1/3 Classical transversality methods in SFT


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1/3 Classical transversality methods in SFT
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Wendl, Chris
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I will give a quick review of the Sard-Smale theorem and the universal moduli space approach to transversality, discuss the relative merits of classical vs. inhomogeneous perturbations, and sketch proofs of the standard theorems stating that the moduli space of regular J-holomorphic curves in symplectic cobordisms is a smooth orbifold, and that all somewhere injective curves are regular for generic J.
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and and if you can have to you thanks to the news that on this board with just a preface of generalities let's let's call this the archetypal problems by which from the perspective of the global analysis differential topology years invited quality as distinct from algebraic geometry where they also like to talk about modular problems so for me a model I problem means this kind of thing you have some space which I'm calling suggestively Jr but right now I'm just going to call the states of data which has some kind of topology on it and for each choice which I suggested we call also Jr In this space we get some based on Monday which also has a natural topology of
course we would like to say that it has some much nicer structure than a topology like being a smooth manifold or if we can't achieve that maybe we move or before and over the course of these two-week Century also hear things about weighted branched waterfalls with boundaries and corners no not for me From my perspective well the 1st thing to say about this model space is that locally you can identify with some kind of 0 center was alright Sigma Jr In sincere it's not precisely that often you have to divide it by some kind cemeteries 2 segment please cite section of about about-face bundle and we already and there's a sketch 1 example of this because you can express all sorts of geometric is in this way as preferably smooth sections of bonnet space fumbles and an instant special context but I'm saying this is true only locally in general so do described the situation with whole spheres where you can actually do this globally but in my experience working with the Hoffa cravats rather the exception to the role and so I'll talk about some more general situations where you can do this but only locally and these symmetries can cause quite a lot of headaches but the 1st problem of course is you want to be able to say that this 0 section here is a object an implicit function theorem ought to tell you that if you do things correctly were there to general ways that you can approach that 1st this article abstract approach where you perturb the right-hand side of the equation you can say something like defined the modular space and Tilda which depends on the data Jr and some extra auxiliary choices that that's the please you in this bond manifold such that Sigma debut madam knew of you so this is for son other sections you part of that same bonds this fumble and the idea is to be true that generically then set of solutions will be a smooth object simply by Resorts Milford right that's kind of equivalent to that of the statement just expressed in local trivialization that aid generic point in the target of a smooth map is a regular value of that man so this is a generic section is also intersect this section
transversely so generic traces of this auxiliary New are special section said Monday is transits to that which means this modular a space perturbed objects defined is a manifold and I even want to say more because we're talking about elliptic problems this section is not just a smooth section but also Fred Hall in the sense that is legalization at any point is a friend home operators so when we have these kinds of transit conditions we get just not as enough over the finite dimensional manifold because finite additional kernels of immunization so that of course very nice when that happens the problem is I perturbed migration to a different 1 which might not be the 1 I'm actually interested that's the problem number 1 there more problems which I think you hear more about Over the course of the 2 weeks because it's abstract approach is sort of the the raison d'etre for the whole colorful project but it's difficult part of the reason it's difficult is that 1st of all this correspondence between 7 0 section and my my modulus things of interest it is in general only local and it also has the cemetery's you have to deal with those in both of those in a sensible way when choosing these perturbations means that just choosing some generic section of our space bundle is usually not enough you have to choose a satisfying some conditions and may be those conditions are too stringent to actually applied Sorensen of animus wh so we want to be with you on the left side of this isn't something like you you want various things of that sort I reserve the right to not talk about that anymore because other people and I'm going to talk about the alternative which is In the title of many forests of so-called classical form of medieval geometric perturbation Methods which is the notion that if you take a generic choices as the data MySpace data then your actual section will be transferred to the 0 section which means that Christofias said section is a smooth manifold and finite dimensional as well because we're in from home context and while the space were actually interested in is that divided by some kind of symmetry that in general will be in order for some oversold is something that looks locally like Euclidean space divided by a finite group and that's what you