's a stabilizer is that group uh then for any gamma in the base in gamma b uh gamma v is also very general so in group terms this means that uh gamma v is normalized by gamma b okay and this is where you win because now this means that uh uh yeah okay uh this implies that if you call theta so now recall that I want to prove that this group is trivial okay this generic stabilizer of the fiber I want to prove it is trivial so I just have to take the risk enclosure and prove that the connected component of the identity is trivial well I have finite problem but I can go to finite detail covers this is no big deal so I look at uh gamma v uh star of a q and then I take a connected component of the identity and uh this tells you that uh this is still normalized by gamma b so this is also normalized by uh the algebraic uh closure of gamma b but we know that gamma b is the risk dense in g so it's normalized by g okay and then uh now we want to prove uh finish as approved by the claim that theta is trivial okay well now prove it uh you get that g normalize this is g the adjunct group at least is semi-simple so up to isogeny let's say uh this is fine because as I said I take the monodromy so this group is really semi-simple so this is really uh an isogeny I have to go to the adjunct group to get the product but let's suppose this is just a stupid product so this means that I can decompose it because of this normalization in theta 1 cross theta 2 where theta 2 is theta and suppose now that this group is non-trivial so what happens uh for my pn map well it induces the composition of the pn domain as d1 cross d2 and so I will get a pn map phi which will be uh which are two components so from s to d1 mod gamma 1 cross d2 mod gamma 2 so now the idea is to project everything on the first factor so look at this new period map this is the first component and then out of this construct a guy of smaller type so now I look at s to d1 mod gamma 1 so this is my new period map and now out of v as v is contained in s cross d hat and is invariant so d hat now is d1 hat cross d2 hat and this is invariant under theta 2 so uh this means that v is of the form v1 cross d2 hat uh with v1 in s cross d1 okay so uh now I can look at the intersection of this new v1 uh with s cross d1 uh over d1 mod gamma 1 so this is in s cross d1 and then I will argue with this so uh as now the claim is that v1 intersected with w1 cannot be contained in a strict weekly special sub-vity of s because otherwise v which is a preimage of v1 contains the preimage of v1 would be otherwise otherwise v intersected would be contained right I'm just saying that uh in these product situations the weekly special uh come from the factors okay so uh otherwise would be contained in a strict weekly special of s then uh and we know that uh s is minimal uh among the guys the bad guys necessarily I know that for v1 intersected with w1 the intersection is generic so I get that the code I mentioned in s cross d1 of v1 intersected with v w1 is actually equal to the code I mentioned in s cross d1 hat of v1 plus the code I mentioned in s cross d1 of w1 so this is a double star so out of my v which was bad I construct its projection of one of a one factor which has to be good because otherwise otherwise I would ruin the minimality I'm claiming that if v1 intersected with w1 would be contained in a strict weekly special sub-vity of s1 then this would imply because v1 is nothing else than the preimage the image of v by projection on the first factor this would imply that v intersected with w would also be contained in a strict weekly special sub-vity of s okay so applying the theorem because I am in lower dimension uh then I deduce this equality of dimension and now the remark is that note that w over w1 has finite fibers not finite but discrete so in terms of dimension if you look at what happens you will see that you get exactly the same invariant as you begin with so this is dimension uh w is dimension w1 dimension of w intersected with z is dimension of w1 intersected with v1 whereas as the code I mentioned in the adequate space of v1 is the co-dimension of v and so if you look at those two uh I guess I erase the first equality for v inequality for v but I claim that you cannot have at the same time an equality and inequality and so if you look at this this contradicts star as soon when you make the computation of co-dimension as soon as phi 2 is non-constant i.e. theta is non-trivial theta 2 equal theta is non-trivial okay so if you make this projection business you look at the equality that you get you compare it to the original inequality and then the claim is that as soon as the second period map is non-trivial then you get an extra term that and you get a contradiction out of comparing those this inequality to that equality so this proves that theta has to be trivial star was the original inequality so the co-dimension of the intersection strictly smaller than the sum of the co-dimension okay so this is everything that you get from this algebra geometric perspective the rest comes from differential geometry so here I want to spend my life on this this is much too long okay so I will try to not give all the details I will just catch the argument because I want to talk also untypical so now the claim is that so remain recalls that we are still arguing by contradiction in all this picture and