 So this time I was going to give some applications, but so let me just recall very briefly what I did the first time then. So here we have the big picture, which is like this, that we have some blob up here, which is called beautiful X. So that's a complex manifold. It could be a domain in CEN, it could be an open manifold, or it could be a compact manifold. And then we have a projection map P that goes from this beautiful X down to a base manifold B, which is also a complex manifold. And for instance, this here could be an open set in CEN, and then this could be a linear projection. So there are many possibilities. There are two main cases. One is when it's an open set in CEN, or CEN plus one in that case, because think of this as one dimensional and the relative dimension in this direction would be N. And so there are two main cases. You think of this either as an open set in CEN plus one pseudo convex, and this is a linear projection down to C. Or you think of it as a complex manifold, and then the typical case, all the fibers here we wanted to be compact complex manifold. And then we can think of this here as T-varys, we can think of those fibers as a family of complex manifolds. So I proved some theorems. I'm not going to go through them again, but I put the slides here in case I will want to go back to them. So this was the first theorem that was about the proper vibration. That meant that all the fibers were compact manifold, we have a family of compact manifolds. And then all the time when we have such a fiber, we think of the L2 space or the Bergman space or some holomorphic objects here. In the CEN case there would be holomorphic functions, otherwise there would be N forms with values in a line bundle L. That's what is intended by this notation here, ET. So this is the space for each T, we have a Hilbert space, in this case finite dimensional. And that's the space of holomorphic sections of this bundle, which means that it's an N form with values in L. This means N forms. And then we can compute the L2 norms, they look like that. We get a metric, we get a family of vector spaces that has a metric. So it's a Hermitian vector bundle and the conclusion was that the curvature of this vector bundle with this metric is positive or non-negative. Yes, in this case they will. So that's a non-trivial fact. So in this case it will be. We look at slightly more general cases at the end, and this is not assumed. Or we have in the CEN plus, when this is an open subset of CEN plus one, the CEN plus M in this case, the formulation was a little bit more complicated. Maybe I do not go. Let's just remember that there was some rather complicated statement like this. And the special case of that was that if we have such a pseudo convex domain, and we look for each slice here, we look at the Bergman kernel function here. Then the conclusion was that the logarithm of the Bergman kernel is plurisabharmonic as a function of both the variables T and C. In particular, it's plurisabharmonic with respect to T there. So in case T is one-dimensional, it depends subharmonically on the variable T on the parameter. The plurisabharmonicity with respect to C is a very classical fact, which is always true. Okay, so now we can start with the applications, and maybe I can come back to those things later if we need that. So I'm going to talk first about the Osawa Takigoshi extension theorem. Then I have to draw a very similar picture here. We should not be confused with that one, although it looks almost the same. So we have a domain here, which is now called the D in CN. And we have a, let me write it down here, and we have a hyperplane, no it's not a hyperplane, it's a linear subspace that's supposedly intersects this. So here is V, a chosen coordinates so that the linear subspace is just a space where the first coordinates are equal to zero. And we have a plurisabharmonic function in the big domain here, and we assume that this domain is not very big. So this is the C prime direction here, and this is the C double prime direction. So this should not be confused with the other picture because there is no, yeah, it will, yeah. So I assume that this is not too big here, so I assume that C prime, some normalization is less than or equal to one here, all the time. Okay. And then the theorem is the following, that's the Osawa Takigoshi theorem. It says that if we have a holomorphic function on this slice here, we can extend it holomorphically to the whole domain. So this is a classical fact, but we can do it in such a way that a certain L2 estimate holds, namely we extend it to capital H, and the L2 norm, weighted L2 norm of capital H is bounded by a constant times the weighted L2 norm of small h, which is the function that we extend on V. And the important thing is that C is a universal constant that does not depend on phi. After this, that's where the normalization comes in here. That's what makes the constant universal. So this is a non-trivial fact. It was proved by Osawa and Takigoshi. I should remember when, but I don't. And there are many versions of the theorem for manifolds, et cetera, for sections of line bundles, et cetera, et cetera. And they actually, it's actually a very useful fact. But I'm going to stick to the simplest case in order to show what it has to do with the theorems that I discussed last time. So one question that arose