get if you taking a finite dimensional manifold and dividing it by the kind of symmetry groups that were interested tho that is minor headache number 1 to be aware of we won't get manifolds in general but we could get more defaults if we're lucky a much larger haddock number 2 Is that often were not lucky in this this condition simply isn't true and there's already talked about that a little bit I'm going to talk about that some more so that's all I'll say about generalities and focus on the 2nd is classical approach over 3 talks on a particular context I want to discuss invited field theory so 3rd In the original propaganda paper in 2000 by which begins on you have somebody field theory presented as a a very general way of defining invariants of contact manifolds and syntactic aboard isn't tree in contact manifold and Wellington do it even a little bit more generally than that which is sort of nice for certain purposes so I'm going to talk about stable Hambletonian structures 1st of all the standards and on dimensional manifolds a stable Hambletonian structure ,comma H and M's clear going to be a pair capital only when which consists of the following her 1st on it closed to form a maximal ranks so you should imagine that is something that arises on any hybrid services in impacted manifold right you just take this eclectic form restricted to a hyper surface it's going to be a close to form of maximal ranks and if I omit the words stable that'll recalled Justin Hamilton instruction now stable is that
it comes from this additional 1 farmland which satisfies the following properties 1st when the of the top dimensional power of Omega must be positive volume forms so this implies in particular that you can take on gun restricted To the kernel of land the land is no way 0 so it's kernel is ahead of plan distributions which I'm going to call oxide as I always do with contact structures because that's 1 important example it could be a contact structure but in general it's just some hydroplane distribution and only makes that had the plane distribution into its inflected vector bundles physicist inflected global structure we think and then the other conditions since Omega has a maximum range on on dimensional manifold it has a occurred that's always one-dimensional that's so-called characteristic line field the 1 dimension of directions in which only goes to generate so but only requires a deal and also vanishes on that same Colonel another way of saying is :colon against Indians in colonel of the land and so this is the definition which has been around for several years and that as stated it's not too hard to understand but I think a lot of people don't really understand what it means so when we talk a little bit about that 1st observation is that if I now take this fundamental manifold and look at a small cylinder constructed out of it close and small interval times missing now inherits a natural synthetic form called the we'll coordinate little are anti-crime only get to be deals and was the only guy and as easy exercise that's going to be a synthetic form as long as the cylinders sufficiently small it might even be subtracted from long larger than that but in general I can only guarantee this is true for once small moreover what is a characteristic line field look like if you stick to the 1 from a family of hyper surfaces that say tens of or is the hybrid circuits odds and sitting inside that cylinder the characteristic line field of born-again restricted some bands of our is independent of our so it's the same characteristic minefield for all are all parameters immigrant-rights family of positions that's really the original definition which I think appeared 1st in the in the book by Holdren standard of stable interest effects that's resident on it's it's the kind of situation in which you would expect to be able to prove something like The Weinstein conjecture that asserts the existence of close characteristics of the characters declined you because if you have them on some Cyprus surfaces in the vicinity you have them on all you the 1st 1st single as an and and In order to get there the world has lost much of that's the was president I factor but the characters declined yeah and conversely if you are given a 1 from the family of hypersensitive that hazardous properties stability of the characters declined sealed then you can always find stable home structure that produces so that's kind of an exercise using the modern information how to write the nor the nation in the context of and nothing but it perfectly good point if you if you don't like this definition for some reason because it's new just think land as a contact structure and Bigelow miracles do and the aggressive as soon anyway so there is From this data we have also a canonical generator of the carriages declined field which by analogy with contact forms of colors of the rainbow field amended petition the arts of H and it's determined by the conditions on again at a mile east of direction of the rape vector field and we normalize it by land the now there is no reason melody and
condition I also say that again in moment so so we have a stable home time structure it defines this hydroplane distribution CSI which might be a context actions and examples but doesn't have to be could also be affiliations we have is rated vector field another thing we can do is since we have a small syntactic cylinder we built out of it we can also just arbitrarily expand that to are times and rewrite my syntactic structure there In the 4 only to find that DVD of fire or planned plus so was fine plight is going to be in the space of smooth functions taking are to pat seems small epsilon interval With strictly positive derivative so