so now the contradiction is that we get the opposite to proposition 2 namely we prove so we obtain a contradiction by proving proposition 3 that in fact the stabilizer of g of z of v is infinite for any fiber v of mu b we were proving that generically it is finite and now we are proving that this is infinite and okay let me schedule proof so you look at the following set so this is a set of elements i so the proof is very similar to what we did yesterday for proving axiom demand in the opinion case so we construct the definable set proves that it contains a lot of integral points by counting then apply pilawilki so that we are sure that we get a semi algebraic set inside this i and then arguing geometrically with that we exhibit infinitely many guys in the stabilizer so this is the idea so i will be the set of q of element g in g of r such that the dimension of the translate of my fundamental set uh as translate of v intersected with my fundamental set is exactly n zero okay so as i said this is similar to the set sigma of lecture three for axiom demand for a billion varieties then as before g of r as canonical real algebraic structure and uh because uh everything is definable here in r and x because then we know that i is r and x definable okay so uh so what we want to do is that uh what we want to say is that if we have uh uh exactly uh an intersection of v with w which is exactly of dimension n zero and that intersect f then we want to argue that uh this uh u intersection passes through many fundamental sets okay so if u is an n zero dimensional component of v intersected with w then we know that for each gamma one f that you meet then gamma is in i so the only thing that we have to prove is that u is cutting a lot of fundamental sets so uh and this is what we do now but the counting is much more complicated than yesterday but basically the idea comes from what we did for axiom demand and it's a bit more tricky because you have to use the orthogonality but otherwise the curvature arguments are the same so uh we may assume u meets f so we fix a point in f intersected with u and we will argue with uh spheres both centered at that point for the standard homogeneous riemannian metric on d okay so uh so let y zero be the image of uh x zero uh in uh d hat this thing has the natural projections to d hat uh okay so uh now the claim is that for any gamma uh in gamma uh the volume uh so this volume will be the volume of the projection uh on d for the homogeneous metric on d of u intersected with gamma minus one f is the same as the volume of v uh intersected with gamma minus one f so this is my definition and uh now we use the fact that this guy is algebraic so a priori is this guy when you think of it it's just analytic but uh this guy is algebraic in uh d hat and so it has a certain degree and what i'm saying is that when i move v by gamma i preserve the degree and so more or less this degree is exactly that volume and so this it remains essentially constant so what you prove is that uniformly in gamma this is o of one so it stays bounded so this argument we already had for x in demand um and so this tells you that if you want to count the number of elements in gamma such that u intersected with gamma minus one f intersected with x cross the ball of radius y zero in r in r sorry s then this is essentially uh up to a multiplicative constant the volume of u intersected with s cross this ball so this argument was also in axiom demand for schimoa variety so it tells you that counting elements is essentially the same thing as computing a volume okay and now you think that you're in good shape because you know that by definition of hot theory this the image of your u will be horizontal whereas the curvature is very negative so it has a tendency to be exponential exponentially divergent and uh this is uh what you want to prove so claim and this is one of the proposition in uh baker and team and man is that uh this volume grows exponentially uh in uh in the radius so there exists constant beta and r larger than zero such that for any closed uh positive dimensional horizontal analytic uh sub r t z in the ball by zero of r then for r larger than this big uh r the volume of z has to be larger than uh the exponential uh beta r and then there is a constant which is uh basically the multiplicity at twice zero uh of z okay so uh so this is this is given by some kind of poincare le long formula uh as well in schimoa variety this is a bit simpler we but uh at the end this comes back to that using uh vang toe theorem so this is the same uh thing okay so we have this and on the other hand so i know that this volume uh will grow exponentially fast now uh remind and remind you that in pila wilkie what you have to compute is the height on the group and you are counting the group so you have to have a height function on the group g of q so you look at g of q you embedded inside g of r and then you embedded inside gl of v r and then you map it to the symmetric space and then you take any norm which is on on the symmetric space basically you take the the i don't know the i mean the whatever norm you want on your matrices okay and it will work and this is uh the height you uh consider and the other claim is that uh phony r which we prove also uh in the axinderman case for schimoa is that uh with uh if you look at um an element in g of z such that