2/3 Lagrangian Floer cohomology in families
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Title 
2/3 Lagrangian Floer cohomology in families

Title of Series  
Number of Parts 
36

Author 

License 
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2015

Language 
English

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Subject Area  
Abstract 
We will begin with a brief overview of Lagrangian Floer cohomology, in a setting designed to minimise technical difficulties (i.e. no bubbling). Then we will ponder the question of what happens to Floer theory when we vary Lagrangians in families, which we will not require to be Hamiltonian. We will see rigid analytic spaces naturally arise from such families; these spaces are the analogue of complex analytic manifolds over the Novikov field. In order to be faithful to the theme of the conference, we will end by constructing lots of moduli spaces in order to see that Floer complexes give rise to analytic coherent sheaves.

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we're the and some clarification so time described the moment we have now were too on spaces Q and I said this is married now in order for them to actually work so this is the final if welcoming you can actually have some slightly conditions let's help and you don't know so the 1st thing I want to do is give some context for what where this module is coming from and more generally what happens if the then there's the 1st part of fully only take 20 minutes or so and then in the next part II said that there is some kind of this is some very severe completion you could do something slightly milder which is still well which has cannot how the various properties but which will no more and no less today give precisely what it means to recover more information so 1st the context it was that was going back and forth at some point I said it but I didn't say no the the point where I introduced you In name of the context the world OK so the context is that so if accused have defined category show called Sky category next next summer closer like the manifold or a maybe it has come down some property if it open has some properties of all of this the state and the objects around you manifold let's make them close Lincoln not in general this category is kind with complexes general this category should really deserve the name category because it is more than these complexes they're going to have some kind of French but the squared it is not equal to the area instead we have which we can call and 0 about that lives inside coaching chains of held itself which in the world of usually identified with some coaching and 0 . also lives in prime prime and instead of getting the square root causes and squared equals something like an 0 l times ex post minus taxes time so invited as a generalization this is kind of already unification work were generally we have the subject together with something on the and this is a unitary local systems the system and I think essentially if you look at what's written in the In the literature it's only rank once all just imposed this year in the it's not fairly straightforward to generalize what they did to the higherranking case so it it means to have a unitary local systems local system is
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rank 1 vector bundle
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for your space and in addition you have some data of apparel transport and parallel transport medical depend on the to be classes of possible talk only depend only
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dependent the on cars up to the left on pads up to the heart of the I have to go to the class so this is classified it's not class all of this is classified by the company which in the red 1 gate prison elements all of the major of 1 the held with provisions of unitary group known for working over the complex numbers I just write like ominous 1 year later basically you want but if you're working with another covering we can't do that this 1 is the set of things that no 1 so we need the analogous thing which is a set of things of Norway won over the over the counter and in that that just said expressions of deformities 0 plus some of the IT and I will and I before it was due were arbitrary real numbers going to plus infinity managed so that it was infinitely greater than we 0 the at and a zeroes in on so that was a wonderful things about guys in the former of group under multiplication you can just invert dash Nunavut everything that's unitary group and this is framework that is discussed this you now you can define complexes In this exercise wonderfully never done this if you have some idea to the final round of the ecology group so try to define I just don't worry about the issue of this novel and this is not what you should try to do hard to define the French clause so it is that Indian hideous so well equipped with a bundle in prime with about Yukon which is defined to be the direct some overall intersection point of pride all halted clover next to Friday products into the picture you have in mind as he hears my health here's my pride this L is equipped with some local system operates a rank was going around well and the fiber at this point callbacks that fiber called and the other Lagrangian is also equipped its own 1 system and the fiber over at this point I called Prime so this vector space is supposed to be just year morphisms from this and now that we're on the right 1 case is just rank 1 thing over land you need the same thing as before you look at your home office desks and you count them appropriately used somehow the Senate United of the theory should be able to do this and that but now now you have to know if the finding of the map from lender to lender surviving number which defined homomorphism from this Frank 1 local system here to this problem look so that's the site so that when you