this is a synthetic form also I think it is simply to migrate to the 1 I wrote down before just such things but the point is that our crossing is the kind of setting where I would like to talk about geomorphic curves and I can do that now bye saying well this space and J. dependent on the state of and instruction going to be space of almost complex structures Javed Khan are crossed and which I will assume always to be there are invariant census are translation action are cross and I want that to preserve all my jails I also want to say that J. maps the unit vector in the art direction To the main vector field and I want
to say that J. reserves I had a plane distribution as well as the syntactic structure is compatible with the syntactic structure on so say J. restricted to size is only compatible think so His alumni which is very easy to prove and so it helps justify all these definitions that is that for all traces Jerry King this space of translation variables complex structures and all functions find as I indicated above their mapping to that small interval with positive derivatives it's not just that only defies infected but in fact it's compatible in the usual sense a problem with this strategic so I can 6 J and I can then allow any choice if I I want and that's important details for defining things like the energy of the home of a curve because we can take that to be the supreme most emphatic area defined with respect to all these different choices of inflected forms so let me give you 2 examples already sort of been mentioned 1st of all it also is a contact form we had a stable Hamilton instructed the offer ,comma Alpha and then this eclectic manifold are crossed them with this kind of simplistic structure I've been talking about I'm not going to fight a status to precisely you have to make some kind of condition on how you define 5 in order to say this but I'm just going to say 1 candidate so this is what's called a symbolic decisions of the contact manifold uh yeah I really should relax the epsilon demanded to see that it has actually the least make plots long the Infiniti but does not worry about the fact that you quickly realize that only of the No I might say this yes on Tuesday the Commission will agree I want distribution to them all where you can integrate behind when it you use you don't and while for the for the hobgoblin distribution to be integral I think that's equivalent to saying that DU and vanishes all so certainly sufficient to do and the vanishes altogether which my next example for a year and has only word havel packed well I'm sorry I take this opportunity to make a blanket apology for my handwriting as they customarily do there isn't that much I can do about it OK come to pass simple it is will of the competitive 2nd example if I am equals the product of S 1 with some of the manifold W and I take W these inflected manifold say I synthetic form little only W and then said capital only going to be that plus the T weds the age so here team is the best 1 coordinate and I'm using some function H they Hambletonian function which is dependent on time in periodic waves
so this is capital on again I take Lambert to be just the this is a case where it's closed due and advanced so now and can check this also is a stable home to instruction and it has a a meeting at which a lot of people will probably like you can compute the rain-affected field for it's simply is the vector field in the team Action plus fields in the W direction which is precisely the Hambletonian that OK so this is a way to turn questions about Hambletonian vector fields on synthetic chemicals even time-dependent ones into questions about its tiny independent the raids like vector field on all dimensional manifold and it gets better because for the class almost complex structures that we've talked about B. J homomorphic curve in our times this USA basically equivalent to a solution on the floor equation In W this is the way that you can do for homology held 24 homology from the perspective of synthetic fuel so now that setting in which I really want to think about home off curves is also a bit more general than the so take a syntactic manifold so here until he local borders and on With stable boundaries so 1 can define stable boundary in various ways the main thing I want to say is that there exists near the boundary of 1 gram of family of hyper surfaces that all have the same dynamics on the same characters declined fields and that's equivalent to saying that there exists this collar neighborhood near implies that looks like a small interval times and plus where the Cycladic structure is just the kind that I wrote down already deal the land of plus plus ,comma plus for some stable Hambletonian structure ,comma get less landed close to current and past and present danger to do the same thing on and minus but the collar neighborhood will look like the office of Interpol OK so this is the definition of a syntactical borders with stable boundary and having done that I can add cylindrical ends compact object non-contact objects that's what we call the completion I did note that by W hat only a hat is going to depend on some choice of functions fly of the sort that I used to define syntactic structures on some accusations and that philosophize not particularly meaningful it's just nice to know that I can do it so the definition is we take Nader cylindrical and the glue that along a minus 2 W without a long and cost too positive cylindrical and and the syntactic structure here we take only a only a 5 on each of the ends and the original w you can choose the function of fire such that remember that I still have this definition on the board somewhere in the middle of our time frame 5 times landed puts on again if I choose 570 equals just the identity map close to 0 then I can obviously cylindrical and smoothly together with this :colon if that's what Click so with that traces emphatic structure and I don't really have to specify fight and is that I can choose 1 on the 2nd part of the world I don't care actually I specifically wanted not specify final only allow of freedom to change so let's said J only 10 to stable Hamilton structures this is going to be in the space of Paul almost complex structures Jr on the completion W hat which I assumed to be everywhere compatible with well let's just say compatible with
