BASIC NOTIONS SEMINAR: The Calabi Problem

Okay. >> Well, thank you all for coming to the interview. It is a pleasure to have Professor Kolina Rajo from IMPA give us a basic notion seminar on the Alabi problem. >> Thank you Alina. So um thank you for the invitation. It's really a great pleasure to be here back at the ICTP. So this is a basic notion seminar. So please if you have any questions uh interrupt me and and ask me whenever you want. Uh okay. Okay. So I will tell you today about the calabi problem. So before I start the math uh let me introduce the mathematician. So this is uh Eljenu Kalabi uh who passed away couple years ago at the age of 100 and uh he left like a very important legacy in in geometry and in particular this is what we're talking about today. He advertised this what is became known as the calabi problem in his ICM talk in 1954. And the goal of this seminar is to explain to you what the calabi problem is. And so here is uh here is the um the problem itself. This is um problem consists on determining which comp complex varieties admit a kaylor Einstein metric. So I will explain all the uh all the terms in this question but okay this is the this is the calabi problem and this is the plan for today's talk. So the the calabi problem comes existence of kalytric is a problem that comes from uh theial geometry and so the first part of the talk I will just review some very basic notions from differential geometry uh but there are some very deep connections with algebraic geometry. So I will spend the second part of the talk uh introducing some basic notions about algebraic geometry and in particular by rational geometry which is the the field that I work on and then finally I will talk about the calabi problem itself and I should warn you that this is going to come from an algebraic geometry viewpoint. Okay. So let's start with uh differential geometry and uh I need to tell you or remind you about the notion of curvature. So in geometry this is probably the most basic notion and um and let me start with the curvature of a say a plane curve. So we can we can define a curvature to to to um as a measure of how much a curve deviates from being a straight line. And now let me give you more you know a a concrete way to to uh to understand the the curvature of a curve. So we have a curve in the plane and then we have a point P in the curve. We want to measure the curvature or define the curvature of the curve at this point. So what we do is we take the circle that best approximates the curve at this point P and uh and then this curve will have a certain radius R and then the curvature is 1 / R. So if you have a straight line this you can think of a circle of infinite radius. So the curvature would be would be zero or saying this curve at this point here this is the circle of the smallest radius that approximates the curve. So there it's the it's the point where the curvature is bigger. So this is a a number but also we have a sign convention. Uh we want to put also at a sign to the curvature and so we make some convention. For instance in this case we can say well if the circle that best approximates the curve is you know above the curve then we say that the curvature is positive and if it is below the curve it is negative. So we have a number but also with a sign. So this is a curvature for a plane curve. Now let's move on and talk about the gausian curvature of a surface. So let's assume we have a surf for simplicity we have a surface in R3 and now we want to measure so let's see is there oh sorry so we want to measure the the curvature of the surface or define the curvature of the surface at the point. So this is what we do. we take some plane that is uh that is that is normal to the surface at that point and when you intersect the surface with that plane you will get some curve and then you measure the curvature of the curve the way I told you before. So you do this for all the planes that are perpendicular to the surface at that given point and then you define the gausian curvature to be the product of the minimum curvature and the maximum curvature of the plane curves that you obtain this way. So this is the gausian curvature and uh so this gives us a a very known tricotomy I call it like the great tricotomy of surface curvatures which is here the first one is a negative curvature. You see that you have a in this in a saddle you have curves that it points in different directions. So the curvature is negative. The second one curvature is zero and the f in the third one the curvature is uh positive. And one remark is that differently than the case of uh of curvature of plane curves the gausian curvature is something that is intrinsic to the surface. It does not depend on how the surface is embedded say in R3 but only on the local geometry of the curve and this is the Gaus celebrated uh uh theorem and to make this precise I will recall some notions of Romanian geometry. So this how do we see uh these uh this concepts in in in modern mathematics? Well, so first of all in Romanian geometry, we start with a Romanian manifold that is a smooth manifold M together with a metric G. So a manifold is just given by by charts and then so open subsets that are identified with open subsets of say RM and then you have uh the transition function. So this is a smooth manifold. It's given by these charts and the matrix is uh it's a it's um it's an inner product at every tangent space that varies continuously when the point varies on the man. So this is uh what we call a remmanion manifold and once you have a Romanian manifold there is a way this gives you a way to measure angles and distance and then you can define curvature just from from this information. And another thing that comes uh also from the remaining structure is what we call a parallel transport. So these