2/4 Universal mixed elliptic motives

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2/4 Universal mixed elliptic motives
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Universal mixed elliptic motives are certain local systems over a modular curve that are endowed with additional structure, such as that of a variation of mixed Hodge structure. They form a tannakian category. The coordinate ring of its fundamental group is a Hopf algebra in a category of mixed Tate motives. This course will be an introduction to universal mixed elliptic motives, which were defined with Makoto Matsumoto, and a report on more recent developments. One focus will be on the structure of the tannakian fundamental group of the category of mixed elliptic motives over M1,1. In particular, we will explain that it is an extension of GL2 by a prounipotent group whose Lie algebra is generated by Eisenstein series and has non-trivial relations coming from cusp forms. We will also discuss the relation of mixed elliptic motives to mixed Tate motives via specialization to the Tate curve and the nodal cubic.
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In the end of the world in and I think a man and we want thank you for you and on the but today I have to deal with the sediment technical stuff that In order for it to makes sense I'm going to give an overview of you know where I'm going and what I'm doing and I'm gonna explain what all the pieces are and it's gonna take me longer than I thought to explain all the pieces they think so Al and the 1st thing to is that I put up the notes from the previous lecture open up these notes and some other references number written words of nodes for myself to try to get a list of straight both in terms of notation other on the other right the 1st .period here's what right so getting in this overview but in order to define precisely and and things safer theory of limit makes limits of mixed Todd structures is somewhat technical Nagano through the context of similar systems over model after model and I can state the definition I don't think they'll get there today but on so In the 1st and 1 was marginalized stack it because and so this essay the stack beseech God reduction and every prime to this is important for us the 2nd thing that so this is the universal it because and we the most basic essentially elliptic motive in this case in this relative case and that's going to be a local system H which will be no 1 ever lower star of Q suitably interpreted it'll have b the euro which mentioned briefly and Chrysler about which I know essentially nothing but use a local system movements fiber over the fiber of marginalized .period of is just as each 1 of you here and so we have so 1 way to think of an 1 1 if we just think even 1 1 analytic wealth lamented in take start some punctured disk and whatever theory of thinking about this inclusion of the center and 1 1 and this would be the cutest so Q a local but if you look inside M 1 1 but it has a unique cost that corresponds to the nodal cubic this the unique cusp here and I just think it as being a neighborhood in whatever theories and I write so it's it's the upper half plane was this guy this is in some of the very tight situations for the analytic situation and Q equals each of the 2 pi I tell and then in the In the more charismatic situations take it to be this year job so it's a punctured disk signed we have tangential .period which is Dedie Q which is an element of the change in space at say cast off and by 1 1 and this tension vectors non-zero at every prime minister no problems so it's essentially this is corresponding to a I'm more or less to the takeover and so 1 failed talk about later today both the Haugen also in this whole situation is that but and we can talk about the fiber of this local systems Over I this change in vector may be hero called for the time being Colts safety of the T so if you calculate this in the various theories this is going to be Q plus Q of minus 1 2 however you interpret these guys here and actually splits to direct sun and the key point here is the key conservation dealers said this is the well I'll say it was mixed others it's 2 very trivial 1 in a way that it's it is 1 In fact in this category there are no extensions of Dubai Q minus 1 so so what does this mean so in the hard situation this is going to be a limit next time structure and it out we so what does it mean in Intel situation but it's it's it's the it's the GQ module Q action on the tape module the takeover of qualitative here over it's an algebraic we closed field so you look the takeover this and you compute the Galois action and decomposes in this way so that you have to twisted intensity with the next we 1 1
analytic With this basic .period this is naturally I think L to it Q I the profile and completion which is which is definitely a cell to the profile and completion and so I wrote these down because we're interested in and local systems so approximately the universal next elliptic motives so it's a a universal next elliptic motive state of Missouri you can put lots of adjectives in here which element a motor-vehicle local systems V Over and 1 1 of he publicly bikers and other things because of lack of technology this again be compatible realizations some variations of mixed taught structure lead sheaves and so on that have various properties at satisfying 1 Is that each so such a motive local system will come with a way infiltration so H. graded H. Wayne graded quotient it is a direct sun all of these simple elliptic motives some of local systems of form symmetric piles of age twisted by are adjusting case this any confusion it's a certain age 2 of them To do that take the symmetric tell them that way 2 the fiber all 1 the the fiber which may be on the school with the plane V all of V over tea so this is going to be the realization Cecil B. some sort of limit makes taught structure tell situation we will respect you you will restrict this guy here abroad when but there are it is all M and when I run into him I really mean mixed take motives of so by this I mean to these again to be variations of hard struck mixed Todd structure and so on and here I mean that there is a mix tape motive and the fiber of the variation of next time structure will be the Hodge realization of the motive of and so on and same with the elegiac story the the these fiber Felicia chief of the TV will be the elastic realization of the make statement I think that in order for this to be comprehensible I have to talk about variations of mixed taught structure and so on I have to talk him into a little bit about once liable to close such throwing and you today just explain what all the words mean the user 1 and well we'll have the local systems well lie over an 1 1 mean anyone 1 buyer simply connected with the well for example he had had he won 1 gold 2 days to explain that that that but the point is simply this that and the local systems themselves down ink-stained that you look at the corresponding fled vector bundles they will extend the new get a connection with a logarithmic singularity and then you have to define the limits extort structure and Bell