 Okay, so I'm pleased to introduce Remy Rebouli from Institute for Year, who's going to talk about the Chandry Transform convex bodies and pleurisobarmonic matrix. Okay, so first of all, thanks to the organizers for the invitation. So as is apparently required by the team of the seminar, I'll try to not say things are too hard at the start, so I'll try to go into trig geometry and explain things as nicely as I can, and hopefully you'll be fine and don't hesitate to, you know, just tell me or ask me a question if there is an issue. Thank you. So as I said, I'll begin with some notions of Torek geometry and in particular Torek potential theory. So we'll start with the very basic stuff. So the setting for this section is going to be that we're going to take X, a smooth predictive variety, which we ask to be Torek, which means that there is a dense open action of a Taurus, C starts in the power of D, and D is going to be the dimension of my variety, so it is a maximal rank Taurus. And I'll pick L to be a Taurus equivalent and polarizing model on my variety X. And the first thing I'm going to talk about is the correspondence between lattice polytopes, so polytopes with integer points or integer edges. I don't remember the English word. And between polarized Torek varieties. So a lot of this section is going to be presenting some correspondence of this type. And the second part of this talk is going to be explaining how we can generalize these correspondences as well as we can hope to. So if you have this data of a Torek variety with an equivalent ampoline bundle, we can associate to it a lattice polytope in the following way. So the first thing that we have to notice is that it induces an action, so the Taurus action induces an action on each space of sections of the tensor powers of L. So I use additive notation here, so KL is the kth tensor power of L. And in turn, this induces a weighted composition of this space of sections according to the one-dimensional representations. So I'm going to denote this set of weights by mk because we're dealing with the rk here. And so we can decompose our space of sections as the sum of this ck alpha. And so these weights are going to be in zd. So we are going to have a collection of points in zd which we can actually notice like the number of these weights is exactly the dimension of hrotl. And we can take their convex form inside rd and this is going to define naturally a polytope. So a lattice polytope, so a polytope with integral vertices was the word I was looking for because these points are in zd. That's the definition of a lattice polytope. Conversely, we can go the other way around. And if we pick a lattice polytope in rd, we can reconstruct a polarized straight variety. So okay, this is not supposed to jump so easily, but okay. Let's just do this one sentence at a time. So first you have the underlying variety. We're just going to take the closure of the image of the torus. So c starts with part of d. Inside the productivization of the direct sum of the integer points of the k-fuminkovsky sum of your polytope intersected with zd. So we just take the integral points of k times your polytope. This gives a set of integer vectors. We can take the direct sum of c with respect to these vectors. And by sending a coordinate z inside your torus to zm0 or m0 is going to be the first element in the set, et cetera, et cetera, we can sort of embed it into this productivization. And we take the closure of its image and that's how we record the Detroit variety. And so we did this on the k-fuminkovsky sum of your polytope. So by restricting the o of 1 to this image, to this straight variety, we get what we call k times the line bundle l-delta. So if we did this just with k equals 1, we get what we call the line ball l-delta in this sense. So this gives all tensor powers of them basically. And so the main first theorem that we're going to have is that there is an actual one-to-one correspondence issue fixed at dimension d between three varieties of dimension d and all with an ample torus invariant, a torus invariant line bundle, and between lattice polytopes in rt. So these constructions are really inverted to each other. Now we're going to get into the more analytic aspects of this. So we're going to define our basic objects to do quite potential theory on trig manifolds or trig varieties. So we pick x and l as before and we say that a continuous metric pi on l is going to be price of harmonic if its current recurrent is positive. So I'm going to denote it by ddc5. And again, can I? Okay. So only the first pose works for whatever reason. And so, okay, you can think of this as a generalized convex function or convex metric in sense because this condition of having ddc5 be positive is a generalization of the condition of having, for example, the second derivative function be positive in the c2 case. So this is a sort of generalization that works really well of the notion of convexity except that you normally can work in domains of c to the power of d but also in general manifolds and it also extends to metrics on line bundles. So we're going to denote psh of xl for the set of psh metrics on l and we're going to denote by psh t of xl the sets of torus invariants, psh metrics. So we're going