are the notions that are going to be important for for later on. So a parallel transport is just a way to move the vectors along a curve in a way that they remain parallel. So I'm just giving this you a very informal uh definition. Okay. So let me finish this part by uh recalling a very important and classical theorem in differential geometry which is the Gaus bonet theorem. So the Gauss bonet theorem relates the curvature of a surface with its underlying topology. So if you have a compact orientable surface like this one so topologically it will be a sphere with a certain number of handles. This number of handles is what we call the genus of the surface. So in this case here this surface has genus 2. And what the gausbon theorem says is that the average of the gausian curvature of s is just given by this formula. So it only depends on the genus which is a topological uh invariant. And now revisiting the great tricotomy for compact orientable surfaces in terms of the genus. So we had positive zero and negative curvature that in terms of the genus gets translated into genus zero, genus one and genus greater equal than two. So this tricotomy manifests itself in in in many different ways. Here we're interested in terms of of curvature. Okay. So now let me uh move to algebraic geometry. So I will tell you what are the objects that we study in in in algebraic geometry. So first of all what is algebraic geometry? So this is uh the field of mathematics that is study geometric objects. So we really interested in geometric properties of objects. But these objects are defined by polomial equations. So polomials carry a very uh a very strong algebraic structure that can be exploited to understand geometry. So usually we have this bridge that we want to understand geometry and exploit the algebraic uh structure or vice versa. Say we want to solve equations uh and then maybe we cannot solve them explicitly but we can by looking at the geometric objects that they associate we can get some qualitative information. We will see some examples uh very of this breach. Um so now what are the what are the objects that we study? So let me first introduce the complex projective spaces. Actually we can do algebraic geometry over other fields or even rings. But for this uh because I'm interested in the calabi problem we will be working over the complex numbers. So the complex projected space is just a compactification of the aine space CN where we add one point for each tangent direction. So this is the complex projective space and now let me introduce the complex projective variety. So let me first start by okay as I said they're defined by polomials. So you take finite set of polomials say in n and n variables and then we can define what we call the aine variety which is the the points of the of the aine space cn where all this polomials vanish. So this will define some some set in CN and then we compactify them inside the projective space and this is what we call a projective right. Let me give you a very simple example plane curves. So a plane curve a plane curve in a fine plane curve would be given by a polomial of two variables in C2 and then we compactify them. So this will give you a a curve a projective curve or contact curve in CP2. Now let me make some I'm making here some underlying assumptions that this is a smooth curve. So that no that the the derivatives of this polomial don't vanish simultaneously at any point. So if it is smooth in fact this C can be viewed as a reman surface. you know at every point if the derivative is different than zero we can use the implicit function theorem to write one variable in terms of the other so this this gives we can view C as a remon surface and as a remon surface it will have a genus right so and but now this curve was given by algebraic data was given by by a polomial and so we can we actually connect the algebraic and the the algebra and the geometry and in this case. Yes, the genus of the curve which we already saw that measures curvature. It's a topological invariant that measures curvature can be completely computed in terms of the genus. So this is what we call the genus degree formula we learned in the first course in uh in algebraic geometry. And so we see that algebraic information the degree of the polomial translates into geometric information namely the genus. Okay. Now let's go to higher dimensions and what I want to do now is to define the analog of the genus in higher dimensions. Now I have X some complex projective variety defined inside some uh projective space and uh what is the analogous of the genus in this case? Well the genus the the what we call the anti-iconical class is just the f so this is the defined as the first turn class of the tangent bomb. So if you don't know what the first chunk class of a vector bundle is so don't worry let me tell you what you need to know. So this is an element of the second coalology group of X and uh what it does is if you if you give me a complex projective curve contained in X then this curve will give me an element of the second homology group and that means that I can compute the intersection between the anticcononical class and the curve and this should give me an integer and what is this integer? This integer is nothing but up to a multiple the average of the curvature of X along C. So again this is what you you should take it is this is again a topological invariant referring a is a first turn class of a tangent bundle. So it's a topological invariant um that can carries information about the curvature. We are going to reenounter it uh later. Okay. Now um this canonical class or anticonical class plays a very important role in the problem in the classification problem in algebraic challenges. So let me tell you what this is about. So what do we mean by uh classifying