doing detailed Tal situation you you have a map of the punctured Diaz say it well so you have a map the punctured In 1 and so and then if I tell you what geometric .period lies about Q yeah this is this is the a geometric point of this guy the fiber of so the fact that this this and this is actually the far-right Siletz yeah but I'm going to explain I would I'm going to explain the hard story not maybe an excruciating him detail I don't know it's not going to be excruciating I have not found but there's some really important point here because you might say To this local system age doesn't look very take note but when it degenerates becomes status I explained it before it's the same thing with the mixed unstructured what it's confusing here is what's potentially confusing is that there to wait filtration saw this 1 coming from the modern drumming this 1 coming from this global local systems right let best thing to do just that I think I've already messed up my numbering and Mogilev ellipticals so I realize a lot of this is well known but I'm going to repeat some of that because 1st orders to establish notation and then we need to be taken helpless because we need to do certain calculations said the 1st thing is I think this is a notation I like and I'm trying to propagate is that this this is equal to the marginalized all of us with projective coach of genus where the everywhere here to mind plus I'll cluster in is positive With .period and I should stress their labels so it's an ordered set in points and I are non-zero attention and although the anchor points of the tension factors have to be distinct and distinct from the other .period Sokolov like this might look like and and now we might put it
in my life .period 6 1 6 2 doctors and then he will find these tensions 1 the 2 up to the I think books yet all of these guys have to be anchored at different points in each of these books and they have to be this point troll distinct and these adjusting and distinct from the so for example if you forgot the tangent vector you you could turn tangent vector into another .period maybe I should say the appoints a Y 1 here and wanted while X wanted the impulse distinct and if you're topologist rules will discuss this later refuted the real oriented blow at each of these points you'd get a surface with boundaries so this is the algebraic way of doing the boundary components that topologist like but so actually I denied the module I point of such a code by underwriters settling like C X 1 X N V 1 and V R all right so we're interested in so what we need for the current situation is and 1 1 and here a typical .period is to genius 1 Covenant .period that similar because of bad always understand it's looking good described and so on and 1 a nice marginalized place that's good to work with that in fact the scheme is on 1 vector 1 2 this is going to be this vector will will be the change in the space of 0 so this topological picture it's a non-zero tensions and want Is this the on a 3 1 lead it's it's 0 x so X is not equal to 0 so it's an elliptical together with the addition of 0 .period let's look at these as 1st of all analytic waterfalls so let's playing in cities such as Intel's positive and so I like to think of this as the 1 July space also framed elliptic curves it's a way whenever you so much alike problems are at least 1 seoul Jennifer the problem and if you doing it analytically what you can do it is new and so what I mean by this this is an elliptical so smooth elliptic curve together with and I apologize for the abuse of notation of where this here's a basis it's a simplex basis of H 1 of the reasons the coefficients why is this the case well if you so if you have such an elliptic curve with a framing of H 1 you can it corresponds to well it gives you a point in the upper half playing namely you just take time out to be the integral of manager of a baby divided by the integral remained a where major is not equal to 0 it's just a polymorphic consistently non-zero billion differential if you go the other away few have attacked in the upper Plains and just take it up to become like to call you and tell the differential I was based in 1 of the planes the number said land equal to if this notation to leaders feel always right and for just feet the commodity land which is the obvious because you get out of tail and the bases here so H 1 of the naturally isomorphic Tolland detail and this has basis 1 on that basis so so I sell to the act on this we all know that acts on their behalf .period unjust acts on the natural way framing found so we have the S L 2 the well the usual actions tell goes to a Tel Plus the of held the and you can also think of it as taking you can also think it is taking elliptic curve with the framing B alright the framing this would be an the elliptic curve with the framing a B D b and 1 1 analytic Watson elliptical of it's just a friend elliptical way you forget the framing the orbits any 2 frames differ by a cell Tuesday so it's going to be the quotient of the upper half plane by a still to Benedict take the although fold quotient very briefly what we will befall quotient of the upper half playing and maintenance work on the upper half that everything an action vessel to see a new act as filters the equilibrium so it's just a fancy way of saying work here the work is still to see a "quotation mark variant like a good example of what that's like it's just the theory of classical module forms right a letter it's becoming 1 of my favorite mudslides basis let's look at end 1 vector-one analytic and 1st of all a dumb remark is that if you have an elliptic curve you have attention this is this is this corresponds exactly to an elliptic
curve and and the major where major is a non-zero homomorphic differential on the end why because the change in bonds trivial so To be if you've got such a differential in Geneva which 1 With your whole differential write to this is the same Zoellick because of the home of the differential but often think of it this way the rights but 1 waited this various ways to do this but let's take the step across stage and I'm going to take it into framed elliptic curves plus a non-zero French and so helm again under this I'm going to take seat I'm going to take it to the Philippines and then the basis of the framing of my lettuces 1 tale that's no surprise and then I just which which differential delay taken just take times dizzy so this is the coordination the Universal cover of each and you can check here I think you have filters and and it will take the the turnout at a gamut of tale but that's the same elliptic curve and then you have to compare the differentials and then you see that it takes about a B indeed see Tel Plus the C on this 1 because of this this said what we have is fed up this induces a map from S L 2 using system Cross page and it will induce a nice amorphous with and 1 to 1 and all so let's talk about there is no God but very little over her part now I haven't I could I am not going to I'll explain the compact visual when I do the algebraic case but you can hear maybe el-Sayed briefly what's up M 1 1 but are you going to glued to all the faults together and going to bit careful but it's equal it's essentially a like this picture here I will take up so S L to see the quotient of the upper half playing by it up here all put this 1 I'm just a question of the upper half plane collided in this map standing to here and this is just the punctured country to him I wanna map down into the group which Opel plus or minus 1 0 1 nite to the cost the upper house playing by the same mistake sickly group of auditor and dividing the spy that's the trivial to action right so that that's as a as an analytical befall them 1 1 by this is given by the my tossed picture so to work on it you would want to sell to the variance his work aquavit beyond this group here and have to match up on this year he said something about so if you have some extra simply connect and GAM as a discrete but you can define will befalls quotient pie 1 of you will befall quotient and say Peterson 2nd this is naturally isomorphic to gamma so here what's peace P is the projection from
