1/3 Lagrangian Floer cohomology in families


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1/3 Lagrangian Floer cohomology in families
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Abouzaid, Mohammed
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Institut des Hautes Études Scientifiques (IHÉS)
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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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so lean and if but I thank you so yes I am I am aware that this is a summer school and that this has the costs and that I was officially wherever that I sent you some but plan for what I was planning to talk about and then I fed them to prepare my talk and it looks nothing like what I told her I will talk about that now now that basically you should think of today I would just give some general philosophy so the other pending much tells you want may not be interference in then and then on Wednesday I will I will try to go back to the plant OK so so that the general the framework by that time this is clear theory takes Legrand genes 2 of chain complexes or maybe the larger groups and what you want to say is that if you have a Lagrangian parametrized guys from space ,comma the speedier then you'll get families "quotation mark chain complexes for Monday groups over and such a thing is basically what is known as the ship now it turned out that there is actually 2 completely different frameworks different kinds of shoes that come out of all this idea the the 1st 1 is probably the most familiar ones which is if you'll only 120 families so in that case but there is the familiar statements that but the homology is invariant under Andromeda Minnesota's and that says no machines that you get it is a local system From and then of course you know when you work at the level of chain complex as it turns out that you don't want to say local system you want to say derived local systems and but regardless the way shooting with such an object that is determined by the value at 1 point by the group were chain complex at 1 point together with 108 moderate data some kind of representation of the fundamental group or change business space something but what I want to talk about this this is the subject of a kind of another lecture course which is not this 1 answer this to what I will talk about is a different situation and that's where the influx of money from on the tangent space please the appointed Prentice basis . 2 on the 1st hole military of L. a mural grungy instead asked about people fibers people with real coefficients enlightenment so roughly speaking if you take run genes than you think about Peltonen verse of the grunge Mississippi there's a transverse direction to the Hambletonian leaves and the transverse direction is exactly are controlled by 1st Call logic and this is saying you have some kind of sliced and transfers and in that case the assertion is that we get something that not local systems but another familiar kind of clause she is we get coherent she on the well now you can start modifying things could essentially on on this for schoolchildren for some associated and so that's that's the subject of the stock to kind of explains this correspondence so given the audience I should start by at least giving some intuition by what what what is a chief so that's so nice you know you have to this not regulatory select state the Sea which will consider the complex manifold while that for a moment let's just take it as a as a complex of brick so that the reign of functions on the stars some familiar objects as hold Norfolk algebraic functions on the stock so that's just Laurel opponents and the simplest kind of Goodyear chief that I want you to think about well you could think about this I think this is a coherent chief is going to be some special kind chief corresponds to confinement provided finally generated modules Over the strain thus ,comma sprinting clear don't have to keep writing but the way they will appear in 1st year it's better to actually think about complexes the perfect complexes or in this case against finite land 3 complexes Over all over this and you go from here to here but taking calls 5 exercise 1 show that's actually true so if you know the Monterey geometry this is something that you know if if it's not then something you should so the of the complex that I will want you to think about and just keeping your mind is the following so what does this mean there's amount from R 2 to itself obtained by multiplying by this by this Parliament this polynomial vanishes at 1 point and 1 interventions to degree once the exercise to again if he knows most beyond his museum there would have to think about it show that ecology From this complex is the right 1 for this so these are the kinds of complexes that show up when you think about flip like my sergeant assertion that no when here we are in this kind of transverse the Hambletonian situation we get cohere says that we should see complexes of this and that they could they kind of control theory and in in the in the direction of stuff the transfers the transistor now the reason that I started with the star instead of which would have been slightly simpler is this picture while I'm going to stay near the top of 2 those complexities which is part time this 1 but that be better thought of as the starts usually 1 things about the stars 1 In terms of 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to just enough to prove that this is era that I want to show where this I'm ringing might appear and this appears when you think about it just 1 Lagrange in which would be our 0 and and we're getting this Lagrangian England run to covered with a base .period into baseball is going to be called Q so this ring are for years yes security cordon the Stars 1 Romeno enough for college but you can erase the course they wanted so but so the point is that if you take the basement space meeting the homology the business then this is easily seen to be this way so if you go back to your topology you will see that this is a major 0 of any space of the basement space of any space is just going to be the group range of fundamental group and in this case the fundamental group Azanian news gives you this description Laurel Pub and when is the product you mean
