 And thanks very much for the invitation to speak here. So the topic of this lecture is complex Brunminkowski theory. That's a term that I have invented myself, so if you don't know it, that's easily explained. And I'm going to explain what it is. And actually I'm going to give two lectures, one now and one on Friday. And now the thing now, the lecture today will be basically about stating some theorems. And then in the next lecture I will try to show applications of those theorems to geometrical problems. So what is complex Brunminkowski theory? I think I'll move over here. Well, there are theorems about the curvature of certain holomorphic vector bundles. And I will argue that those theorems are formally analogous to the classical theorems of Brunminkowski and also Precopa that I will describe soon first. And they also imply those theorems. So they are sort of, in a sense, they are stronger than the theorem from convex geometry, but they are in a complex setting. And they are really from complex analysis. And they have applications I claim. If you hang around for the next lecture you will see some of them. You will also see a little bit of it today maybe. But they have applications in complex analysis in algebraic geometry and in Kehler geometry. So what are they? In order to explain that I will start to give a review of the classical Brunminkowski inequality. So the real Brunminkowski, the first formulation of the theorem is the following. I say that I have two convex bodies. Oh, they are called A0 and A1 here. I should change here then. This is A0 and this one is A1. There are two different convex bodies. And then you look at the Minkowski sum of those two things. That's the following thing. So you just take A0 plus A1. That's the set of all little A0s plus little A1 such that the first one lies in capital A0 and the second one lies in capital A1. That's called the Minkowski sum of those. And they are convex bodies in Rn. So I drawn them over here. So here is like A0 and here is A1 that I think of here as a ball but it could be any convex body. And then when I add A1 to A0 I will get something A0 union all these little balls around here. And so I will get something slightly in this case slightly larger than A1. And the question is how much larger? Well, that's exactly the content of the theorem, the Brun Minkowski theorem. It was first given by Brun in two dimensions I think and then the general version by Minkowski in 1896. And it says that you have a sort of triangle inequality for those things. But it goes the wrong way. So capital, so the volume of the Minkowski sum raised to the power one over the dimension is greater than this sum of those things. So they satisfy a sort of reverse triangle inequality. And you can formulate it a little bit more generally like this. That if you take convex combination of them in the same way as you form the sum there, you take the volume of the result and you get a concave function, concave function. So it says that the Minkowski sum is pretty big, the volume of it is pretty big. And in this case here, when one of them is the ball, it sort of says that when you do this, when you take A0 plus A1, if this is a ball with radius epsilon, the sum here will be like all the points whose distance to A0 is smaller than epsilon. And you get an estimate for that. And that's related, then inequality is related to the surface area of the boundary of this thing. Right. So it implies isoparametric inequality for instance for convex bodies, but it is much stronger than that. Okay. And now you can also state it in a different way here. So what's this different way? There is an equivalent formulation. It looks like this. Instead of taking convex bodies and taking the sum, we look at one convex body in a higher dimension. So this is supposed, as I draw it here, this is going to be, this you can think of as R, and this you think of as Rn. And then you look at the slices of those things. So you take a point here, T, here in R, and you look at the slice about that. So that's the subset, convex subset of Rn. And the conclusion is the same, that the volume of this slice here is a concave function. You take it, raise it to the power 1 over N. So this is actually, you can go from one to the other fairly easily. So they are equivalent in this mathematical sloppy way of defining things. They are equivalent. It's easy to prove one from the other. But I prefer this formulation as you will see soon. And you can even make it a little bit weaker, which I will do by saying that this implies that the logarithm of the volume is concave. That's an immediate consequence, but as a matter of fact, it's also equivalent, this thing here. Because you can prove it with the stronger version with the power here from the log version by using the homogeneity of the begmeasure. But when we soon going to look at other forms of measures that are not the begmeasure, then it's really a difference and then the log version is better. Sometimes this is called the multiplicative version of the Bloeming-Cosketeer. Okay, so they are equivalent. Now, they are philosophically different, I will argue, because say that you want to generalize the theorem like I wanted to do. Then if you look at the first formulation, you would look for a situation