exon gamma Mondex this is the projection so in ordinary topology can use any simply connected spaces are based .period was in particularly in the universe cover of this was fixed .period through properly continuous action then the the fundamental group of space with respect to the universal covering is just a group of DEC transformations in this just as all the full version of that for example .period the fancy X crossed again so and then what about so this tells us that Paul I won of M 1 1 with respect to this project analytic is naturally isomorphic to SLK and empire 1 of M 1 2 1 this maps SL Tuesday and if you think about it we wouldn't take the cost was simply connected space Duguid Paul 1 see stuff you actually get a copy of the here or if you wanna be really fancy a copy of 0 1 I and this group here you can think of in lots of ways 1 way this guy here's the braid group on 3 strings on 3 strings it's also the inverse image of ASL choosing the covering sell to our fundamental the traffic is this is a central extension but now so I'm going to assume 6 is invertible so it's 6 is inviolable In a arranged so this a cumulative ring every elliptic curve I can be written y squared equals 4 executed minus Q X minus so find cats there might be some assumption on but it's certainly true Varzi sarkozy was 6 and voted and so I then courses Is is 0 1 0 riders projective toward the appointed infinity and so what you can conclude from that is that this gives us that M 1 1 over an hour it is isomorphic too a 2 are minors you have to remove the inverse of 0 radius indiscriminate up to a factor of force is equal to you cubed minus 27 the square it's it's essential it's on the times or one-fourth of the screen one-fourth discriminate and said that this is a scheme and so and maybe I should say what is UV correspond to new Vt corresponds to you look because of maybe I'll call this 1 UV which is why squared equals 4 execute minors X minus the plus the differential DX over why that's an abelian differential and then on what is an what's a M 1 1 it's over it's equal to the quotient of GM's the caution of this reply have to tell you Max and acts by multiplying the guy he replied T this looks slightly strange but you got it does look strange when you realize that the same Cove is isomorphic to this with the negative of this differential because their real objective has been dilution takes a point to inverse so what's the GM action so land that would take you and it takes it 2 planned to minus 4 6 and this is exactly the action that multiplies the sky by landed and here I mean stack quotient and
what does this mean that says Well if you wanna work on here you can work GM equivalently up here and again if we want to talk about In 1 1 by what once 1 1 it's equal to H 2 Over time the quotient by actions a I smile all points on the equals 0 adjust the notable but with different differentials a whole set of working genes for use in its way in world Will you're going to have do and you can ride been viewed as a stack and not as a variety of greatness is it's the usual M 1 1 just the and also note that minus 1 extra trivially and everything because if you multiply differential of minus 1 you get an isomorphic could to everything has inertia and this is an exercise and seen some of the notes I put up on the way the Dominican can hunt through some of their sped what's the universal it because of you can I think of this as its 1 2 plus 1 this looks desire that this is his right down the equation on of the corner of the 6 while you and so on and on Aegean it's all look to cope with 2 points and a tangent vector and then you so the Soviet the magnitude and the Judean action multiplies the BN differential very yes that's the universal code this will be the universal curve of a N 1 1 the according to the pothole did I am sorry I In the right and so this is actually a this is actually and I want to do and that's equal to what I like to call it prime prime equal to my thinking but you can also write down the universal appeal this way and 10 somewhere in the so to compare the algebraic what we need are rising related I'll just 1 reason for writing a stanza when given the normalization I use so for me G 2 cakes tale is equal to 1 of the 2 tell all had to be granted equal to 2 but if we work on and 1 vector-one then we need to watch what this is this is the normalized guys so it's minus B 4 K divided by for OK plus the sum and equal 1 to infinity signal to Kate minus 1 the cued to the queues always into the 2 fights Pfizer function here guys yet this winter wrote and it's a module for the way that you can buy from so he tells so it's the usual formula but let me write it like this since 2 pi pi squared alike to isolate out all these visible with tape twists himself out 1 of 2 it the square so I guess what I'm saying here better variable here would be to apologize instead of the plus this sounds and who 1 to and good to and factorial G 2 and plus to times to apply position To the 2 and this is just the usual formula it's just the it's 1 of those the squared plus the Prime I want see and then again I'm writing this down just because I won't have the normalization stew to tell is equal to 20 G 4 tail and Judith retail is equal to 7 June 6