here or here the even in and so here you know the product assuming here well it's Citigroup rank In qualities not concatenation of loops what's the significance of the sea and the fact that for some reason I thought it was easy here I don't want to talk about you algebraic right arithmetical to break geometry but here it's perfectly safe to talk about this overseas so what if you really want to construct this you just can't work with the coefficients that things are a little bit more delicate if you working with integers don't beyond her advantages and redoing it oversees the only so you you can go out there and do things a little more carefully you just wishes work overseas but so I thought I want to explain something married so so everything tribal drawl it has to well willy-nilly just this as 1 but actually almost everything I will say it doesn't matter that some has 1 or not you could use you on the notation will just be accused demand for if I wanna status there and later I will there will be a little bit more precise and what kind of smooth manifold I wanted but felt so what looked tired goal here well I said I should go there should produce for use on some coherent she's right now it's not going to be very clear what it has to do with flex maps that will win that's that's for the for next time around the goal Q :colon side-effects isn't exactly Rongen and held in context so for now and other liberal June then we want to produce module with :colon no that's presentation with :colon sensible way over the change in the basic this and it will turn out for the reason I'm saying this is that again is just about the moment is but this Q happens to be a circle on with one-dimensional situation then progress saying is were producing some kind of go here she and this the granting here happened to be a Taurus then it would happen that we're constructing nuclear achievements he started the end when it turns out actually that if this Q with anything then you're out of this procedure you constructing a coherent she's on the torrents of dimension equal to the 1st college not fought you know in that general situation when accuses general situation this school year she doesn't you know everything that it needs to know about the flare theory of of Alan Q but for tore I there's basically nothing more and so that's why there is this kind of you hear is a little bit of a lie the new producers career achievement it's not doesn't carry all the information you want unless you assume your Lagrange science with only just explain the story from because hiring going to produce the cell this coherent chiefs sexually fairly old work but said that the way you do sell it to me as usual so let's that Ellis transfers to Cuba and then let's consider for each intersection points this space the Liggett on cue which is the space compounds From Cuba Texas so this is our base .period and this is an intersection .period Products Inc so I said I would produce for you ,comma some model over these strains in the basement space and all you do it is you consider 1st you consider a dog can contact their surrender direct some or all acts which is such point of life all changes on the chains in the space of I'm here then you have to shift by something and I just tried the index 12 and if you don't know that as you can just ignore it because then you can just work we just work without any gratings it doesn't really matter too stand this is a model Over the chancellor businesses and how is that if I give you they give you path for next acute and making use new computer itself then of course you can concatenate traders said yeah the different agreed to the ii when I wrote this I use cubicle chains the
it's not I use it so I would have to worry about so this module is like this amount from this time is this To this given by concatenation Nadal exhibits induces a map on chains around and if you singular she'd have to subdivide and I didn't feel like reading about subdivision so I just use chemical change but it's works with Cingular changes as well I don't know what because the this the next and to that of the university Georgia but I think this page Houston health said that indicate what yes figures from the National which is why cubes good because the products options yes any other questions except for here you once but seriously anybody else should feel free to work on so that the battery thank you this is what Nate corrected I wrote here activation of the nearly all of the use the use of the policy you get it all that so we have a module is on the fridge where much over this that actually we don't yet have the differential it is currently at all is the intersection .period in news any flair theory so too built building some kind of 1st theory you have to consider 21 faces of critical made section .period proposed that we have a lot space strips In and I will just write it instead of money I I'm going to assume that people have some idea with this this and if you don't tell just rather picture it was a test I