to build again a new correspondence which will be between essentially convex functions and polytopes and torus invariant psh metrics on polarized toric varieties. So we pick five to be a continuous torus invariant psh metrics on l. And for, okay, to make things easier, we'll just assume that zero is in the interior of the polytope, the translation, we can do this. And by the previous correspondence, r0.0 is going to correspond to a section of l which I'm going to denote by s0, so a torus invariant section. And we're, so to associate to this by a convex function in the polytope, we're going to do this in two steps. So first, if we pick again coordinates z1, etc, zd on the torus inside x, we, we define a function which I'm going to do called philogue on rd in the following way. So philogue evaluated against the log of the absolute values of these coordinates here is going to be defined to be equal of minus, to be equal to minus log of material squared of z. And in fact, this is well-defined because of the torus invariant. So the fibers of this log map with which we compose are just products of circles. And you can see that this map is, in fact, independent on the choice of the points z1, etc, zd with a given image by this log map. So this is all well-defined. Then the second thing that we're going to do is that we're going to take its regime transform. So for those of you who are not very familiar with the, with this tool from convex analysis, it's basically a procedure that takes a convex function and turns it into another convex function which is going to be defined on its gradients or its subgradients, depending on the irregularity of the convex function, which has origins in physics. But okay, this doesn't really matter for what we're doing. You can take this definition for granted. So we're going to define c phi of 8.p. I'm not yet saying where it lies. To be equal to the supremum over all x in rd of the error product of p and x minus philogue of x. So I'm saying by convex analysis magic because as I briefly mentioned, the Legendre transform of a convex function is defined on its set of gradients. And this, here says that the gradient of this function philogue is actually a polytope. And this function is going to be minus infinity outside of this polytope. And it's exactly the polytope delta xl that we defined before. And because the Legendre transform turns convex functions into convex functions, we then obtain a convex function defined on the polytope associated to our polarized rate variety. And it is also continuous if the original metric is continuous. So we have done the first part of this construction. I'm not really going to go into the other way around. But basically you can inverse this Legendre transform to get a function like this. And then you can just lift it invariantly with respect to the torus section to get back a PSH continuous metric on L. And so the second correspondence I'm going to present today is saying that there is exactly one-to-one correspondence between continuous torus invariant PSH metrics on L and continuous convex functions on the polytope associated to L. So as I say, we're going to try to generalize the results later on through more general situations, as best as we can. This is one of the theorems we're going to be looking at. And to finish, so a last little bit of potential theory for the prerequisites is the notion of a PSH geodesic. So well, this might seem weird at first if you're not used to it, because the set of, for example, PSH metrics as a natural, let's say continuous PSH metrics as a natural open convex structure. And so there are just natural geodesics given by genuine segments between metrics. But for historical reasons, we're not going to be interested in this naive notion of geodesic. So in the setting of the Jotian-Dolson trajectory, for example, or just more generally the search for nice metrics, we have this K energy functional, which is a function on PSH XL or some more subset of this that associates to such a metric, a real value. And it is what we call an Euler Lagrange functional for what is known as the constant scalar curvature carrier equation. So the critical points of this K energy functional are going to be CSEK metrics that satisfy some very important PDE confectionary. And so for the rational approach to solving such equations, we're going to want to show that this K energy is going to be convex along certain geodesics. And that is, for example, corrosive in a certain sense, so that there exists a minimizer. And so if there is a minimizer, then there is a solution to PDE. But so this K energy is not convex along these naive affine segments, but is convex for respect to geodesics defined by slightly more complicated initially remaining structure. But this is just a historical viewpoint on this. We're not going to need this. For our purposes, we can just know that the geodesic joining two continuous PSH metrics by 0 and by 1 is given as the supremum of all perisobarmonically varying segments, TEMAPS 2 by T, such that the endpoints are bounded by the metric by 0 and by 1 respectively. So taking the supremum of all this and regularizing it actually gives you a what we call a PSH geodesic. And again, so there is a nice correspondence which says that