only got on the original compact objects and belonging to these bases with respect to the stable how infrastructures that already defined the spaces of translation during almost complex structures on the ends so that now is going to be a and almost complex structure that's compatible with only got half final matter what 5 so and to start talking about the actual jail North Korea's any questions so far that you require this is thing travellers it I was in order that wasn't part I don't have a factor or accept cross the boundary yeah I don't make any requirement that except compatible with the given synthetic form Click so the kinds of always occurs when you talk about our last fall's we take Sigma at close the on surface it's actually said that maybe 2 finally disjoint ordered subsets but and in particular Sigma .period always denotes signal with the 1st finite ordered subset removed at a punctured Riemann surfaces we take that also is subset and don't have to worry about it too often but it gets partitioned into the so-called positive and negative punctures each of those could be empty but I each function belongs to 1 or the other those and now I think about maps from Israel on surface With the various points removed going into my completed Quantico Buddhism and I want to be what I call as emphatically cylindrical which means no that's not right down the precise definition let's just say asymptotic near each positive or negative Concha z 2 or at times the raid boarded Korean and plus-minus rescue Bloomberg I'm missing a whole attitude could be worse the opposite would sound like a medical emergency I think so so what I mean is essentially for this kind of almost complex structure and considering with the the translation invariant being satisfies these conditions on the ends but I have Jr map the vector in our direction to the raid vector field which means or times the rate audit is always a whole market and India so the special Hallmark records that always exists at least in that region maybe not globally and I want to consider punches ,comma Vickers whose behavior near each of those punches approximates 1 of those cylinders better and better as you go toward the puncture and so the major precise it's choose from Hall cylindrical coordinates near each punches and in those coordinates as you go further toward infinity on the cylinder your curve looks more and more like 1 these trivial audits they give time to recover that is the Faina Ireland and they said the theater is thus not to be empty it's just some finite subset right now in it's the doesn't appear in the picture because it it doesn't affect the picture in any way this distress .period but there are meant to be more points all right so the
actual modelers face looks like the following going to be a space of polls Sigmund Jerry Gana theater you where well synergies a close months surface of the road above gamma destroying ordered finance subsets 11 not rewrite that mainly I need to say you maps punctured surface To the completion W hat to a whole logically and asymptotic lease allegedly but and then I just need to say what my equivalence relation is due so basically already said it you could deduce completed the right answer to this from what do so said I will say the tuple Sigma Jr ,comma said use is equivalent to a tuple segment prime five-star general clients of gamma preserving the ordering fighting this fatal you composed with fight for any diffeomorphisms flies from Singapore to Sydney thank there was about yeah this definition still OK if I allowed reporters you are truly terrible but I won't have any theorems yeah we do have so finally to make some use of the mark .period I have an evaluation that sending his modelers space to W hat but I this state the market .period labeled saying that 1 can say that again so this is just evaluates you at this point is well-defined because of the way that I Michael installation that these diffeomorphisms at Madison Smartt points to the sort of March .period but so it's finally stayed a couple of times set 1 says if I have disappointed abbreviate elements of his moderate faced by you and the writing of charitable and saying equivalence class and everything so you is what I call Fred home regular as a conditional have to explain it also includes the condition that the asymptotic category reworded not arbitrarily terrible then neighborhood of you in the Mandalay space is a smooth finite dimensional oversold With isotropic at you isomorphic to the automorphism group of you plodding right down the definition of the automorphism group review is simply the space of all by homomorphic maps of its domain which fixed the punches and enlarge points and satisfy you includes you compose with fire so unless you constant that's always going to be fine and for someone objective crows is going to be a trivial but for non somewhere injective Christopher mostly covered curves if we get this friend home regularity condition whatever it is we're going to have to deal with an awful rather than a manifold because of this I suppose it was the only 1 in some senses the top college onward regularly and anyone had 1 does have to work a little bit to see while you can make this local identification of the modular space with something that looks like an hour before you for and I will at least have time to get to that in the stock and then to probably postpone proof of this for tomorrow but state now let's say let's fix and open subset you with contact closure then I'm going to say the exist subset called Jay you read that right this subset of the space of all I needed to fix some more data in
