A school hosted simultaneously at CIRM and IMPA
IMPA, Rio de Janeiro, CIRM, Marseille Luminy, February 26 – March 01, 2024.
Event page: https://conferences.cirm-math.fr/3185.html
The Brazilian and French communities working on foliation theory have a history of long-term collaborations. To maintain this tradition, CIRM and IMPA will host a School on Foliation Theory involving the two communities.
The event, which will happen simultaneously at CIRM and IMPA, will be a school on foliation theory. Its core will be 4 mini-courses, each one of them with three sessions of fifty minutes. Two of the mini-courses will be taught presentially at CIRM, and broadcast live to IMPA. Likewise, the two other mini-courses will be delivered at IMPA and broadcast to CIRM. Besides the mini-courses, we plan to have six short talks by Ph.D. students or young post-docs, three on each side of the Atlantic.
Please note that the Conference “Brazil-France School on Foliation Theory” will be held in person and will only be broadcasted on YouTube, without the possibility of online interaction. Therefore, it will only be possible to participate in the event by registering and being present at one of the the two venues of the event: either at IMPA (Brazil) or CIRM (France).
Organizers:
Erwan Rousseau (Univ. de Bretagne Occidentale)
Frank Loray (Univ. de Rennes)
Jorge Vitório Pereira (IMPA)
Olivia Barbarroux (CIRM)
Suely Lima (IMPA)
Scientific Committee:
Dominique Cerveau (Univ. de Rennes)
Paulo Sad (IMPA)
Um first I’d like to thank uh the organizers for this invitation uh it’s a great pleasure to uh give this mini course Excuse Me Lucy we don’t see you we don’t have uh ah oh now it’s good thank you so I should stay here and not
There so uh this mini course uh is well I’m not really uh in foliations but I’m going to talk about a subject which is in uh is very important in geometry and geometry in complex and real geometry and so it is my belief that this kind of
Subject could be of use to people INF foliations also uh so I will be talking about um real forms and real structures of complex varieties in this course um there will be today I will just do a classical setting where we just look at complex varieties and Define these
Objects and then in the next courses I will add a group action and talk about the equivariant version of this subject uh and my goal is to State some classical questions and some relatively new results in some of the parts of the this subject so for today where there’s
Not a group action and it’s just definitions I think what is important is the way I Define real forms and real structures there are many different ways of doing it and I will set up a a structure which uh is well adapted to what I want to do here and I hope that
It convinces you that it’s a good way of looking at real structures and real forms so for today the references that I would give would be a book by uh frederi Mongol um it was originally in French it’s now translated into English and it’s called real algebraic varieties and um
Fred mongold had a thesis student in 2016 that is only in French but it’s a very nice work where the description of real algebraic varities is given very clearly his thesis was from 2016 the book was originally written in 2017 I think but the English version is in
2020 okay so this is just to to settle the definitions and what I will talk about today so I’ll start by giving a definition of a real structure of a complex variety so the definition is as follows if x is a complex algebraic variety so it could be apine it could be
Projective it could be quasi projective it can be whatever but it’s a I will Define a real structure on X is an anti-egg involution on X and what I mean by that anti- reggulator is is such that if I I’m going to I’m going to do
It with uh brutally here and then I’ll describe what I mean I look at X I look at specy which is just a point and I look at the map that comes from Z goes to z bar that means that it’s anti-linear and I want this map to
Commute okay so that’s a little weird to say because this is just a point and this is just a point but I want the structure of C to be the same so I’ll say it a diff in a different way so that to be clear that everybody understands what I mean by that
So in other words I’ll I’ll do in other words in the case I’ll take the case where X is apine so it’s an Aline variety it has a coordinate ring and I’ll Define a a real uh real structure on the coordinate ring is just uh a uh is is a is given
By an action of the gallowa group gamma which is the gallowa group of C over R this Gallow group contains two elements one and another element I’ll call gamma and I want it to be such that gamma um on C is just given by bar of C okay in other words I want
Uh that is uh gamma is a ring homomorphism ring automorph is a ring induces a ring involution on CX such that gamma applied to CF for anything it’s just c bar gamma F so that’s what I mean by anti- regular on the complex numbers it’s just bar
