 polarized right. So yeah okay so when now for the background in Kela geometry, finding the kinetic matrix of Kela manifold is a central problem. So the first result is classical uniformization theorem for dimension one. In higher dimension we want to finding the Kela Einstein matrix or more generally constant scalar curvature matrix for short way you know the CSTK matrix and the extremal Kela matrix. So for today's talk we focus on CSTK matrix. So let's recall for Kela Einstein matrix the when the first question class is negative or zero, yaw first of all bound in negative case shows that so that the compact Kela manifold that means unique Kela Einstein matrix. But for the positive case point it's not always exists the Kela Einstein matrix and Tien introduced a notion of case stability and the contract it is equivalent to the existence of a Kela Einstein matrix. Donaldson reformulates the Tien's definition by giving an algebra geometry definition of the Taki invariant and the contract does that it is equivalent to the existence of a CSTK matrix. This is the so-called Yao Tian Donaldson contract. So it is stated that the polarized manifold ML that means CSTK matrix in the first question class C1L if I'm only in ML is K point stable. So for follow case the 10 Donaldson song and the Tien solved the independent is this problem that is the final manifold and it's Kela Einstein matrix if I'm only if I'm is K point stable or in in in the algebraic side the theory has achieved substantial progress the break the main breakthrough you go to Fijita and also Fijita and Odaka which really integrates the case stability in terms of variations by the algebraic invariant that's so-called delta invariant or beta invariant. So people can test the case stability of a follow variety by computing its delta invariant or beta invariant. So let's recall some classical definition. So let XL be a polarized variety, let pair be a form a normal variety maybe small normal variety Y to X be a subjective irrational morphism where we always assume the calonic device the kx is q katiya and a prime device f on the on y for some irrational model y of x is called a prime device over x we always use this notation prime device over x the set of the all prime device over x and for any prime device over x we can define it's log discrepancy as one plus the order of the relative calonic device. So let's recall the definition of original beta invariant let X be a Q follow variety which means that the at calonic device the kx in ample q katiya device and for singularity the log discrepancy is always positive and the fijita and the fijita underneath defined a numerically invariant or the beta invariant this one equal to log discrepancy minus the integration of the volume part so we always denote this sxf by the second part. So due to Blum and Johnson we know that the delta invariant can be defined as the input of x over sx also can state the original definition fijita and otaku which involves some some notion the so-called and best type device by the way to not use in today's talk. So fijita underneath also some other people so that a follow variety is k same stable if and only if the beta invariant is always non-negative and if and only if the delta invariant is x to 1 and the k stable if and only if the beta invariant is positive for all f and uniform k stable if and only if the beta is some uniform constant epsilon times s invariant if and only if the delta invariant is fake because the what sorry I do not introduce the explicit definition of k stability but we do not use this so namely we call some classical definition of volume a big device when they exist continuous volume function on the sorry this castle uh the nirvana survey space and by x which is a real that real wet space whose element is numerical impermanence class of r katiya device for any katiya device as a volume of of the device that is defined to be the new seal seal of the asymptotic growth of x zero and d and if uh if the volume is strictly positive with the such device that is big and our big class in the in this space forms a form a count convex open cone we called as a big cone you know to buy the big x so in general so for net device when those volume of the of the device is is it's self-intersection but for but for big device it does not so to to compare to the volume of a big deviser the books on farry and johnson introduced products the so-called positive intersection product such that the volume of the big deviser equal to the self-positive intersection product i do not want to introduce the explicit definition of this product but uh interesting fact due to the same paper is following the they show the volume function is still one still one the differentiable on the big cone and compute the derivative as this formula the on the right hand side of this part is uh is uh is a coder dimension one cycle and gamma is a deviser you can say as a deviser so the intersection is well defined uh well i'm called deviser i uh sorry not i some people define the slope of the polarize right as mule l as the intersection minus kx dot l and minus one over the volume of l in this paper deva and the legendary computes the donelson-futaki environment of the test configuration associated to a jimmy deviser for polarize right and obtain a new numerical invariant beta which generalize fittas original beta environment so we know for a deviser it's induced uh variation and if the deviser is dreaming the variation is finite generated so we can take the the approach of the of the wrist algebra of the induced variation which give gives a test configuration with the integral central fiber and the deva and the legendary computes this type test configuration and and shows the the donelson-futaki environment of this type type type test configuration equal to the beta lf so the definition is beta lf divided into three parts first one is the north discoloration part and the second one is mule l turns the integration of the as the volume part and then the third part is the integration of derivative part for the volume l we always omit this pesta so so this volume computed on the vibrational model y not x but for simplicity we always omit the pesta and this notation is