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Algebraic and motivic vector bundles

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around we the top and England if you nothing want to 2 In the venison and thank you for inviting me here the sex and my 1st visit to the GSA and it's very nicely but such as those already busted once and I gave the stock at the Klamath conference in the fall so if you were there you've heard this before and I apologize the way I want this is kind of a evolving project the lot of aspects of its Arvind Ozark and John fossil and I'm working on about approaching the problem of constructing algebraic vector bundles using methods of Mohammed happy theory and especially approaching the problem of trying to give a topological vector bundle an algebraic structures so I want to be in the context of the varieties just over the complex numbers and the basic question I have in mind is which complex vector bundles an algebraic brick structures like it is said that there is a really hard problem that's a positive solution that would imply the Hodge conjectures even just in the case of are fine varieties so but and I'm not going to talk about this problem in general that I'll eventually focused on how the kinds of varieties there come as far away from the Hodge conjecture you can get where all the integral qualities of Hodge tight end and hopefully face some some interesting things about that so this story in this interaction really goes all the way back to the stairs conjecture and 2 serious FAC poet when the background that's what are I end up in that thing In that paper series exploring this relationship between vector bundles of space and finally generated projector modules or the ring of functions ,comma space so if there was a topological spaces will be continuous functions and it wasn't an algebraic variety here are fine algebraic variety has just the ring of algebraic functions I'm sorry something happened well I apologize that background so dark I just changed and changed back but anyway so In in that paper in their rights of it's not known if there is any when exodus are fine art space if there is any other finally generated projected models which are not free of course is thinking by analogy with topology where vector bundles over a contractible space are all trivial and this became known of course as Sears problem even know it was just a question and eventually became known as serious conjecture In my guests has that kind of karma and that problem was solved independently in the seventies by equivalency so the theorem was exactly that that if case of field every finitely generated projected marginal over carriers free from just give me 1 2nd I
don't know why that it is so .period so it's still light up in a minute alright so these are just going to be .period
OK so as a survivalist solved independently Michael and Switzerland in this case are and so there was a whole bunch of problems like that and you many of them poor promoted by Hyman Mass and Sierra and others an Adams have identified in his math review of these papers he identified an ongoing program I on this and imply that this was Adams program but he articulated in this matter views saying why don't you can read that when he says that leads to the following program take definitions constructions of the Arabs from bundle theory and express them in terms of finitely generated modules railing and then use bundle theory as a conjecture generating device and try to prove these things about ratings and home 1 of the "quotation mark of maybe 1 of the most definitive conjectures along these lines is what's known as the Basque willing conjecture From and their concerns so I be looking at vector bundles in various categories and Saudis right that K for the set of ice amorphous and classes of breaking a vector bundles and decorate without a break topological whatever and so the basic thing about in a topological vector bundles as their home atop the invariant Tyson of isomorphism classes of vector bundle over space cross interval is the same as the ice morphism classes of vector bundles over the space and the Basque willing conjecture asked that that be true if that's true enough for algebraic vector bundles and I think that we need some hypothesis and I think the most general form conjecture I know of is that when a is a regular ring of finite Nicole dimension then every every finitely generated projected module is extended from 8 in the sense that I rode so that would be the exact analog of the of the thing in topology that vectorbundles are common touching so far as I know as well this problem isn't solved in general but in 1981 just building on the equivalent some proof Lyndell show that the Basques Quillen conjecture was true in finally generated over a field so that's called the geometric case and if you're interested in this you don't know about it but he Weiland has a wonderful book on on this serious problem on projected modules so this leads to if we really take Adams articulation of his program seriously it leads to this question can we study algebraic vectorbundles using home atop the theory that it was sort of a lot of conjectures and theorems were setting up the basics of doing that Apple would happen if he really tried to go the distance and and really tried to use the methods of homotopy theory to study correctable spot allowed abstract would have a theory comes in light everything up so in order to do that we have to have an abstract framework for talking about home atop the theory candor and so I I mean this is come up in 2 of the 3 talks already today but all all of this it also review with a little bit from so set up for abstract settlement have a theory we have been in public really also starts with growth in the bank it's sort of takes starts to take on an apparatus of the definitions in the work of goddamn Quillen and then come on Quillen of course had found himself doing home atop the hearing many different context and he wanted to prove the different approaches