have this generalization that the terminology the pair L the unobstructed if the correspond responding in Europe and Europe grew so again and have to write it an LED another bench now but not know the using again In the usual case you have a complex vector bundle and monogamy is it trends out the Baltimore prison which is known to 1 1 of the introductory session so sorry I'm so now now I have my own terminology which is he is topologically unobstructed funds sorry structure associated to the limit carefully also carried accomplished strut around in there exists the Jag almost complex structure on the Net keep the it was its but I can tell you that it's encountered this 1 .period boundary on part of that is that there is not 1 that that's when it's encountered this on but again if you don't notice exercise without thinking of disks but said that this so that you that's my that's that's definition and the point is that if you want to understand and what generality does this but they can I tell you that this result holds it that's on now the following L practice With this Fla theories welldefined and Cuba is tautological incorrect but so still this this this from a formal .period usage is still not very satisfactory because of their had whole category and here I just haven't this is the of of what not that within a few minutes if know if if it's still defined which you were only unobstructed then you'd have a problem and it is not enough neither ministry I thought I was strong you have to love it he noticed there is nothing to think about it for 1 that's good enough for that 1 opportunities good and you just use that Jane in Europe construct nature use the fact that when you define the theory when you look at your desk you not to use the same J at all .period so when you know and Ellen Q Interactive Media Guild of his different useful so on that that's what we need but admitted that there there's some kind of there's some kind of more general context which I think is important to 2 to realize that this group is it's really just a very small part of the of general strike so environment there is an enlargement of this and unfortunately there's nothing that isn't literature so just some some some here I will write this down there is an enlargement come off so it is a little guy category in which objects pairs L it's not fine drink anymore so this is derived local system which is an appropriate sense unitary which is actually the most delicate part prior to communitarian with the club it's hard to make sense with what maintains it's that delicate to make sense of what that means Though I introduced in King Ocean for that that makes it a reasonable unitary and but of arbitrary arrests so you could say what what example should I have in mind and the example but you should have in mind it something that I already its assigned to any point P assigned to the public .period political next acts in In In a few cases like this and turn this into a Q assigned to thinking the completion completion all of the changes on the space of patents from the base .period to x coefficient because this is the completion of a introduced the last time you know you could be a good nite anything that you just can't take the change on the states of passed from and from
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contact with coefficients land that but they should you last time that's not a very good notion because when you try to clear the minutes but you still run into the problem of not having enough not enough infinite expression to take this completion that puts the on the local system over the of the canonical drive local system that's exactly what's known so in the end you and that meeting a module over this corresponding thing OK please note that any of other during that's delegates to know what it means for that to be a but you can just you can just isn't just funny to be a model this this thing where you just put Cucuta units and other related stressed that Anthony a model of that and that's your notions this derived local systems Over keeper OK so that's that's our the claim there is a generalization and it has the following wonderful feature so the authorities said .period what can we discussed it has to do with my so the facts it's going to certain infinite expression convergence he added because there are so many elements in I said that which is which is the foundation of the of the interest the reason for the introduction of the model and so did his bounds in this sort of thing then if I take the the floor coaching so I need a name for this for a local system and what I will do it I will be something evil I will just soldiers New Englandbased space Q ,comma blank so the flamenco chains of Kill equipped with local system which is the chain's completed 1 accused blank lender with itself the flora and amorphous this is a arranged this is actually isomorphic too "quotation mark was accused so now not now Nunavut is keeping so yesterday I was talking about this guy and we were building modules over this now I'm telling you that this thing appears isn't very large for category as morphism rather some object of which is which is just this Lagrangian equipped with some local system on it so in particular if you have any liberal from given Alex we 2 model can produce that this is a model God this space and we can also produce the far from but putting decoration with the object and this is module this complex and claim that there was some so again appointed I gave you this thing as an isolated thing it's just song group having to do with Alan Q but in reality it's not it's just some more the group in some category and so it is that many additional structure that that you know we won't have time to discuss Lewis was forced to this module is for the equivalent of this much the 1st equivalent vector space and exercise I get enough here that you can just proved that has a complex