Okay so I’ll give you some examples let me just make sure that I’m not what I do here okay I started here put here okay I’ll just um well before I do that I want just want to say okay this is a on CX and I’ll say this action this induces a gamma
Action on X okay because I have it on the on the ring okay so um I’ll just uh give some examples okay um well the first example it’ll be very easy iine space and I can say um mu of Z1 to ZN this gives me Z1 bar to z n
Bar that is a real that’s a real structure on this alha and variety uh on the on the Ring of functions what does it do it’s mu of any polom c i1 to i n Z1 i1 to ZN i n it’ll be uh
The sum of c i1 i n Bar Z1 i1 ZN I it just changes the constants doesn’t do anything else to the all the other variables so this is a typical real structure on Aline SI okay let’s do one where there’s something actually in interesting that happens I will take um
Uh uh complex sphere so I’ll take to uh um X is okay how did I write it here I want to use the same notation yes so it’s a spec it’s the real variety coming from three variables such that x^2 + y^2 uh plus z^ 2al
One okay it’s a complex V variety which we often call uh the complex sphere because it comes from this it has several different real structures let’s just look at someone I can I can take new1 of XYZ can be xbar Y Bar Z
Bar but I can also do mu do to of x y z gives me Y Bar xar Z bar okay now are these is there a is there something but both of these are involutions and I claim that they look very different okay so I have to Define what
That means I’m going to say two real structures okay so in definition two real structures mu1 and mu2 on X are equivalent if one is conjugate to the other that means there exists five an auto complex automorphism of X such that um f mu 1 Pi inverse equals M2
Okay okay so that’s a a definition and what I can I claim that these two are different these are not equivalent okay and the way I will show it is is by looking at another part another way of looking at this thing I would look at real forms and real points so
Um we can’t see very well here it’s what’s the idea of defining it is x² + y² + Square – 1 yes a sphere there’s no sign okay X2 + y^2 + z^2 – 1 yes yeah but then if take f as just the permutation of X and Y wouldn’t it
Uh wouldn’t be that would be a real form but what what’s going to happen is that these will two will be different and then many of them will be the same will be equivalent like if you just if you if you um well I’ll I’ll show you but there
There are different ways of there there are many different real structures on this okay on this uh sphere okay um and what will be different will be the set of real points okay so let me just go on here for a second and then we’ll come back to that okay a real form
Of X of of X is a real variety Z so that’s a variety defined over the real numbers such that its comp its complexification um is just isomorphic to X okay okay so what happens if x is quasi projective if x is quasi [Applause] projective and mu is a real form on it
Mu is a real structure then if I take the quotient it will be a real form okay it’s just a real variety whose complexification is here and the way that we’ve set it by construction X mu1 is isomorphic as real varieties if and only if mu1 is equivalent to mu2 so
My equ equivalence comes from saying I want the real forms to be the same okay so what this means is that at least in the Quasi projective case giving a real structure is the same thing as giving a real form there’s a bje between real forms and real structures I’ll put it this
Way the real forms of X real structures on X if uh they’re bjective if I have mu here I just go to the quotient if I have Z here I just um complexify and so this this gives me two ways of saying the same thing in the case of Quasi projective varieties if
The variety is not quasi projective I might have problems with this quotient because it might not be a real variety so I’ll just stick to the Quasi projective case where things go stay nice okay so now that’s real forms that’s real structures and then I can look at the real
Points what are the real points the real points it’s the fix Point set it’s the element that are fixed by X okay yes if I didn’t get the if I have a real a real variety I complexify it um spec see sorry what is the real structure the real structure will be a
Map that fixes Z and sends Z to zbar okay okay okay yeah I it’s not really written correctly there but okay but that’s the idea if I write it like this I mean that I want Mew to fix the elements here and and not here okay okay so I have my real
Points here uh and uh what happens is that if mu1 and mu2 mu1 and mu2 equivalent will imply that these fixed points will be theomorphic and so if I want to show that two different real structures are not equivalent one way of doing that is to show that their real points are not
