 Okay, so the theme of this talk is how you can apply techniques from nothing but potential theory to study case stability and in particular we are interested in openness, case stability in varying variations. So I want to illustrate how techniques developed can be useful to something like that in order to stick to one of the themes of the conference I will try to systematically illustrate what I'm saying in the correct situation and in fact I think already in that case the statement says something not so easy, more general but good aspects whenever we want. Okay, so I'm dealing with a polarized perspective and we are going to work with the Bakovic space, the patch width which is basically the space of valuations. So it's the Bakovic and the edification with respect to the trivial absolute value that's case number two maybe, as I said in this talk. But anyway, so again, if you don't know what this is, I will not have it in the correct case, you can keep this picture in mind. So the Bakovic space X is X and it's a compact apolitical space that you can think of as a compactification of the space valuations. The valuations here are valuations variety and among these you have these valuations which are set inside the space of valuations and therefore in this compactification it's advanced so it sets with you. So these valuations are the ones that are attached to prime divisors and some variational model of X, they are the more significant valuations. So if you want this space it's a compactification of the space of geographically significant valuations. And we've considered functions on this Bakovic space, the way it works is that every time you have some risky open subset of X and some different function on this open set, it defines value, a continuous function on the Bakovic space. So points in the Bakovic space are, they are not all valuations but they are simply valuations. It means that they can degenerate somehow around some variety but you can think of valuations for simplicity. And then this function defines continuous open sets of each. Open subset of X defines an open subset in the analytification. And this function here defines a continuous function. It's even like it's a valuation, you can evaluate it on the function F that you view as a rational function. But you pass to this multiplicative point of view, multiplicative semi-norms and you get a continuous function. So these conditions here describe the property of the Bakovic space. Okay, so of course you never saw that before. So we'll play for a bit in a minute. But again, as I said in the right case, suppose X is buried. So then I'm going to take some open orbit which has more fit to the torus T. I'm going to iterate this to the end. And as I said, every object here is open like that. It's also set in the analytification. And as Matthias told us this morning, this contains the kind of algorithm with the space, the valuations, and it's applied after the end. Okay, and then, so these would be special elements. They are the torus, they are the torus valuations. It's like a perfect space. And the divisorio once that I was talking about before, so the torus, divisorio valuations, they correspond to the rational points in these small points there. Indeed, that's in this subset. Functions in time in Bakovic space that basically come from the description of the topology here. And they would be called piecewise linear functions because the idea that here we are going to do a context geometry with some Rn in the torus situation. In the general case, you may want to think of the verbiage space as a kind of limit of p-spaces like that on which you have piecewise linear functions. But anyways, the space here is a certain dense subset of the space of continuous functions on the x-and that defines its topology. So you have to place the role of smooth functions to some extent in this situation to do it at the time. As the vector space has generated graph here, so you have functions that are locally in this sense here and some torus can open given by a definitively many functions open set. And you have this induced log of absolute vector functions. This is allowed to translate by constant and take max and also to divide by some in order to get cube vector space. So basically differences of functions like that are going to be p-f functions. These ones look more complex p-f functions. So far we don't see the polarization. So I can say that again in the torus case in the sense of torus p-f functions, so I can say every on R to the n. And conversely, every torus p-f function is obtained from such a function like that. So if you respect the torus situation, this notion here is just a user notion this was in our function on R to the n. Okay, and now the way you use polarization so we have this l, c in the beginning, p-f functions, these p-f functions play for us and the reason I can look at this setting is that they prioritize test integrations. Test integrations, but I should say p-f functions in the space of p-f functions here is in one-to-one correspondence with a set of test integrations for x-f, but without... Now if you impose to the test configuration some ampulness or netness then that will correspond here to LBSH function. So p-f function that corresponds to test configuration is L. We will say that phi at LBSH corresponding test configuration is a number in the torus k. That's, yeah, sorry. So that's a torus k in the general situation. So the p-f functions they parametrize these configurations and here we impose some positivity on the test configurations as we do when we define k-stability then we are dealing with we say that the corresponding functions at PSH. Okay, so again it's much too quick if you never solve it before but in the torus case it comes something that we understand that's quite concrete terms. So if phi is a torus then phi will be at PSH depending on the case and restricts. So we know that the restriction of phi to N arms some is quite enough function and then it becomes convex and it's only after adding the support function the support function. So that's also the condition that describes at PSH functions in complex situations or PSH metrics on the line model we have the same picture like that. So we can directly restrict the functions and how because it leaves in the space. When you say that when we think about the torus case you and you in that case when you talk about k-stability do you mean really mean k-stability with respect to torus? It's not... I mean not so far not really talking about k-stability actually it's not. When you say test configuration yes I mean so the torus case function would correspond to a torus test configuration which is more special than any test configuration on a torus. Exactly, yeah. So we are in the torus case and we look at torus objects that situation but we are considering corresponds exactly to the functions and then are that satisfy this condition here that we can meet