H Cartan's 'Variétés analytiques reelles et variétés analytiques complexes' (part 2)

0:00 What are we talking about chief now? If I refuse to talk about chief, what am I missing? From local point of view, some things. From global point of view, there have been nothing. Why? Why? Consider the situation in the algebraic case. You have a well-defined open set where all functions of your functions are defined. So you can use this open set to have complexification to gluing together some things as in a fine space but consider to have gluing some things in the same way as a manifold as you can glue together two things which are not manifold, not smooth. You don't have charts, you say. So you have charts, fine, but you have to glue them and the sheaf is a way to glue it, to put together the So, the local structure at X with the local structure at Y, they are fibers of the same shift. No, another thing. Consider also, at this time, a big problem was the cohomology. The cohomology, the coefficients of cohomology, are not constant in the point, depends on the points. So you need that. It's a way, shift, to group together the local rings which are different in this case. And it is exactly the same method if you will consider an abstract algebraic variety. In abstract algebraic variety, what is an algebraic variety? Now it is a scheme, but in this time it is a topological space locally isomorphic to a model. What is a model? A model is an affine algebraic saber. For a local situation, you don't need a shift. To glue together, I don't know if this is an answer or not.

2:30 In a sense, you have a family of local rings, which are the ring of, in a suitable system of coordinates, are the ring, are, say, ideals in the ring of convergent power series. And since they are germs of analytic functions, they are defined in an angle. So what happens in a point still have some significance in the nearby points. And you compare, so the generator, so in this way I also define coherence. You look at the ideal at the point. This is the germs of an analytic function vanishing at the real analytic space A nearby this point. This is an A here in an Ethereum ring, a very nice object, and thus a finite system of generators. So you can look at these generators as function defined in a small label of the point X. So you may look at them in the nearby fibers, which are, again, ideals in a Therian ring, which is the ring of convergent power series at the nearby point. and they are still germs and you may ask if they generate the ideal in the near point by point and this is true in the complex case if you take a complex analytic space defined in the same way then the generators of the fiber at the point X generate the fiber of so the ideals in the near by points the local the idea of the vanishing function also at the points nearby this is in a sense obvious in the algebraic case because you have a global you have an idea in the ring of polynomials so make no sense look so you have always the same idea

5:00 here you are maybe different ideas in the same so in isomorphic ring which are the converter panel series and you look if the generators at the point generate in the nearby points if so you call the shift coherent and in the conflict phase it is always coherent but you look at for instance an algebraic set like whitney umbrella you know the lines which go catanum bread or catanum umbrella but maybe it's easier in the the whitney umbrella then you take the ideal at the origin. It is generated by the equation. But the near five points, there are also the ones of the stick. And this equation of degree 3 cannot generate x and y, which are vanishing on the line. So, it is not coherent. So, in the real case appears this phenomenon on non-coherence, And then coherence means that the ship is locally of finite presentation. It's more or less this. But consider another situation. What is a complexification? I would like to make a point to maybe explain my concern. Maybe you remember that during my presentation, Arseneus Schreyer paper has stressed the fact that you have no cone in Arseneus Schreyer paper. And for me it is significant that there is no cone and this point of view will come later. Here what we have in paper of carton is not only the idea that you can have this kind of structure But this ends something else. We have several many things which appear when we are coming with this kind of paper. There are a kind of acceleration here, as you know, of the story. It happens many things, not at the same time, but you have various abstractions which come successively, and which has to come successively,

7:30 which are not coming at the same time, but very closely. So these mathematics are very different, and in a certain point of view, very close in the time. So it's a kind of singular point from a historical point of view and maybe two, several singular points very near and we have to make a kind of decent realization of those points to look at them closely. It's very dense. Yes, I consider that the complex situation was already clear at that time, because there had been the OCA results that were put in a more clear form in Seminary Cartan, and so the complex situation was already established. And the new point of view is to consider real objects and to notice that, in a sense, those phenomena were already present in the algebraic case, but apparently there was not such a deep study of those situations. Maybe we will see the example at the end where there is a real space of R3, which is, yes, such that any function vanishing on him, on it, should vanish on all the space. And then there are some phenomena which were not present in the algebraic case, and this is the first time, I think, they are considered that the Seminaire Cartan is of 51, 52, and the paper is of 57. So, in the mind of Catan, all this terminology is well-defined, well-accepted, for all world. All world! I don't know why we train that to do it. In the French world, around yesterday. Yes, yes, yes, yes, but it's a bit creepy, right?