so there is a bijection between the sets of PSH geodesics of torus invariance metrics in L and genuine affine segments of continuous convex functions on the polytube associated to L through this operation of the Legendre transform as port. So the geodesic joining 5, 0 and 5, 1 is sent to 1 minus T times the C's 5, 0, they would have the port plus T times C, 5, 1, they have the port. And conversely, if you have an affine segment like this of convex functions on the polytube, you can turn it into a PSH geodesic. This is very nice. And so, okay, so I like there is a bit more prior potential, but we're nearing the end of the prerequisites. The last quantity that I'm going to mention is the Morgens pair energy, which might seem absolutely horrible, but it doesn't matter that we really know the full expression. So basically, this is, this can be seen as an Euler Lagrange functional for the Morgens pair equation. So let's just break down the expression and simply awaited some of the integrals of 5, 0 minus 5, 1, so the energy of 5, 0 against 5, 1 integrated against all possible mixed measures involving 5, 0 and 5, 1. So this is what we call the Morgens pair energy. If you know a bit of classical potential theory, it's interesting, multivariate generalization of this. And this quantity is very nice because, for example, it detects PSH geodesics in the sense that a PSH segment is geodesic if not only if the Morgens pair energy is defined along this segment. So this is a very important quantity by whose importance you might just want to accept if you're not convinced. And a last very nice result regarding this Legendre transform is that it linearizes the Morgens pair energy in the sense that there exists a universal constant depending on the dimension D such that the energy of 5, 0 against 5, 1 is just the C times the integral over the polytope of the difference of the transforms of 5, 1 and 5, 0 against the normalization of the big measure. So we've transformed this really complicated expression which involves just these very nonlinear expressions into these very simple and linear expressions. So this is a very powerful result actually. And there are other nice functionals that they have a good tray version of. We're not going to get into this. In this talk, and so what we're going to try to do for now is to generalize these results as best as possible through the case of a non-toric polarized right. So the equivalent of the polytope is going to be the O'Connor body and then we're going to define this generalized resultant transforms. We'll get into this right now. So again, sorry, this is supposed to be step by step, so you have a lot of information at the same time. So let's just try to follow my mouse and do this sentence after sentence. So the first thing that we're going to do is to generalize the notion of a polytope associated to a polarized rig variety. So this is a construction that originates from an idea of Henri O'Connor, and it's the name of O'Connor body. And that's been systematically developed by Ladersfeld and Statsa as well as Kavek Komarski. So a version of this construction is the one that we're going to present here. So we fix as promised just a smooth complex rective variety with no symmetries, with no torus sections of dimension D, and we pick an ample line bubble L on it. And we're going to pick any points P and X. And so if you expand a section of a power of L in local coordinates near P, you can represent it as a power series. So a sum of a alpha z to power k alpha with alpha and D. And we simply define a evaluation on this all possible sections of tensor powers of L, on the section of algebra of L. So this is simply the usual z at a correlation. So the valuation of a section S is going to be the minimum of the alpha appearing in this expression, such that the coefficient AL phi is nonzero. And so this is a minimum which is taken with respect to any monomial total order on and D, which is just a generalization of the notion of the lexicographic order. So you can simply imagine that this is the lexicographic order. So in my notation, I do not make apparent the fact that we have made a lot of implicit choices, we have made the choice of an order, and we have made the choice of the points. But this doesn't really matter. You get objects that are essentially equivalent if you change these choices. So we're just going to denote this valuation by nu. And so the important point is that it's a valuation like function that has rank D in the sense that it takes values in Nd. So what are we going to do next? In what are we going to do next, this comes up. So we define for all k the set of c-migra points in degree k of L with respect to this valuation to be simply the image of all sections of kL against this valuation. So again, just as before, the number of points we obtain is in fact just the dimension of the space of sections. So through this, we obtain as before just a set of points in Nd. And we define the point of body as the closure of the convex pole of the union of all these c-migra points that we rescale. So the rescaling makes it so that we are basically putting all points around the same place. Otherwise, we just have something infinite. So for all k, we look at all the possible images against the