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are if is small based on 1 thing you can do I want to spilled the options it is you can evaluate along Q To get past so we have map from here called evaluation To the space of parts Our from white text from OK so in that this is whenever you will just have to namely that extra life no the modular space itself In case it has enough and that now goes to the this actual space of past from X 2 1 and the way orders and it's not clear my notation is notably with precise for you to know whether goes this way or that way but it doesn't someone with OK so you get you get this passed by the now the main point is that we can pick fundamental change change for the money licenses and that they should say relative fundamental change so that I mean you you you just ensured that they are manifold by should send descended from resulting and then well since their manifolds you can save triangulate them and and then take a sum of all the top dimensional simplicity or fuel subdivided into King's is reason cues and then take off and didn't take a sum of all top dimensions so I will write histories because this number here is basically on the degree of explanations to the group Mike the fact remains that the White that's the number in which this past lives so when when the difference in degrees is 1 OK then this is a constant degrees 0 it's just an element is just counting that each 1 of these modular spaces gets the number 1 or rate minus 1 depending on orientations which we're not discussed OK so so now I have this evaluation of their no wonder what I said is here as a fundamental class it's not an innocent man and I'm saying it's a matter for the removal from across the middle of boundary the top class and homology but it has it has a class in its changed such as the image relative homology generates the class of the manifold rebounded that's basically what happened at White said :colon well to parliament but that is a change you had to pick this change "quotation mark so we have this thing so we can evaluate it I don't know that keep track of this anymore I submit is supposed to be 1 eccentricity blacks 1 OK great so I think so the Lagrangian look at intersections with Q I formed this big thing which is the module over the chains on the basement space and I also look at these modular spaces and now I get some classes in the chains of the space of pads from the intersection .period future From next now this is space we have not when I whatever reviews already is that we have enough funding acute images no rights and destroy the unity of the council cross-selling owning up Q X 2 earlier QNX I think this is what this would visit concatenation this a concatenates at CU but you also have a not only a few the excuse missing everywhere from the Gulf War but in no time out across this which is the destiny of the other and these operations essentially the commutes because you know 1 of them happens 1 of them blockages rather than 1 of them happens here the other 1 happens on the other side it can you could cut it with this 1st or you get name and then the story this and that's the same as it doesn't matter but independent that if you actually saying well you get some Cuba simplicity Chadian and take the product of this Q and a product of this cute to reorder the kids but this is not an essential phenomena OK so we have that we have and this thing here this about unchanged so Our fundamental classes but rather its image under evaluation the final From the chain's on the Net on the basis of the news on the pads from 2 to access to the chains on the pad I'm from here to what was going on in the area of the world with his everywhere money can do here you have to do include the Megan only lighted command the think and then had diverse theory but you're only allowed to take homology Indian OK so the funds is not but now each 1 of these is part of my complex and this direct some of these groups and now I have enough from 1 to the other so this is given differential on the Net is the construction off the the module over the chains that the chains on the basis of basic that is associated 2 2 an exact Lagrangian if you want an expression basically Woodward is saying is take the some of multiplication with modestly does that and so now there's a of the basic exercises please take well To this Lagrangian said the kitchen his Lagrangian intake Ltd this 1 the now Al is part of the property of this area here in this area the said so when you go through this procedure so computer
this quality Of this ML as a model Over the technology on the basis of these events 1 which by the way we already decided was closed and the answer but you are trying to get to is the thing that we saw the beginning the thing is over one-liners the it's it's the quality of the complex assessing the use OK so so far what I've done talked only about exactly wrong Jews I so you may want to do things In would not exactly running immunity that were going to try to do some kind of family for ecology where we're not going to stick to haven't announced at class of surely really get out of them the exact situation even if we're a situation where it made sense to talk about it well the 1st thing you know you never thought about granting in the theory nonexempt non grata non-exec situation is we have to work over the medical field breeders and cases we can avoid that led to the emergence of what was then he knows chairman of the state the year the you get this simple model over the high-volume debasement I mean that you have to say which simple model because in case each 1 is non-zero there is the of pie was on there there's more than 1 I met you get those things go the thing that corresponds to the augmentation ideal of fundamental OK so so so they may say something about the number covering because I was here last week but I'm pretty sure that nobody talked about it it's a talk OK so so that it is best to fix a field now it up with another covering