10:00 It's also German, and it's a German fascination. Claudia, what are we talking about? Obviously not in the U.S. Not in U.S. No, I guess. Second, did you let it denot? Maybe, sir, but at the first time... I don't know exactly, but... I think MacLean was using... MacLean. Enomology. Enomology. Enomology. Enomology. Enomology. Eilenberg was using... When you came to France. In the 40s. I mean, perhaps even in the 30s. I don't know. The problem comes from the fact that the coefficient of homology changes the coefficient of homotopy groups. Now, the coefficient... But maybe it comes from the topology, from the problems that the homology has coefficients not fixed but in the homotopy groups and that is a way to control the coefficient of homology. But for instance to the work of Carton-Siminal, I really realized on the results of OCA, but they were formulated in different formats. So a big part of the activity of Cartan Seminar was to chiffify the results of OCA. And only, for example, I have here the part of Cartan Seminar where is the translation in the shift theory. The point of Michel is important because it is not only a matter to go from C to R, but also to go from somewhere to C point of view.

12:30 Yes, but as soon as you pass from the algebraic case to the analytic one, dealing with manifolds, the structure is locally constant. This is no more true in the similar case. So, I have some trouble with germs, and so I want to know if, when you say that the chief is locally of finite presentations, which is a formulation for germs, is it true for an open covering also? Yes, there is an open covering such. the implication is immediate from the division there is a you need that if you have a short exact sequence if two shift are coherent the third is coherent so you have a finite presentation you have the sheet the camera the camera is coherence so also the camera is finite and presented so gluing together the two short sequence you have a good presentation i think it is really a more abstract thing to say that means of germs are basically presented and saying that the wings of a section of a little open and so if this is equivalent why is it not gives a more understandable definition I don't know man since the catan is lying what can You see, you have the fiber at the point, then you have an ideal in an ethelian ring, then you have infinitely many generators, and this gives you a map from the part, assume they are P, then you get a map from all, they are defined all in a small neighborhood, which you can call U,

15:00 and you get the surjective map from the U to the power P to the surjective on this ideal, and then you look at the kernel of this map, and the kernel of this map is still a shift, a coherent shift, so it is finitely generated, and you get the other O to the power Q, And put together, you decide that an equivalent condition for a shift to pico event is that it is locally a co-curve of some... I cannot say this. And it is by a result of a fact. Okay, so, is it the case that this notion of shift and coherence was appeared in the study of Nansou's variety, analytic and algebraic, and that was the work of CERN? To say that, for example, FESO, algebraic fact, and GAGA was a way of pondering the analytic structure and the algebraic structure in non-singular situations, and basically to say that it was the same and the coherence which was automatically coerence was... Coerence was true in the complex case. It carries from all kinds of help. And now, in the real case, we try to do something that Catherine is trying to say, except that now coherence becomes some property that needs to be allowed to get the results, because there are natural cases in the real case where coherence is not true. I hope you are well aware that you are going to have to write on that. but yes consider it also the the local structure consider the problem of you are still talking. No, no, I'm happy that even after what I say you are still talking.

17:30 If this world begins, maybe I can write the history of a complex analysis in mind. And, no, I can start by the structure of the Susan Sambl analytic of the real section A, page 12. Consider the algebraic case. In the algebraic case, it is not difficult to define a complexification. We are exactly at the beginning of section 8. In the algebraic case it is not very difficult to consider a complexification. Why? Because you have a polynomial and so it is easy to define some things of complex, a complex algebraic variety defined in Cn or Lpn. What is a complexification? A complexification is something of complex and in a certain sense smaller. The smaller one, the smallest. Consider an example as this. You have equations. Consider some things as this. what is the complexification in this point if you take the curve the complexification in this point is not the analytic complexification because in this point from an analytic point of view this is a is not a variety so the complexification in this point should be a point. So, in order to have some things of local and smaller, Kantans define the complexification for germs, only for gels and the complete a global complexification only to the analytic set which are coherence since in them