chain complexes just this 1 and this 1 and I would change the face "quotation mark and American Standard actually isomorphic I just did what I find this is just and you can see it because the local system is based those from cute to something and what is that something at the intersection .period that's exactly how we fund OK so why don't I go through this introducing a margin of just because yes we know so the flow complex where this equipped with his local system which is the local system on this Lagrangian whose values at 2 blank it is something like chains in the space of parts from the base .period OK so so so why did I do this I did this because I you know I just more than 1 person asked me What happens if Q is not In general construction this gives an AntiDefamation off the train on the completion of the change right while will and then in general you don't get to modules cities and help which was defined as a module over this guy I'm not defined as a model this kind was that the final minutes different and that somehow story that I don't want to go to too but I just want to mention it because it's sometimes hard to know that this is just he was the beginning of a very long story and and the other thing that I want to mention is that you should think of this isomorphism here had been having minus the Africans to say it is that this is the analog on Net if it if you use them if you use the trivial local systems Frank wants you just get the ecology of cute itself ,comma now I'm telling you if you don't do that you huge thing you get instead it's what you would expect from this some kind of competition OK but all of that and by that I mean the only homomorphic curve theory you need to know to prove this is the homework occur .period boosters and the rest is altered OK so now we switched gears will return to this To intermediate completion to the relations this kind of serious meaning these kind of logical and l hats UK flocks so my goal is to explain to you usual the theory if you compute the local knowledge between the 2 was granted another Lagrangian plight many points out that the I think this Lagrangian here and now take some other Lagrange unless they have arranged things that picture the Senate we've arranged things so that these areas and areas that then what we know that the political Monty is nonzero it's great we're happy was your vocal logic but if I take so this isn't cure was going to be all but there was 1 there tonight CQ and I knew made by an infinitesimal amount I did some titles my goal is to say something about the floor ecology of this Lagrangian that's been moved a little bit with the orange granted the new rules the theory this is a very delicate
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point if you if you work really hard you can kind of workout something infinitesimal but it's really hard to define stated that tells you what happens local knowledge of this community we actually move away the reason is that you can just see over here and new orange and white the knowledge is nonzero indicated the exercise the body of orange and yellow local knowledge is 0 that was 1 of the exercise last so we were trying to work trying to undo this problem we're trying to find something involving a L & Q that will tell us about L and yes I found that in the usual setting using the local knowledge L with cute and that tells you essentially nothing about the usual flair theory of Alan you so even the L and in acute chips on very close to each 1 of them is 0 1 exercise last time I ask you to ask you to compute the local knowledge between something like this and something like this would area here is an the area here is being any basis in is there as long as a is not equal to the and stood in the way the article stated but so so 1st I will introduce does something that is most natural from the point of view of selecting topology even tho it will turn out that we will come to the site of quite a bit to get something that other Brinkley convenience so do so they want to the find but given signal which is a change OK maybe cubicle but today manages this singular 7 will have to worry about into the into the basement to define the line the signal to the market that I keep and keep worrying Winigan writer or anything it's the maximum Over sea in the domain of the length and cost of those segments of have been so you have this is a family of loops they have with respect to the peak summer money metric like the specter from so what we can do now is making defined by finally I said there is some kind of along dozens of prime completion which wasn't quite as completion the final of the completion of the Feinstein that this is going to change the basement space and I completed that as follows this is going to consist all series it has before to sing my think I there is a change In the usual change coefficients payment Grandfield and now the Forbes said that the limits as land I suppose invented as as I go through infinity on the land I was infinity now instead of the required the limits of the demise of summer limits as I used to infinity 12 and I minus epsilon times the length of equals so there is for a 2nd so the only so this is is actually the 1st observation of the year we of always positive so if this expression goes to infinity the that particular if you don't have negative terms than the land on which infinity that's the 1st observation and of course this includes the usual training With completion whatsoever .period basically because if you find at any time there's nothing to say OK so now comes the 2nd exercise we just this 1 is not for the 2nd size as show that this is very different from Renault well actually tried to show that it's a different agreed or you will run into a bit of an issue because you will run into the issue of understanding why this is French agreed in algebra and so pretend that this is the 1st agreed on the ranch and this