 So I'm very pleased to introduce Quang Thanh Dang from University of Paris-St-Clay and University of Paul Sabatier who's going to talk about Ketterhainstern Metrics on log canonical varieties of general type. Okay. First, I would like to thank the organizer for inviting me to give a talk. And yeah, we hope, I really hope that we can meet each other fresh to face someday. And, but today, I will talk about my work about the Ketterhainstern Metrics on the log canonical varieties of general type. So let's get started. So, first, I'm going to outline my talk and we're going to be starting by discussing some background and some motivation to my work, my main problem. And then we are moving into the definition of the singular Ketterhainstern Metrics on the mildly singular varieties and how to make sense the notion of Ketterhainstern Metrics on the underlying space, which is no longer smooth. And then we discussed, yeah, and then we discussed about the, about the associate degenerate complex Moom Zampere equation, and then studies the continuity of the which solution to the Moom Zampere equation in big co-moluzi classes. And I list here some reference to my talk and yeah, the first one and the first one that is a reference, I think that is the first paper, which is study the singular Ketterhainstern Metrics on the mildly singular Ketterhainstern varieties. But I don't know the paper before, but and then the next one that is the paper, Burman and Ganeshya, and I will borrow their definition. And the third one that is a paper, I, yeah, that is a paper I learned some technique to my, to my paper in the last of the list. Okay. First, I fixed some notion in my talk, and we're going to let it to be a compact calamity for a complex dimension and and Omega is a is a California that is a one one positive form. And then I fixed a big co-moluzi classes. And yeah, I will give you a precise definition of the big co-moluzi class and theta is the smooth representative. And for any close the positive, yeah, I missed the positive there, the close positive one current, I did not angle bracket to be the positive to be the non-Buri polar measure of the the one one current on the manifold. And we, yeah, you can understand this measure. And in the, in the case of California, that is, it's the coincide with the usual moves and pay operator. And my work might. The word my talk today is to study the regularity of the, with your solution. The solution being the degenerate complex motion per equation and F this solution fine is, is very quick, but this have the finite energy. And I will give you some, I will give you the PSI definition of that. And on the right hand side, the density, the density F is of this form, and I, and I will explain you why we study the regularity of this solution. Okay. So first, I, I recall you are some, some notion and some no results of the gallein Stein matrix, the usual gallein Stein matrix on the smooth compact color manifold. So let's, let's, let's discuss about the chun Rishi form that is a one one form given by the minus ID diva lock of the determinant of the metric omega. And the definition of the gallein Stein matrix. Yeah, it is easy. I think it is very familiar to all the guy. That is, if a cali metric is a cali in Stein, if they exist, a real number, such as the, the Rishi form is equal to the lambda omega. Of course, this definition requires that the first chance glass of the, of the metaphor to have the device side. And this is always a case in, in a complex of dimension one. I mean, on any compact reman surface, they exist. They always exist. They exist in the metric, but not in general in a complex of dimension two. So, for instance, if you take the product up to any, to any compact reman surfaces and they exist gallein Stein metric if and only if S one and S two have the same time. And by the rescaling, we can easily to see that they are only three case to keep to consider that is where lambda is equal to minus one zero and one. We can easily to solve that we can easily solve that solving the cali in Stein metric is equivalent to solving the complex motion per equation. And we have, and this the equation is so 14, 14 or 15, 15 years ago by Obama and Joe in the case when lambda is equal to minus one. They, they was working independently. And the case when lambda is equal to zero and on the Calabi now many fraud. The East Cali Cormolo Z classes always contains a unique reticulat cali in Stein metric, but after the work of Joe and Obama, almost 40 years after or more than 40 years, the problem to the K lambda is equal to plus one is has been solved by a series, a series, a series work up to Donaldson and soon and so it's the first-gen class has the, if the first-gen class of extra is a positive and one cannot always so the cali in Stein equation. There's, they don't always exist the cali in Stein metric. And they saw that the existence of the cali in Stein metric is equivalent to the answer pro geometric condition, which is so called the key stability. And I'm gonna not explain what what it means, but let me properly say that the solution to this highly long linear PDE is equivalent to the certain and several and several geometric condition, and which is called uniform case stability. And