where you have a notion of addition or something like that. You have some sort of group structure. You could look, instead of looking at points in order, you could look at lattice points or more general groups or something like that. And you could try to prove Bloeming-Cosketeer in that setting. So that has been done. I don't know so much about it, but that's one line of research of generalization, so Bloeming-Cosketeer. But if you look at the other formulation, you don't need any addition. There is no addition mentioned. You just start with something which is convex, and then you look at the slices of that convex thing and look at the volume of those. All you need to know in that case is a notion of convexity. So that's what I will look at. In our case here, in the complex setting, I will change the usual notion of convexity, which is real convexity. I'll change it to notions in complex analysis, which is known as pseudo-convexity or maybe the Kähler condition in case of manifolds. So that's how we will do. That's the philosophical side of it. But first I will state the function version of Bloeming-Cosketeer, which is the following. It's called Preckopass theorem. So what is Preckopass theorem? Well, then you don't start with a convex body in Rn like I had here, but you start with a convex function. So I write it like this. It depends on T and X. So now it's a function Rn plus 1 on this big set there. And then instead of looking at the volumes, I look at integrals. So here I look at the integrals. I fix the T variable and I integrate with respect to X. And then I define a new function phi tilde like that. Minus the logarithm of those fiber integrals, slice integrals. And the conclusion, you can also write it like this, which is somehow more suggestive in a little second. And the conclusion is then that phi tilde is convex. But that's a convex function. That's called Preckopass theorem. And I should know, but I don't really know. But Preckopass was an applied mathematician. So this is something that comes from, maybe you know. But I have in back of my head something about water tanks and the flow of water. God knows what, but I mean from a mathematics point of view, it's certainly a natural sort of question. You start with convex body. You can ask, do you have something similar for convex functions? But I don't know. But I do know that, well, first, it's stated also. I mean, it's interesting, of course, in probability theory. It says that a measure that looks like this, a big measure multiplied by e to the minus phi, where phi is convex. That's called a log concave measure. And this, I take the fiber integral of such things here. That's called the marginal distribution of this measure. And so the jargon is that the marginals of log concave things are still log concave. So it certainly makes sense in that formulation. And it does imply the Brunminkowski theorem, because you apply this theorem here to the function phi. In the Brunminkowski case, you have a convex body. You define a function, which is zero in here, infinity outside. So it's zero in here and infinity out here. And then you look at those integrals. Then you have e to the minus zero, so you get one. You integrate, those integrals will just be the volume of the slices. Nothing outside, because you get e to the minus infinity. And so the conclusion is precisely the same. So it's a stronger version than Brunminkowski. Okay. So one important thing to say, because first when you just look at the theorem, you say that this must be a consequence. Well, if you don't know the business, if you've never seen it before, you would say that this should follow from Helders inequality. You just apply the convexity with respect to t or something like that. But no, that's not the case. If you try to prove it with Helders inequality, you will get a similar inequality that goes in some sense in the opposite direction. So if you do Helders, you could prove the same thing when you take away the minus sign there and the minus sign there. So that's a simple theorem. But the point with this theorem is that you have the minus signs. And let me see. And for this, you really need to know that phi is convex with respect to all of the variables. So that's important, not just the t variable. And I'm going to sketch one proof of this, because I think it has a lot to do with what I'm going to say in the complex version after. And this is via the so-called Braskamp Lieb inequality. So that was a proof that was discovered by Braskamp and Lieb in the 1970s or so. I'm going to write it down now. And I'm going to write it down in one real variable, which is really the main case. It's a simplification, but if you know it in one variable, then you actually know it in any number of variables. So this is not a big restriction. And so you start with a convex function, no t now. It's just a convex function on R. And you look at, so to speak, the L2 space of functions u, such that are square integrable with respect to this weighted measure. And you take one such function and you assume that its mean value is equal to zero. Integral of u is equal to zero against this measure. Then you have this inequality. You can estimate the u squared, the L2 norm of u, with respect to this measure, by the L2 norm, some