was the following. What is the best constant? We know that the C there is universal. So that was a question of what is the value of C. In a particular case, there was a conjectured answer to this. And the conjecture was called a suites conjecture. If you, yeah, it's probably not so well known. But, and the answer is the following. It was given by Spinger Bloschke and Guan and Shu. They found that the optimal value of C is sigma K, which is just the volume of the unit ball in CK. So that value is obtained. It's attained. If you take just the generic case when V here is just a point, say the origin of the unit ball in CK, and then you take a function there, one, say, and you want to extend it with a small L2 norm as possible. You take phi equal to zero. Then certainly the best extension will be the function that's constant equal to one. And that will give you precisely this bound. So it says that you have a better, this is the worst estimate. So if you formulate it demagogically, you can say that this is the hardest extension problem. This thing, you always get some. You never get the worst estimate than that. That's the theorem of Bloschke and Guan and Shu. And, yeah, so I'm going to describe an approach to prove this, reprove their statement here, and prove maybe something a little bit sharper than that. And this is joint work with Laszlo Lempert. You heard of him, maybe, Thomas? Yeah. Not recently. It was Thomas' advisor. So the idea is this, that we look at DT, which is the subdomain of C, such that the logarithm of mod C prime squared is less than T. So we shrink the domains here. I said that the logarithm of this should be less than T. So when T is now a negative number, it looks like this. I get this is DT here. DT, like this domain here. I look at this family of domains for T less than 0. So we exhaust D by subdomains. And they shrink. When T goes to minus infinity, they shrink just down to this plane here. And you can view this as a special case of what we were looking at before, because you can now construct a domain in one dimension higher, which is the domain I have down there. Beautiful D is a set of tau, C, such that the logarithm of mod C squared minus real part of tau is less than 0. So this corresponds to this picture now. And this is a pseudo convex domain because the function you have there is pleurisabharmonic. It's defined by pleurisabharmonic inequality. So it's pseudo convex. And the DTs are the slices of this domain now. Right? The way I do it. Yeah. So this is how we do it. So we have. And then the theorem with Laszlo is the following, that for each of those domains now, we can apply the Osawa takigoshi, where we can think of the Osawa takigoshi theorem and says that there is an extension here. And we look at the optimal extension. There is always an optimal extension, the one with the smallest norm. We call that HT. And we look at its L2 norm. So that's the index T. That's the L2 norm of a DT. I take, so to speak, the best extension for DT. I take its norm and multiply by this constant, e to the minus kT. So obviously those numbers, the norms here will be smaller and smaller. So I compensate for that by multiplying by e to the minus kT. And then I prove that this is a, we prove that this is a decreasing function for T less than zero. That's the theorem with Laszlo. This is a decreasing function. So what does that mean? It means. Huh? Huh? You mean that T gets more and more negative? Yeah, it still gets more and more negative. Yeah, exactly. What do you say it's decreasing? I really mean that it's decreasing. So, yeah, when, as T increases, yeah, it decreases, yes, yes. So what I want about all is this inequality to hold. So this is the quantity that I really want to estimate. It's the best, when T is equal to zero, I have my original domain. And if I got the decreasing, increasing right, that would be now smaller than the limit when T goes to minus infinity of those things there. There should be an e to the minus kT here. Sorry, e to the minus kT should be there also. Exists? Yeah. Because it's monotone. So I made a mistake that it should be an e to, it should be the same expression here as there. It should be with the e to the minus kT also. And then it's monotone, so there is a limit. And, yeah. Yes, so monotone is, yeah, so, yeah, well, yes, but I have hTT squared e to the minus kT. So I draw this quantity, let's call this AT. It looks something like this then. I said it was decreasing, so it looks something like this, maybe. And then I say that this value here, which is what I'm interested in, would be smaller than the limit here when I go out to minus infinity. Could be, in that case it's not interesting, but so far it's all we claim. But I claim it is not, I claim it cannot really be minus infinity, it cannot really be plus infinity. Actually it will be very easy to compute what the limit is. So the consequence of this here is that optimal constant is sigma k. What is that? Why is that? So the theorem of Lodzky and Guanshu is the consequence of this last inequality there. Because when t is close to minus infinity, I have a very, very thin little strip here. Very thin, it looks like this. So now t is close to minus infinity, I have a very thin. So I want to construct an extension, I can take any extension here. And when t, when this is very thin, I can compute more or less the normal of