in the center of Tian and Donaldson. Okay. And now I'm moving on the situation where the underlying, where the underlying space is no longer smooth, but rather a mildly singular Cali variety and seen a few decades ago. Sorry, since a few decades, the so called minimal model program. The MMP studies the problem of classifying classifying the algebraic many first in in dimension one, two and higher and up to a bright and soon be rational equivalence. Yeah, it's the, it's, it's, it's been sold in a complex dimension one for a few decades ago, and almost, sorry, and almost and almost finished in a complex dimension to buy from the Italian school and, and the compilation of the Japanese school. But in the high in the higher dimension, more precisely in the complex of dimension three, the situation is incredibly, incredibly more complicated. And one has to, one has to engage the picture and considers the Miley singular varieties. And, and we am at defining the corresponding the singularities and, and how to make sense, the Cali Einstein equations, and I'm going to be explaining how these can be interrupted in terms of the complex moon sample equation and that will be degenerate. And then we study these degenerate equations. And yeah, we show that the equation exists with solution and established some partial. These we are dealing with the main type of singular with singularity. We are going to consider a so called the log terminal and log canonical singularities. And I, in this talk, I'm going to considering, I'm going to be considering a normal variety. It's the means that the singular locust is, is a complex code dimension. Ah, sorry, a core dimension, at least two. And the, the variety is core key warrants time, if it's the canonical divisor exists, and exist. Yeah, is the mid send as the key live under. Okay. And, and then you take a local resolution of the normal variety. We have the assumption formula. And the AI there is core discrepancies of the why and we, we say why has the log, log canonical respectfully a log terminal singularity is the coefficient AI is greater than or equal minus one. And for the log terminal singularities that is AI is greater than it is strictly greater than minus one. And for my talk today, I'm going to focus on the log canonical singularities. Okay, and we also say that why is is a variety of general time if if the car if it's the canonical bundle case why it's big. I will, yeah, I will give you an analytic definition of the, of the big nest of the nine bundle in the secret. Okay. And, and of course, this definition is the independent independent of the choice of the local results. Okay. Okay, I'm going to give you some example of the singularities. And we, you can also define. We can also do, we can also define the singularities for a general pair, why and data with the data with data being effective divisor. Okay. And for, okay, on the surface, the log terminal surface singularities are precisely the quotient singularities of this form. Okay. And in a higher dimension, we can also prove that quotient singularities are still log terminal. And, yeah, we have the beside beside beside example. Yeah, you are taking a smoother hypersurface of degree D in the in the complex projective spaces, and you take a five cone over the hypersurface. And this, and this cone has the only kind of this code has the singularities and this, and only has the kind of equal singularity if and only if D is not the exit act. And for, for Calabi yaw log pair, why and data with why being a touric variety and data being the complement of the embedded torus. Yeah, and you can, yeah, we can so that the pair is log canonical. That is an example of real 40 years of 14 or 50 years ago. Okay, now, let's discuss about the singular Caroline Stein magic and how, yeah, and in general, on the, on the Miley singular varieties is not is not easy to make sense of the some objective of some key objective like the turn richie form or, or Caroline Stein equation. And I'm going to give you the first definition constant the richie curvature of the current. And I say that a close positive one on current is admissible if is the satisfy the two following conditions. I assume the known a product of the close positive current is absolutely continuous measure on the regular locus with respect to the logic measure and I assume the log of the density is locally integrable in the in the regular locus and this definition has this definition has me to define the richie curvature of the admissible close positive one on current. And another way of thinking of this definition to interpret the positive the non-pripola positive measure of Omega on the regular locus as the singular metric on on the anti canonical life and oh, and with curvature is the richie curvature by definition above. And then I can, and after that, I can make sense of the Caroline Stein metric with negative richie curvature. Yeah. In the sequel, I will consider sorry, I will consider why to be log canonical variety of general time. So I will define the Caroline Stein defy the negatively curve Caroline Stein metric. Yeah. And I say that a positive admissible current Omega is a Caroline Stein metric. If it satisfies the two following equation. The first