 I'm pleased to introduce Shin Fu from UCI, who's going to talk about Kader Einstein metrics near an isolated log canonical singularity. Okay thanks for the introduction and thanks Professor Yanir Thomas and Jeff for the invitation. So it's my honor to speak here. There are a lot of experts in complex geometry here so today I'm going to discuss Kader Einstein metric near isolated log canonical singularity. Focusing on the geometry side maybe in a moment I will introduce what is a log canonical singularity. So here's the plan of my talk. So the first part I will review background of complex complex monogram pair equation. So first I will discuss, of course we know that one of the motivations of complex monogram pair equation is to study the is looking for the Kader Einstein metric. So first I were looking for Kader Einstein metric on a canonical polarized variety. And by the way in this talk when I say Kader Einstein metric I always assume it has a negative scale of aperture. So the classical result of a Bayou can find Kader Einstein metric on canonical polarized smooth maybe smooth manifold. By co-orderies fundamental zero estimate so Kader Einstein metric can even be constructed on singular variety. On the other hand the Kader Einstein metric complete Kader Einstein metric can also be constructed on strongly suricomax domain initiated by the Chen and the Yao using the so-called finite geometry method. So that's the history somehow the compact case singular case and even the complete non-compact case. Then the second part I will move to and I will discuss my result with my collaborator. So we we consider we considering a construct constructing Kader Einstein metric locally and namely we want to solve a deletion problem then we discuss the geometry of the Kader Einstein metric. And the third part if I have time I will mention the optimal sympathotics of Kader Einstein metric on harmonic case. So let me start on part one. So let's extend the complex manifold with canonical boundal be a positive. Then a question is to ask if there is the Kader Einstein metric on X. So to looking for a Kader Einstein metric we just need to look at the following complex to solve the following complex manual pair equation where theta is the metric in the Taylor class of the canonical boundal and the omega is the volume four. So we can consider the following continuity paths. Namely we consider a family of equation theta plus theta by phi t equal to the wrong side. And here capital F is defined as log theta n to the omega. So a classical result of Aubameyang said that phi t in the family of equation one satisfy uniform estimate, uniform seizure estimate. So more precisely phi t will be controlled by the theta depends on theta c0 norm of capital F and X. So here I want to emphasize that this c0 norm of phi t will depend on the c0 norm of capital F. Then a co-odger has a fundamental improvement about seizure estimate for complex manual pair equation. Here's the setting so let X be a compact manifold theta is a Kader Kader four and omega is a volume form satisfying the following normalized conditions. The volume of theta only is equal to the volume of capital omega. So also we assume the volume form capital omega oh sorry capital omega over theta n to the p is less than capital K for some p larger than one. Then the complex manual pair equation theta plus d d by phi equal to the volume form will satisfy the following seizure estimate. Phi minus 2 phi is smaller than a constant depends on theta and manifold X and only depends on the LP bound of the of the volume form here this capital K. So the improvement is this seizure estimate doesn't depend on the point one norm of this right hand side volume form it only depends on the LP norm of the right side. So with this improvement people are able to construct a kinestimetric or singular right. So here's the here's the setup. Now again X is a canonical polarized but now it's a singular right with kawamata log terminal singularity as theta is a Kader form in class kx. Again we can see the same equation theta plus d d by phi and equal to e to the phi here omega is a volume form satisfying the it's it's rich it's rich form is equal to negative theta. So the equation is the same as before we are considering the equation on a singular right so to solve equation two on the smooth locus of X in a moment we will reduce to a family of degenerate monumentary equation but before I do the reduction before I solve equation two I want to discuss a little bit about the algebraic singularity so what is kawamata log terminal. So for a singular variety if we fix a log resolution pi y to x then we can write the difference of canonical class ky minus pullback of kx as a as a combination of exceptional divisor ei. So when ai here the coefficient ai is larger than negative one then we call it kawamata log if it's only no less than negative one we call it log canonical. So here are some examples example one is we're considering an aphan cone over a quadratic quadrant so it's in fact it's aphan cone over city one so it's kawamata log terminal example one. The example two is we consider aphan cone over elliptic curve it's in fact it's a log canonical singularity so analytically kawamata log terminal means that if you look at this difference of canonical class ai and negative ai larger than negative one essentially means that there exists a local volume four on x when you pull back omega x to wedge bar omega x to the smoothest manifold y then it's it's p larger than one integral that's the meaning that's analytic meaning of kt but when you are in the situation of log terminal so since you allow ai to be negative one so then you can imagine if you pull back this omega x wedge bar omega x then it's uh it's even not l1 integral root here somehow l1 is the borderline case so in other words um log canonical is we we could we can't apply converges a seizure estimate to the log canonical case because here l1 also fails so now i'm i'm trying to solve equation two and on a singular variety so first i do a reduction i fix a resolution pi to x that i pull back to the equation to manifold y y is a manifold it's not singular so