Of the 3 so why am I doing all this so what's the significance the significance as you can embed the tale into P 2 and the way I'm going to do is slightly non-standard it's a wonderful to pi pi To the minus 2 times people tell the 2 minus-3 3 1 this is what you need to do to make the Q theory can out correct the kid around theory so the engage it is the elliptic curve y squared equals 4 executed minors G 2 the next time and the discrimination up to a factor of 4 it is which is because for equals in acute times the surprise the count and the important point here is that the differential here is I DX over why abelian differential is equal to times that's why I put all these factors of 2 pioneer and why did I do that because when we this class to the nodal CU the novel QB is basically 1 with 0 infinity glued the identity at 1 What does this differential become on animal care if we choose a corn and he witches 1 0 infinity these 2 points and 1 here this degenerates to the W the W which is acute around for but the 2 and be 1 of the 2 of you can see just by seeing what happens to the period over at the same period 1 could be because he goes to the loop around right right so we want we want a map from the analytic picture to the algebraic picture and so this is a map into the sea to minus the inverse of 0 so this is equal to and just takes the tale goes to see to the minors to 2 1 to and this induces a nice amorphous from this gives you a man from 1 1 and better here this is a complex picture the picture right but growing more than enough on 1 July and let's talk about the local system so remember what we have here we have the universal appeal of over In 1 1 and H is in our 1 airflow star of queue with a suitably interpreted in whichever theories so I'm going to be however the I'm sorry this that this this is correct that should be yet this year because the cost of this by to use this this has where miners to assess what Sorensen's white minus forces would fall so this is a map this map a vessel to Ziercke will vary with the central re-election on the side right so go through all the realization so will discuss all realization separately so how bad is the most trivial just topological local system and with moral authority disgusted so I accepting a play a little game you .period greater well-being so the fiber of over his age 1 of Q but that's equal go home a H 1 2 Q stresses and now we have so if we play so you know we're get a lift and so then we can talk about the fiber over a particular elliptic curve so let's look at with got the a B this guy he is appointed the upper house of playing as friendly and so let's take let slip a duel the jewels dual basis also H 1 of it's the basis of age 1 duel to these guys but this guy here is isomorphic to age Wall 1 of each by 1 grade Ltd so under Plonka a duality I'm a duel this is equal to line of the be duel is equal to a so in there's no difference in the other theories this can be attained twist but I'll still use this basis so I'm on so what's the ESL Tuesday action so so we have a cheer so what I want to think here we have we have H over it's a local system to trivial local system on an H is really just
equal to say a few times a check plus a few times check that this huge times by conquering duality plus few times so I'm going to think of it this way than the news commentary and so what's the Excel to the action so best-seller Tuesday is going to act on everything in sight to the local system Downstairs is just a local system upstairs with Nestle to the action and a B C D this is this a little exercise a minus V into a line a B so exacting on the right frame so checking on the left of the mumble like it said here the quotient is the local system page over and 1 1 so that's the simple case this is not just me but will he's right so what what does it mean that a locals see this guy here is a space we noticed the jail line is simply connected so what'll systems they would be boring so what is that 1 way to think about is it's a local system on the this at all fold universal covering together with an action vessel to the rest you been well although full quotient so you when I say quashed I'm always going to mean stack questionable befall quotient that when when we're on him 1 vector-one it's going to be OK because that's a genuine variety of so what about Hodge well it's just the same but with more information services can be a variation of Hodge structure the hot realization of H is the most canonical example the variation Hodge structure on the planet so canonical it's a typical In the sense that it's in Wilfred Schmitz work under reasonable Hodge structure the Singleton SL to all but and this is the most the fundamentalist cell to so so again weighed we want to talk about the vector bundle associated to H 2 this is going to be tents Ravello no always be vague about those were on the upper half it'll be over half and so so this is this is the associated flat that the whole thing vector bundle should we can take a little break here if you want polls Cafe tried to admit to do the hard story so this this bundle has action and canonical 2 sections of the local system are I had to revert to the Nabil Amr Nabil equal so want to write the hard story we need to write down the Hodge filtration and from this will extract some features so we need so 1st of all I should say if we look at each 1 of the this is equal to F 0 this contains F 1 which is equal to 6 times say the abelian differential this contented to equals the thing to describe as this 1 so we we to describe it's 0 of age and this is a dumb little this isn't done little calculation so on the detail cautioned by the letters this is the destroy usual picture here here's a here's the 1 his tail just look at this and is equal to a duel plus tale the duel that way using Ponderay duality so Angel was minus B I'm going to let W bowl W B the section all of age detective so will just take the value of it are W at 10 hours going to read this it's going to be to pile times this guy bonds B +plus tail so this is again the reason for this to buy eyes so that it's a Q Russian Of the spot despite the defined virtue maybe the right I can write a mine too I'd be last longer Q because the 2 to pile guide tells a lot it is cute so just take logarithms and so let's look at the derivative of W it's equal to DQ overcame Cuba so this done calculation of the somehow fundamental this derivation arises here and so I want a formula for the connections so frightening right we can't still abusing the framing and B. within the frame a and W so this is overrated this simple reasons for doing this will become apparent that but 1 that said this is adapted to the hot filtration this is going to be here this is 1 of and you can see very solemn offered this W's all Norfolk linear combination of a and B so this is a whole Norfolk some bundle so this is a in in fancy Hodge there this is always true view of a family varieties you at the corresponding local system of quality the Hodge bundles very all more frequent here just which is written advanced revealed to do by hand and so so we're