valuation of the sections of kL. We scale them by 1 over k. And we take the convex pole of this. And what we obtain is a convex body. So it is a compact, close compact set with no empty interior. Sorry, convex compact set, most importantly. And this is why I define to be the open curve body of xL. So again, this depends on this valuation, which depends on the data of a point and an order on Nd. But basically, if you change these things, the open curve body that you obtain is just going to be different in, let's say, trivial ways. I ask you a question, Mr. Is there, because here you only get the convex body, not the polytope necessarily? Yes. Yes. Are there easy examples where you don't get a polytope by something? Actually, most of the time. Okay. Almost all the time, you don't get a polytope. So whenever you're not on a tariff manual? Yeah. So it can really, like the boundary of this body can be really arbitrarily bad. I mean, it can be absolutely awful, so you can't integrate in the boundary, for example. Right. So a lot of work in the field is also to try to understand when the open curve body is nice in a way. Okay. Very non-trivial question. So on the heels of that, may I ask another question? Is it known when the open curve body is a polytope? I don't think that there would be a general result. I think there are some partial results that you might find in the work of Alex Boronia, for example, and Petra and McLean, who I'm mentioning after this. But as far as I know, I'm not aware of any results where this, where you can actually check if it's an actual polytope. And perhaps another smaller one, making sure you don't go off track. Is it known what convex bodies can be or concave bodies? Yes. That's a paper actually that I'm going to mention just at the end of this slide. Okay, good. Thank you. So, okay, so I can start with this example then because it's the relevant one. So the key point is that the correspondence is no longer invariable. So if I have a convex body, I can't immediately tell if it is the open curve body of some polarized variety, nor can I usually tell if it's unique, if there is one. And the best result that we know in this direction is this result of Boronia, also Vanu and McLean, which says that actually the set of convex bodies that are concave bodies is countable. And as you can imagine, there are many more than countably many possibilities in general of convex bodies. Of course, basically none of them are concave bodies. And I do not remember exactly how deep they go in the paper, but I think you can at least have some, if you look at it, there are some characterizations of when a convex body is in a concave body. So the example to tie in with what we discussed before is that in the trade case, so everything is exactly the same in a certain sense. We deal with data at order one, but the important tourist environment sections are defined at order one. So the set of similar group points is just the set of weights of the tourist action. And if we pick the correct order, in this case, it matters a little bit. We recover exactly the tourist polytope from the first section. So at least it recovers the construction that we know when it should, which is nice. And so now we're going to be looking into generalizing the logeant transform construction. So this is based on the work of David Bidnishson. So we begin with picking Phi, a continuous PSH metric on L. So again, there are no more assumptions on X and L than just the usual ones. And so the key point is that to Phi, you can associate different norms on spaces of sections of tensor powers of L. So you can define a subnorm whose square is just the square evaluated against the section S is the supremum of S squared against E minus K Phi of X. You can also associate to it an L2 norm. If you pick a symbol in form mu on X, so the square of this norm evaluated at S is the integral of S squared against the E minus K Phi d mu. And so we're going to use these norms to construct our generalizers on a transform. And due to a simplification of David, actually we can just use error norm in construction that follows. So for notational convenience, I'm just going to write norm K Phi instead of specifying error value form or supremum. So the first part of this construction involves defining a fine space of nominal sections of KL, in a sense. So if we pick a semi-migropoint alpha in degree K, so just the evaluation of a section against this valuation that we had, we define the space TRK alpha to be the set of sections of KL, such that in local coordinates near your points, it has a monomial lowest order term. So you can actually see this as a class in a quotient space. So yeah, this is just the space of nominal sections in KL such that the evaluation of S is equal to alpha, which is our semi-migropoint. So this is what we define as DRK alpha. And to define our generalized general transform, we proceed as follows. So for all K, we first define a function CK on the semi-migrop, degree K. So to an alpha in the semi-migrop, we define CK Phi of alpha to be the log for conventional reasons of the infimum or this defined space of monomial sections of the evaluation of the such a monomial section against our norm associated to Phi in degree K, which you can see as, again, the