20:00 in section 10 proves that if a sub analytic subset of n is coherent then as global equation so you can consider the complexity of the analytic space in CN but now the local structure now an analytic sector is defined only locally only locally so in a point can you can use functions in another point you can use other function and maybe this function are not extendable to complex In a certain sense it is the beginning, the germ of the beginning of some or minimal structure not polynomially bounded. The last example in Cartan papers proves that there are very strange situations in the real case considering some infinity function which when one extends these functions to the complex you have a position essential singularity this is the the problem in the in this world so Qatar need a company needs a complexification to define also other properties the idea is that a real analytic space is in some sense a real part of a complex space at this moment it's not us in the algebraic case we are very far from the real spectrum with the real case here is not so independent from the this is an article of several complex variables this is the in in the real case restricted to LN more or less is not so true what is the containing the real germ and such that, for each olomorphic function, f olomorphic, vanishing

22:30 A on the real germ, then F vanishes on a field. This is the idea of minimality of this germ. This is a notion of complexification. A little remark on germs, effective point of view. What is a germ? A germ is a geometric object, no? It is not an algebraic object. But what is really a germ? But what is a germ? A germ, from a geometric point of view, is a point, with some history, but it's not so well defined. It's a point. all the paper is take a point in a germ germ if you have a chance to look at the point so is in a certain sense the cut down the trees control controls in a as real object an object which is empty more or less it's always is exactly as the Taylor Spansion. It's a point with all the history. To have an algebraic treatment of this is a little bit more complicated. One can interpret a point as a morphism in some in some rings of formal there is a way that place exactly this is the same notion of place in algebraic it is a place it's a point with this it is a long history I think that the idea of seeing a job as you are seeing it that

25:00 It's a bad description. It's a monad. It's not a ponad. It's a monad. It is an equivalent class of open... It's a ponad with something very infinitesimal around where there is some behavior. It's no concrete geometric logic. I think it's more concrete if you see this in this way. We actually see it in a non-verbal world. It's an infinitesimal neighborhood. Just an equivalent class, you take all two couples of U, S, where S is analytic set, and then you consider equivalent to open sets, if they coincide in a smaller neighborhood of the point X, and then the equivalent class is what we call a GER. So, in any case, we can think about a concrete, no, analytics in a nation. But you have a representation. Can you make neighborhoods smaller and smaller? Smaller, smaller. What do you mean? Smaller, smaller, smaller. Yes, of course. But in any concrete situation, you can look at a concrete open set, because you take generators, and they are defined in smaller neighbors and generate, in a coherent situation, even smaller but true of effect. They are what they are. A germ is a germ. So what I wanted to say is that in Whitney also, we have the notion of conflict solution We have some global definition of the set with these equations. And then, so... In our situation, we look only at the right, and the globally defined function gives you everything. Yes. So what I wanted to say is that in a way that the same phenomenon which is the fact that there can be some local transport, real dimensions, we see it, so for us, the other day, it was the fact that we had this...

27:30 So we have this BI star, which is pretty contained, I mean that's what we really want to do, but there is this notion which is that the dimension of the complex dissipation and the dimension of the point, of the rear cartons are not necessarily the same. And it's also clear what happens in this motion. Posso scrivere l'ultimo esempio? Five minutes, five minutes. Catan proves that the complexification in the sense of germ, I ask, may I ask you that? Proves that the complexification in the sense of germ exists and is unique. The starting point of Kertan is take a point, take a representation of the germ, take the generators of the stalk in these points and take the germ. Then, take the eight point, take generator of Ea, F1, Fk, this is Neterian, so one can ask the right number of generators, take this, F1, 0, in a complex neighborhood, take this analytic set, and take the germ induced by the A. This is the notion of Kertan. And Kertan, with this object, proves all properties. Why use this?