Dorph um okay where’s my example that I wanted to put here did I want to put after yeah do it a little bit afterwards okay so one classical question is given uh complex variety x what are all the equivalence classes of its real forms or real structures so so I’ll do that determine
All real structures up to equivalence of X um okay to do that there are two things one has to do one is existence is there any real structure I won’t talk too much about that today because the examples I give will have an obvious real structure and
What will be more interesting to me is the second part if one real structure if at least one exists how many are there what are all of them or or how many are there now this is a classical result of uh uh where the way one
Can figure out how many there are is with a chology set so we suppose um mu0 is a real structure on X and and we consider uh the gap the action of gamma on complex automorphisms and the one I’m going to pick is one given by conjugation by this mu0 that I’ve given
Okay um okay fi okay okay so gamma sends fi to Mu F mu conjugation by mu mu0 is an involution so this is conjugation mu0 is anti-g f is regular mu0 is anti-g so this is regular okay sorry what what’s the question what is gamma say consider this is my my my group
Gamma has identity and this one other element gamma I have a gamma is the gallawa group of C over R it has two elements the identity and Gamma so I just have a Z2 action where this is this is my involution okay so how can I okay so this gives me
Um if I have mu0 fixed I have an action on this group I’ll give you some examples in a moment and then I can say um the set of the real structures will be given given by Fe zero such that Fe okay well let’s put it like this any real
Structure use one on x is of the form of the form F mu0 where f mu0 f mu0 is the identity right anytime I want if anytime I have I real okay structure I’ll put it here in other words if in other words that is if if mu1 is a real
Structure then I can say mu1 mu0 is an automorphism of uh X so I can always write it like Fe music Z but I have this condition because I want it to be an involution and what is this condition this says Fe times the action of gamma on Fe is the
Identity that’s just exactly the condition of of being a cycle so I’m looking for all these automorphisms such that f mu0 f mu0 equals identity this is just the one Cycles in the group gamma to Art X this is gamma of this element here and when are they equivalent okay F1
Gamma F1 mu0 is equivalent to 5 2 mu zero if and only if well there’s got to be an automorphism so there has to be there exists a in the automorphism of C of X such that s inverse 51 mu0 uh s and Si should be 52
Mu0 but what does that mean I just move the things around that just means that um s inverse F mu0 SI mu0 equals 5 2 this is Si inverse and this is acting by gamma on s and this gives me exactly the equivalence um um defines the equivalence giving so H Z1
Sorry H1 of gamma on X is simply Z1 of gamma or X modulo this equivalence okay so it’s just a terminology it’s a way of saying how to find all real forms what I need to do is look at at this automorphism group and look at all of its um
Uh the elements of this form and when they’re equivalent so I’ll give you some examples to start we’ll start on aine line there we know very well the automorphism group it’s just apine linear things and it’s not hard to show that up to equils the only real structure is X gives
Xar there the real form is just the apine real line and the real fix Point set is the real line is here so that’s for A1 that’s easy two when I find two space it’s the same result is true but it’s much harder okay the group of automorphisms Here is known
To be uh Amalgamated product of the Triangular and the apine uh linear um Maps so um one finds here also only one real structure so this is a result in 1975 by Kashi what happens for n higher we do not know we do not know if there are other
Real structures on a and three so I’ll write this question in a different way um just for for n bigger than or equal to three we don’t know so a a way of saying this question is let a be a real finitely genrated ring such that when I take a and I
Tensor over C I have a polinomial ring of n variables and at least three the question is is a isomorphic to R X1 xn surprising to me that we don’t know that answer to that but we don’t the the reason it’s difficult is because the group of automorphisms of Aline SP space
Starting at n equal 3 is very complicated okay okay so let me go back to my to another another example which will be more important to me yeah where no idea no people think try try play I’ve played a lot it’s not but I think many people have and well it’s
It’s difficult because well you need to have other ones you have to have the automorphism group has to be big enough but then when are they equivalent okay we didn’t listen the question what was the question the question is is here do you see the question here not the one