algebraic varieties? Okay, again let let's go to dimension one. So in dimension one let me tell you how we classify variety. So as we already saw that given a project a smooth projective curve then it has some genus associated to it. So we we fix some discretion the genus and then we put our curves uh in in these boxes. So we classify them according to the genus and now for every fixed genus we have what we call a modelized space which is an algebraic all right an algebraic object that parameterize all the curves of genus G. So this is how the classification goes in in uh in dimension one. In higher dimensions however there are way too many isomorphism classes. So we don't have expect to have a classification like this. So we will the classification problem in higher dimensions is slightly different. And uh so we will not first of all we will not classify up to isomorphism. Instead we will define a new notion of equivalence and we will try to classify things in in um with respect to this new notion. So let me start with an example and then I will give you a definition. So this is a this is a this is a construction that appears in algebraic geometry all the time which is the blow up. So the blow up so I've illustrated here the blow up of the projective plane in one point. So what this is is this is a projective variety that is obtained by so here I have the the P2 the plane and it's obtained by replacing one point by a P1 of tangent direction. So here I have all this P1 of tangent directions through this point and so in the blow up I I remove the point and I substitute it and I replace it with this whole P1 of tangent directions. We see what is important in this example is that if I remove the point from P1 and its inverse image in the blow up what I have is an isomorphism. And so this gives me a very important notion of the notion of rational equivalence to projected varieties X and Y are projectively equivalent if they have dense open subsets that are isomorphic. And so for instance the Pichu is isomorphic to the blow to its blow up in one point. And now this is how we can uh describe the classification problem in higher dimensional algebraic geometry. So in the first in the first step given a projective variety we want to find a simplest representative in its bational class. So this is what we usually call the minimal model of x. And then once we know how to do this, the second step consists in construct and describe modelized spaces of minimal models with some fixed invariance. So today we will be more interested in the first step and uh this is achieved by what we call the minimal model program. So this is a very important theory in higher dimensional algebraic geometry. So the goal of the minimal model program is given a projected variety to find a simplest representative in its brational class. So for instance, if you want to compare the blow the P2 with its blow up, it's very easy to to to accept that the P2 is easier is simplest than its uh its blow up. Okay. So in dimension one there's not much to be done here because uh in dimension one uh by rational equivalence is the same thing as isomorphism for smooth curves. So this was already done by reman in the 19th century. Now in dimension two uh this uh this minimal model was achieved by the Italian school in the earlier 20th century and basically you you only have to understand blowups. So if you give me a surface so here in dimension two so if x knot is a given surface then I will find the blowdowns I will you know find find uh p1s that I can blow down to points and eventually I reach a surface that cannot be blown down further and this is what we call a minimal model. So this is very well understood. Now in dimension three things get much more complicated and then there was uh a few decades of work of many people and then eventually I just here I just gave them I just cited my but I mean actually a lot of mathematicians worked for here but Mory was the one who completed uh the program in 1988 and that actually yield him the the fuels medal in 1990. So this was a a very very difficult um uh a very challenging uh problem. And now in higher dimensions the program is not completely finished but a big advanced was made by beer carini hiken mccerna in 2010 and that gave us a minimal models for most important uh most important cases. So now we can say that we have you know a big chunk of the minimal model in any dimension and as a consequence of the minimum model every projected variety can be constructed from three classes and so the the the canonical class here plays a very crucial role in the minimal model and then after so it it usually does not have a well- definfined sign. It's not positive, definite, negative, definite or or or zero. And so every step of the minimal model, you try to make the canonical class more definite. And in the end, these are the three building blocks for projective varieties. So the ones with positive canonical class, which correspond to negative curvature are the ones called canonically polarized varieties. Canonical class zero, these are called calabial type variety. And then I will explain to you why the name calabia is given to these varieties. And then the case where the canonical the anticononical class is positive. So this would be canonical class negative or or positive carvatures are called funnel varieties. So this is the observation I told you before. The sign of a canonical class is usually not well defined. Okay. So now I will finally get to the calabi problem. Maybe it's a good time to stop and ask if there are some questions. Okay, if not then uh then let's move to the calabi problem. So again we want to understand which compact complex varieties admit a kaylor einstein metric. So now I want to explain these terms and so first of all what do I mean by a complex