30:00 Catan is obliged to do this, since consider this example, but this example is in the part of Francesca. No, it was the final example, so maybe there are some notions which were still we have not seen. The example is the enumbrella where in the stick there is defined by a function with essential singularity. So you can complexify with the equations. You are obliged to use another representation of the gel. but now I prefer to spoon the answer and says what Quentin does. Quentin, this is a list of properties it's more or less a stretch for over the cannot prove that the complexification in the sense of germs exists is unique the the stalk of the complexification is exactly the stalk at the point a and the complexification has good properties in the sense that respect the the decomposition in irreducible components since the germ is a germ attached to an ethereum ring one can do the irreducible components as in the algebraic case so the complexification respect this this means that if the germ has an irreducible component decomposition at this point then the complexification has the decomposition and any object is the complexification of this point, of the irreducible component. And Cartan proves also that if a germ is irreducible, its complexification is irreducible, and gives this definition. Take a germ, a complex germ,

32:30 be a this term is a complexification if is the complexification of Israel part for instance take this the germing used by this function is not a complexification, since the real part is a point. And this is not the complexification of the point. Now, Cartin has a good, under Guillemet, good notion of complexification, so Cartin defines the dimension. The dimension of the The dimension of a germ, of a real germ, the dimension of a real germ in a point, the real dimension of this germ, is exactly the dimension of AAC. there what is the dimension of a complex germ for carton in the article in the paper there is no definition for this object but it is clear that it is clear that in the the idea of carton is that the dimension of a germ of an analytic of a complex but many variety is a the dimension of smooth points as analytical this is one can deduce this exactly so you look at page 13 in the English version, the last line and the beginning of page 14. And you look and there is this notion of a p-dimensional analytic, complex analytic manifold, which is dense in the set of, in the point set of the complexification. So the dimension of a complex germ is the

35:00 you get when putting away the singular set, which is known to be a less dimensional analytic set. And for the French edition, it's 92. The commencement of the page is 92. You take a complex analytic set in a small open set, then you take the singular part, It is not everything. So the remaining part is a complex variety of dimension P. And this is the dimension of the germ, the complex germ. And then you define the dimension of the real germ as the dimension of its complexification. And Cartan proves that there are nearby A. So, in any neighborhood of A, there are points in which A is a smooth analytic variety of real dimension P. So, in the complex case, all the neighbors, except the singular set, is covered by an analytic manifold. In the real set, there is just one piece, but which approaches the point A, where it is a real analytic variety. Look at the Whitney umbrella. So, the smooth part in the complex is dense, in the sense that it is everything except the line. In the real part, it is just the outer part, which is a real and it is manifold, except in the line. So, it is not dense anymore in the neighborhood of the point. And Cartin states also that a complex, an analytic complex, irreducible, is a complexification. This is important for the next theorem. If, and only if, the A intersected at M, the real

37:30 part of the germ is irreducible and of the same dimension has been of dimension P where P is the dimension the complex dimension of BA there is there is the And now, the coherence, the coherence of the work. So, the first thing that Cartan observes is that coherence is a local fact. If you look at coherence you have to see that the generators at a point generate also in the nearby point, so it is a local concept. And what he proves is that a germ is a coherent germ if, and only if, so let's say, first of all, coherent is a coherent germ. a local fact. Secondly, it depends only on the germ. And then the condition is this, EIA is coherent, so take, let's say, an irreducible, yes, irreducible, then, it is coherent, if and

40:00 only if, take the complexification and look at the germ that this object induces in the the nearby points, they should be complexifications. So, it induces on nearby points, nearby real which is a complexification. And then, if AR is not irreducible, then it is coherent if and only if all its irreducible components are still coherent and the main example is that one cone, which can be considered in C4 or in R4, excuse me, C3 or R3. And you see this This is a cone over a quatic, which has a triple point, since when you look, so put z equals one. Can you draw the picture? No, the picture I cannot make. It's very easy. It's very easy. I want to do only algebraic consideration, so, this is this, this is the x plus y equals