this is this is defined as this one is derivative of the volume along the path direction of pesta kx so by by bookstone farry and the Johnson's result this is well defined because when x is is suitable this one is big so this is when t is small this is also big so this is well defined i always you know to the sl environment as the integration for of the of the volume parts for simplicity and the way to you know to tau lf as the field of effect in three shoulder of f it's defined as a silver x as the volume l minus x f is striking positive we can state a useful number and they use it again repeatedly for any big divisor and any prime prime divisor the integration of the this this is the derivative part by by bookstone farry johnson's the theorem this is this one is just the derivative part by by this one which equals to the n plus one and the integration of the volume part so if xl is called a value at the same stable if the beta l is non active and and the xl is called a value at the stable if the beta l is strictly positive for any long trivial prime divisor well in which the long trivial means the valuation associated to the lf using long trivial and the uniformly value is usually stable if there exists some uniform constant epsilon l such that the beta l is that epsilon sl so in devon and johnson's paper the steam paper the they show that they show that the case stability with respect to the integral test configurations is equivalent to the variative stability over the dreamy divisors so uh motivated by the have a quick question yeah what's an integral test configuration uh sorry uh integral test configuration means the central fiber is the integral it's just it's just has a one one component i see i see i see i see so it's not quite equivalent to case stability just some simplified version of case stability yes yes okay okay thank you welcome so for for any ample divisor we can define a uniformly variate the stability threshold of l it is defined as i use this theta l as the skill such that the course the beta l bigger than the x sl so if this beta l is strictly positive like it is equivalent to the uniform variative stable so in fact when x l equal to the x minus px the final case we know we have the theta i equal to the delta delta invariant minus one this is the main motivation to study the theta invariant so my my theorem is this uniform variative stability is an open condition which means the uniform variative stable entity locus is an open subquam of the ample cone so where you can also show by the steam steam technique this this uniform variative stability special the theta invariant is continuous on the ample cone for a rough ideal if we fix the ample divisor which is the uniform variative stable and choose a constant epsilon l such that satisfy the steam quality we fix any norm on this real space and define an open subset ill your epsilon so we want to show there exists a small open neighborhood ill epsilon l in this ample cone such that for any element in your they exist on the constant delta l prime satisfying the beta l prime bigger than the delta l prime dot the sl prime so for so this it's it's a surface surface to show the following two estimates first of all is the difference of the beta l beta invariant is not not bound by minus epsilon and the times l prime well f is a continuous function and such that epsilon go to zero when epsilon go to zero and the second one is this sl f is bigger than s minus epsilon the sl prime well s minus epsilon is is a continuous function also continuous function and which go to one as the epsilon go to zero indeed yes so beta l prime equal to the this one and first the term by by the something this one and then the statement one by this formula steps equal to the this one so we plug in the formula seven gets this this this in quality and when epsilon go to zero this one goes to affect the constant and this one go to zero so this is a positive constant so name me name me name me start a new new new subject so the following i want to start is evaluative stability for the j equation so let xl be a polarized manifold and h be an ample light bundle and therefore it affects the kinematic chi in the c one h and omega in c one l and we consider this j with the constant c omega five n minus one equal to the constant c omega five the many of them mirror omega five well the constant c is a topological constant we need to go in both sides get the c equal to the times the integration over the mirror and the which can be can be computed by the algebraic invariant by n the ample ample i bond h intersects the l m minus one over the volume l i always denote this one by the mu hl some twisty can be valued as some twisty slope mu l but it's well known that j equation is the the critical point of the j functional defined as as the minimum attested minimum energy mass the c times the minimum energy sorry i do not write the explicit definition of twisty the minimum energy and the minimum energy is to the you can see the derivative of this this one is the twisted part of this this this this mirror and the derivative of this minimum energy is the minimum mirror so for any test configuration curly x and curly l well a test configuration roughly you can see as a six star six thirty generation of a polarized variety for a generally definition it's it's a flat family from an undimensional n plus one dimension variety or scheme go to a complex number of c such that each non-zero non-zero variable uh isomorph to the original xl the l is a relative same ample li bond so the way we recall the non-community jh functional the definition is a twisted non-community twisted minimum pair uh minimum pair and mirror uh functional minus the three times non-community minimum pair functional that the explicit definition is defined by the by the intersection number first of all is the new star rho star h times the l bar and well rho is this this morphism when i says dominating which means the test configuration curly x has morphisms from from its form to the x times the p1 sorry i so the say the x p p1 which means x times p1 and this this this morphism is a projection the l bar means the uh chemical compact compactification which which obtained by add the