log of a quotient norm induced by this norm K Phi on the defined space of sections from before. And we're going to just take all the values of all the CKs and if you scale them, so there should be a K minus 1 here. If you scale them, this defines a natural set of points on all the semi-migropoints in your conquer body and take the convex hole on these points, which you can see immediately is a convex function C Phi on your conquer body. And the key point is that this naturally generalizes the general transform in the sense that we have these two properties that we recover from the degree Ks. So first, yeah, it corresponds to the usual general transform of a tourist invariant passage metric. If you have a tourist action on Excel, and if you've made the choice of the Mexico graphic order in your evaluation. And again, it linearizes the motion of energy, which is a very, very strong result. So again, the energy of a zero against Phi 1, just this universal constant times the integral on the conquer body on the interior of the conquer body of the difference of these generalize general transforms of Phi 1 minus Phi 0 against the normalize the big measure on the conquer body. So this is a very, very essential result. And it's, in a certain sense, a bit surprising that it works well, given that, for example, again, you can't invert the correspondence between convex functions on the conquer body and VSH metrics, you can't really go back. So having this is actually very, very strong. And so the first result of mine, I'm going to present in this talk. Can I ask you just a quick question before we move on? So it seems that this is injective, this transform in the sense that the transform of a continuous pure subarmonic metric is determined. Determines the metric. Am I right? I'm not so sure. I mean, I think it might be possible so that the fibers of this method are not single terms. I mean, you lose all the information by doing this. You lose all the information that's two degrees higher than one to order higher than one. Great, I see. In the Tori case, it's one to one, but there we have a lot of symmetry as well. Well, here we don't have a lot of symmetry at all is the thing in general. So we basically try to project them to something that's symmetric, but by doing this, there's a lot of things. So that's why this is a very strong result is because we can still recover some very essential information, even though we lost a lot of things, because the amount of energy is also in a certain sense, just information up to order one. So another thing that you can do, which is my results here, is that you can also linearize your physics as before. In the following sense, or rather, detect your density. So if you pick a PCI segment, team up to phi t, joining phi 0 and phi 1, which are continuous and PSH and L, then you have an equivalence between the statement that phi t is geodesic and that the transform of phi t is an affine function of t as before. So this is again a result that tells you that even though you lose some information that is transformed, you can still at least partially go back. So you can't necessarily reconstruct a PSH geodesic from an affine function like this. But you can tell whether a segment is geodesic by looking at it. This is an equivalent. So this is basically the best sort of result that we could expect in this generality. So I'll say a few words about the idea of the proof. So again, sorry for assaulting you with this much information at once. So step by step, if we pick a smootable informant x as before, the endpoint metrics the fine norms as before. So s k phi i for i equals 0 1 charges the integrals also yield to norms associated to phi 0 and phi 1. And so what we can do is that these are two Hermitian norms. So it's a classical theorem from spectral analysis that you can always find a basis which is orthonormal for the first norm and orthogonal for the second norm. So we're going to do just this now. We're going to know this basis by s j. And using this basis, we can define for all t a sort of a segment of norms, which I'll denote by norm k t. So at all t, this norm k t is the unique norm which is orthogonal with respect to the basis of j and at which we prescribe the values. So sorry, this is for all j here. So again, so an element of this basis, the value of the norm, there is no square here is just the value of the norm at time one to the power of t. So you can really see this as a genuine segment of norms, sorry, log a fine segment of norms in a certain sense. And the key point is that this is, this norm k t is not equal to the altitude norm associated to k phi t who had done this. Sorry, this is the, if this works, we wouldn't have to do anything at all. But this is absolutely not true at all. So this segment, we don't really know how it is really difficult to work with it. And this segment is really easy to work with. So we like to have it. And then we can just take this Bergman kernel and turn it into a metric, which I'm going to write phi t k, which is just k minus one times the log of, sorry, this is a sum. I am too used to our comedian stuff. This is a sum of the s j squares of e minus t log of the, the exponential of the log of this essentially. So this is just a weighted Bergman