the translate of your fundamental domain by this element crosses uh s cross uh by zero uh of r then essentially uh the height of the element has to be uh of the form exponential of o of r okay so here you have an true exponential and here also you have this uh exponential here and then out of this you get that you control what you want namely you see that so concerning this claim i believe that if you work just with c m then you have an inequality for any radius you have some inequality yes so basically the strategy what was done it was it's classical for c to the n then it was done for a non-positive curve uh bounded uh bounded symmetric domain by vang and toe and then this claim is the generalization you have to think that the horizontal the directions in pure domains they look like bounded symmetric domains of course this is wrong because this is not this is a totally non-integrable um in system but in the horizontal uh directions uh you have uh very negative curvature so this comes from that and where the fact you have to take small how large enough comes from is it necessary for bounded symmetric domain or no it's also necessary for uh bounded symmetric domain i mean okay i have to check i'm sorry i didn't check uh yeah i had other details to check i didn't take this one um no but i guess at some point you need probably you need some additive constant and then when you take a radius large enough you kill the constant i think there is something like that when you really apply uh le long formula around this but okay um okay so out of these two claims you get you now you compare them and then you get the proposition that for any epsilon larger than zero the the cardinality of the set of gamma in i which was a set that you want to prove is big intersected with g of q such that the height of gamma is smaller than t uh as then is something which will be bigger than some t to the epsilon right because you are comparing this exponential with that one and then uh now you know that i is definable so you can apply a pilawiki intersect with g that or g q uh do you intersect with g q or g i don't care g z is enough here you are really counting with g z so um but anyway the inequality in the right direction so this is no problem uh by pilawiki it tells you that this i contains a semi-algebraic c containing infinitely many integral points in fact many integral points in particular you need at least two integral points okay uh okay so let me try now to finish the proof that the stable the joint stabilizer has to be infinite i'm taking a bit too much time on this but okay now that we are here okay so now you argue uh you have uh you have this i and you have proven that you have a semi-algebraic curve something in it so now there are two cases the first case is that suppose that uh when you translate your v uh by these elements then this is equal to v for every c in c but then this means that v is stabilized by uh now c you know that c contains infinitely many integral elements and now you know that uh uh well okay i should have started by things that i i i assume that uh there is no torsion in the monoramy so i take i pass to a finite et al cover and then okay and so uh but then a gamma is stabilized by a non identity identity uh integral element and so this guy is infinite and so i see that v is stabilized by an infinite group and so i'm done so we are done in that case right this is the trivial case where i created a curve and by accident it happens that this curve is stabilizing entirely v then obviously i conclude that the stabilizer has to be infinite but of course i remind you that this is not enough because in the first statement it was only for almost all v uh the stabilizer is finite so this statement itself does nothing for you but so now we can assume can assume now that c v varies in the Hilbert scheme with c in c okay um so now the claim is that as you know that c contains at least a non trivial integral point uh you know that um uh this implies that uh phi of c v intersected with w is not contained in a strict uh with a strict weekly special uh for uh all c but cant be many because when you apply it to c0 which is this other non trivial uh uh element then as w is stabilized you see that this is the same thing as your hypothesis on v intersected w itself and so it's not contained in a strict proper and so now you just get a countable uh bad guys as there are only uh countably many families of weekly specials okay so you get this and now you look at the picture so you compare the action and the claim is that uh there are only two possibilities so either uh there is no fixed n zero dimensional component u of uh c v intersected with w as c in c varies so let me make a picture to clarify the thing so you have this w this diagonal then uh you have your v so for simplicity i will make the assumption that the intersection is just a point now i'm assuming that this family v is moving so this was my v and there is some other c v right and uh the guy that considering so this is c v and here this is my u zero okay and now i'm assuming that there is no a fixed uh such component so in the picture this means that i have picture like this not only v zero is moving but uh let's call it but uh this u is also moving okay so uh so let me just write v intersected w although this is still one component and then what you do is then uh you