one. Yeah, it's more. It's like in it's more like the usual one. And the second one that this is a sufficient coalition guarantees guarantees that the Caroline Stein metric is equivalent to the degenerate complex moves and take with. Okay. And next. Can I ask a quick question about the station. Yeah, why I can't remember now why is that what you get when after you do the resolution, or is it what you have to start. Yeah, why in, in my talk, I am, I am considering why to be a normal variety. We go to Stein normal variety, which has the log canonical singularities and such as the is the canonical bundle case why it's big. Okay, got it. Okay, go ahead. Let's keep that. Ah, okay. Yeah, let's me make a few comments about the above definitions. And around 10 years ago, we can. Yeah, we have another definitions in the sense of in the sense of the French school. It's easier guest and Zegah here. They first they define the, the so called adopted the measure. And this definition is independent of the choice of the section of the non zero local section of the RKY. And then they, they saw that. And then they defy the Caroline Stein metric is a solution of the, of the following complex moves up a question. And they also saw that on the, on the local resolution, we can, we can write the, the adaptation measure of the following. And this is, this is a product of the Zi of the model of that ice to to power to AI. AI there is a discrepancy coefficient. And we can easily to see that why has the log terminal singularities if and only if every adopted measure mu has the locally finite must near the singular, near the singularity of why. And with, with that definition, yeah, they can, they could equally defy the positively curve Caroline Stein matrix on local final on local final varieties. Around, yeah, three years ago, yeah, publics on our friends, you know, okay, now we are going to, we're going to be in a decrease, we're going to be a decrease in the degenerate complex. I recall you the setting I am considering in the secret. I'm going to let why be a local canonical variety of the general time. And we, we have seen about that, the Kalein Stein, the existence of Kalein Stein matrix is equivalent to solving complex moment. And we will work, we will work on the local resolution up a wide and then we are pushed down the solution on the base. We can prove that the existence of the singular Kalein Stein matrix is equivalent to solving the complex mung zhampa equation that will be degenerate, because you see the density on the right hand side is degenerate. And I also, yeah, we also remarked that. The big less is guaranteed is preserved by by taking local resolution. Okay. And, yeah, of course, the canonical, the boom back of the canonical live under K why is still big by taking local under under local resolution. But we have to. Yeah, we have to be careful that the density F is low longer is is not integrable when some coefficient AI is minus one. Yeah. And to solve this degenerate complex mung zhampa equation, we need to, we need to give no sin, a good, a very good notion of which solutions and and this let's add the Puri potency theory. And back man and a studio guest Sega he around 10 years ago. They introduced, they introduce a new good notion of which solution and introduce the variational approach to, to solve the degenerate complex mung zhampa equation in the big co multi classes. And they, yeah, they, they use the variational approach. Yeah, the idea of the variational approach is to construct a critical point of, of an convenience functional, but I'm gonna, I'm not gonna explain what they do. But first, the, the, the main thing of my talk today is focused on the regularity of the with solution. Okay. And, okay, today, we will study the regularity of the with solution. Okay, first, we assume, we assume the solution, we assume the equation exists with solution phi with finite energy. And, and then I gonna be studying the regularity of this solution. I will show that this unique solution is continuous on the ample occurs outside the local canonical locus, the local, the local canonical locus, that is a union of the, of the exceptional divisor, the union of the support of the exceptional divisor with the coefficient ai is equal to minus one. And some no result before in the case of the Kawamata local terminal when ai is truly greater than minus one. The density, the density there is a density is a product of, of the SI, the model SI to power to AI is belongs to the LP and around 10 years ago in, yeah, around 10 years ago in 2014. The my with his collaborator collaborators, they saw that phi is continuous on the ample locus of theta. Yeah, I'm gonna give you the definition of the ample locus but you can, you can think about, you can think that the ample locus, which is open, which is the Zariski open subset. On the, on the variety one, a variety, it's sorry. And if the theta, yeah, this is if the theta, the reference from theta is additionally semi positive or is classed to be a nef. We have a result to show that phi is smooth on the Zariski open subset. And it's, and by consequence, and consequently, they exist a unique, sorry, they exist a unique singular Kalenstein metric on and suggest on the regular