kernel. Or if you are used to this notation, it's just an issue in the metric associated to this norm k t. So this gives this metric. And there is a fundamental theorem of Bergman that says that you can approximate geodesics with these segments phi t k uniformly in time and space. So the C0 norm of phi t k minus phi t on x times zero one actually goes to zero as k goes to infinity. So we have a uniform approximation of a piece geodesic using these Bergman segments, basically. And this is a very, very strong and important theorem. And that's why I use in my results. So roughly this takes place in three steps. So the first thing that we do is that we define a second Legendre transform till the C of phi t, which we compute by using these these Bergman norms instead of using just the usual all two rules of norms. And so the first thing that we do is that we show that this transform here is going to be larger than the original original transform. At the end, the alternatives to be the same. So we have this really specific transform that just works down to geodesics, basically. And so to show this inequality, we use the uniform for a result of Benson. Because basically we're going to use the characterization of this C phi t using these sub norms. And so I think these uniform estimates turn into estimates for the sub norms as well. That's the kind of thing that we do is that we show a formal estimate, which is not a convexity, just convexity at the end points. So we show that this new transform at time t is more than the segment of the original transform of phi zero. So the segment training the original transform of phi zero and the original transform of phi one. And so the key point here is that this is a because these Bergman norms are easy to work with, we can prove this rather easily. And combining this with the first point. So that this is more than this, and this gives that this is more than this. This is a formal estimate. We have not proved convexity of this, unfortunately. And then we just integrate. And so by the results of David, we know that the integral of this is going to be, in a certain sense, this is going to give a difference of monoparagies. And here, this is also going to give a difference of monoparagies. Because phi t is geodesic, the monoparagies is going to be a fine. Along phi t, so the integral of this is going to be a fine. And we're going to see that basically the intervals of these two functions, which are comparable, are going to be the same. So this implies that these two functions are actually the same, which basically shows the proof. So I'll try to go as nicely and non painfully into the North American aspects of this. So sorry, how much time do I have left? You've got 17 minutes. Okay. So maybe maybe a 16, because I would actually like you to just go back to the statement of the theorem just one more time. Yes, very quickly. This one. Is that the one you presented the proof of? Yeah. Yes. Yes. Thank you. Yes. Perfect. Thank you. Yeah. That's fine. Thanks. So basically, if you're a bit scared of the word North American, you can just understand that it's going to be a way to capture the singularities of this configurations or general access configurations. So if we begin with picking what we call an ample test configuration for XL, which is the data of a flat relative family currently X fiber over a one with a relatively ample line ball currently L with respect to this type, which we end up with a sister action lifting the usual action C and identifying the favorite one with our original Excel. So this is what we define to be a test configuration. And by the work of Armstrong, they show how to sort of canonically associate to this data, a price of ramanic geodesic rate. So it's like a PSA geodesic segment, except that we go to infinity here. And the geodesic condition just means that on any compact interval in the whole fine the restricted segment is also geodesic. So that's okay. That was written anyway. And the way that they constructed is using so Bergman arrays, which is basically the ray version of the segment. So we constructed when we looked at the results of Benson. So this geodesic prey associated to XL, so currently L, I'm going to denote by a big five currently L. So we'll try to not get too lost too much in the notation. So for the North American things, so basically it's back to work of Bergman, Buxom, and Janssen and also others. So given a ray like we had before associated to its test configuration, we associate to it a North American metric five and a currently L on an object, which is called the verification identification of X, which I'm going to denote by Xn. So we're expected to be three absolute values. So to say a few words about this construction, basically Xn is a way to compactify the sets of divisorial valuations and X as a genuine analytic space in a sense. So the underlying set of points is just the set of valuations on function fields irreducible varieties of X with the topology of point-wise convergence. And with respect to this topology of point-wise convergence, the set of divisorial valuations lying over X, so associated to variational modes of X is dense in this space Xn. So this is a very specific case of a more general construction that works for any variety