to the same home atop the theory where the saying you want to prove the differential greeted algebras modeled topological spaces up to rationally equivalents and differential greatly optimism Colwell was all entity needed an abstract framework for for making sense of them and there's a lot of and Dan :colon spent most of his working career can a boiling this notion down to its essence and nowadays at least in algebraic topology we think of the view that we think of the basic arena of abstract color theory just consisting of a category and a subcategory of morphisms that you intend to identify as equivalences so the languages the terminology borrowed from home atop the theory but you're really just localizing her in some sophisticated sensors just the way Sara localized abelian categories so in this language you have a category and subcategory week equivalences and you consider functors called home with happy frontiers that take the week equivalences to isomorphic and there's a universal home atop the functor that's characterized by would be well it's almost obvious Universal property given a home atop the photo there's a unique front to making the diagram commute and if you're clear thinking about
categories are higher categories this might seem like kind of a rigid thing To request because you're characterizing and arrow into 2 category and so it should only be characterized up to the Contracting Group will use something but it is the difference between this rigid thing and perhaps the more complicated that natural higher categorical statement is just it is just the 1 of asking that the objects of the home atop category be the same as objects to the category anyway that was just me editorializing it's not going to play an important role and the universal ,comma tapping phone I keep losing so a classical example is topological spaces and the class of weak equivalences are the ones that induce Isa morphisms home atop the groups at all possible choices of base points and that becomes classical home atop the theory and the home happy category is the usual home atop the category of CW complexes so and it mentioning that just cannot locate the terminal origins of the terminology for you but also to introduce some notation but nobody really in their right mind ever writes out that symbol and usually it's abbreviated with square brackets so whenever in all these worlds context talk about abstract homotopy theory artist square brackets and some subscript indicates become are it's only logical algebra also gets absorbed in this abstract home atop the theory world in fact Quillen called abstract homered happy theory ,comma topical algebra groups myself the
media doesn't a matter but there you take the category of chain complexes possibly bounded below and the weak equivalences requires the ISA morphisms in your intention is to regard this only study functors that's and cause the ice morphisms to buy some offices so as far as doing whole of theory of algebraic varieties goes on I think the 1st place that's set up and the frameworks laid down is in the thesis of Ken Brown and I didn't put a day down but that think also goes back to the late 1970 and he has paid his thesis was published in the paper called abstract ,comma Tapie theory and generalized chief cohomology really the main thing there is he wanted to set up a world where she'd nology was home atop the classes of maps into some kind of Unilever McClain space suitably defined and to be able to talk about generalized cohomology like we do in Homewood before the end so I this isn't precisely the category he set up but I it's convenient and connects better with the ones I want talk about so so I want considered the category of smooth varieties over the complex number and from and I'm sorry but that's the that's the basic Grundy topology Greta basic category and I wanted just consider the categories simplicial preaching on that so contrary advantage to simplicial set and so that that's just that doesn't have anything to do with the the notion that with with recovering you want your honor recover you want some sort of local to global principle you want to recover the sense in which your smooth variety was built out of smaller pieces and so you end you you you you you don't impose that relation in achieve condition you add that to the week equivalences so the week equivalences are the original simplicial week equivalences and then for technical reasons which plays a role throughout the talk but won't play a role at the level of detail and giving the news than its neighbor topology so weak equivalents as of the of this name it's hyper coverings which expresses the way in which local things accumulate to global things and the the simplicial week equivalences switches what you was the whole point of putting in the simplicial appreciates so this leads to what you might call out of what I'll call algebraic homered happy Fiorina decorate them with and Helge and this is in this world you can formally put in I remember McClain objects for any Schaefer I appreciate the even of abelian groups and calculate hyper colleges chief College in terms of the home happy category OK so and then there's the Motiva ,comma top the theory so I'm bats basically the same except you take the algebraic week equivalences any force that fine line to be contractible so we take a lot work now we want to add to the week equivalences all those maps of the projection maps from the product of any X with the ah fine line Texas and so that's more difficult with the theory then so in varying of algebraic ,comma to studying how much of the founders of the algebraic category would just be things like sheaf cohomology comments systematically studying home atop the founders of the motivate category would would be studying things that are 81 homers happy variant in a systematic way so there is we might call realization phone desire that the motivate equivalences contained the algebraic equivalences so for formal reasons there's a functor from algebraic to Motiva ,comma Tovey theory and for the complex numbers since the 2 that are fine lines contractual and topology there's also a realization from Gerta topology of I might have maybe a better notation would of course Vinci upper hand because it's the underlying analytic it's in the variety to the underlying analytic OK so that's the basic setup of abstract ,comma atop theory and I said I wonder if the question was Can we study vector bundles he in that using abstract homotopy theory sorry but Hakim dark again this or whatever so enough I the so I don't really I thought I was to change all until light they didn't really mean to say you know only the discussion of abstract ,comma tapping periods can have a life back hopefully that'll that'll hopefully will go back OK so far so in topology and refuel few ever group a classifying space for principal G bundles and um Siegel introduced a simplicial definition which I think is pie familiar to everybody In topology principal G bundles up to ISO morphism is given by home atop class of maps into BG so the problem of writing down the set of ice amorphous and classes of K dimensional vector bundles is the problem of determining the homered happy classes of maps into some classified and the same thing is true for formal reasons in the can Brown algebraic home atop the theory of saying more about that in a minute but that that that really just unpacks to the local description of vector bundles in terms of charts and transition functions or from and so so you also have a say in a statement like that there's a universal vector bundle and vectorbundles becomes a home atop problem but home atop the isn't doesn't involve on Monday deformations parameterize tries by that are finalized so anyway with with that in mind let's just abstractly defined motivate vector bundles to be the set of maps in the Motiva ,comma Tapie category from X to BG OK so so right now that's just an abstract definition and there's the realization match between those categories give me a map from the set of eyes morphism classes of algebraic vector bundles to the 1 adviser morphism classes of motivate vector bundles and 2 topological vector bundles OK so 1 can study the problem of take the problem of giving a vector bundle on a topological sits on the on the analytic variety underfloor underlying a complex region projector variety what could take the problem of finding an algebraic structure on it and can ostensibly factor it into 2 separate problems but in fact this is a good relationship between these arms that was stolen turned to so the 1st thing is there's a theorem of Morrell I'm so Morrell published his theory in 2012 but I think it goes back quite a before that and then Marcus slipped playing and then later on Gaza mark oil alignment is bent on a much simpler and more direct proof of this theorem but the theorem says that for smooth are fine varieties this map of the map from Alba algebraic and motivate vector bundles are the same thing and you can think of that as as a as a as a sort of really definitive version of the of Lyndel more even McQuillan deer that Windows theorems saying that at least the there at least it has a chance when theorem says that algebraic vectorbundles is 81 invariant handle anyway this is kind of a it's it's an improvement of Wendell sphere and men in some sense of the more definitive statement that that you might wish for OK OK so that's so that's the case of smooth are fine varieties for projecting varieties but you know there isn't really much of a chance of projective the set of eyes morphism classes of algebraic vector bundles on projector varieties isn't even defamation invariant that so I wrote down an example here but about which is to replace pretty elementary and save it just take in a few takers and we take no 0 of 1 will take the loyalist sequence on P 1 so their rights over 1 plus old minus 1 hour idea as an extension involving I I write the trivial bundling is an element is an extension of all won by all minus 1 and I just moved the axed class in a straight line to the origin and I just wrote down explicitly to do that Mac is you defamation from all of
1 plus all of minus 1 of trivial bundle so probably everyone in this room knows that kind of thing very well anyway so that's set is not is not a anyone 1 invariant and so on and so there's sort of no watch no hope really to describe it doesn't mean it's sort of a non-starter you're not going to describe algebraic vector bundles in terms of motor home atop the the however so this is a question I don't really know the answer to that I don't really expect this to have a positive answer but I don't really know the answer and that is that the most naive answer would be well that's all you do you you knew that the set vector bundles is just algebraic vector bundles Margalo the module the equivalence relation generated by identifying vector bundle over X crossing 1 with its fibers over 0 1 1 of this seems quite strong to me but I don't know I don't really know what counterexample to it and you'll save this would have a lot of if so this would imply for instance that every motivate vector bundle had algebraic structure and again I don't really know a counterexample to that either but I mechanisms insults to convene a meeting and talking about but when you and putting it up as a naive gas they don't really expect that to be true but it would be how it would be nice to even know To understand there's quite a bit better look at OK so nevertheless there's something you you can do and by exploiting the G Girondins device so I just so if he is the easiest example I know of to explain it take are fine and space minus the origin so that's that's defined by can algebraic inequalities and that's not a R find variety but if you give them if you give an algebraic reason for the exits to bees not be 0 so if there is if you take this rainy add some wives that 1 then and that is enough I'm variety and that's a tosser for a vector bundle over are financed space minus the origin and so it's a week