also interested in the local system S & H and here SNH tens of 0 it is isomorphic to its end of page I'm sorry I forgot to write down the connection here so with respect to this trivialization the connection is D plus ADD W times chew over Cuba so this is a formula for the connection on this bond In here it's just linear algebra to get from this situation today he believes was writing down on here the Hodge filtration is longer where going to have S & H will be equal to a 0 0 will contain F 1 will contain all the way down to alright down to earth and this'll be spanned by 2 In minus pity W to the people consultants will be spent by W to the end you take all the wheels of degree greater than or equal P & W and so these all very home more thickly in the connection is given by the same formula the of yet this means symmetric power so there is a guarantee of basic these imitate twist will be a basic simple elliptic motives were going to be interested in extensions of in various categories like the kind of variations of the said this important and so there's some important observations here the 1st is on that the connection takes if you have a section of F P of S age you get a section of F P 1 1 work 1 and I a right and 1 1 but not for long Of the cap for this at these of forms with a logarithmic pole of the Caspian Sea the connection has a singularity and this is called this is if it if I put so this is the 1 the standard axioms of variation Hodge structure and skull Griffis transfer Celtic and again it holds in greater generality and we're going to be imposing conditions on the variations with this condition didn't variations of structuring the structure and that the section for so the intersection for almost no pay at a the equaled the the equals 0 and a equals 1 equals 1 B 2 this is a flat in a product and it satisfies the remand by linear relations is flat With respect to the connection and satisfying right wing and a concern this is called a polarization Jernigan dwell on it the word polarized variation Hodge structure and in reality people often drop sometimes dropped the term polarized polarization that plays a key technical role without 1 of our resolve that hold and so this is also polarizes the it's in their own polarized invariant by linear form and they'll automatically satisfy the remand by Lanier relations what SAN age With connection With it's hard filtration and also this guy here is a polarized please check if you're a professional and and basically the the axioms of what a Polaroid variation contract Rogers generalize these we want need any other polarize variations of Hodge structure other than these guys here and sexually of wait and because every fiber of the local system with the hot induced heart filtration is a hard structural weight so so the whole family hard structures that move fall more frequently and so on and they don't move too fast at Griffith transfer sell the says because this is slow food movement in the U.S. right is this led variations here in derivatives not too big so let me quickly say something about the the so yeah I take a quick and dirty here but salute of accused say and this case fueled points and 1 1 Q and so we get an exact sequence of fundamental groups would get high 1 of M 1 1 over Cuba that's the geometric fundamental group here I Nicolas .period this will map to hi 1 of M 1 1 of a Q this is the most fundamental group and the format G Q the Kellogg group of Cuba over Q the elliptic curve gives you a section because this is part of speck you map back in here and was set up
base point of this induced by elected the and so this tells you this group here is just isomorphic to GQ samurai product supply 1 all of M 1 1 Over Q and this is this group peers also isomorphic to the usual fundamental group Pro finitely completed and now what's what's a and a local system here will be a representation of this group and what representation will be for the middle group here and hire 1 of M 1 1 of the queue b it affects all along a Each 1 of the of acute I so we can think of this as just of plywood 1 of these of accuse so we describe this last time into Q L and so you can figure is an action of the semi direct product Havel's despotic tells the geometric poetic well just text on the like that just the fundamental representations filled in and you have to have but also the Galois group acts on this guy as well and together they give you an action the semi direct product we can also do it by looking at the top of CIA coming from the L torsion points of the universal but it has decided an idea so anyway so this this Aladdin Italy's and then let me say something briefly about the kid around theory at 1st I was going to leave it out but it turns out to be very useful for the same reason indeed on the mix tape case is that it gives you a very good fiber funded it's the best fiber functor from mixed electric motors and and solve briefly sketched this here so I'm going to do it over and 1 vector-one and on right down into the gene invariant connection and therefore will give me a Q around story on in 1 1 right where I want that 1 so again the trivialize age over I got of acute and how we going to do that to promote this is going to be a 2 Q minus indeed the inverse of 0 we have to section I'll call them S & T so 1 Over so this is a point in here the value of the UV which corresponds the elliptical y squared equals 4 executed minors you X minus and so on the value of team and new V is just going to be DX over why this is clearly a Q rational section and best of UV is going to be the differential the 2nd kind Ex TAX Overlie and and also if you wanna write so this is actually the sorry this is going to be the way that this is true but by the end of this guy here is going to be 1 of the 2 P tale this differential the 2nd time secure rational and these guys you can calculate their come prior to pile on the front so they linearly independent monitoring fiber both defined over Cuba this tells us that age is equal to 0 of them 1 vector-one times AS plus 0 of M 1 1 1 of the factors that and not have to prove it an exercise is that with respect to this trivialization the connection is plus my 112 did indeed of last 3 Alpha 2 decades which is visible and I have to tell you that Alpha is so where Alford is equal to the 2 near Gainesville minus-3 3 DVD you indeed is always is equal to you cute minus 27 convinced square and so this is clearly defined over to and if you not only that it sold what's GM invariant I run up by 1 writer down I wrote up the weights of everything under the GM action but just to get by said up to the gym action of multiplies land multiplies this by land and will multiply this by 1 of the lander and just look at everything else I think you you had weight minus foreign V had weighed minus 6 and so on look at 11 sewage June variant so at descends connection a Q rational connection on M 1 1 of the key ah I'd say that this is so the end of the set up an even finer proof of this since it's so it's not had just an exercise but I have these notes on the elliptic KCB equation in the last section there I work this out together with what I really wanted which is another cue rational connection so Oh sorry this a cancerous missing right here right there yeah it's in the nite which put on the Web later today