42:30 zero is something as this but in this point there are two the the the important things is in an angle of these these are two complex conjugated lines the cone is important sorry so the cone is important the contact need irreducibility, and irreducibility between analytic and algebraic, if something is algebraic, the cone is irreducible, since, because of Chow theory, since the section is analytic. Assume you are a complex algebraic geometer, and you look at the section of this surface with it equal to 1 then you look at the tangent core at the point and you see three lines this means that the curve has a triple point three branches going through the to the origin in the complex in the in the complex because you see the smallest degree part which is the product of three lines it is the tangent cone, the smaller form, the form of smaller degree is exactly the tangent cone. So you have three lines, this means three branches, and when you do the cone from the origin, so this is on the plane z equals one, then you take the cone from the origin. so you you see three sheet no no sheet is another word so three local components which goes through the line because they have distinct tangent planes so in the complex case you have this triple point and you see it cannot be a a complexification, because this is not a complexification, so the real set is not coherent, because in the origin this one is the complex, the equation for the complexified germ is just this one. Right, because this is irreducible, the important thing is that this is irreducible, analytically irreducible. the ideal of the real set at the origin is generated by this equation so this

45:00 equation is also the generator of the ideal of the complexification but the complexification in all the points of the line get three reducible components and if the set would be coherent all those three components should be complexifications and one is the one corresponding to this tangent not those one because they are complex sheets branches which have just so they do not have a real part of immersion tool so even if this set is pure dimensional dimensional, because a feature that we imagine in a non-coherent analytic set is the dropping of dimension as in the Cartan umbrella, or as in Whitney umbrella, and so on. In this case, there is no dropping of dimension, because the dimension is everywhere too. But you look at the composition of the gel of the complex space, and you see that not all components are complexifications and then the set is not coherent there is a stick embedded yes in a sense there is the stick that appears in the other umbrellas here is it's just inside the close of the... Yes, so it does not appear as a dimensional phenomenon. Scusi, solo una cosa. Questa è irriducibile, proprio una reazione tra l'algebrico e l'analitico, perché se fosse riducibile... Yes, and the fact that this set is irreducible is because this curve is irreducible. And if you take a cone over an algebraic irreducible curve, you get something which is irreducible, not only from the algebraic point of view, but also from the analytic point of view. Every component will be essentially a cone, a projective object. So a component will be an analytic set in the projective space. But an analytic set in the projective space is algebraic.

47:30 And then there are the final considerations of Cartin are very interesting because it is the beginning of a theory, in fact, it is the beginning of the theory of definable sets in a ring of analytic functions, so it is the main object of our research at this moment, and particularly There is a characterization of sets that are definable by global anti-funcias, so it is proposition something proposition 15 and he states three different three conditions which are proved to be equivalent and that they are like this Let's take an analytic subset into our N. Now we have something globally defined, in the sense that it's an analytic set to the N, and then the following properties are equivalent. The first is there exists a coherent ideal shift, I, in such that A is the support of quotient. This means that this shift is zero outside A. Yes, it is zero outside A. The The second one is the A is B intercepted with Rn, where B is a complex analytic set in

50:00 in a neighborhood of rn in u in cn. Admit a global complexification, exactly as in the algebraic case. This is the real part of a complex analytic set defined in a neighborhood. And the third condition is a is the zero set of finitely many equations. And so it is easy to pass from one to the other, because if it is the support of a coherent ideal shift, you may take the complex analytic set defined by this E, which defines a complex analytic object in the neighborhood of r to the n into c to the n, which has as a real part, exactly a. Then if there is a coherent ideal shift, so there is a complex analytic object which induces A, and since for a complex analytic set, any complex analytic set in C-N can be described by finitely many equations, this was proved by Grauert, and there is an indication of the proof in Cartan's paper as a note. Then you take the complex analytic equation for b take real and imaginary parts and you get a finite set of of equations defining a so a implies b implies c and then you get back from c to a since you can take the shift generated by those functions which is coherent by definition because it has global generators so so this is the you see these are the sets that now we call global analytic