variety? Well a complex um so I will explain the first the kalor condition. So the comp what a compact a complex variety means just the same thing as a smooth manifold. Um but now instead of uh having charts identifying open subsets with open subsets of RM now I use cm and and in in a way that the transition functions now should be by holyics. So this is what we call a complex manifold. So it's given by this charts. And now one way that we use to actually uh to grab this complex structure is the following. So this is uh often called the almost contract structure. So if you look at the tangent space at every point, we can look at the multiplication by i. And so this gives us a linear a linear map in each tangent space whose square is minus the identity. So multiplication by minus Y and this is a way to capture this complex structure and so we will use this J uh later on. Okay. So now we have a compact complex variety of dimension M and then we capture the complex structure by looking at this linear map J. Now let me define what I mean by a tailor metric on X. There are many different ways of defining given the kalor condition and I will choose what I find the most uh geometric way to define what a variety what you mean for a complex variet to be tailored. So you can always put a remmanion metric on your given complex manifold. So now you see your complex manifold will be a real variety of dimension 2M and so you can put a remmanion structure remmanion metric there but it may or may not be compatible with the complex structure. So what do I mean by compatible with the complex structure? Well first condition I want the matrix to be invariant by this complex structure J. So this is very natural and and now comes the actually the the interesting important geometric condition. You also ask that this complex structure commutes with parallel transport. So this is one way of defining I will give you another definition later on but this is one way of defining a kalor metric. It's a remmanion metric on your complex manifold that is compatible with the complex structure. A note is that a compact complex variety may or may not admit a kalormetric. And when it does admit a kalor metric, it may it will actually may admit it will admit an an infinite dimensional uh space of kalor matrix. Okay. So when a complex variety admits a kalor metric, we call it a kalor variety, kaler manifold. And let me give you uh some some examples. So the first of all is the the the complex projective space. This is what we call the Fubini 2D metric. So it's actually if you look in the text in in books, this is actually a metric that is given in terms of equation something very concrete and then one check that it satisfies the kalor condition. So CPN is a kaler variety. And now if you have a complex projected variety inside some projected space and you restrict the Fubinis to the metric again you get a kala metric. So any algebraic variety is automatically kaler and for uh a kaler variety as I said it may not admit but if it admits a kalor metric it will have an infinite dimensional space which I will call h of kalor matrix and then a natural question is are there some canonical kalor matrix on x so if it admits an infinite dimensional space of matrix Is there some special metric that I can choose? Again, let's look at the example in dimension one. So we have X a compact rim on surface you know given again this is the tricotomy in terms of the genus. If I want to write it in terms of the curvature this is the tricotomy and it turns out that given a compact rimon surface it admits a unique tailor metric with constant gaus curvature. So for instance if I look at the um at the Taurus curvature is zero. You see the curvature is zero in the sense that the average of the curvature in the bars is zero. But we see here clearly that there are points where the curvature is negative and here points where the curvature is positive. So the curvature actually vanishes. But I can put a matrix on this uh surface with constant curvature equals zero. And this is usually um when we write the Taurus in in this way. So in this model here the curvature is actually when we do this identifications the actually the curvature is is zero. So this is the canonical metric in this case. Okay. So now we want to generalize this notion of a canonical metric uh to say higher dimension. So now I have a compact complex uh manifold and now I want to now redefine the kalor condition in a way or the kalor metric in a way that it's more tractable. So remember the kalor metric is a remmanion metric that is compatible with the complex structure. So now if you give me a remmanion matrix on X I will define a real true form in this way. So it combines the matrix G and the complex structure. So this is a a real true positive definite true form on on X and conversely if I if you give me such a such a true form I can recover the matrix G. So it is equivalent to look at the matrix and the true form and in this language the kaylor condition becomes very simple. It's just that you ask that d omega is zero. So that the form is closed. Okay. So this condition being kaler is that this form this real two form is is closed and once this is a closed t2 form uh it defines a element in the second coomology group. So now I'm viewing this as a deraomology. So it gives me a class there. Okay. So this class will be important. Let's keep it this way. But now let's look at curvature. Now I can define another shoe form on X which is given by the Richi curvature tensor. So we're starting with the richy curvature tensor which is what I said that in higher dimension um will measure the curvature of your variety. It gives you a true form. So that is also closed. So again it gives you an element of the second coalology group. And then what is you know the higher dimensional