That you made but that the one that the person asked if I had a guess of which way what is the answer I don’t have a guess I don’t know if it’s true or not I have no feeling of why it should be or shouldn’t um okay so that’s those are
Aine spaces so now let me go back to my complex sphere so I’m going to write it C XYZ coming from the usual this one is a usual way of writing it uh and this way I’m going to write I before I was doing structures just to to do it a little differently
I’m going to do it with real forms L now there this up to equivalence there are four real forms of this and what they are they come from Spec R XYZ x^2 plus or minus y^ 2 plus z^2 plus or minus one and I claim they give me four
Different varieties and they are all different they’re not equivalent and to see they’re not equivalent one can one can look at the real fixed points if you say plus with a minus here then you get a real sphere if you say x^2 + y^2 + z^2 + 1 = 0
You get the empty set as the real set set of real points if you say um x^2 – y^2 + z^2 + 1 = 0 then I just pick x and z however I like and I have y and I get two copies of air R okay it’s not it’s
Disconnected and finally if I have x^2 – y^2 + z^2 – 1 I get a cylinder so they’re all topologically different which means that the real forms are all different or the corresponding real structures are all different one has to work a little harder to show that there no no other
Ones but it’s not too hard so for something like that we know and I will well I doubt it’s today because I see the time going but I will be interested in two special cases uh of this of this variety here I will be interested in the real sphere and the cylinder
Okay okay so I’m going to rewrite these two varieties so S2 over r with respect of r x y z over X2 + y^2 + c^2 – 1 and H which is spec of R I’m going to write it slightly differently UV and Z where UV + z^ 2us
1 U is x + y v is xus y over two okay and so these are two different real varieties whose complexifications are the same not surprising a hyperbola it’s like a hyperbola in a circle okay okay and the last example I want to give here is
Not an apine one but a projective one I’ll do projective space just this projective space okay okay p uh c n okay and I claim that it has two different real structures or real structures or real forms if n is odd and one real structure if n is
Even and I’m actually going to leave this as an exercise for you but I’m going to tell you how to do it okay so this is this is one that we can actually really calculate the reason is we know the automorphism group of PNC okay the automorphism
Group it’s just gln see modulo you know so PG so if I want to find a real form that means find a matrix I that that uh so I have I have my obvious one I have um I have uh z0 ZN which goes to that’s one real form
Always and then now I look at automorphisms and see can I can I compose that with an automorphism to get something different and the answer is I can if and only if n is odd and I can then uh change this to minus Z1 bar okay I can make it go
Too minus Z1 bar z0 bar minus Z3 Bar Z uh two pardon two three Etc I just by pairs I switched them and put a minus sign it’s easy to see the these two are different because here the fix Point set is a PNR and here the fix Point set is
Empty so that is easy right so to see that for for n being odd there are at least two is easy but then to show that that’s all you have you have to look a little at these automorphisms and say well if you have an automorphism with this one I call
That the The Matrix of the automorphism will be a and it means that if I go a a bar it’s going to be a homoy and I have to look at that homoy so I’ll leave that as an exercise and we can discuss it tomorrow or in
Otherwise okay so that that is a a nice picture here so um the the topic I want to talk to you about now is that well for many years many people knew lots of examples of real structures real forms of um of varieties but what they didn’t know is can one
Find one variety that has an infinite number of different real structures okay and and the answer we know now is yes but it’s quite recent so I’m going to tell give you a list of some different examples that were given and um the first example was in 2017 by Li and their
Projective of Dimension at least six the um 2019 Dean and Oro gave families of examples families of projective examples of dimension at least two so they brought it down um but their examples in their examples the ideas they weren’t looking for infinite number of real structures what
They were looking for was projective V varieties with interesting automorphism groups and these these varieties that they construct have an automorphism groups which are discrete but not a finite type and because of the structure of these automorphism groups they got uh as a plus infinite number of real structures
Because the automorphism groups they could show that when you have uh the discreetness of the automorphism uh gives that these projective varieties are not rational and it’s because of that that um about in let’s say we’ll say in 200 uh 19 or 20 I’ll put 2020 to be precise um ad D Jean