52:30 sets because they are definable in the ring of analytic function and so only because B implies C for is a consequence of theorem B of Kerta exactly And finally I want to give this example we were speaking about before, which is the following one. So you take this equation, so it is z times x alla terza, where This is just an equation, where this function appears to be, look at this function, in the real domain it is an analytic function outside the points where the denominator vanishes, which can be extended to a smooth function putting zero at those points. so this is a smooth function but the set here is analytic because in a neighborhood of say zero zero one where this function is not analytic the set is reduced to the line so it has analytic equation So, this is an analytic set in Art 3, not globally defined, but because this function is not analytic. But if you look at this function in the complex domain, you see that this cannot be extended to a smooth function in the points where it is not defined, because they are essential similarities. No? It is not meromorphic at

55:00 this point. So, consider the set S' which is defined by the same equation except the plane z equal to one z equal minus one where it is not the the function a is not analytic so same equation same equation in c3 minus the plane okay Then you observe that this at zero, at the origin, is just the complexification of the set S. And now Cartan's claim is any F in R of R3, such that F restricted to S is zero, the only function which vanishes on the set S is the zero function. And it is clear that you have to use the fact that this function is not moved in a complex case. And the argument of Cartan is the following. So, I may erase. I don't know. So, this set has no global complexification. this cannot be globally complexified and also it has not global equations and so it is not definable in the ring of analytic functions let's look at the proof so first of all if such a function exist, it should be holomorphic in a neighborhood of the real space, into C3. And if you take this open neighborhood, rather small, you may assume that the set S' is not singular outside us, all its singularity are in the plane x equal to zero, so just to have some

57:30 situation. Excuse me, this is important, now this is open, for the night, for the night, the sun Now consider the regular part of S' called M, this is the regular part of S' of S prime, of the regular part of S prime, which is adherent, so get the origin in its So, here is the origin, and you take the S prime, which is the complexification of S as a germ at zero, but it is defined in all the C3 minus two planes, so it is a global object. and you look at the connected component of the regular part of this complex set which is approaches zero zero zero and then the first the first remark of Cartan which is done by joining points without with analytic acts so it is not so difficult is that the point zero zero one one of the bad points belongs to the closure of M. So that correct component, it is something around here, but goes through, and this also cannot be made the drawing, but it is, can be any point nearby, this point can be joined to point nearby zero zero zero, remaining in this M. So, our function, which is supposed to vanish on S, should vanish on the complexification,

1:00:00 hence on the smooth part of the complexification, hence on this connected component. And then, by analytic continuation should be zero on the closure of this component, F, the complex extension, maybe I put. And now look at the function, which is this one divided by Z. And you see that this function, so look at this function, it is no more a meromorphic function because of the presence of this object, so it has still an exceptional singularity at Z, But on the set M, it is a function of X and Y. Yes, it is. On the set M, it is. And then you use Picard's theorem. What says Picard's theorem in one complex variable? that in any neighborhood of an essential singularity, the function takes all the values of C, except maybe one, which is called the lacune. I don't know how to say it. it's called it okay so this means that in any neighborhood of one this function as a function of complex variety takes this value fix x and y then fix x and y in such a way that this value is not the lacuna the lacuna the exceptional value which is not obtained in any neighborhood of one Then this function gets this value infinitely in each neighborhood of zeta equal to 1. And this means that the positive take x0, y0, fixed, and look at the line defined by them.

1:02:30 this line meets the set m in infinitely many points which accumulate to the point one okay since f vanishes on s prime should vanish also in this sequence of points which goes to one it is the f is defined on the real line so on the real space so it is defined also in in zero zero one and then you you have a an allomorphic function on a line which has a not isolated so the zero set is not made of isolated points so our f f tilda should vanish on the whole line but the lines so you see every line except this one which gets the exceptional value so f should be identically clear. And this is the proof. I'm not sure to have convinced anyone, but it is the argument. So the argument of the arcs maybe is a bit... Not so clear. It is very easy, but you have to look why it works. But it is just an argument of joining two points by an arc. The second one, which uses the Picard theorem, I think it is the more interesting, and suggests developments, because, just to begin the discussion, I would say that I know about this example from the very beginning of my mathematical life, But I didn't ever go really into the proof. I was just supposing that an additive function which vanishes on the real should vanish on an essential singularity in the complex domain, so it would be zero. That's all. But it is not so. It is more involved with proof. you have to consider to get the true proof you have to do some way and maybe so such a function could not be, say, in a polynomially bounded or minimal structure. This is the future.