equivalents and then if I might out the evidence action of the multiplicative group then I would get 1 of bundle are fine spaces over projective space and in fact and then you can restrict a close effect is a quite general story here but if you have a closet projector variety there's always an are find space bundle and where this J of accident are fine scheme and that of axes is equivalent to acts in the motivate ,comma Tapie category because those are fine spaces are contractible and it's also smooth access smooth because it's it's is a risky locally trivial bundles OK so so in anyway the upside is every smooth Variety's weekly equivalent to a smooth are fine varieties is and so if you knew the problem of whether motivate vector bundle has an algebraic structures this is equivalent to a purely algebraic problem because motivate vector bundles on extra :colon are the same as which is the vector bundles on and you want to device and those are the same by the fear of Morrell and Ozark can only 1 went to vector bundles over the issue on the device and so Mike the question I was asking sort of really comes down to the question of whether a vector bundles on July 1 of devices extend to the base space and I don't know but I don't know an example of 1 that doesn't see easy to find the mother and the descent that vectorbundles can descend in many many ways but and not I knew it would be nice to get to have I have hands on an example of 1 that doesn't but OK alright so that's that's a little bit of a tour of the work of the easy relations over the loss of the known relationships between algebraic Motive Inc and attribute vector bundles and so now when I say I want a study vectorbundles doing home atop the theory what I have in mind well so in any abstract home atop the fears anytime you set it up it comes to you with its own kind of internal notion of homered happy groups and its own kind of internal notion of cohomology and there's always there's a scheme rod kind of obstruction theory you can always at that sort of also always you can always exploited always relates that I've cohomology of acts of coefficients in life ,comma Tapie group of that supposed to be an X term maps in the home attack the category from X to watch so in algebraic home atop the theory if we were to look so this obstruction during gives you an idea of what these categories how these categories are kind digging into the theory of vector bundles and so this gives you another way of saying why the algebraic home theories is just a formal it's really just kind of formal the home atop the group searched the home atop happy sheaves of BG OK artist G OK 4 nines 1 and the trivial otherwise so this just becomes the statement of the obstruction theory just becomes a statement that vector bundles as 1 with G L K coefficients so there's nothing that doesn't penetrate into the fury of vector bundles at all on the other hand Motiva typical atop the theory does so all of the fundamental group of BG OK and Motiva ,comma topic areas Jia 1 and 2 is the 2nd Mueller will take on rarified numerate Kayseri in them there's a little bit of information known about the higher homered happy she's but it's it's it it's a little complicated and it does lead to an interesting obstruction theory story Filed reject this year he was still applies to all pie to BG OK yes thank you so you think you yes not to right yeah I colony KDB yet at least really yeah sorry about that thank you so I want talk about this obstruction vary from but so this instruction theory this will apply if you're studying them vector bundles over smooth are fine varieties and that was something that was studied pretty intensively by Griffis said others in the 1970 is using analytic methods so Griffis Griffis idea was you you take up to take the under the a complex vector bundle bye bye Grauer steering has a unique hold more fixed structure and if you can give them and so you have a whole worth a connection if you can make that connection algebraic you have an algebraic structures and so he studied using value distribution theory the growth rate of the connection form as you go to infinity and he found kind of a series of obstructions to algebra algebraic sizing vector bundles on at these give you a different series of obstructions that a very homotopy-theoretic and very kind of discreet ceiling and 1 of them things we haven't really worked out but I think would be really interesting is to understand the relationship between the coal nology with coefficients in these homered happy she's and the value distribution theory that the Griffis was looking at that because there they're not and there's a kind of this but agreed that I obviously related to each other at all so moral and other can fossil investigated problems of splitting free summons off of projective modules over regular ratings In low call dimensions using these methods and John and Oregon and I to the week this was sort of more proof of concept at this point in anything because so the other thing from that era in the seventies was a sort of philosophy and I don't wanna call it really is but there was just a sense that if the Chern classes of a vector bundle were algebraic the bundle was algebraic and and that's probably because of that program was designed to get the Hodge conjecture and the idea was once you all they wanted was the churn classes to be algebra but so we started dug into this and and just produced an example of a smooth hyper surface in P 1 crossed for which the compliment