uh contradict the minimality of your triple by replacing v by the union of a old c in c of c v okay so uh when do you do that you increase increase this increases both dimension of v right because you have one more dimension here uh and dimension of the intersection so here you have to be convinced by the picture by one right i mean if i'm in this situation when i take this family i get one more dimension for v but also one more dimension for v intersected with w by one so uh this means that uh you are lowering the type right because you are not changing the base which was s you are not changing the difference of those two things but you are changing uh uh the last one so uh you are lowering the type and you get a contradiction um by by for this new family that's contradiction to the minimality of the type that you had and uh what is the other case well the other case is a different picture where uh uh or there exists such a u or there exists such a u so what is the picture now you have your w and then uh you have your uh you have your v here and uh when you move your v then you have this family uh cv so your cv will be of the same shape but uh here you have a fixed u okay you are moving the family of of your v so this is cv this is v so you are moving along c but it does not move u so they have to collapse like this and then uh you uh do the other procedure namely you uh take the intersection so replace v by uh the intersection of a c of the cv uh and then what you do is that you are lowering uh the dimension of v of course so of course in these pictures is kind of bad but because this is exactly you but uh without changing uh dimension of v intersected with u and so you are also lowering the type because the dimension of s is still the same but the second which is the difference becomes smaller um okay and so this is this is basically uh so you get the contradiction so this is a rough sketch of the proof so now you have to try to check all the details okay so uh now what i want to do uh is to move to uh typical intersections so the question is we are as before we have v over s uh we have uh uh phi from s to d mod gamma and again i will assume that gamma has no torsion for simplicity we'll see why later then we we have these hodge lockers of s uh v tensor that i defined this is a countable union of algebraic sub varieties of s if you have a countable union of algebraic sub varieties you can ask what is this i see closure are are there any noise geometric property of this so this was my original question so sadly enough it happens that with the i'm a geometer so with the method that i have uh i cannot touch the points and anyway as i explained in this setting uh the special points uh you do not see anything you don't understand really what they are because they are not necessarily cm points and even if they were cm points you don't know that they are defined of a number field so there is no galois orbits anywhere in this business so uh i will restrict myself to a positive component of the hodge lockers so y in s is positive so what does it mean well this is a bit stupid if uh just so we take an irreducible sub variety of s and i think it is uh positive if the image has positive dimensional so in some sense you can just imagine that this map is an immersion i'm just considering sinks of positive dimension in s okay um then what is the positive hodge lockers so uh we know that this positive this hodge lockers is a union of special sub varieties and then i just take the union of the all strict positive so i just add positive in the definition special sub varieties so this is still a countable union of algebraic sub sub varieties of s and i ask the same question but for the positive lockers and the tom which is i think kind of surprising at least i was not expecting it so by myself and uh dino scar is that uh okay i will make an assumption to simplify the statement suppose that the joint for the group is uh as a joint simple group so i don't want any product uh business so uh sorry a joint is simple okay and then the claim is that this apriori countable union is either a finite union and then either this hodge lockers of s the tensor positive is a finite union of strict special sub varieties and in particular so it is algebraic or it is arskid dense and there is nothing in middle you cannot create interesting geometric sub varieties of s and as i said i mean this result is new even for uh shimua varieties so as i said uh you take s in ag principally polarized uh a bit of moduli of principality borders g dimension will be in varieties close the irresistible and hodge generics so that you have this uh that the joint important group is a joint just a simplistic group and then uh what is the hodge lockers uh positive of s well this is just uh s intersected with the correction of all special sub varieties of ag and you keep only the positive components and the claim is that this thing so you are in ag right you have your s and you take the intersection of that thing with all the special uh things but you keep only the positive component so the picture is very bad because i'm not able to take positive dimensional intersections but anyway so uh this positive guy is either s1 union sn where si is of the form s intersected with some shimua sub variety of ag or the arskid dense so you can compare this results to uh in some sense more