is the case or if you know how to make this a diffraction grating algebra just use whatever you do to the key factor that the length of also we multiply to chains ships as their top you can relate that I will let you think about the world of intersection equal to the length of the attack and the length of the boundaries of his is 1 I always get wrong so it's the maximum sentence is less than or equal to who can the new and yeah people do that thing yes but grandma used it to prove that there are no manifold with infinitely many contractible genetics are for example I mean not basically things like that he did on the field of space so actually this is the addition of very good question this problem working over this spring probably carries a lot of information but I don't I don't know how to extract a lot of very delicate information in order to extract the argument that still look so comes on maintaining the wider introduced this because there is a proposition which is due to groans the Solomon Jake that the right references that student only and then there is slightly so that the case is not literature but I know that higher had gave in talks where he was doing the rounds of the theory for immersed Lagrangian explained basically version this and there's also a paper ballot does something which is not quite directly applicable the right again and says that's up to 4 weeks I tell you there exists a quarter along you greater than 0 for any homomorphic straight with boundary and Alan cute the Norfolk of about just right instead of doing that L U in modular space trip why with some intersection points prior to that I think they had assumed they were transfers OK so that for all such things we have the quality the the strike meant like a word of the length of the ban you would like to say that it's less than the area but of course I don't know if that's the case but if I multiplied this excellent and so this is called a reverse in rejected the the thinking and why is that OK this is reversed and size of parametric equality in basically says yes and I'm not going to prove that our say why Howard through referring to jail I it basically says that the Of those you can do it you cannot have a very long you cannot have curves of very long boundary and very small area please no subsequent you are thinking of the things that go into improvements monotonicity in which in fact the inequality goes the other way and then and then you get squared but that's like an estimate for things
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very small area this is innocent of the very large for probably yeah sure you can ask a feature you want to know the absolutely that's what I said That's what has grown a new you can use this information to detect usually for them on the Freedom face you detect she does it by saying there is some homology class I don't didn't the you units had some information about how it is used yes you guessed it some cycle appeared disappear and that can be Atkins OK so great and so I just making sure that I had missed everything so that's that's the that's the statement and the location of the site is to show that this the expression some Over here To the Member sorry data maybe this is basically there is a great times the evaluation of Of the modular space of curves and classes at 10 1 the cost base that would boundaries next lies in this completion How long Of the space parts from text to water here is where the epsilon is chosen as is Grumman on with so the connection between the geometry and the other rockers here so you must time said the fact that these groups but the fact that this and have had that I constructed as a module is the fact that it's welldefined subversion brought compact I should think of this as some kind of refined version were of compactness not only did I tell you I tell you something about the there's only financial compactness for faces curves and Bund area but actually also tell you something about it as the area goes to infinity what happens the line and once know this refined version girl compactness I can go down from here to here smaller smoldering OK so now I have to do so gymnastics extracts of it here as it trying to trip the president is very intelligently but so what does this have to do so particular lenders continue to the 1 where once you know that element lies in brings you just play the same game as before and political we have something like that at all the facts which is the fight over this spring model fine and and actually at this stage we can start talking about the connection with flocks but it turns out that it's better to to you you slightly so that this is extremely this is an extremely delicate but ring to work with like for example no it may have like that that the amount of energy that you need me to represent different from all the classes will be different even if the classes line thing components so that's what thought it's it's really it's really delicate but so what we're going to do it is His work with some slightly less refined grains OK so so this notes given any conclusion the ratings like this where we have seen that Absalon correlated Q inside something else came so that American do anything you do any more than just going to make my life easier to imagine that some other completion which is called Prime Time about to explain will be very special kind we can again we get a module come on in and had prime and model it was the model and had over this meeting just like answering it 10 this month With this over this and it will again be perfect and basically this is a free modules of them when you tend to yield for Fremont still so now the goal is to find find such kind different regions which are slightly easier to describe 2 the final completion let me start with us let me note that Sigma me again the simplex and images precarious and thing is the dimension into some fixed component of our those