has algebraic churned classes but it's not Algebra algebra sizable and so I think this is the we haven't done it for this example but I think this would be a really interesting 1 see quiet from the point of view of value distribution theory that this bundle Frank to bundle doesn't have a whole lot more algebraic structure the history of the world have if you make new EU Yeah yeah yeah yeah this hat is is important that its two-dimensional if I make the rank bigger it'll be compelled to rise of all but I think that's right yeah I think that's where is out of question was OK so so as I said I really do what I think so that's that's a little bit of a tour of canned 1 thing that's going on ,comma but where were right where we're starting to see some interesting project is progress is in kind of a kinder varieties that are as far from the Hodge conjecture can be so that's the problem of defining algebraic vector bundles on projective space and line bundles we now also the 1st interesting case 1st interesting problems to classify the right to bundles on projected space and time but the course very famous problem there's hot Sean's conjecture which says that if any is greater than 5 every rank to bundle was a sum of line bundles and there's something I'm going to mention it because it plays a role In the story and telling them so I don't know what to call this a good caught the Grauer Schneider problem problem and it says that every month stable right to bundle unstable in the Mumford sense every unstable right to vector bundle on piano is the sum of line bundles Sylvester and 4 and on and on I'm calling it a problem conservative well because there was a paper and then shortly after the paper was published there was a mistake found and and the proof is never been repaired but impacts the timing that paper impacts the story and telling OK so so if the churn classes is 0 that implies the vectorbundles unstable now I am so in the 1970 is there or so With this this hot conjectures round the Grauer Snyder theorem was a feral and so then the question turned to some topology is still produce some as topological vector bundles with no cheering classes for instance and so Larry Smith did l were restated and um Bob Switzer the kind of 10 meter elaborate table of vector bundles topological vector bundles on projective space through a range of dimensions and I think these are all right to bundles so ElBaradei's using homered heavy theory constructed some right to vector bundles on TV and with no chance classes and at the time of writing that was those were examples of topological vector bundles with no algebraic structure and some but now I that the current state of the art is there still no known example of a topological vector bundle on projected space that doesn't have an algebraic structure and 1 question we might ask is do the regional is this was that Rhonda the responders actually have algebraic structures now I OK so so when we tell you a little bit about the respondent's cell so in topology you study the home topical as the maps and B U 2 in terms of the Cold War allergy of projective space With coefficients in the home Tapie groups of of beautiful and those are by long-exact-sequence the fact that we have whatever that those are the only Topinka was 1 dimension down of U 2 or S U 2 or S 3 so what you need when you try to understand right to topological vector bundles on complex projective space of the odd homered happy groups of the 3 spheres now the 1st odd home atop the Group of the 3 sphere of here at the prime P is in this dimension pie for P minus 3 and Ellery east and just took that element and that may be obvious vector bundles you take this complex projective space of that dimension to P minus 1 collapse out the lower dimension no projected space together for P minus 2 sphere and go over by this 1st by chosen generator of that group and and then galleries proved that these were nontrivial for all p he proved he determined To what larger projective space they could extend the answered the kind of questions you'd you'd want to know about the and those bundles of course have no cheering classes at least 4 Pt something 3 maybe even bigger because they vanished on the complex projective plane inside so that's predicted have no algebraic structures OK and so on the Saluda predicted and so now I wanna go back to the sort of way I was setting this up in ousting these bundles have motorbikes structures and if they do can we get those 2 have algebraic structures and I so In order to do that we need to we need to come of see if we can do the same home atop the same kind of we need to study the same obstruction theory and motivate the Motiva ,comma topped the category so Montana ,comma topping theory there's there's not just at best and for every there's an S P Q and I learn this stuff from Morrell a cooking and Boulevard sky to live in the late nineties and was taught to use these 2 indices so the first one stands for dimension and the 2nd 1 is kind of a half what you can think of is the Hodge weight some people nowadays and if you're 1 of them I forgive me some people use a different conventional those so so that a 1 1 sphere is GM or that fine line minus origin in the 1 0 sphere which you should think of his religious ,comma tutorial Is that fine line 0 1 into the Athlon lines contractible amenities that's really just the server the circle in the simplicial direction and more generally S Q is governed by taking the smashed product of these with each other I but the ladies come up in more geometric questions is that fine and space minus the origin as dimension to an minus 1 and there's a little bit of Hodge weight of and in there an n-dimensional projective space model hoops that sources that supposed to be a P & minus 1 that's the 2 and hemisphere OK so motor-vehicle McHappy groups and and there's a topological realization functor before and under that the these byGrayden spheres they just go to the underlying topological sphere with of that dimensions so there's a map from a B In Motiva ,comma topping theory 2 just ordinary group pies so what you're expecting when you look at this the way you expect so long familiar elements in the home atop the groups of spheres lift the Motiva ,comma the theory pioneer vessel the you can get the hot maps but then there's a lot of more the elaborate discussion did things at home atop the Serie where you using the fact that some probative things a 0 in your building some higher-order product and there's room in there for the Hodge weights for there not to be a consistent way of lifting the Hodge weights so when what you're looking for the wheeling for when you're trying to show that something fails to lift from all ordinary topology Demotte today ,comma Tapie theory the service on the only thing you have to look for is some it's me and I think of it as kind of elaborate refinement of a Hodge conditions and it's it's it's somehow I don't know a great way of expressing it but you know what you're looking for is there not to be a consistent way of putting a Hodge weighed in to some construction now I actually had prepared I wanted to show you a construction of this element in FY pie for P minus 3 but I can do that on the board later but I decided not to do it I've done it and it doesn't I found it doesn't