Fenberg and myself gave examples where uh X is aine rational of Dimension at least four so what’s nice here is the rational um soon after uh Dean U and Ozil made their results better and they could get um uh rational projective of Dimension at least three now in all these examples there’s
Infinite number of real structures but it’s still countable and the to the nicest result we have now 2029 is of anot who gave a surface it’s an apine surface with noncount with uncountably many different real structures and this comes from giz gizatullin surfaces now now um I um would like to uh
Describe all of these results are quite difficult to describe this one I can give a sketch and that’s what I’m hoping to do probably the next time I give a sketch of this proof but so for today I just want to I have still three minutes
Something like that is that what I have okay so I’ll just set this up I’ll set up what we will do to make this work okay so I will tell you what uh wait hey it was right here okay I should have left it here okay well okay I’ll do it
Differ the variety that I’m going to pick so it’s just a sketch of this of the Aline result sketch of proof of DF from so the variety I’m going to pick is going to Simply Be This complex sphere cross a to c can’t get much easier than that okay
And what I’m going to do is think of this as a fiber bundle where this is the base and this is a trivial bundle and I’m going to uh uh put real structures on this Variety in such a way that I will find infinitely many that are different because the topology of
The real points will all be not the two by one all not diffeomorphic let me just say a couple comments and then tomorrow I will give this this um result um so first of all the group of automorphisms here is huge there’s no way that I can pretend
That we have all of the real structures there can be ones that mess up this fiber bundle structure but if I look at the fiber bundle structures there are reasons to think that I have all of them okay uh secondly I will the ideas of how I’m going to do it tomorrow will
Be quite Elementary so going uh but behind the scenes the reason that it works is because a very deep results by myty Swan um let me just give you I I I think that’s what I’ll do just in this last minute I’ll just tell you the results
That that they gave that we will that are behind the scenes here myty from 1969 said that any rank two Vector bundle on this space here um splits into a trivial factor and a line model now I in the end don’t need it because I’m going to do it but I would never
Have guessed that it worked without this result of M okay so that’s one thing that we use um and another is um well what we’re going to do is we’re going to look at uh Vector bundles in three settings complex rank two Vector bundles real algebraic and topological and well there’s
A how do we compare the real and the and topological it’s a result of okay that’s the last thing I’ll Do For Today of bar andan okay on the sphere okay bar and O tell me the following um all real forms uh okay uh you take uh any uh topological Vector
Bundel over the real sphere is realizable is real realizable algebraically they do more than that but that’s the part that’s going to be interesting to me so I’m going to look at the topological Vector bundles over a sphere compare them I’m going to make them algebraic and make that algebraic part
Be a real form of uh complex one and well that’s for tomorrow okay thank you very much [Applause] are there any question any question here yeah there’s a question there yeah I don’t think so I think we can thank uh Lu again this question have one okay so if I
Understand the existences is always guaranteed the existence I didn’t say anything about here that’s another story u i mean you they don’t always exist but are they are they example of of of complex variety with no real oh yeah sure I’m do some elliptic curve where there’s too many complex things I I’m
I’m sure well I I don’t have one off hand I have many with with uh group actions that don’t but Um yeah I don’t I don’t have one off off hand but you could certainly construct them with um with non-rational curves and something so thank you uh maybe uh another question uh if you consider uh uh muzo given real structure and you de it by taking a an automorphism which is in the
Component of the identity right can you find other uh sometimes real structure well they’re of um I was wondering if it is a question of connected components well I mean that the examples of anab Bot you can you can do that but she’s moving a point that she blows up and that changes
A little bit so well I okay I’m going to I’m I’m going to say that usually that does not work and that’s why it was so hard to find examples where there uh uh infinite numbers of them but I think that it isn’t clear to me that you can never do it uhuh
Okay so if there is no other question so let us than you see again okay [Applause]