locus that is the satisfy, it satisfies the Kalenstein equation in the usual set. Okay, now, before going to my main result, I will give you some break ground and some, some background on the three percent, three percent theory. We still let, we still let it be, we still let it be complex complex Kalen manifold of complex dimension and a co homology class. One one co homology classes on farm is said to be big. If it contains a Kale current, which means that they exist a positive close current tea, such as the tea dominates the dominates the California. And in particular, for any live and draw L, we say that L is, we say that L is big if is the first in class is. Yeah, that is analytic analytic of the big live and draw analytic definition of the big. Okay, the next item, that is the definition of the ampoule cast. The ampoule cast of the big co homology classes. Yeah, we fixed and father is a big co homology classes. And we let the ampoule cast of unfair. That is a Zarisky open subset of all it's for which they exist. The Kale current tea is belongs to belongs to alpha with analytic singularity. We will not focus on what analytic singularity is but you can, you can understand that the Kale current tea is smooth around it. Okay. A little bit definition of the, a little bit of a brewery sub harmonic city. Yeah, we call a function of upper semi continuous function you is brewery sub harmonic. ID bar five is positive in the center currents. And we, then we have the definition of Gazi brewery sub harmonic. Did, did is locally the sum of the smooth function, the sum of smooth function and the piece of a brewery sub harmonic function. Yeah. That is a PS hat stand for. And for, for a big co homology class and far, I'm going to fix a smooth the representative theta. And we say that five is a theta PSR or theta brewery sub harmonic function, if it is a Gazi brewery sub harmonic function and theta plus the ID bar five is a positive in the center current. I think you are very similar to this definition. And I did not the envelope of the any boring function as by be theta as is a supremum or theta PSR function lying below as and I am an if as to be the constant zero. We, I will denote the V theta is, is a envelope of zero or envelope of some constant. Okay. And the factor, the fact that V theta is a theta PSR function. And which has the minimal singularities in the center of the money. It means that for any theta PSR function fine. Phi is more singular than V theta. And it means that there exists a positive constant of C and phi is is less than theta plus C. And in particular, V theta is locally bounded on the Emperor Lucas. Because the, because the for any for a big Cormolo Z class the unfound, they exist a current, which is the smooth on the Emperor Lucas by the work of books song and V theta is a is a list singular in the in the set of all theta PSR function. And so V theta is a cost less singular than the than the potential with analytics singularity. And this, and this one is locally bounded on the Emperor Lucas so V theta so is V theta. Okay. Now, I'm going to recall you the definition of the non puripura of theta PSR function, and we can extend it, which can extend this definition for any closer positive current. And, okay, first given given a theta PSR function fine. We take, because the V theta minus T is locally bounded on the Emperor Lucas. We can buy the work by the bed for Taylor Terry by the bed for Taylor Terry, we can defy the complex, we can defy the most unfair measure of theta plus the DC must find V theta minus T. On the on the sub level set the fire is greater than V theta minus T. And you see, the, and we defy the, the non puripura most unfair measure of fire to be the limit, the big, the increasing limit, the increasing limit of the of the sequences of the, of the, of the positive measures. And after that, we extend, we extend this measure trivially honest, because the, because the Emperor Lucas is a Zaris key open subset. Okay. And from now on, I did not am I fine to be the non puripura measure. The non puripura most unfair measure of a fight of a theta PSR function fine. And I defy the volume of the volume of the, the volume of the theta or the volume of the big come with the class and far is the master of the is the master of the most of no non puripura most unfair measure of V theta. And, and this the definition coincide coincide with the definition in algebraic geometry when I'm far is a first chance class of the big life and by the work of books. And we can check that the total mass of any the total mass of any theta PSR function is always less than or equal or less than or equals the volume of unfair by, by using the monotonicity of the witness. And I will let, I will let the, the class the E. Do not the, the set of theta PSR function with full most unfair must. That's mean the, the total mass of the, of the most unfair measure of fine is equal to the volume of the unfair. Okay. Now, I'm going to restate my main result. First, we assume, we assume we have a unique, we assume, we have the weakest solution. So which solution to degenerate a complex moves and pay equation. And then we study the continuity of this solution. And I, and, and this, and this theorem implies the, the