advance suffix also almost complex structure J. not but the type described the tender exists and there subset so this is a subset of the space of all j is of the type describes with the extra conditions that day matches chain knots everywhere outside of this special subset you so special subset you we can call the perturbation domains were going to allow ourselves to change the almost comic structure but we need fixed everywhere else so such that for all day In this chair ideal it's finished it around here every somewhere objectives kurds you In the modular space with respect to J the passing through my special subset capital you is that alright here we have said that he and I need you and there's subset literally means coal needed so there was hope that will all yet I never understand residual it sounds to me like it means the opposite side of user with some people call it 2nd category which I think is also not technically right if you look at the actual definition yeah I think thank you that protection so you well summoned Sunday some kind of sensible general habit will prevail hopefully I say bear subset and most people don't seem to mind so the literal meaning is accountable intersection of open and what would have contains accountable intersection welcomed the verdict Poland was somehow you think you're assuming you know but the regular well that's in in attempt to I am that's true thank you OK so the boys out of reach some of the right let's just say that the that 1 additional condition you need in there too is that the said all the close trade 1 its both still Hamilton instructions and ungenerous so under those conditions the interval In other words you would have thought at least of features stable hammer instructors a contact forms static condition which is generic consistent result he hearing more the were also find more spot 100 of grief so talk a little about the functional analytics set up for the neighborhood of let's say I have a specific element of the modular space complex structure Jane art and the map is called you not to do too last year the injected inside you this is someone that is somewhere and those 2 things are equivalent which isn't not so hard said so what I really should ask for is that there is injected .period mapped into you and as a it's a good point because if I were working in other settings where I have boundary with Grant and under conditions of something but then it's not necessarily true that just because the current somewhere injectors that it's injected .period and dance so then I would have to be careful but here injected appointed Odense if the current injected there you go to the end of the year right people the this is taking place on the side of it this way because this this 1 the world the words thank you this is called Democratic backward technique so I need to actually write down some kind of manifold of maps that will contain the solutions I'm interested in this is something that's kind of standard in the case of closed the matching close remand services and the 1st punctured I guess it's considered sufficiently standard that nobody really feels the need to go through the details from the right papers but it means there is no paperwork and 1 doesn't Intel's really is for us think In Mateus starts his thesis I think there may be details about possibly the only place in the over now it's that was what he does he may he when he doesn't areas is basically just lifted from Francis is OK so I'm going to talk about they spaces Sobel of class maps from the painted surface to W hat the class WK piece so I want to have a caterer is of class locally locally that would be the definition and it's not hard to express but globally was this mean that the easiest way to say it is the use of the form along best of age where else is going to be some smooth maps which is not just as emphatically that literally cylindrical near the ends
a lawsuit that in moments h is going to be some sections "quotation mark class wkt locally along back so the pullback tens of modeled along S so In contact subsets this is a sufficient definition but I need to save more specifically what happens at the end so but I don't know at the end you have to have the I know I don't I don't care if only to direct so far only 1 working I don't have to require really anything about 2 0 because I'm going to perturbed in a certain region and I'm only looking at home off occurs a pass through the area 78 puncture In Cold War think cylindrical coordinates I can choose coordinates STD Research in a hostel in there and they just call it a positive have cylinders positive punctured negative tests different it's the negative function and then the requirements are that the map and just looks like T. S. plus a constant gamma the meeting was a constant in a cylindrical an hour plus minus times and plus-minus so here C-1 and C-2 artists Crandall constants and you know because I haven't I'm about to say that outside well again I will be valued in this 1 you can add constant the government a very good .period that I thank you for that it's again I'm going to be a teacher at a quarter that of the raid field and then when I say about age is simply that in the cylindrical coordinates were also I'm assuming I've chosen some kind of sensible asymptotic we translation invariant trivialization of this a tangent bundle I want to have a page on that and be of class W K P on the cylinder that has and so I haven't said anything about what came I'm not really going to make any requirement but about that accepts the Times piece should be greater than 2 because Sullivan betting tells me W K P injects continuously into C 0 so these maps are at least always going to be there were continuing signs that