overcame the field overcame so you would you want to say would really want to do is we want to work over law all series over overcame so something like OK and then you do power series all over its managers a team here and then University I think that's what we want to do but unfortunately this would work if every homomorphic this we ever encountered have integral the venue we would use the area would up the powers of tea but that's not what happens in the theory you can have things Chery real area so instead is that in doing this we allow real exponents so the number of field will be carried on this this subscript double dropped and so it consists of things like this because this is the infinite pop possibly infinite thump these coefficients living K. that's why we introduced that these exponents are real numbers that's why I was started with the interval case and they satisfy 1 condition which is the condition of the says here anyway you can write down infinitely many new cannot things here which are infinitely an inventory many times but only the power of key goes to infinity similarly here you can have things which are infinitely many terms but only if the limits providers goes to infinity both lender because that's that's the ring that we have to work so on and although there there's very naive and naive expectation which would be what you would expect if you have thought about the range of the they're trying to extend into the situation and that we should replace replaced should work with the change on the basis of space With coefficient this and sometimes it works very well so that drove the 1st images they want if you haven't seen this but what it means to extend was supposed to work over the strength but it means Over there I wrote an expression for the differential should look like and I and now In the same thing as above the differential is the sudden and what I need to do to break up this model is space in settlement be classes the the disks made whined around the space in different ways and as the wine around the space in different ways they will pick up different areas I think this expression here is just the area of the corresponding disk and then multiplied it by what we were doing before consent of costly break and work with things in class this Fla contradictory is or whatever move he strips and all this is the case that standard thing that new inflow theory when you are not in the exact situation is still use basically a modeled faces of you introduce away this thing isn't it the president said he was on the other day he before it was like I equals 1 so let me just show you always remember is that of tour again just think about this and with duration of it's interesting take the game the cylinder again this is cute but now I want my under the grant to not be symmetric matters on her snobby Hambletonian isotopic to it this is not only reason number they and this is now 1st mission number they then what you're supposed to show again lowered my numbering scheme resulted from the show that this module corresponds to the quality all home so I
take it take no Landau I declare all series and land them there upon the heels of another and then brought home commander and now my differential it has to do something it's cheap we to minus the Internet they kind of look at figures 1st and think about is this example and see that you before we had 1 line the because we just said everything before that the notices you and so so ended the college's long great so it looks like we have for extension outside of the also the the exact situation but I lied to you this is expression is not final the and in fact following the release to me amazing fact it's true so let's take this elements to looking unless the form held in Cuba but it's another thing to fix with the former held by Hambletonian isotope this is for so the module associated to hell the at in this very special case and is invariant under Hambletonian isotope is it helps intersect Q demands moment if you look at the there is no easy Hambletonian isotope which in fact melt makes Ellen Q disjointed you can take Helen movements by Hamilton studied and you will get OK maybe that would have pushed it too far from a little bit closer to queue in reality that it's a I and then of course complex 0 there's nothing to do there's no intersections but up until that point you can do all the deformations you want them and you can as long as the intersection .period you can compute this complex and the exercise to show that they're all actually isomorphic as modules over these it was the wrong so this problem it's it's it's not as it is a problem but this is because of the war the well everything you would get again isomorphic complex the committee was initially winning the 1st 2 as long as they intersect at work here can you see in the front of what it well that's 1 way to say this but it's certainly not by any means not a finite expression assuming it's not an element of this race it's not element of this chain complex it's the knowledge of this chain complexes you have change with coefficients note would you are telling me and Of course this happened the other thing some of the other theories as well is that we should actually complete this race this is the next word in this time so to get a well-behaved the theory that is behaving general In this case and reason it works by no problem but really what we should do when we write such expressed or what we think we should do when we write a check for expressions the you convergence mission complete these change would coefficients unless so that's right that as C Of the number I don't think you know so what is this is very consistent expressions of the form some of them can be 1 2nd the can be that to some of AI To to