classical results coming from differential geometry so uh there is a a result to izad and then generalized by chai to any shimua variety but let me just give it for ag of course there is a statement for any shimua varieties which tells you that if you take this positive lockers then uh in fact you can prove that this is uh analytically dense so dense for the odds of topology so yeah analytically dense if s is very big the co-dimension of s in ag is smaller than g minus one and for chai you have a numerics that depend only on the ambient shimua variety so this is one comparison you see that here there is no condition on the dimension uh but of course the conclusion is just for zaizki uh and uh you can also compare so this is to compare to andre art where the statement is that if s contains so you are not looking at intersection all right you are looking at trivial i mean really atypical intersections as i said then set of special sub varieties of ag then s is special itself okay so this is atypical intersection this is typical of course not completely typical in the sense that i need that the intersection is at least of dimension one but is not atypical in the sense that you are not having an excess necessarily okay so uh how do you prove such things no this is the main problem if you give me an example i'm completely unable to tell so take a family of hypersurface of p 24 or whatever uh i have no idea of course i can make some conjectures once more i am expecting that uh the closest you are to shimua varieties and the closer you are to the density that depends uh like here but uh uh i don't know it's not completely not clear at all this is a big problem but the problem is that usually it's very difficult to compute a moment for the group so the question was are there any nice example where you know in which situation you are so what i'm saying is that uh even here in ag it's not clear i mean i didn't spend too much time trying to construct examples now i'm we'll try to come back to this because i think this is really in fact interesting but seems to be really nontrivial to decide uh okay so let me try to explain uh what happens so uh for simplicity suppose you start with a variation of attraction of weight zero then what is the picture that you have so you have s and you have v and so v is your fiber bundle uh autonomic fiber bundle or algebraic fiber bundle over s then uh okay take your fiber so you fix the point s in s you take the corresponding fiber v s so what happens well first if you fix a point class lambda in in that fiber you uh uh you have geometric speaking you have the flat leaf patting through lambda so locally this is biomorphic to s but of course there are monodromy uh problems that it will come back uh under the monodromy so v of lambda is by definition the flat leaf of lambda and so this is a subset of v a priori really disgusting right because if lambda is not a rational class or a complex multiple of a rational class then usually the monodromy will have accumulation orbit so this means that this leaf will come back and accumulate to the original one so this is really a bad topological set okay um then uh the second thing that you have which is important is uh but you have your hot filtration so you have a linear subspace fi which is your hot filtration and so for each uh such leaf you can look at the locus where vi that we'll call vi of lambda which is the intersection of v of lambda with fi okay and then i will call si of lambda the projection so it's just a projection so this is subset uh inside s okay so uh so as i explained well what are the good points the good points is that um um uh yeah maybe i'll skip this uh the good point is that uh si of lambda both vi vi of lambda and vi of lambda have a complex structure a chemical one right and the reason is that you have to think of those guys as uh embedded in the etaly space of your local system this etaly space is an enormous complex analytic space and those guys are complex analytic uh sub-ities there okay but of course uh so this is a good point uh but the the bad point is that if lambda does not belong to uh the project here uh vq inside pvc um uh then vi of lambda and of course sorry vi of lambda uh has no chance uh are not c analytic sub-ities of uh vi respectively fi of vi right there are complex analytic in the etaly space but not in the holomorphic bundle because of this return uh by the monromy so and uh you have to compare this in fact if you look at what katani doing kaplan proved they prove that uh in ur in the situation of projective so katani doing kaplan they look at the case where i is equal to zero and lambda is in pvq and then in that case the hot locus of lambda is exactly what i denoted by v0 of lambda so this is the locus of flat transport of lambda which are hodge so the project onto uh uh uh the hot locus of lambda which is uh with my notation s0 of lambda uh so this this is the locus where some determination of lambda becomes a hot class and the true result of katani in kaplan so i explained you that we using tame geometry you can prove that uh this is algebraic but what they prove is that really uh this guy inside v is algebraic and that uh uh uh this guy inside s is algebraic and moreover that this projection is finite right so what happens is that for the flat leave of a rational class the monomer would be discrete so