there so let me mention a little history so In the abstract Wilfried Schmid sometime about 76 prove the few have any polarize variation Hodge structure it has a limit makes Todd structure he assume it came from geometry I the Wilfred Schmidt and so then and a beautiful piece of work and then prove the thing if you In the case of geometric origin although there was a gap it's it's correct but there was a gap and various people fix the gap but ultimately I think it's by Saito but so is the geometric case that you can explicitly Rotterdam and in this case we can't do it because things so simple so I'm going to do about but and that's why didn't put limits of hot structures in the definition of a polarized variation to automatic by the water Schmidt the case of mixed on structures you have to talk about the limits on and talk limits upon structures than limits of mixed on structures and then you can probably ready for lunch with a right so I am and I'm just going to do this in the context of ancient symmetric palace a right so let's so the question is what's so what's what's the issue here of God the staff is sitting inside and 1 1 and you can think of this at Waco thing about this analytically grind so this is really the upper half plane mind 1 0 0 1 including into S L Tuesday so this is the puncture Q 1 happens when we didn't know the 1st thing is we won all understand the behavior of this local system on the stuff so I said a little while ago those can explain why we switched the framing from a and B to a and W so let's check the Mona drumming his what's the monogram in the local variable cute so just to be very clear Q is equal to the to the 2 apply and so what happens here we can just as you go if you're in the cutest QI you go around from some point you go around like this the city's but what's happening in the upper half blind here is that some Tal here is going to tell plus 1 and so he is 0 here here's my 1 so this will be we do have tell so we've got we've got here is a and here is baby and now what happens when you go around the disk is a baby goes to this 1 which is a plus it's it's not a surprise that we already wrote down the formula for the actions is this B to 1 1 0 1 the IRA which equals a plus B B rights or got this local union Poland wanted me the 1st thing to both a and B urine against a note so this is 1 reason we switched to WB is clearly not invariant is check it out by running out of the sorry last 2 In that view yeah but should thank you it's a guest at a and B goes a plus B you see from this picture here he's young and so and you only ages W it's also just clear it's just the 71 billion differentials determined by its period on so it it out it's very because W takes the always takes to by ion thank so what's the saying it saying that so age on duty stuff is just equal to all of the star a plus after that so if you have a bundle it's trivial on a punctured disk and you have a framing that gives you an extension to the entire so we're going to extend 2 Of day we don't standard so this is just going to be equal to 0 duty a plus part of the Delhi so now there are many ways to extend because I couldn't stand I could have used the framing some power of 2 times and some have 2 times W but this 1 is a canonical choice and delaying was the 1 I guess to realize this so we extend the connections so when you extend the bundled together Merom connection Solis calculate what it is and I should say similarly 4 S and and ate it extends just as the instrument repairer of this guy OK so what it and so we already know the formula for the connection in both cases Nabil as equal to D plus a D the W DQ over Cuba so what we notice about this this status I watched the residue 0 of Nablus it's equal to this Endemol morphism it's a good it's equal to this standard amorphous and witches a D W and should think of this as I an element of the and morphisms of the fiber of the 0 so I was going that age this is the fiber of 0 or I could put it on Paul H. but so it's different people think of limit makes Todd structures in different ways I think of the limit mixed Hodge structure as living on the Central fiber of this extension this is related question and this is this is this equation has a regular singular .period because the polls just a simple Paul so these 2 conditions alone uniquely determine this expansion abstractly it's just the fact that the connection extends to Merom off a connection with a simple Paul and no podium residue and that's called leans canonical expansion OK so the next point is that what we had started out with basically the age of the day that this gave his age he's stuck with the connection that we
extended that to age With the with a mere a connection nobody all the different now this guy he has a hard filtration by home or fix a bundle we to make it more interesting I can put it now we know what the next thing is that we observe that the Hodge bundles extend through some bundles this is what's important we take the correct extension of predict the wrong 1 they want piece of bundles so observed there F P of age stock exchange To homomorphic some bundles just the same way just give basically UW Type 1 0 in a tight but only after so after of his down of age but is equal to the sum of all of the times a to the In minus J W To the Jay Wenger is greater than or equal to people as visibly as some bundle this year thank you right to this is a some so this was in the abstract case this is what Schmitz no potent Norwood clearances says the Hodge bundles extend to hold more fix a bundles of the canonical extension so what's the limit makes Todd structures so so when we got so far so what we get so fast and so far we have with God S & H so the equals the fiber also S & H plan of 0 to somehow that's to me that's where the limit makes told structure should live right there and we also have a hunch filtration we have the the law it's just it's just I F P of NHS just to be up F P all of this FPU this and age which is the fiber 0 right just but the extended the fact that the Hodge bundles extending to counter the Hodge filtration on the central fiber so and you might get optimistic and think that all this would define a Hodge structure so we're missing 2 things 1 ECU structure this is where the tension going to come in and to missing away filtration so let's solve deal with these now time is all right so I so let's just say that but said an equal to a D the W and this guy is an amorphous on all of S and thanks minimal office h and therefore an animal for some of its nail polish animal fizzled and so a little proposition here is said it is a mail potency In morphism all of some that displays the uncharacteristic 0 then it is a unique and call W the daughter of V but what and all of W is contained in a W and minus 2 this filtration recalled the way filtration and the 1st properties that and and the 2nd property is there I if you look at say a end to the can a it's going to take graded these it's going to take it you'll certainly get a map here for every case and you asked that this be an isomorphism and so this thing is centered at 0 and let me give you a very quick enough approve it but I'm gonna just we want to see the idea is to sell the proof is but the 1st thing is without loss of generality but don't know the French equivalent is without loss of generality a it is a Jordan blocks that's can look like zeros and then just once an end to readers write it down so you've got a basis here and for single Jordan block eliminated nontrivial greater questions either in order or even parity depending on the degree of no pardons but this 1 is right at him so and once you got its existence uniqueness is easy and you can write it down in terms of kernels of powers and so on you know what you go in and out of yeah this is just undergraduate linear algebra this is a