the land of the town's subtlety to the land i'd times from behind where all where elements of the chains In the case of space again with provisions in our ground field it just like before before we did but the elements of our ground field and to the island and I just replacing zones around fueled by by change and land on a real and limits to the goes to infinity commander on because it wasn't the so so great so this is the right thing we think and it's it makes it makes not expression makes that differential differential is well defined if we similarly Yevgeny similar things similarly whether news completion on change from Q to acts some but that's it now as a model the requirements we and that is that the story is just 1 of the many different eventually the flight I think it is yes here you can have an infinite series but you can only have finitely many in the sense that this part can be infinite but you have to take 1 of these inventing the multiplied by 1 element of the chains and take finite sums of that and here it is you in the cultural moment in the history of the modern and international I don't know anything about the telecom all year but I suspect the answer is yes and I think I wanted explain to me more after that talking so so we're happy we think we're happy but because because this kind of you know seriously says this module clover this change in the basic space but not now I I can say that is a different ways but Tom and good this In invariant "quotation mark isomorphism type all of this module is a is invariant From the holiday next to the class all of them it was his things are now what
is it doing so you may Mancino the invariants like there's another invariant which you could have thought about more basic just you you do that you them I completed when you didn't complete and so what workload is pointing out that there is another exercise is that you're supposed to show that after completion yet I was that was going to be I was gonna come back to this but at the end of the group that we were playing with In view of the of is exactly what did seduces them the exercise I'm not worried about you and it's 0 so the other warmer Michigan variance in the history of the virus the sections of rule to monitor when doing anything that sophisticated and 1 of 1 section of the research which you are taking this year section in your streaking into smaller SUVs you take from house smaller charts that's more centered around a while the winning wood so that the site missing the other way Q these with Rogers them drawing action corresponded .period so we have some points and another point it's fine a distance away yeah yeah section yes that's right sold Missouri that this is a very reasonable said as of now that I that I completed at or before completing yes it's obvious that it's well-defined because the values that have been the source of the funds disillusioned some from of compact tells you that the only way that you can have infinitely many yeah components of your model spaces is there is the young is if there is in area that infants so it's an element of that right 1 is that the letter was signed by members of the In the area of the world would it also said it would also be and in many cases the the yes the only way the debates that there are infinitely many Bates has that only financing many of them Have you will get a little bit of candidate named many of us here at the end of the matter is that the form of these things across all regions and all these years OK I think it is we can talk about it but I missed them saying that the powers will add up and will just you don't have a unified the negative direction and that's why this this is OK that's the extent of the injury was notified the fact that this element is that lies in that completion you have to remember something that to get a job there were without a specific elements that the factory lies in completions gram of the political in she was a and so what's so that using whether they stay here here I said you can take this module over the chains and its cause isomorphic top over the changes in invariant if the president as some of you drink a lot easier you have to say Karzai's isomorphism OK so I kind of lost a little more basic invariants which are basically the vertical montages of L of that's another more basic and married but also are there more basic and the audience which are its HQ and you equip it with some local system like a representation of the fundamental group then they flew commentary these are the ones that may be our most familiar to American but CF here and the nature the point is that this group is integrating over all of them the knows about all need information 20 years so that the above module about all of it In the only 1 all right every representation is not from playing 1 but I actually have changed so that already has all 1 of the group poems and contained life and so that sounds good so now we have our coronary but it's kind of completely useless for the task of understanding you know what happens to phrenology and families because as we thought about the fact that this is what is otherwise of this exercise facts managers 4 you I think B if it is not going to be basically tells us that you can't use it to extract any information about what happens to Lagrangian is when they very nontrivial In In there Hamilton you can use it for other things but but we can't use it for this task but now so so so we have an issue so if if we if we want to use the ring which actually the attack and tells us something nontrivial even when the the Rogers around Hamilton in established for example in this case that's the 1 we from the beginning then we don't know that it's not that we in general even have a well behaved well defined differential because they them the differential that we write downs infinite and the things we Laurel but if we use the 1 that always works then we get a bunch of groups there 0 and we would like for them not to easier to what you say think it would be nice know that so I believe the answer is mere cemetery but I can't and this is not a lecture and you just it's amazing that it's true :colon think so now I will explain why you can't sell how you can salvage the situation so this is compliance fundamentals and he as yet the fact that this conventions before 1 0 and the of this year so far and that's so it is that some momentum in this series of hearings with when I completed so before I said take take the chains on the basements with coefficients inland and then I said all well