defined over a normed algebra, for example. So we analyze the five with respect to the trivial absolute value on C, such as the absolute value, which is one everywhere except at zero. If we did this construction with respect to the usual absolute value on C, the one that we all know, then we just get the amplification of X in the sense of Xn, so the analytic space associated to X. But so basically, when I'm going to speak of Xn, you can just remember that it's basically just a compactified set of divisorial valuations. And that's really the only subset of it that we care about. So to define this function by L and A, we just basically define it on this set of divisorial valuations and extended by density. So given a divisorial valuation, we basically lift it to the evaluation on the test configuration in a certain sense. And we define this evaluated at this valuation to be a generically low number against this lifted divisorial valuation, which corresponds to a lifted divisor. So a low number can be seen as a sort of analytic order of foundation, if you're not familiar with that. And so there is a more general construction which associates to any PSH array, no event geodesic or no event of the type that we had before, a function phi na. So basically, you can think of this construction as capturing the singularities of phi t as you approach the central fiber of any test configuration or your initial variety. So this is what we're going to call an Archimedean PSH function on XI, so LPSH function because it comes from L. Okay, so if this is a bit confusing, it doesn't matter, it's mostly to just explain the objects, but you can take this for granted as well, if you wish. So among these PSH arrays, there are some particularly nice and important arrays which we're going to call a maximal. So we say that array big phi is maximal if it is the largest in the class of all PSH arrays, which begin at the same initial metric or below it as your array. And if they define it, so okay, so this is the class arrays which begin below phi and which have the same non-Archimedean function as your array phi. So you can see this as having singularities prescribed by phi. So basically, to make a maximal array, you just take the supremum or in a certain sense, the regularized supremum of this class of metrics, and what you obtain is essentially an array that has optimal singularities in the sense that they're no worse than the values of phi. What's nice is that you can use them to really capture the slopes of infinity of some functionals. So as I mentioned before, there is this conjecture which is partially no longer a conjecture which relates the existence of, for example, a CSK metric and the positivity of some object which should be purely algebraically defined. And these arrays are a way to make this bridge because as we suspect, for example, the slope at infinity of the energy along a maximal array is exactly the non-Archimedean k energy of the associated non-Archimedean PSH function. So you can actually, we know this in certain cases and if we know this in all cases, then we can prove it to be basically. So this non-Archimedean k energy is constructed by an intersection theory that I really don't want to get into right now. But it's basically integrating the, so the entropy parts of this functionals are going to be integrating the log discrepancy function against a non-Archimedean non-Archimedean measure associated to your initial metric, that's your non-Archimedean metric. But so anyway, what matters for our talk, which doesn't deal with YCD at all, is that so we can solve this problem. So if we pick a continuous non-Archimedean function, phi NA coming from L, then there exists a unique maximal genetic ray emanating from phi zero. So we start at phi zero and with the given similarity data. And as a particular case, the rays of long stone that I mentioned before are maximal. So this is just an exposition. You can think of these rays are rays with optimal similarities and which make the bridge between complex non-Archimedean data, essentially. And so the last results that I'm going to mention here, and I won't attempt to prove it, so basically by doing essentially the same thing from the work of Buxom-Chen and then generalized by Chen and McLean, you can define and generalize the genre transform for non-Archimedean PSH functions for metrics as well. And the last result basically says that so if we pick phi to be interested in the ray, then it is maximal in the sense of before, if and only if. So it's transformed on the Ogunkov body is a fine, but this is anyway given by genocidity. But the maximality is given by the fact that it's slope. So the slope of the transform is going to equal the non-Archimedean transform of the associated non-Archimedean metric. So it's going to be of the form c of i0 plus t times c and a of finite, basically. So this is a way to see this non-Archimedean limit as a fine limit in a certain sense. So this both, so this gives more insight into the fact that this maximal rays really realize the non-Archimedean limits in a certain sense. And yeah, this is a very nice way to detect the maximality of the rays as well. Okay, so that's all I have to say. Thank you for listening. Thank you, Remy. So I'm going to stop.