L2 norm of the derivative of u. That's the Braskamp Lieb inequality. And I think it is like something like 1972 or 1973 or something like that. So it's a Poincaré inequality. Poincaré inequality means that you estimate the function with its derivative. And then you need some sort of condition so that you rule out the constant. And that is this condition here in this case. And it is also the key word here is that it is actually a version of Hermander's L2 estimate for the d bar equation. But now it's not the equation d bar u equal to f. Now it's the equation u prime of x equal to f on the real line. So if you think of, you want to study this differential equation, which is probably the simplest differential equation in the world. And you decide to use all these L2 techniques, etc. To study this, then you might arrive at the theorem here that Braskamp Lieb inequality. I think this is sort of interesting and nice in a way that you can really do something. You can find something simple. You can find something interesting about real simple things. So this says that the equation, this is the real variable version of Hermander's theorem. You can solve the equation u prime x equal to f with such an estimate here. You put an f there instead of u prime. So that's it. Okay, once you have this. Now we, yeah, yeah, yeah. So I'm going to sketch the proof of Precopa from Braskamp Lieb. So you want to prove that this function phi tilde that I defined, that it is convex. And so the natural way to try to do it is just to compute the second derivative of phi tilde. And after some computation you arrive at this formula. Phi dot means the derivative with respect to t of phi. And phi double dot is the second derivative with respect to t. And phi dot with a hat is a constant. It's the average of phi dot with respect to the measure. And then you get this. You want to prove that this is positive. Now this function here, phi dot minus the average of phi dot, is a perfectly legitimate u. It has integral zero now. So it qualifies as a u in the Braskamp Lieb inequality. We can choose u in the Braskamp Lieb inequality as this. And then we find, so we replace this now from the Braskamp Lieb, replace it by something bigger. And we get phi dot, and the prime means the derivative with respect to x. And then just plug it in there. And now this integrand here that you integrate is essentially the determinant of the Hessian of phi. Determinant of the Hessian. If you put them on a common denominator you get second derivative with respect to t multiplied by second derivative with respect to x. And then minus the mixed derivative squared. So it's a determinant. And since the function is convex with respect to all of the variables, the Hessian is positively definite. So the determinant is certainly positive. And therefore you get the function of phi till the second derivative is positive. So it's convex. So that's it. So that's a neat prove. Once you have proved the Braskamp Lieb inequality. And Braskamp Lieb inequality really was discovered after the Hermander L2 estimate for d bar. That's another interesting thing. Maybe I'm discussing too much, but if you look at, if you are, if you're a complex analyst, you're interested in Hermander's L2 estimates for d bar. It's a very important theorem in that business. And it's a theorem that has had a sort of a strange career because it's been generalized from the most difficult situation to simpler and simpler situation. It's the opposite of the, of the usual road. The first avatar of Hermander's L2 estimate was the Kodera vanishing theorem, which is about compact manifors, etc. Then you got to domains in, in CN and then you got to finally the real case here in Braskamp Lieb. So sort of the opposite direction. Okay. Now we go to the complex case. So we saw that Preko van Brominkowski follows on Braskamp Lieb inequality and that's the real version of Hermander. So you can ask them what theorems you would get in the complex setting by using Hermander instead of Braskamp Lieb. That seems to be a reasonable question. It's not exactly how I started, but in retrospect, this is the most natural way of thinking of things, I think. Okay. So we just say hi here for those who don't work with complex analysis in general. I'm going to talk about plurisabharmonic functions and they are subharmonic. They are functions in CN now that are subharmonic on each complex line. So here is CN. If I have a function such that when I take the restriction to any line in CN, it's subharmonic on the real line, then it's called plurisabharmonic. And that's the same thing as saying that the Hessian, this complex Hessian is positively semi-definite. It looks very much like the condition for convexity in real analysis. You can also say that it means that this differential form is positive. And the definition of positivity of forms is such that it means precisely this. That is the thing there. So IDD bar of phi is a positive form. And then we need a counterpart of convex bodies and they are the pseudo convex domains in case you're looking at domains in CN. If in case you're looking at manifolds, they are maybe Stein manifolds. And such manifolds, such domains, they are defined by the condition that there exists a plurisabharmonic extortion function. So that