analog of the Gauss bonet theorem is that no matter which class you start with the richi true form is always the same and it is nothing but the anticcononical class that I defined before. So this is again I said the anticcononical the class measures curvature and this is independent of the kalor metric you start with. So omega could be different correspond to different classes different homology classes but the reachi curvature true form is always the same and it is uh it's multiple of the anticcononical section. Okay. So now I will finally define what is a kalor einstein metric. So first of all the kal we already seen the kalor condition. So if you have a a kalor manifold we can compute the uh the richy curvature two form and we say that a metric a kalor metric is scalar einstein if the richy curvature true form is a constant multiple of the metric and then we can renormalize to ask that this this uh this multiple is either minus one zero or one okay so This is so what a kaler ein metric is. And now the kabi problem is asks which compact kaler manifolds ad meet kaylor einstein metric. So we clearly see a necessary condition. You know this formula here gives us a a necessary condition which is well we want that the curvature is a multiple of the metric uh but the metric is positive definite. So this will tell us that the the anticcononical class or the canonical class needs to be uh positive definite. So this is a necessary condition. So and then if you have a kalor isomemetric then it is essentially unique. So in this sense it is canonical. Okay. So as I said this this uh this formula here gives us the necessary condition for the existence of a kaler Einstein metric namely that the canonical class is uh has a well- definfined sign. So a necessary condition for for for there being the kalerin metric is that the canonical class is either positive or zero or negative. And this is ex exactly those three building blocks from algebraic geometry that we saw before. Canonically polarized variety, calabial type varieties and honor varieties. Okay. So let's look at the calabi problem uh separately for these three classes of varieties. So the first theorem comes by in in 1976. He proves that for those uh kaler varieties with with the zero canonical class then it always admits a kaylor matrix and now this manifolds are now known as calabia manifolds for for this reason and then shortly after independently Obama and yao proved that for canonical class positive that is negative curvature again uh we always get a kalinis matrix And now what about funnal varieties and then things get much more complicated and much more interesting. So for funo varieties there may or may there may or may not exist a kaylorine metric. So the first obstruction was uh observed later uh early on by Matsushima. So Matsushima observed that if a phono variety admits a killer isometric then its automorphism group must be reductive. So if you don't know what this means let me just give you an example and a non-example to keep in mind. So if you look at let's look at GA and uh GM. So GA is just the the group given by the complex numbers with uh addition and G gm is just the invertible uh complex numbers with multiplication. So this is reductive and this is nonreductive. So the precise definition comes from um come from from related to representation theory but maybe having this examples in mind is enough uh for this talk because now I can give you examples. So here again calabi problem now we already saw it for canonical class positive and zero. So now for me the calabi problem is about funo varieties and we want to understand which funal varieties admit kala is symmetric. Some examples again CPN the projective space again the Fubini study metric is already uh Kaylor Einstein and in fact any rational homogeneous space is also Kaylor Einstein now some non known examples uh funnels without Kaylor Einstein metrics and then we will use Matsushima's uh criterion and we can look at the blow up of P2 in one point and in two points. So I explained you already what the blow up is and for blowups of the plane they are very well understood and see this is their automorphism group is also well understood and see the automorphism group in both cases it contains a term here with a GA which is nonreductive. So for these two surfaces the automorphism group is non-reductive and therefore they do not have a kaler einstein matrix and maybe this is a good moment for me to tell you about the classification of funnal varieties. So if you have a if you have if when we study funnel varieties if you fix a dimension there is a finite number of deformation families of funnels in that dimension. So in dimension two for surfaces here are all the funnel surfaces which are usually called delpzel surfaces. You can have CP2 the projective plane. You can have P1 cross P1 and the blow up of P2 in at most eight points in general position. So these are all the funnel surfaces or delpzo surfaces. Their automorphism group is very well understood. In particular, these ones, the blow up in one point and two points are the only ones with a nonreductive automorphism group. So all the other ones have reductive automorphism group and a very important theorem by Tan and Yao in the 80s and early 90s that is gives you a classification of sunosurfaces that admit Kaler is metric. Basically they proved that so this two cannot have Kaylor Einstein metric by Matsushima and all the other surfaces are Kaylor Einstein and this was actually very difficult I mean I wrote it this way but this were several papers this is for each for each of the uh the funnel surfaces they studied the problem and so the last I think the last paper is a paper of Tan in 199 is of 78 pages so this is really really hard stuff and uh and so one you could think that okay maybe