basement this is a complete labeled bye so that is a decision that this expression the length of the is greater than or equal to the minimum length of the current government appearing here which is itself greater than or equal to the minimum Of the length of the for gun which is homologous lying in and so that's what we're going to do for allowing 2 instead of using this very refined things were every chain carries the information of its length into contention that were doing this slightly less refined thing where every chain carries the information the top of the class and we become complete with respect to that we forget all that it only has to knowledge so given that I have used this when it happened so let's assume that we have more on each 1 of you in the eye and I know yes it's intentional its intention so given and on which 1 of you with our coefficient would do or and went on to the final word to define completion associated to just ignore the easiest way to do so it true now that we have now there is but you don't actually being used this way later on mixed annotations that norms on page 1 and let's the inside of here need a set of elements enormous unable to 1 and then I will define no
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my new completion and this is now finally somewhere will find the kind of place where algebra
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appear here the flux will make sense so when the Finder completion With respect to the
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United States in terms of polygons conduct substantive 1st
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promoter of this place space efficiency and this is defined to mean that expression of the will and I'm hopeful where troops but let me call them signal light can wear but so Singh on I the island chain in the space of part of the class of going so far as things before but now my condition instead of being something about the length of signal it's just something about the minimal length of well not quite representatives of that because the representative the corresponding homology Boston that will be the next I was to infinity land by minors the pairings "quotation mark With I'm sorry about the please and a cost on those 2 possibilities for all so again but just make sure things are clear although I wish think them Taiwan but the cause of all the our business image and for someone wishing I have a whole polygon P in 1st college in particular I have elements they have elements it is Penelope's here which lies in they are element of 1st pairing between collagen ,comma wanted OK so again what is supposed to be what is supposed to show this and in fact we have inclusion I have peace From a huge this contains the completion of the projects along so for any camps on this is true whenever the the enormous sufficiently small complete so the intuition here is
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that you know when the norm was very small then the units the set of unit norm this an element of enormous financial loss is very very large in dual space set of unity and interviewed is very very small subpoenas very very small neighborhood of the origin and particularly the so small that it's smaller than this In this Absalon neighborhood that we have policy here and so we have great so so I promised that there now finally we will see something of this picture which I just erase something but floor so if you know anything about flux it's that it has something to do with the chopper was so recalled that is something you have an element of danger from 1 which is sufficiently close to origin can we then there exists a engine which is in fact the Rongen what about all the accused him which is OK so I'm going to say despite the diesel light music notation I want they the Infiniti closed 2 Q where how close it is to keep his hat depend on how close piece the origin the meeting closed to hear such that the book between Q and he is given by the canal having defined flux so so what is the flux of such and such in nearby Lagrangian well that's just the rather onedimensional case the worst flow from 1 lives with his brother 102 cases so let's assume that there was popular and he was sent to the and they were those and they're both Rongen the kind of close to each other there's no global middle valued nearby there's just some kind of neighborhood of so what can I do well I noticed that I'll try to draw this curve here and of course and were working completely locally there is a continent corresponding curfew section of it in the 1 thing yes so I can connect them by cylinder and the cylinder the because essentially it now my number so I can associate this number a nontrivial aerialists ,comma A1 or has an areas called it 1 of the areas you know we did the same thing with the other curve With going races launched the firm on the curve here and then you connect that cylinder and on some other things you have a pair of real numbers it was created to so far I say said given elements Page overall in the basis on the basis of Kurds so and this this and this element is invariant doesn't it doesn't matter whether I taken such is the represents if you get stuck the meeting he will use its toxicity and that OK so that's the Lagrangian that I want to think about to know we have a bunch of module so we have we have a thing that is