form enough the 1st 2 cities if in the there world during encounter there some saying everything is polymer every Colombia does not extend 1 on any of this would have try the what we're gonna do here is so we have a the W and now we want to get so this is going to which takes SNH into itself so this gives us a way infiltrations which will call W a D the W but we we want do 1 other thing this guy centered at 0 we wanna shifted so centered at the weight of the the variations and I wanna once John Conway gave a lecture and the middle lecture he just really big guy dived onto a table in front of the room adjusted to get people's attention so the best I can do is to hot again going in shift so this this guy's centered that 0 right that what isn't symmetric about 0 we shifted shift to get up and I know you know you put some square bracket with something so that its standard end which is equal to the haulage weight vest in nature it also happens to be DSL to wait but it's the haulage way it's the weight of the variations so now what you have is you're going to see that a D D. W this is going to go from here on and plus all of this in age it's going to go isomorphic leading into G In minus K S Sojourner symmetric about the Hodge was anyway so why do that will see in the 2nd he's got to others as the new so I'm going to use mn I'm using it this is it so last week I called 1 Wade filtration and and this is a it's and and use and because the monarch drumming the cycle 1 of the longest-running wait for so we have a local unit pardon moment from a few takers logarithm it's in fact this this operating here it gives us a way filtration and we get we get away filtration and so on the I have a proposition doing things differently from the Knudsen so what we've done now is we have With got away filtration To put on middle 5 Soso foe of God got vector Vector-Space namely the central fiber of the canonical extension we have a limit Hodge filtration which is what we get from the extended Hodge bundled with another way filtration gonna give it to structure and so far I haven't used any tangent vector species which is worse this is the hardest this is the Honchul filtration so we have all infiltration way filtration now what about the queue structure an elder saying we we will get 1 just a preview for inch V In the tangent space 0 of the and I this Smith deviated from minorities Florida how we gonna do this it's can look kind of strange but I'm so when playing an idyllic will do 4 viii equal to D DQ and maybe I will explain the general thing here otherwise this is gonna look so strange because this variation so special but I need the same construction from extort structures where it ain't so can you never know what's going on so it yet OK so son Limoges talk about the general situation so we're going to have say a V all over the disk With you put money me like we do here we have unique but Mondragon by the no . in the potent matrix of coal and there were gonna get V Delta over indeed that's the corresponding flat bundle the guarantee its canonical extension and so 1 of the key features here is there the residue of Nablus right in Nablus has since I should put him in the reversal of a has a regular single-point has simple poll at stake you equals 0 so in other words as a regular single point run so now but there are various ways you can think of is the way I like to do this is to use the standard old-fashioned this is a good book by parcels all was and what you have to safe but you can always choose a frank so you can always trivialise this parcel named the fiber of the 0 so so it's going to be the fiber of 0 so when I trivialize I'm going to trivializing using the fiber of the 0 2 although these days a different fonts this is blackboard boreholes this is calligraphic Anderson's Roman 4 this is this just an ordinary V this is script right script for
this is black 2 and so now so and I think this is more or less in the wings book on differential equations but it's also you have to read about 3 pages in the book but there exists a true realized all of the above so the ball is going to be isomorphic to d cross V such that the connection when you got a trivialization the connection you can write was 1 form is equal to In times DQ of acute and this is constant and this is in fact the residue and its sequel to the residue and 0 we can always do that that's as a result of the debate but what does this do for you so let me draw a picture here's the deal this is the across the dishes he would are Central fiber and we have some sections like invariant sections will extend through and those with long and drumbeats of behaved logarithmic we they blow up you know they behave badly but what does it do it here and let's suppose I Q E 1 in and it's so what sorry the president of the year the refugees on the fictional yes sorry horizontal sections the flat sections so what this old but what is this true realization later do it allows you to compare the fiber over 1 the fiber of 0 vis-a-vis still this is still very of 1 which equals of fiber all of the by the way that the technical people think of it and everybody can we can confirm that the thing in different ways but I really like to think of it downstairs they like to think of things on the universal cover you know people in like Wilfred Schmid all these young guys like Paul Steiner look at this thing but anyway so what you going to do is we're going to use this true realization to put the kids wouldn't take the cue structure here on the point of this disk corresponding to the tangent vector right DBQ corresponds to 1 you might say whatever ones nodding he you can always do 204 sufficiently small tension vectors and then they get a formula I'll show you the following 2nd up unless someone in will build and this gives you accuse structures so we will use the trivialization but the structures walls to 2 get an eyeful offers a linear isomorphism from the 1 into the 0 and if this has accused structure this is going to give us a queue structure over here so there is a key thing so what we will do it in our case it can be absolutely trivial this is by the the way I think regularizing period and its equivalent I think to what everyone else regularizing period right so let's do it for an set so let's do it let's take a and B is equal to age will trivialization already has this property with God H. crossed D Over D so anxious just going to be see a plus C. W I the connection that is equal to do plus a dw times DQ of acute 7 this is constant we've already got a delay truth no 1 these nice trivialization when this is constant so now let's ride out 1 of what's the flat structure and we know W was equal to minus 2 b plus log Q I don't and I have to turn around walk you times and so what we did is that to apply I'd be it is equal to minus W plus Q right now what trivialization it's just going to amount to setting long queue equal to 0 when he said Qiu equal to 1 get rid of a too so sorry H of 1 is going to be its Q a plus Q the but the fiber of once and the trivialization says that's going to be our structure practice Hamas's right and so but this is actually equal to Q a plus a 2 times W Is it due equals 1 To do you have to put a 1 1 2 2 . 1 that right because it's going to be is by said if this is 0 W over 2 by Isaac be up to sign and that this this is coming around because we've got it's really a queue of minus 1 so this is actually structure on the middle so what have we got we've got so what's this so what's the