have real troubles making this a good definition of the "quotation mark we will be ready for signing getting to that and getting into this this is where the union of the 2 that is you know yeah but I'm going to do well you know what is going to be a map of
this phenomenon going to consider other maps of this form those close to you but I still I haven't localized yes this is still a a global it's all of wouldn't want to have some sort of general wait he will help In addition when ruling out maps then don't have decay at infinity To sum very sort of conclude at this very controlled map and has this precise cylindrical form I'm only considering maps that have asymptotic decay to a map of the form so I don't want them to
moves around non-trivial amounts arbitrarily Florida my father seems kind of pre-empted the next low variation I have to put on this and I will be able to explain why it is said that that's the say for a parameter Delta I can let and going call this indeed is the actual loss manifold that'll worked so this is going to be again you Uzbeks belong as contagious as above but the condition it should be slightly different going to require that the norm the age for nothin nominate itself that phenomenon on past cylinder of the to the Delta last times is fine so that enforces an exponential decay In addition to the decay of the W kicking on already requires so I'm not going to tell you where yet why this is necessary I will tell you that eventually I will tell you why it's possible which is whole this fact don't I guess it's there it's sort of a hard 0 1st version of it approved by the Office of the standards and the other version of this contacts since then there is the asymptotic audits on Monday January then for all Delta's sufficiently small all asymptotic me cylindrical I was just folklorist kindergarteners space defined belong to this bond mindful that is the map involved in defining belongs to that manifold has the asymptotic the exponential decay as long as Delta sufficiently small the president of the rocket was to vote those around him know that there should be you that they had a plus or minus I want a minus sign of other negative function the point is that it enforces
asymptotic decay is the exponential decay rather than
exponential growth or or something now working extensively said in 0 amount of time I have left the blood lead my question stands the world has a lot of it is that the remarkable work this that is the last thing we 3rd don't have with yet does know that to say is properly I would have to say that given a specific set of asymptotic the various Delta sufficiently small so that all consider Eisenhower to those audits satisfied decayed condition so Anthony trial and at least get to the definition of regularity in some form so what I'm going to do now I define a class of maps and want to work with I also have to worry about the complex structure and the domain because that's not fixed it's not fair some sort of God-given data to work with but I allowed to vary in different points in this model is this a different complex structures now I'm looking at just the neighborhood of you're not with complex structure not which the system visible from around the world is so let's just worry about the complex structures that are close to Jane and 1 way to do that I don't have to consider everything in its infinite dimensional space complex structures but I was requires a runner cover all of them that are not equivalent by diffeomorphisms eye candy about ties with respect that's at least a nice smooth manifold so when going to do is say let the team the smooth finite dimensional family of complex structures on sigma which I can if I want to I can even assume that their fixed near Damir and theater another thing that I can just take our models of a Sigma and fix punctures in 1 particular place several optical places fix all supermarket .period in particular places I don't need to allow them to move around because of I want to get all the different conformal structures up to equivalence with these punctures and .period all of them can be represented by some complex structure that have the punches in March points in that particular play and that's just a question of choosing it if amorphous so I'm fixing down theater in place on Sigma but I'm letting the complex structures vary and this is going to be a smooth finite dimensional family know that parameterize a neighborhood of the equivalence classes the J. not entitled space what's that was it was like those used by the end of the summer you know almost all the user and the .period if this camp but I have to bury the complex in the context of this section of the handling of the on the part of the world both of them and also they're not called a few more effective there are more than 3 months .period you know even on the series have to do this OK so some dramatizing a neighborhood enticement space which is the face teams segment of fatal Of all complex searches and Sigma divided by the identity component of the diffeomorphisms group preserving fixing the country's allotments and so the classical effect that this is a fundamental manifold so I can choose some sort of finite dimensional space a complex structures profit and and there are various sensible ways to do this talk about tomorrow if I have time there a quick learner let's also assume just for simplicity at the moment that the characteristic of the punctured surface might the mark .period negative also talked about this a little as the stable situation so that the situation in which the automorphism group of the domain with its punctures and my point is finite does not absolutely necessary assumption for what I'm talking about but it does make a few things easier citing in particular saying and there exists a choice this objects the I'm going in the future call this a tightening sliced simply parameterize is a neighborhood of taking the space that is invariant under the action of at the time of the automorphism group of domain so 2 so having made