here is the pullback omega x equal to the sky just because the krt property then the aj will be smaller than one then the right hand side will be a p larger than one integral that's good um then we do a perturbation for the equation two more precisely now we pull back the kiloform pi theta then it's then pull back of theta is the only seven positive so i add a little bit here omega here this little omega is a kiloform so s is a positive constant plus d by phi s to n e to the phi s for the right hand side i only i also do a little bit perturbation namely i here i for the divisor here i plus s and for the divisor on the denominator i also plus s now this equation when s larger than zero is a perfect smooth equation on a smooth variety then um by the co-order is estimate uh we need a we we have phi s uh the C0 norm of phi s is uniformly bounded um yeah i should mention it's not a direct result for by a co-order because if you look at the left hand side the reference form pull back theta plus s omega is also degenerating so in co-order is a result he fixed a kiloform at the yeah i should mention that so with some additional work to deal with the degeneracy of the reference form theta plus s omega so finally this leads to the following theorem let x be economical polarized or club b yaw single variety with k-l-t singularity then there's a kinematic or smaller smooth slow cast of x by acedule gauge the reachee in the general okay so let me pause for three seconds is there any question okay maybe this is the one no so let me continue so on the other hand we can construct kinematic um complete kinematic which initially initiated by Chen Anilow so let omega be a strongly pseudo convex domain with most boundaries and there's a unique complete kinematic omega k e on on the domain omega they developed a so-called boundary geometry method and quasi-coordinates to deal with so um i just want to do some remark about this construction so first um first um for this theorem they can construct a reference metric which is already uh close to client in some sense then the omega k e the kinematic constructed is somehow a small perturbation in some sense of the reference form so the remark is um if we just consider a complete uh kinematic on complex manifold it's it's all it's always unique it's unique also uh if we um use boundary geometry method when we construct a kinematic the kinematic metric of a tender is only known to be equivalent to the reference form it's only known to be equivalent to the reference form so by using Chen Anilow's method there are more construction about the kinematic on um quasi-projective manifold so here let x bar be a smooth projective manifold d is a smooth type surface or a divisor sitting x bikes so let error define to be the canonical class plus divisor d which is ample then um we can construct a very good reference metric the so-called casting graph is metric on um bar x subtract the divisor d so on this quasi-projective sorry maybe it's not necessary quasi-projective okay it's an open manifold so omega is uh we can do a kinematic on this x which is defined to be the rich curvature of the ample amount of air minus i d bar log log one over a square so here locally if the divisor d is defined by calling the z n equals zero then a square is roughly the n square e to the five five is uh um is a metric associated to the lambda d so let me mention if we forget about e to the five and if we just look at the minus i d bar log log one over a square square then it's exactly the punk it's exactly the punk rate metric on the puncture disk if we just look at this part the construction is by Sebastian here on the yaw then in the in above setting then there exists a unique complete kinematic omega k e which is omega plus c value okay so this is part one i review the classical result about kinematic with negative scalar culture now i'm going to discuss my result so i consider x p is isolated at low canonical singularity embedded in the theorem so i fix uh caliform which was defined as i d d bar the square here z square is just the z is just to coordinate all the n now i cut a neighborhood i call it u which is defined to be z smaller a so this a is uh is uh not important i just cut a neighborhood of the singularity p now here's the result that you'll be uh be a germ of isolated low single low canonical singularity as above then there exists a phi which is chi psh satisfied the following condition so first phi is smooth outside p uh phi also from the manual paper of sorry i forget to write down what's the equation here but it's a it's a relative kind kind of symmetric and for any longer for any epsilon like zero there exists a c epsilon such that this phi construct is satisfied the following estimate so in particular log sigma d uh would be uh negative infinity so i don't so our c zero estimate somehow is not bounded uh but it's better than any log pole so i'm three uh this chi plus d bar phi is calium time and four uh phi goes to an active infinity so here i want to um maybe emphasize item two so the solution we get indeed doesn't have uniform c zero estimate it says it's better than any long pole so since today i want to focus on the geometry side so let me just do a summary math about proof so for the proof uh again we use the uh perturbation approach namely i fixed the resolution i'd pull back the equation to the resolution then i do a perturbation but the the trouble appears since in the log canonical setting we don't have a uniform c zero estimate for the perturbed family of equations so um but uh luckily we can we can somehow locate somehow we can locate where the bottom is of uh bonus of a phi first somehow it's exactly the place exactly the log canonical locus so this is uh our person somehow is a barrier construction so um then with uh with a c zero estimate then i can buy um um then i do the boundary c one estimate then i use blocky uh global c one estimate for minor pair equations and then c two estimate uh c two boundary estimate and then the c two um c two a global c two estimate of uh of yaw maybe is the lucky thing here is although the phi is not uniform bounded somehow our modification of blocky's