theorem I introduced before, because the, when I rise density on the right hand side. I, of this form of a, I write F to be the exponent. So, minus five where, where small fire is a Cassie pressure function. And, and the Z, Z belongs belongs to the LP spaces. Yeah, because the tool because the tool able. The tool able to deal with the log canonical locus. I decompose, I decompose the exceptional divisor into two parts. The first one is a log canonical locus where, where the coefficient AI is equal to minus one. And the second one is a log terminal locus, where the density j belongs to the LP for some be greater than one. And in the, in this theorem, I just proof the solution to be continuous, but I really hope to get a more higher regularity. And some idea of my proof. Firstly, I will prove five is a five have the relative zero estimate five is a greater than some data be a shop on some side minus uniform constant. And then I'm going to look for a decreasing sequences of theta pressure function, which is, which is a contrast to fight and fight chase is a continuous on the open subset. And then I will show that this the convergence above is locally uniform on you. Okay. So, for the idea of the first step, I gonna use the, the color say idea. First, I recall you the definition of the relative most ampere capacity, the capacity of E of the boring subset E to be the supremum of all the integral moons and parameters of you where you is a data piece of function between the psi minus one and psi. And I'm going to set the ST like that. Again, I extend the colors a ideas to show that the function ST is right continuous and satisfy the, the following inequality. And then by using the classical lemma, we can show that the ST is identical zero for the big enough. Yeah, that is a lemma of the day Z or three Z or Z. And I've for us is a decreasing right continuous function and satisfy the following inequality. And we can, we can find some uniform constant only depend only depends on the constancy there and suggest the ST is equal to zero for 40. greater than a infinity. Yeah, that the power of this lemma is a very basic and you can find it in some, in some reference I list above, but I let me tell you some factor of the constant a infinity. Yeah, the constant a infinity is precisely equal to two plus the T zero. And for T zero is a constant satisfy as the T zero is is very small. And of course we can control. Yeah, we can control the a infinity. Okay. And we apply this the lemma to show that we apply system lemma to solve that ST is equal to zero. And you see the formula of the ST that is the capacity. Yeah, that is a capacity of the sub level set the fight less than a psi minus T and where when ST is equal to zero. We can show that the fight is a greater than a psi minus a almost everywhere. Yeah, we can show that fight is a greater than a psi minus a outside of the purple set and hence everywhere. Okay. And by the work of book song. I can, I can pick. I can pick a database that function, row zero, which is smooth on the apple of course. And that is, yeah, that is a success that theta plus the IDD bar row is current. And then I said, yeah, as I then I find a small a small constant a suggest the psi is equal to a fight plus the row zero is, yeah, is It's still the theta PSA function and suggest a theta plus the IDD bar psi is still a current and you will, you will, you will easily to see that psi is locally bounded on the on you, because the fight is the locally bounded and row zero of course. Okay. So for easily, I said that the new that is a ZDV and B is a to power minus one. And I have the following inequality. First, the inequalities is, is a consequence of the comparison principle. I refer you to to the paper of the book some a studio get and see a he or a series, a series paper of diverse the design loop. And on the sub level set the exponent, exponent so up B psi minus five is always is always greater than one. And we have the, the second line is very easy, but you see B phi minus B V theta is not actually a theta PSA function or Cassie PSA function in general. So the, the idea is the idea to control to control to control the interoperability of the e to power e to minus the B phi minus B V theta. That is, we take the envelope. And we show that the envelope up B theta minus the B V theta is a omega PSA function and set and we, and then we control the singularities of this, of this function. And the last line is very easy is that is, that is nothing the Honda inequality. And the fact one by the volume capacity comparison, we can show that the volume, we can show that the measure mu is always less than the constant. Time the capacity of the time the capacity to power four. Omega that is the definition of the capacity in the Coliseum paper, but that is a special key when that is special key when I think theta is equal to omega and psi is equal to zero. We can we have the definition of capacity omega. And the fact to we can control the capacity with respect to omega with the capacity with respect to theta and psi, because the function psi are certified that the theta plus the ID bar psi is a current. And we, we first we take you the, the candidate function, that is a omega PSA function between minus one and