trivial fiber at infinity uh this part of c we know c is m mu h and this one is the self intersection of l bar so with with the xl is the uniform j h stable if there exists some constant such that the non-community jh functional is the grid side equal to the h zone your jh functional non-community jh function the definition of this one is l times l p1 n n times l p1 sorry the l p1 means the rho star l you can say as rho star l minus the n plus y the self intersection rho l bar uh so motivated by the logita this criteria for kelan's time equation we hope to study the uniform value uniform j stability by in terms of variations for any prime device we can define the little j h invariant as follows an s h f minus s invariant well s invariant is this one of volume times the integration of the volume parts as before and s h invariant is the integration of the derivative part this one this is the positive section dot the device h by the boxon fiery johnson serum we know this is a derivative of the volume so same you know with the delta delta invariant we define the volume value of j h stability threshold in terms of s invariant over s h invariant over s which is positive of course which is positive so we say the x l is the variatively j h same stable if the this invariant j h is non-active and the variety uniform or variety of just stable if they exist some uniform constant such that the j h invariant is greater than equal to x n times your j invariant well j invariant is defined as tau l minus s invariant well recall the tau l is a pseudo effective threshold so of wise need the uniform variety of j h stable is equivalent to the gamma gamma invariant is strictly be that constant this constant c and it's greater with recent gauchan's work we can obtain the uniform or variety of stability implies uniform sorry uniform j h stability implies uniform variety of j h stability i is the part is the direction of variety of criteria in other words in other words if the polarize variety together with gauchan's work if the polarize manifold has a unique solution of j equation then this invariant gamma h is strictly be that constant c so for for so witness for the ideal and prove we would naturally cause the theory of test curve developed by the rostrand of the institute my my notation of test curve follow follows the uh min chun xia's paper uh test curve is uh my five dots from the real number to the model model potential well model potential means the psh well psh means the uh omega psh function uh the model potential means is the psh omega psh function with a golden singularity sorry i do not introduce the explicitly the definition so such that the psi is the curve in this this dot and psi is the up same continuous as a function um on the x times r and when r goes to mass infinity the psi goes to zero in l one sense and for i like enough psi r equal to minus infinity the rife rife picture is is this the next set is the i plus as the infim r such that psi equal to minus minus infinity so we know for each prime device f it induce a z filtration f on the section ring r well r means the section ring of xl it's given by the f lambda actually for lambda is non-native the well filtration means the farming of wet space of this of this space for lambda new non-native this space defined as original space h zero xkl and when lambda is non-native the space is h zero it's l kl minus lambda f and therefore this this field this filtration is induced i model test curve psi top sorry psi r i is psi i equal to the equal to the p as a i model envelope or psi r so i do not introduce the explicitly definition of the i model which means go to your life singularity that's the model model model potential the explicit definition of psi as this one the silver upstream is continuous as a star means that i've seen up semi continuous the k over one one over k the silver is allowed as the as the square now well i just means the smallest metric on l such that the first transform equal to the omega and so a simple a simple calculation we can we can solve for suitable r uh so in in this interval and there's a generic and the number of psi r along the a is the greater than equal to r so the definition is this generic and the number is the number along a generic point so since the psi sorry psi r is a i model by by davos and schaust's work we know the uh and i need to come around and uh mass equal to the algebraic mass algebraic volume the asymptotic growth and which is a great thing to the the uh asymptotic growth of the skr rk this space since the its elements we can easily take which is the square square integral with respect to the this metric and for this can be easy check equal to the volume l minus rf so for converse direction we have this the following lemma for any prime divisor and omega psh function uh which uh satisfies the the generic and the number is greater than equals to x than the volume of this l minus xf is greater than the measurement of omega f omega five and the positive intersection products this one it creates the tested American measure so by by this in quality and and this in quality we know we know the the main rampant mass of psi r equal to the volume of l minus rs so we can compute this main rampant energy of the test curve the definition is the firstly in quality uh firstly in quality which equal to this one we replace this by this one and so the main rampant energy of the test curve equal to the ascii environment and we can also the test curve the definition is the first first uh equality by this this one is due to the lemma which equal to the ascii environment so the test field main rampant energy of the test curve psi is smaller than equal equal to the ascii environment so let's recall the inverse redundant transformation for a test curve take its inverse redundant transformation we get maximum geodescary since the psi is a model test curve this by is a dominant uh damage and the maintenance paper recent paper which course uh well i do not introduce the explicit definition of maximum geodescary which means which corresponding to violent energy non-commediometric uh which means can its use can be can be approximated by attributed to delta but uh i uh a family of test configuration i do not