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like the colleges ,comma week you were going to this guy has been upgraded To this end the hat and mediation particularly open and not and then you have the floor ecology of L. the this guy walk to just the same construction was :colon and we have something which we didn't have before which should be called which has a name called in the capital and now the here and so on we have enough time to give a precise think 194 started with a statement the of mines using for carriers swindle so that there the Germans I'm not made it's best to write about just you unsuccessful and of course the point is that this was defined only using and if peace is sufficiently close so if piece of invisible so this is defined using only Q and when I was defining as object there was no let's talk about the flocks and how you can produce some Lagrangian close to it myself by pushing it away and its once the neighborhood and on Hambletonian that didn't appear until I wanted to explain to you why this is important on the way to the point the thing is that the rules more than the correct way to say this is on detention space there is the norm which is sufficiently small so that in the unit ball with respect to this nor do you have this determined for all small schools in that duty by so restated so so that determines that for all he said he capital piece OK so what do I mean by that determined that well in the same way that I hear I suggested that once you've defined as for this like you haven't for any prime by just 3 OK so far what I want to know is that there is is that there is a natural and there is a natural isomorphism from the chains no I'm not completion right now although we could complete Vermont and chains on the basis of space secured a lending dropped this piece pointed out that annoying and to the change on the basis of squeeze of keys this is just saying kind you have have a locally you can just identified locally trivial picture so you can judge the train on here will give you a chance here just thinks wave topologically trivializing the picture book so but many facts and I'm sorry he's needed to give this an isomorphism tightening and Ltd so In particular this is a lot like this sorry things writing everything the wrong way to do this and all from the completion To this completion with this the full completion here this was more for them and it looks like wearers do you want to buy apologist for switching Manitoba Oregon now we have inside here we discussed that we have this while the thing so the main thing is you going to say produced this from this by using this month but that's not right where is almost right not right almost right what you need to
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do it is correct the need to correct this by the flocks I need to correct this but lost so that so the correct while the best way to write it is to go back to the components she is a component thank he wanted the Americans the union of components representing 6 by 1 of the all representing weaving just use all the elements and given Holger cost and I want to not less To itself by multiplication might seem to the pairing between Alpha and are element feed for school and then I applied office if you decide
56:36
multilateral component then you get what we wanted we had the desired not from on the completion Peter capital to completion and changed President Kim went to allow me to tears Q :colon colors that something called site no catalyst because I now finally I can tell you what the precise wording of the theory has clusters the as the capital P is sufficiently small then there is an equivalent me no intensive products please where is there trains In the that space I familiar he killed over it's kind of strange things this is some kind of tens of product a major during practicing right version of game and that says that this this this and this now solves the problem that I started I told you that my goal was you know here's my mind you here is mine that was used for cover here's my I call you absolutely no for keeping I would like to know and like to recover if the ecology of about With QB knowing only something of local knowledge about Cuba and and and this stuff this is the former source so so once again I failed to say anything that I promised talk about but the next time I will try to go down in abstract and only talk about Q which is which are torn some states In his uniform and much the loss Oh yes 0 because the area is in effect yes yes tells you that this thing 0 because of weakening demand but actually what it can do this time is that you can be in a situation where the floor quality of purple over purple orange and Q In the usual definition is actually 0 but orange and yellow is known as the year of the renowned on 0 out of the just discussion because of the new said the I want to call it something like that yes this is trying to control hello How long are that appears when you allow infancy but that may be only quarter look the you know there defined up to mean that the 1 that I'm talking about the just do the ones that in a Weinstein neighborhood coming from the graph construction you the ones on such things as the political close to each other more and more are in the end that in the end the moduli because government depends on nothing only depend on the Hambletonian certificates this guy but in the construction used the representative that has seen fit to you mean the proof of this you don't we can't prove this for the for and advantages has only seen small you have to use the 1 that is actually infinity we will find I think that at this well anybody would have try to prove this by careful estimates so I'll explain tomorrow with the swindle in which is what I think has been this when you would have expected that what I should do is go back to Grumman Solomon and do some very careful estimate initial estimate of kind of how does this affect transfer Celtics modular space that's what you would expect as the energy grows up but uses to control the quantitative terms Celtic but in fact you have to do this you can just use some trick about what about of the space and almost complete story and openness of the space of almost complex and and you just don't have to do any estimates the United that essentially implies come through with no with no With no numbers with no quantities but it is inviting dozens of at it looks like b