gradient estimate also works here so finally then we can solve the equation on our neighborhood of x for any smooth boundary so another marker is uh here i'm choosing reference form as id by this square so you uh you should regard it as a smooth form because um somehow it's the it's the it's the restriction of telemetric or a smooth and in the space so the since this is the reference form is is a standard one then the geometry of a kind of kind of symmetric in fact uh should be encoded in the singularity of this okay before i go to the geometry so uh maybe is there any question maybe a quick one do you allow log log terminal singularity sorry do you allow log terminal singularity yes yes okay yes so uh yeah we allow log log terminal singularity so but here i'm considering only isolated singularity and the reason is i needed the boundary the boundary of u to be strongly through convex because i need some classical result i can see two boundary estimates but if you do a resolution of this singularity p uh i allow there are several some components with with coefficient negative one and some other components with coefficient strictly bigger than negative one it's a lot now move to the geometry side so um now i'm going to discuss the uh the kind of symmetric near isolated logarithmic singularity maybe let's just focus on the following example i mentioned before so this is uh this is often cone over elliptic curve or you can also regard as the total space of a negative lambondo error on a torus d with zero section constructed contracted so this is um this is the example so on one hand uh in fact i guess this guy has a name people call it hyperbolic cusp on one hand we can explicitly explicitly construct a complete kind of symmetric near singularity p so here is a picture so if we look at this singularity x defined by this algebraic equation in the euclidean in the euclidean topology it should be the this guy on the on the on the left hand side but for this guy in fact it's uh it's um um it has a complete kind of symmetric so the the picture on the right side is uh is a shape of this x under the kind of symmetric uh the origin is pushed to the it's pushed to the infinity so okay so uh yeah please keep this picture in mind because so later when i say uh a complete kind of time i mean it's completed towards the singularity it's completed towards the singularity but it's in fact incomplete uh here on the boundary but i somehow i only care about the behavior of the metric towards the singularity so this side is complete and this side is incomplete but i only care about the behavior here so uh there's a model metric on this harmonic cusp where essentially it's the it's the it's the ball quotient it's the it's a Bergman metric which is invariant on some group action so there's a in this example this kind model uh i call in fact is the harmonic metric for this harmonic metric there's another way to construct it it's just you choose a a metric on this negative lambda and whose rich form is who um the negative of the rich form is a rich flat metric on od then you can use some od to construct the hyperbolic metric on this on this x so then then i ask the following question uh so now we already have a good um chi model which is kind of instant i already have a good kind of symmetric here uh if i have some other kind of symmetric so can we compare um a chi model and chi problem that's the question so a priority so if you have two kinds of metric on this set um since it's complete you'll even don't know it's equivalent or not right because it's complete so in the following talk i am to compare two complete kind of symmetric on the set now here's the result so suppose i have two kind of symmetric on u minus p subtract p so here p is isolated logarithmic similarity then let rx be the distance to the boundary of u and r prime x is the distance to the boundary of u under the metric type prime or g prime so the boundary of u uh it should be here okay so uh r and r prime are the distance to the boundary with respect to uh these two different kinds of metric so if these two metric are complete towards p then i have the following aspect uh for the for the volume ratio so namely i for the volume ratio i have it's larger than 1 minus c over r prime smaller than 1 plus c over rx so in particular if r prime x is very large and rx is very large then this volume ratio will be almost one okay let me let me also remind you what is the situation r prime and rx are very large so rx is defined to be the distance to the boundary so when you point the p goes to the on your point uh moving moving towards the infinities and it's uh it's moving away from the boundary so this theorem just says that if your point moves towards the towards infinity then the volume ratio when it goes to 1 here i only assume that these two metric are complete so here's a proof in fact it's quite simple so now i fix a point q uh such that its distance to the boundary is larger than 2r now i do a standard Kato function uh okay as as follows then um this Kato function satisfies it's equal to 1 on the bq arbol and it's equal to 0 outside the bq uh bq 2 arbol then by Laplacian compressances we have the Einstein condition, Kata Einstein condition then by Laplacian compressance then we get the estimate of the Laplacian of this Kato function as follows now since we have two Kata symmetric let phi be the log volume ratio let phi be the log volume ratio then trivially we have this inequality trace chi id bar the log volume ratio will be larger than this guy so then i let this Kato function times this log volume ratio i define this time to be capital h so since we have a Kato function we can assume h obtains some positive maximum at some point q then at this maximum point i calculate Laplacian again following so since at the maximum point the gradient urge and vanish then we get hq is indeed bounded by this card then at the point of q uh at the point of q we fixed at the beginning as Kato function is equal to 1 so capital h is equal to phi then it's smaller than this card that's that's the estimate so this is estimated for one bound if you switch the row and the time and type value we