1:05:00 This is the future. Perfect, no. What do you think? What do you want to say that there is the last example, and it's proving that it's not because it's self-contact. So it's another example when the same situation... No, the example, the compact example is not correct, that the stick maybe is a line, contains a line. So it's not correct. But it is easy to build an example, a compact example. So this one is compact. What is it? No, c'è una retta. There is a line inside. So, x equals zero, y equals zero is contained. The last example at the very last line, but there is a compact example in a famous paper of this passion project. In general, in any case when you have a non-compact example like this, you may manage in a way to get a compact one. But obviously, the point of this example is that you should not think that it's a problem of compactness. No, it is not a problem of compactness. It's just because the points, the bad points, come in a finite... So, at a finite distance from the two-dimensional part. So, it's not a question of compactness. Yes, thank you. Do you think that the proposition 15 is a culminating point of the paper? Where is proposition 15? It is a point of paper. It is not the culminating? It is not the real aim of the paper. No, no, no, no, no, no. It may be. No, this is the starting point from a lot of other papers, but it is a sort of survey, even from beginning. In this proposition there are two equivalent characterizations, and the two last ones are without shift. Yes, you have a global object.

1:07:30 Yes, this is the situation where you may consider a global object. In a global object you don't need a shift. But look, one moment. When you find an analytic equation for a set, this does not mean that those equations generate the shift of all general vanishing on the set, the real structural shift of the set. But it is a more civilized object, because it is. And also there is a remark, in a sense one could think that if you think of an analytic set as an abstract object, and you would like to put it in some Euclidean space. So, the fact that it is not global, it has not finitely many functions such that it is zero-set. It is not, because this example is inside A3, in principle there is no restriction to get as bad as you want examples inside an Euclidean space. They are not not embeddable, say. Bad singularities are embeddable. Locally by definition, and then also globally. It's just a question of looking at the Zarisky tangent space and if the dimension goes to the infinity. of course you can better as soon as the dimension of the risky dungeon space is bound that you may be better as a closer to the space I have a question, but not the one you asked. What part of this is anthropological? What does it mean anthropological? Locally, it is, so, it is what means that it's right. Well, in the algebraic case, because one of the processes we have seen is you get to look at the complex analogies in the algebraic case

1:10:00 and have completely methods which are not depending on the real analogies or complex analogies. And here we have a lot of things which are really depending on the in the fact that we are working on here, air, not as algebraic objects. So, for the coherence you may find as many as you want, algebraic objects which are considered as analytic set are not coherent. This one is not algebraic. And the analytic functions are more strictly related, maybe, to the complex case, since, for instance, things are the result of Whitney's theorem, approximation of Whitney's theorem. Without the complex, it's impossible to prove the approximation of Whitney's theorem, since you need a strong convergence. In principle, when you find an analytic function as a limit, with a limit process, as a limit of some sequence, you have to get, before in the complex domain, get a sequence of holomorphic functions defined on the same domain, and they are convergent, then they converge to a holomorphic function, and the real part is the analytic function you were... otherwise analytic function of dance because so every every function is approximated by energy function this big difference between the complex and the real case so what is the question yeah yeah the guy is the combination between algebraic and and if i remember the the result is that if you consider an algebraic variety you consider the shift of analytical functions generate a the the analytic structure is generated by by particular algebraic functions this is complex it's complex in the real case it is

1:12:30 It's not strictly true, but it's a remark of Malgrange in... Malgrange is in English, it's Toujeroen, because they are in English. and this is in French. Margaret is in English? Margaret is in English. And it is in French. Well, it is proven that a smooth function that belongs to an ideal, which is generated by analytic function, then it is... A condition for coherence. So this paper is the proof that GaGa can hold in a real case. In a sense. In a sense. GaGa, I mean, this kind of connection with geometry. GaGa is Geometry Analytics Geometry. Sure. Obviously, in the real case, because of the phenomena of the basic proper dimensions of the way you start, and then you have some embedded in the motion. But Organization 15 is not a way of saying that the definition of sous-ensemble analytique is not good. I want to add a consideration. This point of view is exactly the point of view of Semi-Analytics, of Voyasevich. What is Semi-Analytics? The semi-analytic is something which is semi-analytic everywhere in the ambient manifold. It's exactly the same definition. Locals. Locals. Locals. Locals. It's exactly this definition. So, why this definition is not good? In the complex case, yes, yes, I know. I know. It's not countering. No, no, I try to understand. No, but even if you define, even in the algebraic case, you understand that, you have the steep of a smaller, you have the parts of a reducible algebraic surface,