extent structures on this alarming is to page 1st of all 2 D and Q 0 by the way I should say here this on the queue structure here this is an important remarks and this is a little surprising when you look at the construction because you have lots of choices the the queue structure on V equals Aviv 0 depends only all do you In other words the parameter the 1st order if you change the parameters if you think about it little exercise you actually change the trivialization the only way you preserve history realization is if you did a silly changes toward a white multiplied quarter by constant but if you take any nontrivial change a quarter to go computer trivialization and when you check everything out sounds hard it's actually easy but you find out that everything depends only on the parameter the 1st order I think so here the mixed Todd structure nature associated DBQ is well but we've we've got HQ now it's going to be equal to queue please the vote to pass and now is that we've got a Hodge filtration HC it is going to be see a plus C. W this is going to be F 1 and if you stare at this you'll see we're gonna Split Hodge structure this saying that age DBQ is equal to a Q plus Q of minus 1 right so this is spanned by the door of the suspension W and actually made acute around which is equal to and that the parts banned by the which is equal to 1 . 2 and the part and it splits visibly Split so this is a very basic calculation if you wanted to do a different tangent vector I want do it you would replace Q by if you want to you do landed D DQ just replace the parameter Q by Land Land Q and do the calculation will get a on Split extension here OK says a question what to do last time I stopped on time and in France's evaluating no longer life it's hard to listen for 2 hours let me see what I only vote Philip that I'm not going to ask any limits of mixed unstructured another 20 pages here but let me just mop up this and I'll stop limits of contractors and next time of the limits of mixed structures which is on OK so race to operate set but what I call h when I write HDD Q I mean the limit next on structure associated with that change invectives skew plus Q of minus 1 and this is essentially spent the nite but if you look at the same age it's just equal to the instrument repairer of HTG Q Everything you bags well on the summit palace and this is just going to be Q 4 0 0 plus plus few of mines in and and this'll be spanned by W to the end this by aid to the end and things in the middle by things like aid to the jury I don't want to on so this is also displayed right and maybe also state this year I question before I do that I should say Why is this reasonable is itself it's in the and on women's you a to the JMW to the amongst yes or no loss in view of the UN if you can usually associated with the use of violence the appointment did you take saying I put a date to be announced at the local level we didn't write anything alright so is this reasonable so instead of writing the whole thing and so let's let's look at the Universal elliptic curve of the day I I'm
so his cell I think it's a little like this the family here and is 0 so so this is essentially the takeover but this time space may be alright so basic fact that appalled and topology varieties is if you include 0 that's the fiber of the 0 into the D. This is a hung atop the equivalent it's an elementary even just seeking to squeeze it in but I I give you a rigorous argument under rigorous lunch so I think in this case it's elementary can write it down but in general I that's a little trick on the 1 hand is trying to brought into general affected generation so I but we also have to be 0 2 0 maybe all right on its meaning the non singular part is this is isomorphic to see stuff that's the smooth locus of the nodal cu so what we have there these zeroed in on singular includes into these I want to say this includes into EDD queue and I'll explain what I powerful give this guy and then this guy the maps into we D which is how top the equivalent to 0 so just suspend belief a 2nd pretend there's this fiber that until is you actually write this down the Intel situation so this is a sea and this is a call to P 1 mod 0 identified with infinite what and here I am let's look at just H. low 1 for each of 1 we can do it I went let's look at a chapel 1 of 0 it's equal to a protection even equal to the EU's 0 when we do all this standard and you can imagine it's a general construction weeks constructors but basically this class is coming through past enthralled .period which has tied 0 0 and then we're going to get a map here into H 1 of the D DQ and then there's going to be a restriction maps into Page 1 of stock which is equal to z of minus 1 so In the limit you think this guy should be an extension of this guy vizier syrup that just a heuristic reasons that this is actually a property of the limit make Todd structure that is compatible with all these things you know it's built essentially on the single fiber in from backing off infinitesimally and maybe the last thing all say before lunch is all tell you how I think of the fiber of DD too so how do I think it's so it and this is probably related to log schemes on I'm too old to no avail of and if we looked at so I want I have described the fiber of so this is so what I'm going to do is undertaking the real oriented blow on 0 an infinity of P 1 what is that that's just really equal to the real oriented blowups of which I discussed last time is like this that his infinity and 0 and have replaced them by the by the circle of directions 1 of the hostages who are all the and real oriented thing and now I'm thinking here instead of doing this suppose I have attention vector here so the Hacienda gives you a map from the tense a product of the space down here to the tangent space here so what you can do is over the song and take also the disk blown up at the origin that's the sphere anonymous so here's the disk the real oriented blocks every point on the circle corresponds to the origin but they tell me all the different directions and how my going to construct the fiber over over Dedeke you so Dedeke you actually has a land that actually arithmetically this picture's OK because we can't double triple into this village would have bad reduction were we have much choice some only going do it is going to identify the 2 ends of this In over Sadie the I faded times this understand rotation by and you can see that style includes into this at everything except the circle and you can see it also maps to the 0 0 at all I get is 0 I just collapsed Sokol and I actually wrote down the equations for this improves and get a nice family here I mean works for stable families of clothes but I rode it at the end of these elliptic KCB at some manuscript where I pack a lawful to stuff that I don't wanna put in papers but I think should be out there .period there so I think of the limit makes Todd structures being on this guy .period


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