bias down on the battery side and moreover by the bowman bowman uh buxton and johnson's work and chelise's work we know the non-commediated function of a non-commediated main rampant function of phi equal to the radial function of this one and and just did a main rampant non-commediated main rampant function this one equal to the corresponding radial function and by damas and the shafts paper this one equal to the main energy of the test curve and by recent work of inter shafts uh the test field it's also holds for the test field main rampant energy so the g-i-t invariant is the great sign equal to the the test field main rampant energy of test curve minus some constancy times the main rampant energy of the test curve which equal to the radial function corresponding radial function and well they exist uh sequenced the smallest non-commediated matrix which induced by uh test configuration curly x and l clearly l which strongly converge to phi uh that's the which means the non-commediated j-functional is converging converging and euro j-functional non-commediated j-functional also converge so the g-i-t invariant is greater than equal to the limit of the j-functional of this test of this family of test configuration by assumption of j's uniform test stability which greater than equal to the epsilon and the limit euro j-functional of phi key so this one is converged to this j-functional of phi and can easily take we can easily take the non-commediated j-functional of phi equal to the j invariant euro j invariant of f the way shows the easy easy direction so for the remind time i i want to talk uh uh application of value to the stability so next question before you before you go to move on to the next topic yeah so uh at one point you mentioned this result of Gao Chen yeah you said uh by his result if there exists a j extremometric then you are stable in this gamma sense right yes it it maybe you said it already but i have to ask is the other direction known uh the the countless direction up up to now i i cannot show the constant direction okay but do you expect it yeah yeah yeah okay thank you that was my question okay so uh let's recall some basic definition of toric priority so let's ask sigma is a n-dimensional projective normal toric right associated to phi and sigma well n is a lattice of all one parameter subgroup of the torus and the phi phi means uh means uh connection of cones such that each cone in the phi is generated by finite many elements and the intersection two is also in this line for any cone for any two cones and if you know time is the deal of the lattice of the character of the torus and sigma one is a set of the row of the reals of the of the phi uh means dimension one cone so for each real row in the determinants prime device d row and element u row in n so for any ample line bundle window it's uh corresponding to a four dimensional lattice polytope p whose normal phi is the sigma so for name you name you write uh m4 divided by l as uh sum of a row d row and the corresponding lattice polytope as m and a p l is m in the mr real width based mi such that in the product m and your row is greater than mass a row so uh we can show the for x l be a polarized toric variety and assume x l is k same table that the volume of l is less than equal to the this maximum of real of a row times the one plus n over n plus one mu l times the tau l as well as is is our exceptional device of blow up through a small point so we can easily check this inequality is a scaling environment under the multiple of l so we can assume normalization condition mu l is the one the upper bound that becomes to the this one uh by uh balloon and johnson's results we can we can explicitly write this this environment but for simplicity i just do not eat and this this type result is the first concept is confided by fijita the fijita gives a famous result for final case and in the same way same paper he gave interesting no bound of the volume of l minus x f so volume l as x f is greater than this one l intersection self-interaction section minus x times n and so the integration of the volume is greater than this the integration of this part so which equal to the square root n of volume l and over n times plus one times volume l sorry sorry sorry and yes the the proof is simple so it's well along k same table implies the origin of fijita in tacky invariant vanishing and by Dover and Legendre's paper they show the variation same table if and only if the tacky origin of tacky invariant vanishing for a topical variety so we know the x l is the valiantian same table so beta l is non-active so we take f as exceptional divisor of the blow up of a smooth point induced by some effect uo0 and can buy the classical theory of topical variety we can show we can compute the log disc square and state to n so right l has this sum and we know the kx is minus the sum of d row so but by by the non-active beta so the the first part is n times the volume and the second part is n equal to s l greater than direction part and this one can be right and this so we times this constant and uh this one so this one is greater than equal to the origin intersection products of l so by the use for lemma we know this one is the integration of volume l minus xf so n plus one times n plus one so can be can be computed as this this one because because because of the fugitas no bound by by this inequality so simplify it so we get this result this this time well this well this inequality is due to a result of books on faria and johnson which states the the the position section with a effective device that is non-active since it can be computed as a restricted volume and so this volume over the maximum of a row is less than equal to one plus this term and this part we know the sl environment and the total the equivalent by coefficient n over n plus one so we show the result this no this this volume bound okay i i think i stopped here thank you for your attention thank you for the nice talk uh are there any questions about any of the students i guess that's a no or maybe second no questions okay good uh well let's thank the speaker again yashiyong thank you very much for spending time with us thank you very much all right thank you