go so well for in in in this format but if you want to know about it or tell you about it that but I want to tell you 1 deep theorem that goes into it so you have to get started somewhere and where you start in home atop the theory as with the element and pi to end of BU generator for the K theory in the 2 ends so that by the on stability comes from an element in pie to pie To End of B U N or it's an element in prior to and minus 1 of you when you start with that element and that element has degree and minus 1 factorial when you Matthew and down to the ends fees so this is this is the key you start with that and then you do a bunch of things and I just up the 1 who do so there's a lot there's some tricks some localization this system boilerplate stuff in home atop the theory that you can imitate Motiva ,comma tubby theory but the key thing that gets you going is this map from the spot sphere into the unitary group and that's provided In algebra bye and by this beautiful theorem was useless thesis called and factorial fear and that says that if you have them a unit of modular run a sequence of elements of a range that you could be the 1st relevant and by and matrix with determine 1 I know you don't get you don't get to have be the 1st row of matrix of with determinant wondered if you raise the entries to enough powers where the products of the powers is In minus-1 factorial then there does exist a unit modular matrix and that that's what gives you this element this lifted this element in pi to win minus 1 of you and that's what makes that algebra and so this mismatch between and factorial and minus 1 factorial just desire started counting with 1 instead of 0 OK so anyway you can use this you can use this secret construction which I'm willing to tell you about that I haven't I haven't told you about it you can use this construction and if you follow that you wired up with a typical lists of this 1st element of order P amp I for P minus 3 2 spheres so that has order P it's got dimension pie for P minus 3 and I Scott Hodge wait to now if we go back and look at what we want and we wanted to imitate Reese's construction by mining out a lower dimensional projective space and that gives us a little so the dimensions trust me they have to work out that the number to pay attention to is the hard way behind Wade was supposed to be 2 P minus 1 and we got some hard wheat to peace so the Hodge weight is wrong and you might think that there would be a way of exploiting this to show that these don't lift to motivate effective but saw the fear as they do at the end of the day they live to motivate vector bundles and the reason is that Salomon had finite order and in Motiva ,comma tapping theory there is this the operator run that lowers you can lower the Hodge weighed on the torsion home atop the groups and it's pretty easy to explain what that is so that so let's take that let's take that long among Take the degree and map on S 1 1 and I'm going to call it off so that mn is the more space for the Mott the degree and that puts the mapping co-owner the degree and map on S 1 1 and I I can so I over the complex numbers so I can choose a of just take a linear map in a 1 that connects 1 2 were chosen and through the unity and that gives me a map From the others fear S 1 0 internists and 1 1 the focus of those so that's an operator right if I have an element in Taiwan 1 or higher of order n I can composers with this and get an element in PY 1 0 and these are compatible with and and what is this realized too and topology and so that circles wrapping and times around the origin and that line is my line from a 1 1 0 1 it and in topology that's of course ,comma topic to just that arc which goes once around the bottom circle so in atop theory this row realizes to the standard inclusion that just says so let operators that that starts with an element of water and and says so that I can just there was using was an element I don't care any more than an order and it's just restricting so map from the 1 thing is isn't as an element and I know home atop the event times and just forgetting and all homered OK so that gives you this operation realizing to the identity map and that lets us a lift to respond tools to motivate vector bundles so I so this is something so these are rank too projected modules over that ring I wrote down over the degree 0 part of that really I wrote that I have no idea how to write those down as projected modules we know that if you add a free module to a may become free of ranks 3 but I don't know but we can't go through every step of the construction and producer was in fact every vector bundle every topological vector bundle I know From Switzer's chart from there I think I can lift we can lift too we vectorbundles over those you want device so so this method gives you it produces a lot of New modern right to modules over these projected models over the this you want with device but the problem of descending from this you want device to protect space that could be where that seems hard and um I don't have anything good to report on that it could well be that we've traded all but 1 hard problem in 3 different ones but these in but nevertheless these so this there's a lot of new rank to projected models over the years Over these rings so so