should exist a plurisabharmonic function inside of here if it's pseudo convex that goes to infinity when you approach the boundary. Yeah, so that's the counterpart of convex sets of bodies. And finally for compact manifolds, they can never be Stein or such a thing. But then we have the Kähler condition, they are Kähler manifolds. And that means that they have a Kähler metric. And that's a Hermitian metric which locally can be given as the Hessian of a plurisabharmonic function. So the fundamental form of the metric, something like this, should be IDD bar of phi. Locally where phi is a plurisabharmonic function, then it's a strictly plurisabharmonic function, then it's a Kähler manifold. So that's a little bit surprising that this is a convexity assumption, but that's a little bit the conclusion of some of the things I'm going to say. So now you can look at the naive generalization of Precopa. You would take a plurisabharmonic function instead of convex function on Cn plus 1. And you construct the same function phi tilde here. And you ask, is it plurisabharmonic in T, or subharmonic in this case then? Instead of convex, that would be the natural generalization. And the answer is no, it is not always subharmonic this thing. And that was a simple example by Chiselman, which is the following example here. You take n equal to 1, so C lies in C. This one lies in C, and this one also lies in C. And you look at this function here, which doesn't look very plurisabharmonic, but it is. If you can rewrite it in this form, then this is plurisabharmonic because it's convex. And this is the real part of all the morphic function, so it's even pluriharmonic. And then if you compute phi tilde, you use the first formulae, they integrate with respect to C. It's very easy to compute the integral because you just integrate the Gaussian. This one goes out and you find that phi tilde is equal to this, which is not then subharmonic. It's rather the opposite. It's superharmonic. So this doesn't work. So that seems like bad news then, but it turns out that we have to look at things in a different way. So we have to change now, instead of looking at the volumes of sets, we look at, we think of volumes as the L2 norms of something. The volume of the set is the L2 norm, squared L2 norm of the function 1. And those pre-copy integrals, they are also weighted L2 norms of the function 1. And in the complex setting, we use L2 norms of holomorphic functions instead of constants. So that's somehow how one gets a theorem that does hold instead of the Navian realization. So we're looking at such things, either L2 norms just plain L2 norms over a domain or weighted L2 norms over some domain. So that's the counterpart of the volume then. We have to plug in a holomorphic function, which is sort of natural because it means that we are not only replacing convexity by subharmonicity, but we are replacing the kernel of D, which are the constant functions, by the kernel of D bar, which are the holomorphic functions. So it's really quite similar and parallel, the whole thing. Okay, so now I'm going to give the first version of the complex Brunminkowski theorem, it's up there. And actually I'm starting with the most complicated version. So this is the least elementary, but it's really in a way still the simplest thing. So we let beautiful X be a complex manifold, five word over base B, which is also a complex manifold. So I can just use the same figure here. So this is now the manifold base here is B. And this is another manifold X, beautiful X. And I have a map from beautiful X to the base, which is called P. And I take a point T here and I get the fiber XT, which is equal to P inverse of T. That's the fiber there. So that looks pretty much like this was a convex body. And I took a linear projection, but now I take a holomorphic map like this. So let me see what I wrote here. So we have a surjective holomorphic map. And yeah, XT are the fibers of the map. And before we have the convex body. So examples of this is actually, let's look at the second example first. The second example is a trivial vibration. So you take beautiful X is just the product of some fixed manifold. And this one, so you take a fixed manifold here. This is X and you take the product with B and you get some sort of cylinder like this. Beautiful X. And you have B here on the projection map. So that's the simplest case. Otherwise, in the more general case here, you can think of this as a family of complex manifolds that are indexed by the T in the base here. And such things occur in algebraic geometry. For instance, when you have algebraic manifolds that are defined by the polynomials that define them in the predictive space. So that's a very common object of study. And I assume that P is smooth. So in this jargon it means that it's a submersion. First we assume that it's not really necessary in the end, but to state the theorem in a nice way we assume that it is smooth. Which means that it's certainly smooth because it's holomorphic. But it's smooth in the sense that the differential is surjective in every point. Yes. And then the fibers will really be manifolds. And then we assume it's proper so that the fibers are compact. So we get a family now of compact manifolds which may be a constant family. Still