get the normal this log volume ratio so it's a it's somehow a localized argument i think the proof is simple but it turns out it's the key result of our paper so now i have the estimate for the log volume ratio the next step is to compare the metric so here's the result again we consider a germ of isolated logarithmic majority and now i put one more assumption as to um the model metric hi has some good curvature property here i'm calling uh i call it bounded geometry method um bounded geometry of order k but you can just roughly regard as it's um it's up to cover it's very close to the liquidity metric not very close to it's equivalent to the liquidity metric up to account then uh for any other completed kinds metric GKE on this set we have the following high order estimate so here nambla is taking with respect to the model metric omega k minus kind model um is has a following decay estimate so here i want to also improve the proof is not very hard so let me assume i already have a model metric kind model with uh now we already have a good good metric kind model with bounded geometry so if we have if i have another kind of symmetric type prime then using a kind standard condition i can write kind prime as the kind model plus i dd bar long volume ratio and by previous previous uh previous result i stated this low volume ratio is already bounded is already bounded and in the second step i solve the following traditional problem uh on the set so i think already a complete model metric plus dd bar phi to n is equal to e to the phi chi to the n here i have a obituary most boundary condition so the hybrid of 20 albana jump method and second is kaffirini column neural books brak uh then i can solve the following traditional problem so let me mention if you will solve uh this equation by using bounded geometry method then a byproduct is phi going to be uh bounded up to any high order derivative with respect to this model metric so i'm three uh by showing some uniqueness of bounded solution of the above equation here so i show that the bounded solution of this equation is unique and then combined with uh item one i know that for any uh completed kind of symmetric type prime it's indeed uh from one solution of the traditional problem so this is the this is the place where this phi is bounded used because i can only argue the bounded solution is is unique so combine this two i know that uh and i'm sorry i know that for any complete metric type prime it's indeed from one of the traditional problem one of the solution of the traditional problem so item four as i said um a byproduct of the bounded geometry method is you know that chi indeed has a uh high order derivative bound so that's the proof so any questions maybe let me pause for us okay so um now we get to the following uh estimate for any fixed kinematic which has some good curvature property and for any other kinematic uh we have the following uh high order derivative estimate but the decay is not the decay on the right hand side is not it's far from optimal so um in a special case uh yeah joint work with uh Han and uh Shunmin Zhang we somehow obtained the optimal uh decay estimate for this for this uh for this function phi so here is uh uh the result so uh record that uh i have bolly cusp another way to look at how bolly cusp is you take a torus you fix a negative lambda on it and you construct the zero section then you get a bolly cusp if you move the thing if you if you move the single point that's a half bolly cusp so now i fixed our transmission metric on this negative lambda and then you'll be a closed turbulent neighborhood with uh uh with the zero section d in this um here i should say the total space of this lambda sorry just uh let omega ke be a complete kinematic on this um turbulent neighborhood subtract this divided d or you can see it as a subtract the singular point that's the same as they are the same thing then uh let phi be the long volume ratio this uh theta h is uh so here i should explain what is this h so in the we know that in the in the copy uh ansa's construction if you fix a metric h on air if you scale this metric if you scale this metric h then um it doesn't affect its reach curvature so in the copy asa there's um um it's it's it's it's uh if you scale metric h then uh the copy asa the result metric is the same but you are choosing different metric on air now here um theta h is uh somehow the yeah maybe let me maybe let me continue then they exist at constant c and the data such that for any k we have the optimal estimate for the for this long volume ratio so here we have phi plus n plus one lot one plus c x uh you take a derivative with respect to theta h uh since o x is uh it's following k negative oh two plus one four and i think k of two and have a exponential four here this little x is in fact uh roughly the inverse of the exponential of the distance function to the boundary so this guy gonna be when x go to zero if you look at this guy uh gonna be very very small because um this is e to the negative uh infinity somehow excellent and with respect to the distance it's it's it's more like the double exponential decay because as i said x is almost the inverse of the inverse of the exponential of distance distance to the boundary so here also lambda one is defined to be the first eigen value of the plus on the torus d let me mention this red guy this red guy why we have this red guy so this red guy this one x is close to zero then log one plus the x is it's just like a polynomial right when x is close to zero so this polynomial part is indeed from the uh the scaling on the metric on the lambano air so um in other words if you choose uh if you scale the metric on this lambano you get the different phi the button you've done to change the kinetics metric so this one tells you that uh if your modulus is scaling on the lambano or you can or in other words the automorphism on the lambano direction then this phi has a very fast decay okay this is a remark okay then uh i'll stop here so what's the time oh i i didn't expect we can do very good with time thank you for a very nice talk um i'm gonna stop the recording and then open up for questions