you have a nice topological structure but when you restrict to the v0 this is even better get something finite over its projection okay good so uh now uh to prove the theorem well in fact we come back to that initial question of understanding uh the vi lambda for all lambda not necessarily rational this is the main ingredient in the proof okay so uh there will be two ingredients there is a global algebraic statement and a detailed study of the vi of lambda and then of si of lambda so uh theorem a which is a global algebraicity uh and for me it was really surprising is that for i in z and d in n star you can define just that vi larger than d as being the union of all these terrible leaves so you take any complex point in your uh in your uh vector bundle and you look at the union of all the vi so the union of uh this terrible uh thing so suppose you are in class here and then you continue it a bit okay but you know that you have monodromes so those things are not algebraic and you look only at those which are at dimension at least d so this is the meaning of this parameterization okay of course at the end i will be interested in the case d is equal to one but uh so this is a subspace in fiv and it projects to si v larger than d so by definition this is just the projection of that guy and this is inside s and uh the claim is that this is algebraic on the nose which is crazy at least for me it was crazy so if you take each of them is disgusting but if you take the all union uh and this is algebraic i don't want to uh okay i guess i will not give the proof of this and i clean this has nothing to do with hot stories this is a statement about algebraic flat vector bundles so algebraic bundles with algebraic flat connection uh no no there is no humanity this is purely algebraic geometry and but the corollary of this which is very important is the following so suppose now that you look at vi larger than d and you take the intersection with the local system uh the q local system and then you take the zaski closure then of course it has to be contained here because this guy is algebraic and so it means that you are saturated in positive dimensional horizontal stuff so this guy there exists to you the way of writing maybe is that you have a zaski open dense u such that in fact u is saturated in uh ah let me give this definition uh fi x uh included in vi larger than d intersected with vq i wanted to go fast but the result is that i don't have uh another required notation so uh where uh and so this ni f fi x is a union of irreducible c analytic component of uh your uh v of lambda intersected with fi through x so the union for x in u so there is this a bigger zaski open set dense uh there so that through each point if you look at the intersection it is at least of dimension larger than d okay so this is what you get out of this so uh uh now uh what is the second ingredient is to study uh si lambda and then what you get is uh tirambi uh that tells you that this si of lambda zah is weakly special so i don't know those guys are disgusting in those si of lambda projection of intersection of my flatlit with fi they are really disgusting as but if i take the zaski closure there is no choice this is weakly special and here this is basically uh a direct application of lax axel in the man for a variation of obstruction because you see that uh upstairs in the fiber bundle you are really taking something algebraic projecting it and taking the zaski closure so uh okay so the corollary uh that you get out of this uh so this is a corollary of corollary a and tirambi two uh is the following is that there exists a u now downstairs uh zaski open and dance uh in uh the zaski closure of si of vq so this means that now i consider only the rational class uh lambda in vq then i took the corresponding vi of lambda for the rational class i project and i take only the components of dimension at least d zah which is also saturated such that uh u is contained uh in the union for x in u of weakly special sobriety so you you have a very strong uh um statement telling you that u is saturated in weakly special sobriety of dimension at least d so yx uh inside si vq larger than d uh zah so where yx is weakly special of dimension at least d passing through x so using this global algebraicity plus x in the man you finally get this that this said that you thought you couldn't touch when you take the zaski closure it is saturated in weakly special and there uh you are almost done so let me maybe give the argument for the end of the proof so proof of theorem uh four which was this theorem uh there that i already stated in the first lecture let me give the proof okay so you look at your hard jokers and you take only the positive component okay so by definition this is what i've denoted um here by s0 vq tensor larger than one right this is the union of uh uh all the projections of intersection of flat leaves of rational class in the tensor uh product of my original local system such that the flat leaf intersect f0 in dimension at least one okay so this is my guy and then there is a finite first finiteness result of delin which is classical algebraic group theory that tells you that notice that here i'm really arguing with the local system v itself here this is an infinite dimension local system so there is something to be proven but what delin proves that this locus is