this thing assessed only has finitely many chains of command infinite you the power theory the trumpeter maybe even that each chain only finding any actual change although they I'll have to go the was singular honest as 1 of whom is always there was no intent to sell has area which monitors the coming series that there is this continuous I don't know where no adults didn't get you very that no infinite series shops but what happens is if you try to prove invariant you can only proven variance over the the other I don't know how how to better incident I should have a better answer the question but I don't have 1 but so the yes but it's having shown you that this user of this new can still do something yes yeah right so I assure you that you would like to work over this range and in net example you get something very nice which is you get this Lagrangian the news about the other Lagrangian that are not actually rooted isotopic too let may know the nose of a very delicate way but it still gets you and tell you you information about but that construction doesn't make sense for the theoretically because in general you have to write down had to work over the completion we work over the completion then get here and you lose nice thing that you just the richest and most of more for the nice thing you could get this module it's non-zero In fact you can recover the Al From that's computation From that modules in knows everything about ecology we have role later began to finish my so so so so now and talk about the case governmental theorem a friend of the crime in the somehow hidden stable but that it's on its way the so Socialists say that we have since then have Rogers returned transfers man there exist so this is a very weak form but it just in this language would I was explained part of a completion also the change on the basis of specific you so let's call this completion prime completion .period which said which is intermediate between this 1 and this 1 and both of these restrict containment what this 1 doesn't have to be stripped the other 1 is also that this is planting so that module and the prime which is again obtained by the same procedure Rabin during except do it over this thing is wonderful and that means nothing so I want I don't have time to explain this and maybe that this shortly on Wednesday will explain a version of this may be slightly different language but then this is the main thing that makes family analytical module this 1 is you can't work with in general but it usually gives you nice answers a non-zero groups is 1 you can always work with but it gives you many 0 groups and this says you can make things slightly better better why not completing all the way up to here with the slightly lower year of it means that that expression this expression followed to the pairing of Plato made a fundamental flaw on this give me an element of again analogously and the completion of the cases ex-wife it is also all while I can't I can't assert that this will be known on 0 because it can you so right now it's just you can you can just as the assertion simply is there is a strictly inclusion here the results of the I this and which is another example formatting level for it with this write an example for it and you quicker than just let this Primedia we can of utilities any completion for things to be the statement is when you look at it expressions it looks to me like you have to allow infinite series but in fact there is something that you can you can always cut down your infinite series by a little bit there's something that you don't nursed the sum further constraints on the series from flair theory which means that you don't have to go you always have to do this thing is very large fluid from the well being of so so that's something I will explain on Wednesday but roughly speaking this thing will allow you to do you know if you now have the defoliation by you can't buy these miscues just 1 member of foliations than you can try to Apaches groups together it's easier to Batumi than it is to patch that may be the 1 thing that that I don't I don't want to here with the use of under all Hamilton and it will not yes in fact this is the size of this completion it directly is related to and how much Ho for energy it's invariant so anyway so that's I'll even at this very unsatisfactory stopping point and will continue on Wednesday the was that here because this is the 1st of all "quotation mark patrol could be cute and it could also be it as long as you often find there is no reason to work out with we've complex numbers mean Over characters in here I can work sometimes it can be fun if you it will work did real example yes I mean yes yes yes if you work in a few work on complex to I for example there is no reason to work characters sincere the complex too you could do this and winter others that that but all feel that once known very far from you find the and the organizer the you know people who have to 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