this choice refinery right down to the point of 1 we now can define a smooth section of symbolic space bundle of the domain is going to be the product of a tightening flights with my bought manifold amounts section is called the word so the fight over J ,comma you it's going to be the space of W K minus 1 key sections the bundle of complex and the linear maps from the signal J 2 fullback tangent bundle I hope that's the right number of parentheses if not somebody will surely let me know there are going to take Jr ,comma you to the obvious thing to plus the Jacob of its new composers little J. if you tell me I need to put a one-half in there somewhere I will tell you to get a life the finely so however that my own PhD students need to be especially careful in moments so you keep I go back and forth so the dilemma that the automorphism group that view that's under CEO said of this section but I call the obvious 5 started Ugandans such that the neighborhood a view in the modular space is in by directors correspondence with a neighborhood of the equivalence classes that I should say neighborhood of you not not less space with the equivalence classes on what is not clear you're not in this 0 said model that action by the with this so now when is this really said by the implicit functions there an NJ is going to be kind of a fault near you not with isotropic equal to the automorphism group that is if that section is manifold which is going to be true if the legalization of the section this rejected so that's all that some linear operated from the direct some of finite dimensional space tangent into space complex structures with well let's just let it as tangents face to the bonnet manifold which is some space sections of the pullback tendon Bible class that UK peak yes I would say to the the idea I thank you very much all you have to this is the moment where I said I thought you people care about mathematics basically so I wanted to lease and with having written down this operated right so fragile regularity means specifically I need to finish the sentence of the is rejected this so regularity means just that these operators rejected I'll talk more tomorrow about what 1 needs to do to make sure that the operator is subjective the result is that we identified local neighborhoods of modular space with listing which is a fundamental manifold abided by when I'm not going to say anything about the proof was lemon except that if you stare at it long enough it becomes it's not that so far the from the loss in the in the water in the form of loans 1 and 2 a man of the people of the world of value ,comma by acting as a good question if you want to show that something is regular well the answer is you you I've never been in a situation where I had actually choose tie for flights the concretely but it has to be there in the theoretical set up because if it's not there right now and he would obviously be sufficient if the realization this operator just mapping on this basis sections to this space sections is rejected that would suffice and implied that rejected the problem is that the it's not true as often as you want it to be there you have this additional finite dimensional stuff in the domain which sometimes actually makes a difference between the operator with a finite dimensional Corcoran 1 subjective so bad infected cells if I have time to talk about automatic transfer salary tomorrow hopefully it's a really crucial detail there the criterion used to prove automatic canvas does not work if you leave out this whole discussion
entitlement space because you need to be have the freedom to vary the complex structure in order to get as much as authorities this this is
not the only 1 of the things I could probably come up with the afternoon offers you and I have a discussion tomorrow that you will 1 of the core of the issue want it was the 1 who prior to the final examination found all the effect of all of them this operator would not be from home and that if you think you can think they were in the public eye of town so far I'm fixing the value of the day I'm here again variations the Norfolk and Western blot on the news he has meanwhile is full of you that it is likely that many of the other on because the nearly you find at 1 time families where you generic and was used you would can get there elements were were allowed to go in the heart of soldiers who were in that there is not what you think the right now I mean I the question why you get cancer study for simple curves for a generic term will address tomorrow so far all of talked about what criterion actually gives you some sense of the modulus space and how does 1 go about proving the only thing I can tell you that the time of the problem Over the course of events in the history of the event or not is accused of the change that you'd like to hold auctions all of that is going to be right there waiting for you you go on and on and on and on and on they have in their view of the vary from 1 of the few reality show where most of the people growth going into so


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