what about three folks so I gave you a list in dimension two and actually I could give you a list also in dimension three so in in the works of Skovsky and Morim Mukai they classified all the funnel threefolds and there are 105 deformation so there is a list and there is actually a table if you if you're interested you could uh you could search for phoggraphy. So if you go to phoggraphy, phoggraphy is a it's a a website where you they give you a list of the 105 funnels and gives you you know their names and address you know and everything everything you want to know about each one of them. uh so they're they're quite well understood and probably at that time it would be something inimaginable to actually solve the calabi problem for these manifolds and okay so let's restrict to this case here can we determine which one of the three-dimensional funnel varieties admit a killer Einstein metric and just a just a remark to see how complicated things get starting in Dimension three, there are some of this deformation family that simultaneously contain members that admit scalarism metric and members that don't admit scalar isometric. So this is something that we did not see in the surface case but already starting in dimension three starts to to appear. So yes sorry you mean that this deformation process >> the fact that having a kometric is not stable under it's not >> it's very subtle >> these two things are not compatible with with each other I mean okay but this starts to happen in dimension three >> it starts to happen in dimension three okay so before I okay so I oops let me just discuss some analytic tools before I will I want to tell you about the algebra geometry tools. But let me discuss some analytic tools and those actually have to say that I learned here at ICTB when I spend a few months here in 2019. Claudia organized a working group on uh on Kaylor Einstein and then I learned many of these things back then. So if you have a kalor manifold let's fix x and now I'm fixing instead of the the matrix g I'm fixing the true form omega. Then as I I told you before that there is an infinite dimensional space of kal of of kalor einstein matrix. So this is how uh how one usually writes this space is called the space of kalor potentials. So this is giving you all the matrix in the same coology class all the kalor matrix in the same coomology class of the one you started with. And this is an infinite dimensional um uh space. You can put even you know can you can put some some structure. It's a homogeneous space a hyperbolic infinite dimensional space. And now a kale or systemmetric would be some some uh some point in this space and and then what this is one way that you analytically can make sense of the scalarizing metric. you define a what we call a mabuchi functional an energy functional in this uh in this space of kaler potentials and a kizite metric corresponds to a minimizer for this mabushi function so maybe this is this is what we need to know from the analytic side and and now I'm going to move to algebraic geometry but I want to keep some I want to um I want to keep a bridge with this so let me just keep here this H. So this is the this is the space of Kalor potentials in a given uh kalor class. And now comes the notion very technical I have to apologize but okay this is what we have uh from algebraic geometry which is called the K polystability. So let me define you what K poly stability is and trying to keep a connection with the with analysis. Okay. So first of all we start with a funnel variety for which we want to solve the c the the kalabi problem and then I will define what we call a test configuration on x. So for X so it is a test configuration for X this is first of all it's a family parameterized by the ained line such that the general fiber outside of S the fiber is isomorphic to our given final variety X but then over zero I allow some degeneration. So and here I put a singular point because you really I really allow this degeneration to be singular. It could have many components. It could be a mess. But the important thing is that this is an algebraic family whose general fiber is always X and that the special fiber it will be anything. And moreover there is one more data to a task configuration which is a C star or GM action. So I have a natural GM action on A1 which gives me by just reparameterization of multiplying by lambda. And then I want an action a C star action on X such that this all the everything here is equivariant. So this is the information of a test configuration. Now let's see how it relates to uh to this space of potential >> this categoric. >> So this is the this is the total space of the family. So over each point I have X and then the total family is what I call this calligraphical X. So my x here would be uh would be um to our given funnel variety x. Okay. So now suppose we have a test configuration. If you embed you can embed this there's an extra data that allows us to embed this in some projected space. So if I embed them in a projective space so let's say for for t time one I embed it in a projective space and then on the projective space I can restrict the k the the fubinis to the metric and this will give me a certain point for t equals 1 this will give me a certain point in the space of kaylor potentials now if I embed this family this whole family if I embed it as a family and I pull back the fubini study metric for each t different than zero. What it gives me is a ray in fact a jodic array in this space of scalar potentials. This is corresponding to this test configuration. Well, not any ray but you know sort of an algebraic way because this an algebraic array because this family here is is algebra. It's assumed to be algebraic. And now we defi one defines the Donaldson futaki invariant and the way that it was