1:15:00 which are from dimension one, to dimension one. So it has nothing to do with analytic functions or anything. So it's the kind of thing that Lissner is identifying, that the complexification, so you want to take, if you have this irreducible step in a smaller dimension, then when you look at the complexification locally, you want to get the same thing as if you look at the local equation, and nevertheless it's the way you see. And that's the phenomenon between real and non-calculation. It is algebraic, so it's not analytic. Yes, it is. But how do you connect analytically the point of the stick with the point of the canon? No, keeping inside the real. Yes, the peace-wise analytical. Peace-wise? Yes. But there is not a non-analytic function. This is related to another paper. This is a Nash component. There is some objects which are connected, but not Nash connected. But this is another paper. I want to add another thing that locally is good. For instance, for the local case, you can use all the art and land properties, machinery for the germs, words. if you want to treat these things only global for the global case you don't have any at the land properties, any in the space that not work well, just partially. Perhaps also I can mention that in the algebraic case, in the real algebraic case, things are even worse because the main point in this paper by Cartin is to show that for coherent analytic sheaths, you have your own A and B working.

1:17:30 In the algebraic case, you may think that you have also considered you have a shift of regular function, you are strict to real parts, but this situation it was, because you have not turned a nor b, you may not turn a nor b in this situation. So actually, from this point of view, you have these commoditical tools, which are very useful in the complex situation, which you can still have them in the real analytic case. Of course, it's always for coherent shields, but even, yeah, always working with coherent shields. The point that the ideals of some sense, maybe not coherent in the real case, but always a chromatical, we have good chromatical properties in the real case for analytic coherent shields, But in the algebraic case, it's no longer true. In Nash Function also, you have no terrain A or terrain B for Nash Function. and that was one, well, maybe that's the point that there are no good technological properties, even if you are working with Nash functions, not speak of shift of regular functions, but there are no good technological properties for shift of Nash functions. Well, this was remarked in a paper by Hubbard that shows that the cohomology is a complete mess with Nash functions, but not what we would like to do. And this, for a long time, was the conclusion, more or less, was that there's no use to consider

1:20:00 function. You can do nothing with them, no global global scope. But it's a long case, sorry. Another question, in the real algebraic case, is there a constructive method to check coherence of the issues of parallel? so if you have some dimension dropping in an irreducible case it is not good also if the dropping of dimension is not so visible after the cone I mentioned before are those the only But, for instance, you need to know if the germ is irreducible or not. We are not experts in a constructible model. So, I think I always offer a reusable model. But one thing that the paper of Catan is the very beginning of a lot of things. not only real analytic subset, global analytic subset, maybe also the paper for Yasevich, there is, in germ, there is all things now analytic subset, sub-analytic, minimal structure. In general, there is more or less a lot of things inside, and maybe it's the first paper of this time, okay, written in ashes, in this time it's more or less the first. Also, there is another paper, immediately sequence, the subject is the irreducible components for analytic subsets of RL, which made progress, but the paper is after this. For instance, if you consider, as I just say, a global analytics set, in the sense that it verifies one of the three properties that were on the blackboard.

1:22:30 Then you may look at its components. You say an analytics set is irreducible if it is not the union of two pieces of the same dimension, say. such that they are analytical each one is not the only the only thing but then those components need not to be global so one thing is to decompose into analytic components in the sense of re-analytic sets another thing is to decompose as global sense. If you want to see very wide analytic, real analytic subset of LN, look at the book of Narasimha, chapter 3. When the lecture were big were large the number five or eleven it's a form of a human not a human there is introduction to analytics space there is the preparation the progression here and the dates a section on reality and the component and a section of wide analytic examples. Forse erano questi tre.