zero. We have the, this is the inequality and the function psi plus the dental zero you is a candidate function is a definition of the capacity. And this is very easy. And by fact one and fact two, we can show that the new new two power one half is less than the constant C time the capacity square. And the next one, we have to control the, the envelope of the be five minus the be V theta. And we can show that this function is a omega PSA function with the full motion by mass by using the technique of the diverse design Lou. This is the fifth one in their series, published on a current journal, and they can, we can show that the envelope of the T five minus TV data is a omega PSA function for all T, for all T greater than one. And then we have the easy, we have the classic inequality and using monotonicity of mass of the witnessium, we have the following inequality, and then we, we take a be going to plus infinity, we can so that the left hand side is a greater than the mass of the omega. So, yeah, it's, it's terrorist. This, this terrorist, the function be omega, the envelope of be five minus the be V theta is a omega PSA function with full motion by mass. And the fact for for any omega PSA function. They always, they always has the zero long numbers. That's why we can control the singularity of the envelope of be five minus be V theta. And this, this is thought is put by get in the year he 15 years ago and generalized, and then generalized by Davos, the DNA is a new 10 years after. Because the envelope of be five minus be V theta has the zero long numbers, we can, yeah, we can. So that the integral of the exponents or minus to be omega be five minus be V theta is finite. And then we can so that the constant a in the step in the step one is a uniform and depends on some quantity I list. In the step two, I gonna find a decreasing sequences, which is the continuous on the Emperor Lucas and decrease to five. That is just the consequence of the, of the two words of Burman and the money with his collaborators. For any for any as for any a smooth function as we can so that the most unfair of the envelope of us is dominated by the by a smoother by the smooth volume fall. By the work of the dummy DNF get farm Colisei and Zegahy, we can show that be theta as is a continuous in the Emperor Lucas. And back to my problem. First, I take a sh is a sequence of a soup smooth function is a Nick is a second of the negative smooth function decreasing to five. And then I take the envelope of this sequences. This this sequence they is still a decrease to five. And by the terrible bow fight chase is a continuous on the Emperor Lucas. But actually fight chase satisfy more higher regularity is the say that be theta of any smooth function is. It belongs to the class the sea one and fun. It means that the first is means that the laplacian of the P theta ash is locally bounded on the Emperor Lucas. By Burman and the money, but we have to be carefully because the dilemma one the player 12 in the in that paper is not the two, and I refer you. I refer you to the paper denies I and company last year. Okay, the step three, it just the consequence of step of step one. I fixed up size a, I fixed the lambda belongs to a zero and one, and I set the size a. Okay, and the function size a plays a symbol upside, and we do exactly the same in the step one. We, we still set the function as day and we show that this function is a right continuous and satisfy the inequality. In the lemma, and we can solve that the fries is a greater than lambda psi plus one minus lambda 5j. But I should mention that the ash J is going to zero as J tends to fit infinity because the in the in the lemma I list above we can we can control the we can control the the constant a infinity. Yeah. And then I take your J goes to plus infinity, and then, because the side is a locally bounded on the on you, and finally, we can show that the, the comparisons is uniform on in lock uniform on any compact of you. Okay, let's me. Last slide. Let me talk a little bit some further questions. Yeah, I, I can prove fire is a continuous on the MPO Lucas the outside. Outside the local record lockers outside the local record singularities, but I hope I can prove fire is harder continuous because we have the we have the result before by the domain with his collaborators. But yeah, the next question is to study the asymptotic behavior of the calenstein metric near local record lockers. And, and we have now we have now are some recent words by currency and who and data for strong, they can contract, they can contract a calenstein metric near the isolated, yeah, near the eye on the near the isolated local record singularities. And, and the last questions is coming from the work of Davos, the design new, and they, they prove they prove that they exist the unique solution, satisfy the complex moves and pay a question with the press guy singularities. And I hope, yeah, I hope when the small fine, the small fire model tire singularity and the density F, they, they are quite good enough. And we can get a more, we can get a more higher regularity of the solution. Okay, and sorry to over the time and I think I should stop my talk there and it's time for questioning. Thank you very much.