originally defined by uh by tion is in terms of weights of this action and then later Donaldson reinterpreted as a product of coology classes. So this is the I'm not right in the definition it's not necessary but I just want to tell you what it is. So this is going to be a rational number and what it is is the following. If I have this this ray and I look at the limit of the derivative of the mambuchi functional along this ray this is exactly this Donald sono talking variant. So this is let me just say it again. This is the uh the limit of the derivative of the the mabuchi functional along this ray when t goes to when t goes to zero. And so therefore if you want your fun variety to admit a killer is metric the mabuchi functional should have a minimum and then therefore the mabuchi functional this the dominance of invariant has to be greater or equal than zero. If you want a minimum, the derivative at infinity cannot uh cannot be negative, right? So otherwise you would not have a a minimal. And so this uh this is uh sort of an easy implication or at least a natural implication. And then the big conjecture so let me define okay before I give the conjecture let me define K poly stability. So a a fun variety is K poly stable if for every non-trivial test configuration the Donald s ofotakin variant is non- negative and it is zero if and only if the special fiber is also isomorphic to x okay so if a kar if a x ad meets a kaline symmetric it will be k poly stable so this direction is sort of natural and what was The big conjecture for many years is the the converse. So this was called the yao tan dono conjecture that says that if x admit meets a killer einstein metric if and only if x is polystate. So what this is saying is the existence of a killer metric which is an analytic condition by nature is equivalent to an algebraic condition and this is now a theorem. So this was proved uh by bloon son and then also there's work of tan from 2015 they actually now so now this is a theoremic existence of kaylor einstein metric is equivalent k stability so see this was about 10 years ago and then algebraic geometry started to work with this notion of kal stability and there was an immense amount of mathematics that was developed in this last 10 years so I'm not I'm going to just jump to the to the to the final statement. Uh but what I'm not telling you here is a lot. So so algebraic geometries in particular by rational geometers started to un to to look at this notion. Soon it became clear that the minimal model program is deeply related to this notion of case stability. So works of perkar have been used to give algebraic criteria for case stability. But I'm not I'm definitely not uh telling you anything about this. But let me tell you what something that happened in 2000 and in in 2020. So there was this uh this workshop in AIM which was called case stability and related topics where people got together to discuss um case stability and then in the morning there were talks in the afternoon there were uh research groups where people tried to work on specific problems and then uh I joined a working group that decided to look at the problem of K poly stability or existence of kalerine metric for funnel three-folds. So we wanted we understood that there were enough boos that had been developed from the algebraic side that made it tractable to actually look at the calabi problem for three-dimensional funnel varieties. And so this is the this is the group the the funnel team. So it's myself, Anamaria Castra, Ivan Cheltov, Kento Fujitta and Sophiaos Jesus Martinez Garcia, Constantine Shremer, uh Hri Sus and Dispanatan. So when we started there to work at this problem and then we worked uh for one year during the first year of the pandemic we worked very hard and we actually proved our main theorem which is for each of the 105 deformation families of threedimensional final varieties we determined whether or not the general member admits a killer Einstein metric. So here is the summary for 78 families the general member admits a kalorism metric and for 27 families no member admits a karismetric of course in the first case there is a more refined problem which is understanding which ones in that family admit a killer metric and which don't and some of many of the families we know everything but there are few families that we don't yet know exactly which ones are killer are and which ones are not. I should also mention that we did not have to do the 105 families because previously 65 cases were already known. And what I mean they were already known is because they fell into some general criteria. For instance, if you have a product, the product is scalar einstein if and only it each factor is scalar or for instance torque varieties. There are among this 105 there are 18 that are toric and for toric variety as usual you have a combinatorial uh characterization of of uh of fun well there's some polytope and then the origin is exactly the the very center of the polytope so so many cases were already known there's was the work of claio on on finite covers so many many cases we could apply previously known criterion but there were 40 that we had to work and it was like really one by one and we were using the new techniques that had been developed which I'm not telling you about but in the end we finished and this was not possible to publish a paper with this result so we actually published a book so this was the the book that was an outcome of this project the calabi problem for um prefolds and I guess this is what I wanted to tell you today so thank you questions. So can you tell us a little bit more about the the deformation process and why it does not talk completely with the existence of Ken metric >> um >> understand correctly this is not >> yeah this is not yes so as I I'm glad that I said