that and you will see exactly that it gives you the value that you have there. If I got the e to the, I should have had the e to the minus kt there. So the monotonicity is the key to the problem. There is some convexity, yes. So I will get to that because that's the way we prove the monotonicity. So, yeah. So now I want to sketch the proof of this, but I'm not going to sketch it in general. I'm going to sketch it in a special case. So we take the special case when k is equal to n. So that means that the variety here is just the point. Then I can draw a new picture perhaps. Then the domain is just this. I have a point here which is the origin, say. And the dt, this is d. This is dt. Something like that. Looks like this. And yes. And then I want to extend a function, a holomorphic function on the point. That just means the value 1. I want to extend it to a holomorphic function with some estimates. So let's see. This is really a Bergman kernel estimate in disguise because if you remember the Bergman kernel, the Bergman kernel kt for the domain there was the supremum of this quotient. h at the value 0 divided by h, the norm of h. That was the Bergman kernel. Now I can normalize, so I can choose always h0 equal to 1. Here I get the 1 upstairs and I get 1 over the norm. So to estimate this norm here, it's the same thing as estimating the Bergman kernel. So the theorem I want to prove is equivalent to saying, let me see here. So now we get to the convexity that you asked about. Now I'm going to use the logarithm of the Bergman kernel. It's a convex function of t. It's actually a subharmonic of t, but it only depends on the real part of t. So it's a convex function. Yes. And the theorem is equivalent now to saying that, let me see. So the theorem on the slide is equivalent to saying that this function is increasing because the relation between kt and the norm is 1 over here. So instead of looking at what we had before, we multiplied the norm here by e to the minus mt. Now I get the Bergman kernel multiplied by e to the plus mt. And I want to prove that this is increasing now, this thing. I don't know if this is too confused or is it roughly correct. And now the convexity comes in. Now the convexity comes in because this function here is convex. I add nt to it. It's still convex. The logarithm of the Bergman kernel is convex with respect to t. I added something linear, it's still convex. And moreover I claim that it is easy to see that it is bounded when t goes to minus infinity because then you just need to estimate the Bergman kernel over essentially a small ball in here. And if you have any function which is like that, it's convex. So it doesn't look like that now. Look at that function. So it's defined on the negative half axis. It's convex and it's bounded when you go out to minus infinity. Well, it has to be increasing. So that finishes the proof in that case. So still some... No. So this was actually Laszlo's argument for the suites conjecture. And then I suggested that we should prove that it works for the entire formulation of Savataki Koshiti. This was just a special case from the right. It was a point. But... And then in order to get the full of Savataki Koshiti theorem, when you have higher dimension here and not just a point, we have to use the full formulation of the theorem, not just the theorem about the Bergman kernel, but the full formulation when we had those measures and arbitrary measures, not only Dirac measures. But the principle or the argument is just the same as what I wrote up there. So that somehow gives you the best constant in the Savataki Koshiti. And it also works in more generality for manifolds, et cetera, but I presented the simplest case here. I get back to... Hopefully at the end I will get back to Savataki Koshiti for manifolds a little bit later. Okay, so I leave this. I go to the next topic. So it's some example from Kehler geometry. So this was about Brunning-Kovsky for domain C and the first one. And now we look at... We look at compact manifolds now. So I'm going to this situation here when this is a proper vibration and all of the fibers here are compact manifolds. But I'm only going to look at the special case of it first. I'm going to look at the special case when this vibration is trivial. So I have a cylinder. So I have a manifold here, which is called X. And then I take the product of X with U here. So this thing here is X times U. So all the fibers here, I take a point T in U. All the fibers here are the same. They are all X. So I have a constant family of complex manifolds. And then I also throw in a line bundle. So I take a line bundle on X here. And then I let it define a line bundle on the product here just by pulling it back by the projection map onto X. So I take, so to speak, the same line bundle on each fiber here. So this is L underlined. So then we have a family of manifolds. And over each of these manifolds we have a line bundle. All the same also. Nothing really changes. So what is changing is a metric on the bundle. So I throw in now a metric on this L underlined on this bundle. I have a metric on the total space here. And this may change from fiber to fiber. So I can think of a metric on L underlined as a curve of metrics. So for each T here I get a metric on the fixed manifold and the fixed line bundle. So the only thing that changes is the metric on the line