with a torus case. When I mean the torus case is when XL is a torus or I have this right here that test configuration are also torus. Yes, so I'm assuming that three is torus that means if you want one way to say it is that it corresponds to a torus test configuration. Exactly. And then as we know this condition here is when the joint transforms on the polytope and then we recover the correspondence between torus test configurations and context is what in our functions are. So case A victory can be described as a condition that bears on all NPSH BF functions the positivity of a certain invariance that we want to view as the analog of the k energy in order to say before that let's describe that as a situation. So we have a version of a momentaire of a raker in this context so I will denote by H H as an M the space of the analog PSH it's the space that corresponds to the operations that I said so to a function like that and associating can associate using calculus on the published spaces or various other approaches that depends on them so this is a divisional measure so it's a probability measure on the published space that has simple it's just a finite context combination of masses at divisorial valuations so it's basically taking context combinations at divisorial valuations corresponding probability measure like that so this correspondence can be described can be defined in terms of what the VIs and MIs are so I'm going to do that now but I can say that this is going to be described in terms of intersection numbers or using a more sophisticated algorithm to probe in the calculus so if you just think of the three cases then this is just the usual real amount of the raker in this space here that corresponds to that TORIC case integration then we said that this function satisfies this complexity property that we've got here this displays in our function complex function so it's a so I mean that in that case the support is contained in an r that consists of TORIC valuations and you are just computing the real amount of that measure of the raker which is the function which has any constant functions in the FDSH so we can take it to 0 for example and in that case it's just one dirac mass has two real valuations it's a special measure in the TORIC situation it corresponds to the origin of the raker so they are normalized they are probability measures if you believe that the formalism works as in the complex situation then this momentum per operator has a primitive space if you take direction of derivative they are computed by integration against the corresponding momentum per measure so that's the momentum per energy can be expressed in terms of the real amount of transform in this search we have a supplement it's a uniform piece of energy of this function normal it's a norm like on the space H that can be produced with J and dimension 4 so it has a not exactly sure but on the left hand side with my formula is there any kind of formula for what you say for what they want to vary absolutely there is a formula that looks exactly the same as in the complex situation or if you want in terms of test configurations it would be the top step in the section of all those things I mean it's I'm really we don't know a lot about this function here I just want to give a rough flavor so this is not a real energy but a complex condition of case stability it's more or less equivalent to the deletion-futaki invariance of the test configuration and it satisfies a certain specific formula the part AX is the largest frequency function but remember that here we have just some measure of this finite complex combination of point masses at Dirac at divisional valuations we are just averaging the log discrepancy taking some average of the discrepancy of these valuations so this is one down and there is another which is the derivative of energy so everything depends on F and so when I say that when it's primitive the function is here we can also differentiate the deletion F so that this is good so anyway so there is some explicit formula like that where this is the HP part this is the energy part and we see that the HP part only depends on the motion pair measure so the game we want to play is to the energy part also as a function the energy part property that it's translation invariant it translates function fee by a constant to get the same motion pair measure and then it becomes so in other words the motion pair measure of a function an HPSH function fee determines the function up to a constant so that means that every time you have a function on H that is translation invariant which is what we want to do in here as a function of the corresponding but this is what we want to understand a bit more and then we have if we do that then we get for the energy that now only involves the space of HPSH functions that depends on L with the space of deletion measures that does not depend on L anymore so that allows us better visualization so in other words we want to do a formulation of the testability condition and think of as a variability characterization and so it's very easy to introduce the following notion so if you have some probability introduce the energy of the measure of L so it's for the potential theoretic energy in the context situation as the normal transform of the normal energy so it's a quantity as the finite energy and these mutations emphasize that it behaves a bit like the space of on the space of the measures so it's context by construction and so the definition is exactly the spatial measurement the direct mass of the future of the equation and this is introduced in the context of the variational resolution of the normal transform equation because the function P H computes the energy of the energy precisely when used in normal so as I said this energy could be infinite but for so every that is in the ring has high energy if that makes sense it's enough to check that one direct mass of the equation okay and in the third case this also has a nice interpretation in terms of optimal transport that goes back to robot's paper I mean it's the same interpretation as in the complex case okay so let me improve so we are considering this energy again this is depending on how the energy so what we show is that the dependence of the polarization the function the energy viewed as a function of the polarization is a C1 differentiable function and furthermore the value corresponds to this H then the derivative the derivative of this function as I said we want to get a description of the case stability in terms of the corresponding measure so what we get here is that if we get a description that not involves only measures that lives in this space and if that does not depend on polarization the final thing we do is that we show that this quantity here which again depends on N is also continuous with respect to L that allows us to get a version of openness of case stability in terms of measures like that stop some