that I'm going to say things from the algebraic perspective so from the I have to say that from the analytic perspective I do not understand it much But it turns out that from the algebraic perspective, we have this notion of modized spaces that we love so much in algebraic children. And for fun of varieties in general, it's not you don't have a nice model space. For instance, you could have uh a family. Let me see if I remember one easy example. uh you could have some you could have a family for which all the fibers except over zero they are funnel the same funnel but over the origin it's a different funnel or a different something different so this cannot live in a model space because this model space would not be separated I mean you can have a family that uh that has different limits and it turns out that the and this in a sense this is related right because this you could see one as a deformation of the other and so what happens is that this uh well stability is exactly the condition that you need to have a modized space. So the central fiber that you would get in the middle is it should not be in your modalized space. So if you only consider K polystable funnels then they actually this is a more recent theorem they actually fit in a nice modular space meaning that there that deformation is actually um allow allowed in in algebraic choice. So in a sense in a in a sense these things are connected but now how to interpret this from the analytical point of view I would not but one thing that one one easy case one easy instance of that is using Matsushima's obstruction so there are some of these families for which the for the general member y the autotocorism groups finite and reductive but for a special member it will be say some something that has GA so this is easy to construct and So you see that it cannot have a kalar isomemetric in this non-reductive case. So this is not yet it's not preserved unfortunately but that's life I guess. Sorry. I the general member what do you mean like specifically it's >> I mean that all of each one of these deformation families they will have um they have a certain description for instance they are the blow up of uh P3 at uh some points and then you there's some some description and then general means for instance if they come from the blow up of P3 in seven points then this the seven points have to be chosen generically so if shows them special then it it may fail. So by the you know so you reference these many algebraic geometry techniques and uh developments of the past 10 years. So do those involve translating this notion of K poly stability relating it to the geometric invariant theory for example you know >> so it seems that this more complicated some people have been doing this you know as a as a fun game for instance you look at cubix cubix in PN so these are funnels and then for cubics or hypersurface in general you have a git theory so for three-folds the git stability coincides with EK poly stability but already for cortex it does not. So it's it seems to be more subtle than than geometric invariant theory. Uh can you tell us a bit more about like so there was this uh cable stability which was about uh test configurations and release and the problem of finding the uh a minimizer for the function right so like can you tell us a bit more about the space because I just want to imagine where what kind of phrase with what space like what what would be the issue there like find the equivalent Uh you mean you mean here? >> Yeah. >> Okay. So as I said the these rays here that come from test configuration are very special. Right. I mean this is an infinite dimensional space. So most rays do not come from test configuration. And this is what is so impressive about this result because it tells you that you only need to look at say jodasics in algebraic directions in your space. But I have to say that for for for this space you have to ask cloud. I cannot uh I cannot tell you m tell you much but I have to say but maybe I what I can say is a little bit about the techniques. Um so this is not what we do right we don't do we don't do test configurations in uh when we when we prove one result like this we don't really look at task configurations. Now in these last 10 years that have been uh for instance invariance that one can that one defines that gives you say either positive or negative criteria. So for inance there's the alpha invariant the beta invariant the delta invariant. So these are invariants that are defined in terms of algebraic geometry uh in in and tools from the minimal model program that somehow capture the existence. So if you for instance one of the invariants that we use is defined in terms of um of exceptional divisor over some blow up or some model of your variety. So if you and then we define a beta invariant for that and if the beta variant is negative there's no kariz matrix. So how do we translate it from if you have find such a divisor for which the beta invariant is negative you can associate a test configuration that is going to give you something negative. So it's it's a it so we don't use this directly but we actually use algebraic geometric invariance that are built to understand this notion but as I said I I did not tell you anything about about this this any further questions for and we thank you very much again. >> Do you get to talk to the students? I believe it's >> I mean are we >> is there a close session between Carolina and the students now after the basic notion seminar or no? We can decide willing to say 10 minutes. >> Sure. I'm also here until the end of the month. So feel free. I'm in room 122. So you can also feel free to stop by. >> We wait for 10 minutes for you in the coffee table. >> Okay. >> In the shoulders can >> stop the recording >> probably. students. Sorry. >> Okay. Okay.