bundle. So I have a curve of metrics on a fixed line bundle over a fixed manifold. That's what I have. What? The metric. The metric, yeah. Semi positive. So I mean that the IDD bar of it should be greater than or equal to zero. So could you take a metric? Here, like this? Yeah. So not really because it has to be a metric on the line bundle. And then if I just take a metric on U here, it would be constant on each fiber. No, I mean pull back from the line. So I have, I could do that also. Yeah, take a metric. But then it would be a constant metric. Then everything would be constant. And then it's not, that's not interesting then because then nothing will change at all from fiber to fiber. It's still allowed. It's still allowed. Okay. And then I can take IDD bar of those things. IDD bar of this. If it's strictly positive on the fibers here, that will be a scalar metric on the fiber. So I have a curve of scalar metrics on the fixed manifold X. And now we specialize further. So we choose a very particular bundle. So we take it to be the dual of the canonical bundle of the fiber. So the dual of the bundle of N0 forms. So I assume that that is positive now, strictly positive on the fibers. That means that the fiber X here is a phano manifold. This is a phano manifold. So I have a family indexed by T in U here of a scalar metrics on a fixed phano manifold. And now I was looking at the fibers of this. You had my vector bundle. They were ET. And they were equal to the space of holomorphic sections over X of K of X plus L, which is now minus K of X. So it looks very trivial. And indeed it is very trivial. This is a trivial bundle. So I just look at the fibers here of the holomorphic functions on X now, in this case. And they are not so many. They are only the constants. So this is a trivial bundle. And the only holomorphic sections are the constants. But still I can do that. So that's really the simplest case of the theorem, if you want. And then you get this theorem. If you now assume that the variation here is semi-positive. So this means that if the IDD bar here should be strictly, it should be greater than or equal to zero on the total space. So it should also be pluralist of our monocle with respect to T. And now I can look at this expression here. A metric on the anti-canonical bundle can be identified with a volume form on the manifold. So I write this somewhat elliptical here so that this stands for the associated volume form. Or alternatively, you could think of it as integral. You take one. The bundle was trivial. It has only constant sections. So one, for instance, you take one squared e to the minus five. And you integrate over x. So that's what I mean by that. And then the conclusion is that if I form this integral here, I get the phi tilde of T. And the conclusion is that this is a subharmonic function. So in particular, if it does not depend on the imaginary part of T, it's convex. So this looks a lot like preco-pasteorem. I mean, if you change x to r and or c and here, it looks exactly like preco-pasteorem. So it's a preco-pasteorem for Fano manifold, if you want. OK. Still no application. We just know this. And I should say that this particular case of the theorem is fairly easy to prove directly. But that's not so important. So there is a converse to this. You can ask this function phi. Yeah, so let's assume here now that we are in the last case here that the metric only depends on the real part of T and not on the imaginary part of T. So then it's convex. And I'll say that it is not really strictly convex, but it's actually linear. So it's linear. What can we get then? Well, the conclusion is that if it is linear, and moreover, you assume that phi T is bounded, just some technical condition, but this is actually quite delicate. So we assume that it is bounded. Then there is a... The conclusion is that then the situation must indeed be trivial. So then there is a holomorphic vector field on manifold X, such that all those calormetrics, they are really the same after applying the flow of the holomorphic vector field. So nothing really changes. Yeah? Yes, yes, yes. So in the previous theorem, it's enough to assume that the anti-canonical is semi-positive. But in this... Well, in this theorem, I want it to be positive. So I really want... I think I do... Maybe I need to check. Maybe it could be. But let's assume it is strictly positive along the fibers. But the variation in this direction does not need to be strictly positive. It can just be semi-positive. It is indeed not strictly positive, but actually only linear in this direction of the integrals. Then as a matter of fact, there is a vector field on X. I can compute the flow. There is a flow associated with that vector field. I apply it to the forms. And applying this, we see all the calorforms would be the same. So then the situation is trivial. So in case you assume that everything here is smooth, et cetera, this follows more or less immediately from the formulas. But if you don't assume that you have any regularity, this is dirty somehow. But still it holds. The only assumption you need on phi is that it's locally bounded. So no singularities. No serious singularities on phi. Okay, so why is this useful? Well, one can use it to prove generalizations of the Bandou-Mabucchi's uniqueness theorem for Keller-Einstein metrics. So when you have Keller-Einstein metrics