that's that so that's the the idea so that's the main result is that the from that at least the Rees bundles and there's a lot of new wrecked Motiva connected bundles on projective space but that was a really the main as part of the main thing I want to talk about so I told you that where you're expecting something to go wrong in lifting something from ordinary home atop the theory to Motiva ,comma Tapie theory is that the Hodge weights starts a sort of their start to be something inconsistent about them but there's this well operator and as soon as stuff starts to have finite water but you can shift the Hodge weight down as much as you want and this makes 1 wonder if there's a way sort of general idea that might be the show that you can always lift things in certain cases from a topological things to motivate ,comma public theory and there is so this is a kind of hypothesis I guess how would even think of calling it a conjecture call the Wilson space hypothesis and it takes something that I think is 1 of the best-kept secrets in homotopy theory and the conjecture is that the analog holds in Motiva ,comma Tapie theory so I'm just going to take a couple minutes and describe it and so on ,comma tabby theory there's this remarkable class of spaces because even spaces so there even in the sense that there ought homered happy groups 0 and they only have even dimensional cells no that simple there's a whole bunch of really not obvious things that come out of this so for 1 but you know that it turns out that the even homered happy groups have to be portion free so if the space has finite type those have to be free abelian that's not obvious from the definition they turn out to be infinite loop spaces and in fact there is trying once there's there's and there's a lot of Wilson's space and even space that starts the server minimal 1 for every even sphere and and I believe those have unique infinite loop space structures and everything is built on world this this theory it has it has a remarkable rigidity so examples of these spaces are infinite dimensional complex projective space 1 that's a Kasey too the EU the classifying space for the infinite unitary group and the sq so I want to just point out we know all these are even the having groups we calculate for some reason the cell composition comes from algebraic geometry and we we have algebraic solely compositions for CP infinity BU and exposed the S so the 1st example where you don't know anything like that is the fiber of the universal 2nd Chern class so topology that's called BU angle brackets 6 but it's not a very useful I mean it's a notation that tells you what it is but it doesn't not suggestive of anything else but that space has only even dimensional cells and I don't know an algebraic geometry reason for that you I'm sorry New even here you know that's the 1st 2 into yeah that's even for those of you who is also a good word for word of her over the phone the the rise world How do you know what you are asking how do we know that's even you have just you so what goes into that is knowing the spectral sequence and knowing the college environment McClain space in the action of the EU's steam run algebra Lineboro McLean's space etc. so I'm about to ask the same question it Motiva ,comma Tapie theory where we don't know and the knowing that even would tell you the college Abkhazia for you can go back and forth so it's it's it uses that you have to do the 1st was a good the or so it has even Howard happy groups that's obvious right has even sells because I calculate its cohomology and I find it's it's free abelian only an even degrees In topology that's enough to have and have even sell the composition OK so Steve Wilson proved this amazing fear that the classifying space for complex scoreboard is amusing even space so the fact that the 1st condition that its home at Happy groups are even that's Mueller's computation of the complex could borders and bring on this it is Was it is a complete surprise I'm an effect that's true for every even suspension positive or negative of and you and I and the Wilsons basic pothesis that I wanna make Motiva ,comma Tapie theory really involves all those suspensions I just didn't I just kept it simple and I kept the statement simple for these purposes so 1 so this also has a highly computational prove it's been tricked out so that its short now but it's it's a very sad that there there there is an unknown geometric reasons but the kind of part of the state what what is the use this 2 0 space of the complex could borders and spectrum useful to think of that as modular space of 0 dimensional Staveley almost complex manifold moving through cobordism so those are so if you really go through Thomas theory these are 0 dimensional manifolds embedded in a big complex vector space and as they move they can transform through call borders and and I asserted that weights and make you kind of think Hilbert skiing I don't know what a real relationship between this and the Hilbert scheme but it's there should be a kind of Hilbert some sort of stabilized Hilbert scheming models for this space and the cell the composition would just come from the poorest action on the big are fine space just like it does for the Hilbert scheme of points in the plane so what there should be so this is that the policies don't have a good enough geometric model for this to to prove this theorem but it it it would be that would be 1 way and that would be something very very useful to have and it does feel like the art of the well I can said some kind of some variation on Hilbert scheme