really a finite union of s0 of some irreducible representation of the m40 group appearing in this very big tensor product so there is some finiteness issue here but this is no problem larger than one for the i finite dimensional inside the tensor okay so now you apply the corollary b what do you get well you apply it to each of those finite piece uh so there exists a u uh which is Zajski open and dense in hl of sv tensor positive zah such that uh for all x so we get that that guy is saturated in uh weekly special such that for all x in u there exists wx weekly special of dimension at least one contain in this zajski closure i know that my zajski closure is basically saturated in this weekly special okay so now what happens well either there exists an x and this is what i cannot control to answer your question either one of those miraculous guy appearing thanks to x in demand is a full s so and then i'm done this means that the zajski closure has to be s itself but you know i mean in this x in demand business you don't know what you get out you just get a weekly special but it's hard to control exactly what happens or for all x in u uh wx is strict so weekly special strict okay so now i didn't use yet my hypothesis of simplifications that the generic mum 40 group is simple so as mum 48 of sv uh adj is simple any uh wx uh which is weekly special and strict is contained in w prime x special and strict right this is a description that i give you of uh of the weekly special is there is a product situation and as i know that the global space is not a product itself then it has to come some to some special strict but now you have the you have the situation where those special by definitions are already in the set you are considering so uh but such w prime x is already in this odd lockers of positive dimension by the very definition of what is what it is to be uh special so uh this means that in fact you have that this zajski closure of the hard lockers contains this zajski dense open set u which is contained in the hard lockers of sv okay so this tells you that those two things have to be equal and uh so and so this guy has to be algebraic in the finite union of special sobriety questions you look dubious right i mean this is what is surprising apparently you have families of uh weekly special but because i'm able to extract out of it the special a special which is itself already in that space then i'm done okay uh i'm even in advance but uh i think i will stop here i don't want to give more details on that questions if you don't assume uh but then in fact i have a complete description i didn't want to write it the theorem is more complicated but basically uh you get a product situation it tells you that uh you are coming from uh a product situation okay if you want you want me really to write the statement so it's not written in the paper but after that i realized i can i can also do it so maybe i will make an so the theorem is so in the general case uh you have the following situation you take so the theorem is that you take z an irreducible component of the closure v tends to be positive zah okay and i want to describe this z and then the claim is that then there exists uh unique uh decomposition so first you uh you get a product decomposition into two factors so this induces a period map uh phi one phi two from s uh from s from s to d one mod gamma one because d two mod gamma two okay you want to search that you get also a projection so basically what i'm saying is that everything will come from a factorization of s the projection of s into some s two such that uh uh phi two factorized through s two i'm saying that my pn map is product and uh one factor uh comes from uh i sorry such and now uh now you can describe z in terms of those data and so three okay i should put commas around theorem because i think i've proven it but i didn't write it so okay so but i think this is correct and three uh there exists an irreducible z two in s two such that z is in fact the the preimage of z two so this is inside s and here you have your projection to s two uh this is a cartesian product and now what are the conditions such that so now the condition will be on on describing z two right now i've reduced i've reduced the problem of describing z to the description of z two and so now what happens such that either z two inside s two special for phi two okay or uh z two not special yeah up to now i know this is not much in s two and contains at most finitely finitely many the maximum positive strict special subvarities special uh for s two but contains a zaskid and set of special points for phi two so basically the answer is that if you do not make the assumption that the mem forte group splits or that the mem forte group is simple then you are back to the original problem anyway of trying to describe the situation where you have a zaskid and set of points the only improvement being that you got rid of of the positive dimensional guys you know that you have only finitely many and uh well you can construct examples this is what i told you but uh the only examples that i know in some sense they come from shimua varieties so i'm not so happy with that right i mean you have shimua varieties where you have zaskid and set of c m points but no positive dimensional special subvarities so you can take this factor uh being that one okay but all right