on a Fano manifold, there is this classical theorem by Bandou-Mabucchi that says that if you have two such metrics, there is a holomorphic vector field on the manifold with the flow, such that if you apply the flow to that, you will transport one metric to the other. So the Keller-Einstein metrics are unique up to the action of holomorphic automorphisms. That's the Bandou-Mabucchi theorem. You can get the proof for that now, and also even in a more general situation as I haven't come to in a while. So this was first observed by Robert Biermann, and then it was generalized to a singular situation by all those people there. And then it was actually used in the work by Chen Donaldson soon on the existence of Keller-Einstein metrics because intimately related to this is the so-called Matsushima theorem about the redactivity of the automorphism group. And this allows for a singular version of the Matsushima theorem. But let's forget about that. I just mentioned this as applications of the proof. So let me sketch very rapidly how this goes, then it's a little bit embarrassing because there are some real experts on this matter here, but maybe I should... I'll sketch it rapidly. So we have to introduce first in order to understand if we have to introduce the Montchampère energy of a metric, and it's defined in this way. So the classical definition is that you define the E of phi. E is the energy of phi. It's defined up to a constant by saying how it varies when phi varies. So the derivative of E of phi with respect to t should be this thing. And actually I probably should have a plus sign here if I want to have it consistent with what I get later. Let's be generous about that. So you define this function of the Montchampère by saying what its derivative is. You can write down an explicit formula also. And then you cook up the so-called ding functional after ding that discovered it, which here would be minus L phi plus the Montchampère energy. And L phi is precisely this logarithmic integral that I had before. Now, the relevance of this for Kell-Einstein matrix is that Kell-Einstein matrix are the critical points of the ding functional. If you take the derivative of this thing with respect to... If phi depends on t and you differentiate with respect to t, say the derivative is phi. If psi, you get the derivative of the first thing here will be integral of psi here with a minus sign. Ah! Yeah, too many minus signs. Maybe I was right as it's written here. So you get this and this from differentiating the logarithm. And then we define the Montchampère energy so that the last thing holds here also. If I get the signs right now, you will see that this is zero for all the choices of psi if and only if e to the minus phi is equal to omega phi raised to the power n modulo constant. And that's exactly the Kell-Einstein equation. It means that you take the logarithm of this and you take dd bar, you get the Ricci and then you get omega phi on the other side. So that must be equal. So this is the way of writing the Kell-Einstein equation. So Kell-Einstein matrix are critical points of this function. And then there is a notion of geodesic. I'll skip over what that means. And it follows from the theorem that the ding functional is convex along geodesics. Because, yeah. And then there is a theorem by Chen. In this case it's easier because you don't need a full force of Chen's theorem, but never mind. Any two points in any such matrix can be connected with geodesic. So the ding functional, if you take a geodesic that connects two critical points, you get a curve there. The ding functional is convex along it. The two end points, the derivative is zero. So the only possibility for this is that the ding functional is actually linear along the geodesic. So here is a Kell-Einstein metric. It's not a Kell-Einstein metric. The ding functional is a geodesic between them. And the ding functional, d of t along the geodesic will be like this. But the derivative at the two end points is zero. So it must actually, the ding functional must actually be a constant. Ding functional must be constant if it's convex. And therefore the L functional must be linear. And then the converse theorem says that we have all those automorphisms that relate to metrics. Because the converse said that if the L functional is linear along some curve, then the curve must arise from holomorphic automorphisms. So this way you get the uniqueness of Kell-Einstein metrics of automorphism. And the point, one point with this proof is that it applies with less regularity. You can study, for instance, a twisted Kell-Einstein equation where this function here can have singularities. So it includes metrics with conical singularities, for instance. And it also applies to Kell-Erich's solitons. Something which Janir asked me to study. But actually it works perfectly well for solitons also. So that's the way you get the theorem by Tian and Shu that way. So you just change a little bit. So in the case of twisted Kell-Einstein metrics, you look at this function instead. You take e to the minus five, you add the function psi there. And in the second case, you don't change the L functional, but you change the energy a little bit to so-called Shu energy, which I will not go through here. But you can prove uniqueness for solitons and you can prove uniqueness for up to automorphisms for twisted