kind of model OK so I want to make the exertion and the there can ask about the exact analog and Motiva ,comma tapping here in the summer a little bit of guessing but I think that they have lot of those 2 conditions is that pipe to and minus 1 on VAX is 0 for all and and that it had it's built it has a motive Excel the composition of those fears so I stated that in kind of a clumsy way there but let me just say anything with an algebraic sell the composition has satisfies part to soapy infinity BSL those are all examples but it's not known the first one we don't know Is the fiber of the universal 2nd during class so for that so that I can do this when we don't know and we don't have and it's mostly because we don't know the cohomology of that Of that Eilenberg McClain space or Abkhazia to 3 so that's 1 the first one we don't know if it's even I we do know answer that the Wilsons space hypothesis is that the that that classifying space for algebraic aboard even and I really mean that smash as to when and for any now that's known to satisfy 1 so that was a year of morals and mine long time ago and I think there's been a lot there's been other proofs anyway those those groups of the motivate borders Inspector be 0 the other thing I want to make a comment that they started talking about being over the complex numbers and in all the proves the nite I needed role so I needed to have some roots of unity in particular you could keep track of those Paolini minus 1 to be a sum of squares for various reasons but this this theorem so you could ask is this true over the real numbers Peru is it reasonable to have guessed this over the real numbers then you would want the Thierman apology to be compatible with complex conjugation so that my children I've checked at hand so that the fear that topology that's true the the analog this is true in z to equity area home atop the theory so I don't I don't know what I think this to be true and I can't willing to run with it and but the indication is that this would really be true I think 1 would expect this to be true over Q or maybe even overseas if this but I think anyway this seemed like that this there should be a field of definition issue with this OK so what that Wilson's space hypothesis let you do is build a resolution out of the motorbike Islander McClain spaces Kasey and 2 and so on what you do is you take the and you take what part is called unstable Adams Nova cover resolution Wilson space hypothesis says you never leaving the haven of even spaces when you do that and then the overly broad sky the Posner ,comma the sliced tower for any 1 of those spaces around is a product of these is the associate agreed is a product of these Kasey too and so if you get a resolution in terms of specific motivate complexes In ordinary home atop theory that's not such a big deal abelian groups can be resolved into free abelian groups that not much costs but in motor-vehicle atop the theory that olmert happy she's can be any strictly 81 invariant chief and its they can't always be resolved in this way so so that's the the Wilson hypothesis is that it lets you make a resolution that you wouldn't know how to make an so so I've been kicking this thing around for a while in a couple years ago where are the bodies of 2 years ago you were creation asked me about it this sort of these compliments of the discriminate variety weather finally generated projected models over the free you run that through the Wilson space hypothesis you find the prediction is that they are and then he got was able to find a pretty elementary algebraic proof of that but this was you know this isn't like Jarrett predicted the redshift of mercury or anything but it predicted something we knew this was sort of a reality check for so the plight of the Wilsons space hypothesis is MCI put this pyramid "quotation mark just because we have a written down the proof involves a lot but it involved bringing a lot of sort apparatus of home atop the theory in the Motiva ,comma Tovey theory that we've we sort of and we understand I want to go but I just put in the quotas we haven't really written paper we have written it all down but if the Wilson space hypothesis holds that there is no difference between them motivate vector bundles on projected space and topological ones and that would that would actually be true not just for projected space but for anything with an algebraic cell decomposition so Cross money and blowing blowups along linear things I mean all that into a class of varieties with algebraic cell the compositions Savino difference between beatific and topological vector bundles so that at least give you lots of vector bundles over these Juwono devices and and not too bad but then you know I don't know about actual projected ones so I think I've had my time limit but I'll just leave you with this diagram and about this problem so you know the Wilsons basic officers implies this 0 Azerbaijan action and and we can lift a lot of things to it that gives you vector bundles on those you want In the Juwono device and then the question about vector bundles on projected space becomes it's possibly difficult problem and if if the could you come back to this later the of this and this month it the "quotation mark the the other 1 the results of the British and chest municipal the book and the replacement from among all registered this sphere national rejected the tradition and we should and must make a collection called "quotation mark may also have to realize some of you was With expectations there should be a revocation of license for this is that it is already in the sense that he lost his collection are involved in this region due to its local because of this but this is as it was kind of is that in the paper than some would want to use OK you are asking later he he could help me member of find that out that sounds interesting that the smell similar thank you on the other it was that have them
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Titel Algebraic and motivic vector bundles
Serientitel Séminaire Grothendieck 30 mars 2016
Teil 04
Anzahl der Teile 05
Autor Hopkins, Michael
Lizenz CC-Namensnennung 3.0 Unported:
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DOI 10.5446/20729
Herausgeber Institut des Hautes Études Scientifiques (IHÉS)
Erscheinungsjahr 2016
Sprache Englisch

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Fachgebiet Mathematik

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