Kell-Einstein metrics also, using this Bruno-Minkowski or Placo-Passo argument here. Okay, is it more or less okay? Yeah, so I go on. So this was a trivial vibration. Now I look at the non-trivial vibrations. Yeah. No, no, no. So that's pretty nice. In the real Precopa case, there is a counterpart of this theorem here for the real Precopa, which says that if you have... So in the real Precopa, you have a convex function of t and x. And if you have, so to speak, equality in the Precopa inequality, then this function has to have a very special form. It has to look like you take one function of x and then you add t times some vector. This is equality in the real case. You take one function that depends only on x and you use the t variable to translate it. That's the only possibility to have equality in Precopa's inequality in the real setting. So this is formally very similar to them. Here you have the flow of a constant vector field. In the other situation, you have a flow of a holomorphic vector field. So somehow all the time the translation is this. You change constants to holomorphic things and then you get the corresponding theorem in the complex setting. That's it. Okay. And now we look briefly at a non-trivial vibration. And we have the same situation. We have a line bundle over the total space, et cetera. And now I will assume, definitely, that i d d borrow phi is strictly positive over the fiber. Because otherwise it will not work. Otherwise it will not even be true. The conclusion I want to write. First, it will not make sense. And second, it will not be true. So let's say now that this b here is a part of c. This is an open set of c. So the base is one dimension. I can think of it as a coordinate patch. It's easier to state them. But now we have a lot of machinery here. So here we have a vector field on the base here, which is dt, which is just d dt. A holomorphic vector field on the base. Now a lift of that field is a field up here. So I have this vector field here. It points like this. Now I can lift it. It's going to look like this. And that it's a lift means that if I apply the p to that, I will get d dt there. So the differential of p acting on the vector field up there should be equal to dt. That's a lift of a vector field, not holomorphic. So it cannot be chosen holomorphic in general. So it's just a real vector field. So therefore you compute the d bar of it to measure how far it is from being holomorphic. And then you get something that I call kappa. And that is codera Spencer form. So if you take the co homology class of that, it will be the codera Spencer co homology class of the vibration. So if you can choose it holomorph, somehow the vibration is trivial. Infinitesimally trivial means that you can choose to be in a holomorphic way. But now, so the kappa will be one zero form, but the values in vertical vector fields, vector valued zero one form. And by Schumacher, building on the work of by Sioux, there is a canonical lift of this vector field up here. I will not go through how that is obtained, but there is a specific way that you can choose to lift here, depending on the metric data. So but I will not go through that, but there is such a canonical lift. And now again, we look at the same vector bundle, the vector bundle of holomorphic sections over the fiber, and we have the norm defined as before. And now I can compute the curvature of this. So the theorem says that the curvature is positive, but now I can compute the curvature. I can write down an explicit formula. So first I define a function c of phi, which is equal to this. So here I take d bar of phi on the total space and I raise it to the power n plus one. And here I take it, raise it to power n and I wedge it with this. And the quotient between this form and that form is called c of phi. It's going to be some non-negative function. And then we have this formula here. The curvature is explicitly given by this operator. So I take... Yeah, so I introduce u1 is the... is this the Cudela-Spenser form. It operates naturally on the form u. I get u1 and I take the... the telemetric defines the Laplace operator. I take this thing here, the Laplace operator, apply it to u1 and et cetera, et cetera. So that's what the theorem says. It's an explicit formula for the curvature. So why is that useful? Well, this is useful first. So this is the formula. So if we assume now that i d d bar of phi is greater than or equal to zero on the total space, then this will be non-negative. So assume now that the curvature is zero. Then both of the terms will have to be zero because both of the terms are non-negative. So c phi must be zero, and u1 must be zero, which means that kappa must be zero. So the conclusion of this is that the canonical lift is holomorphic. So the vibration is trivial. We can use this holomorphic vector field to trivialize. You change the vibration by the flow of the field. You make it into a trivial vibration, and you can also do it in such a way that all the metrics and everything fall over long. So you really get the situation that you asked about. It's really this trivial situation in disguise. It's a trivial vibration and the line bundle and the metric they all are constant on the fiber after a holomorphic change like this. Here. Yes, if this vector bundle has zero curvature, then both... I have a line bundle L over this space here, and then I push forward the line bundle, or rather the n forms with values in the line bundle. I push it forward, I get the vector bundle down here. If that vector bundle has zero curvature, then everything has to be trivial. Everything has to be trivial. So I think this is related to Torelli-type theorems. But let's come to this later. This is a little bit more general so far here. So this is about any positive L here. Sorry? Yeah, so that's what I'm coming to next, actually. So in the special case now, so now we assume that all of the fibers have positive canonical bundle now. So for instance, if they are one-dimensional, they are Riemann surfaces, positive canonical bundle means that the genus is greater than two. Then I can use that as my L. The previous theorem worked for any L. But now I can apply it for the line bundle, the relative canonical bundle, which is a line bundle on the total space that restricts the canonical on each fiber. I can apply the theorem to that. And in that case, there is an argument by Schumacher. I will do that quickly also, which the conclusion is that in that case, if I choose the phi to be the Kellr-Einstein potential on each fiber, then the C phi will be positive. That's what I want to say here. So IDD bar of phi will be positive, in that case, automatically. So I don't choose, I choose the curve like this. I choose the curve for each xt. I take the Kellr-Einstein potential. Here is another Kellr-Einstein potential. And I, so to speak, I put them together, and then it will be automatic that the variation with respect to t will also be subharmonic, so it fits into the theorem. And this is due to this theorem of Schumacher. That's a non-trivial fact. So we can apply the theorem, and we can get that this space here has positive curvature. So now I put in... So here I have like n forms with values in the canonical bundle. If you think of the case when n is equal to 1 so that the fibers are Riemann surfaces, these are the quadratic differentials. So then it's getting closer to this business, right? And actually, let me spell this out here. So the Cordera Spencer formed, they lived in this space, and this space is isomorphic, too. If you, it's isomorphic to this, n minus 1, 1 forms with values in the anti-canonical bundle. You take a tensor product with that, and you will get that this space is isomorphic to that. This is not entirely trivial, but it's not the deep fact, it's just the computation that this space is actually equivalent to that. And then if you take duals, you will get this space. Here I have hn minus 1, 1, and here I will get h1n minus 1. And then I will say that the space where the Cordera Spencer form lives, that's the tangent space of the Teichmühler space. And here we are talking about arbitrary dimension so far, arbitrary dimension. So I can look at the Teichmühler space of canonically polarized things in some diffeomorphic class. Yes, so far. So after this point, we have arbitrary dimension. But now the next line I take n equal to 1. So now we have Riemann surfaces. We have families of Riemann surfaces. And then you get that this space h1n minus 1 becomes h1 0, which is hn 0. So for n equal to 1, it becomes this space that we knew something about. We don't know anything. Those spaces, my theorem does not apply to those, but it does apply to this one. So it's okay when the fiber dimension is 1. And then you can think of this, it's the bundle of quadratic differentials. You can think of that as the cotangent bundle of Teichmühler space. And the fact that this has positive curvature at the Teichmühler space itself has negative curvature, which is all first theorem now. And if you look at the explicit formula that I had, it's really exactly, you can translate the formula and you get Scott's formula in this case, when n is equal to 1, and you choose this line bundle on those metrics, et cetera. But you still have to do some computations because if you look at it like this, Richard said that it looks a lot like Scott's formula, but it looks a lot like Scott's formula if you know a lot about these things because formally, you still have to do some computations in order to translate this, too. But it's really the same formula in that case. So if you want to, you can view it as a generalization of what he had. Yeah, to end with something very quick again, which is related a little bit to what you said before about plurigener, et cetera. So we still have a non-trivial vibration. But I don't assume it's a vibration now. So I assume it's just a surjective map. So I don't assume that it's a submersion or anything. So it means that I have things like this. I have a total space here. I have a map P that goes down to some base, B, it can be higher dimensional now. And mostly, it will be a submersion. So the fibers will look like this. But at some points, there will be singularities if it's not a submersion that may have singularities. So what can we say in that case then? Well, I'm maybe a little bit ahead of myself. We have a notion of Bergmann kernel for such things like we had in domains in CN. But the Bergmann kernel will now, if you see, it will depend on choice of trivialization. And it will actually not be a function, but it will be a metric on this bundle. The Bergmann kernel, in this case, will be a metric on this bundle, the twisted relative canonical bundle. And so this is defined, the way I did it is defined when you have a smooth vibration, the Bergmann kernel metric. And then in joint work with Mihai Porn, we extended that to only surreactive maps. So if it's just surreactive, then it's mostly outside the sub-variety, it's a submersion. So we can take the Bergmann kernel metric. But now the theorem says that we can extend it to a singular metric on the total space. So we get a singular metric of positive, no, of semi-positive curvature on the total space. So the Bergmann kernel metric defines a singular metric of non-negative curvature on this bundle. Bye. I started with a phi, which was a metric on L, but now I get another metric. Yes, which is not good. I should have called it something else. Ah, sorry, sorry, sorry. Okay, so I was right. So I allow also, I actually allow the original phi to be singular. So I assume that there is one fiber that has finite L2 norm on, there is one fiber on which there is a section that has finite L2 norm with respect to this original phi. Then this metric, the Bergmann kernel metric will be, will not be identically zero. And it will define a genuine singular metric of non-negative curvature. That's how it was. Yes, yes, yes, yes. Yeah, something like that, yes, yeah. Yeah, I think so, yes, exactly. Well, I don't claim that it should be, that there should be some L2 condition on every fiber. I say that there should be some fiber over which there is a section that has finite L2 norm. No, no, phi is defined, phi is defined over the total space. Phi is defined over the total space. Yes, so I start with a metric, I start with phi, which is a metric on, which is a metric on L. And then I get the Bergmann kernel, I get the phi Bergmann kernel, I get the new metric, which is not a metric on L, it's a metric on the canonical, twisted canonical plus L. So I start with a positive metric on a bundle and I get a new positive metric on a new bundle, on the twisted bundle. I need to have a metric on L to begin with, yes? I think so, yes, yes, yes. Yes, I think in that case so. And it's also, you can apply it to, say, elliptic vibrations on a codera. So you can have multiple fibers, for instance. And then you will have also minus infinities or minus infinities on the single fibers. Well, but the point is somehow that if the morale of the theorem is that if L is positive, then this is positive. So the relative canonical always gives something more positive. So somehow, one consequence is that you can start with, say, the canonical here and then you get twice the canonical here, then you can feed in this one here, you get three times the canonical and you get positive metrics on all multiples of relative canonical bundle this way. For instance. What? Yeah, then it doesn't work. So this theorem is mostly interesting when you have some fibers of general type or something like that, yes? Well, unless L is very positive. L is very positive. That there should be a section on some. Yeah, a section. You have to assume that. Because otherwise, I mean, the Bergmann kernel is defined in terms of supremum no sections, et cetera, or something. If there are no sections, there is no Bergmann kernel. Yeah. Yes, yes. But the remarkable thing is somehow that you need a condition only on one fiber to guarantee this. And then you will get something for the whole thing. Yes, so for instance, you can use this somehow or there are variants of this you can use to construct because the invariance of plurian is very much. So maybe I think my time is running out here. But the invariance of plurian is very much tied to the construction of metrics. You want metrics on M times the relative canonical bundle. This line, M is some large, well, maybe not even large, but you want to have metrics of non-negative curvature on this. And this somehow is one way to construct such metrics on this bundle. And once you have positive metrics there, you can apply the Savataki-Gorshi theorem to extend things from a fiber to other fibers. So I just end here by saying that this was extended by Pohn and Takayama who showed that even in this case where you just have a subjective map here, you can take, you can still take the direct image of this thing here and you get a sheaf in general and not a vector bundle, you get a sheaf. They defined metrics on, it's a coherent sheaf. You define metrics on coherent sheaf and they get such metrics with positive curvature or non-negative curvature. And this somehow, this extends, but this gives a metric approach to a theory of a feedback on weak positivity for vector bundles. But that will go too far and I know too little about it, but feedback has a notion of weak positivity for such direct image bundles, and this gives a metric version of that. Because algebraic geometries, they don't speak about positivity in the terms of some curvature tensor being positive. They say that it means that there are sections of something or the other. But here you get genuine such thing. And then finally they applied this to the Itaka conjecture, but I don't think, so this was applied by, this kind of thing was applied by Junyang Xiao and Mihar Pan to prove a special case of the Itaka conjecture for codera dimensions assuming that the base then was the torus. Then they could use those metrics to prove it, but I will not, I don't have time to go through that, so I'll stop here. Thank you.