interesting case when it's a constant family like this. And then I assume the convexity assumption which is that the total space here is scalar. Total space is scalar. And now I want to have my e to the minus phi and in the case of compact manifolds it has to be, we have to involve the line bundle. So L is a holomorphic line bundle over a beautiful X. And we have a positively curved metric on the line bundle which I write as e to the minus phi. So this means that locally the metric is given as h e to the minus phi where phi is plural subharmonic function just as before. I suppose here that there are some people that work with complex manifolds and some people that don't. And those that don't work with complex manifold can just listen to this on the intuitive way and I get back to domains and nonline bundles a little bit later. And then we look at holomorphic sections instead of holomorphic functions over here I look at holomorphic sections of the line bundle and we look at their L2 norms. But how do you define the L2 norm of a section of a line bundle? I want to integrate. I have a section of a line bundle and I want to define integral over Xt of u squared e to the minus phi. I would like to define that and it seems that one would want the measure there to integrate against. But we have no such measure available here no God given measure to put in there. How do we handle this? What do we put here as a measure? It would be very important what one puts unless one circumvents the whole problem which is what I'm going to do. I circumvent it in the following way that we look instead of holomorphic n forms so we think of u as a holomorphic n form with value in L. So u is a n zero form something that looks like u locally it looks like u is equal to g times dc1 dcn like this. And then I can easily define the L2 norm because I just say that the L2 norm is I take the wedge product of this form with itself I get the volume form and e to the minus phi guarantees that this is a global expression and then I have a constant here which is there is plus or minus 1 it is there to make the whole thing positive. Oops, was that really? Ja, that was it. Okay, so now is the first theorem then. So the first theorem is this that if we have precisely this situation here we have a smooth proper vibration we have a Hermitren holomorphic line bundle over the total space where the metric is smooth and plural subharmonic so we have this it means the curvature of the line bundle is positive and assume that the total space is scalar and then we can find that we can look now instead of looking at just the family of volumes here I look at the family of vector spaces which are, which is there I write it as E t is H zero X t K X t plus L so what does that mean? It means precisely that I take a fiber X t I look at, this means that I look at the holomorphic N forms with values in L over the fiber that's what it means and then I have the L two norms for each fiber and then the conclusion is that this is actually a holomorphic vector bundle when this moves here and it has a metric which I have there and the conclusion is that this metric has a curvature which is positive the curvature of the metric is positive and then there is some addendum here that in complex geometry there are different notions of curvature of positivity there's a stronger one, Nakano positivity and even that is satisfied here but I'm not going to insist on that so that's the, that's the theorem okay so what does this have to do yeah but then you really need a magnifying glass or yes so the telemetric does not appear that's sort of interesting also that the telemetric makes no appearance here it's actually the same thing if you look at Hermandres L2 estimates if you want to do it on a complex manifold and you are dealing with N1 forms you have to assume the manifold is killer but there is no telemetric in the statement so it's really but you really have to assume it so it's a little bit like Alice in Wonderland with this cat that disappears and only the smile of the cat is left afterwards so we need the smile of the telemetric is here so I just want to say here that in the language of algebraic geometry this vector bundle is called the vector bundle that's associated to the direct image sheaf this means the direct image of this sheaf up there and so that's just words that will not play any role here but in case somebody is working with such things this says that the direct image bundle of a twisted relative canonical bundle is positive so a relative canonical bundle is a bundle on the total space that restricts to the canonical bundle the bundle of N forms on each fiber and you can define it like this but it's not important here I just want to say that Griffith proved this theorem when L is trivial so you don't have a twisted bundle a twisted canonical bundle you just look at the N forms themselves and then Griffith proved that this it was a part of his theory variation of hodg structures that this bundle is positive and this was emphasized by Fujita and Griffith's motivation was different he wanted to generalize the notion of a period map from Riemann surfaces to higher dimensions so he was interested in the N forms themselves in case of Riemann surfaces you have one forms period maps are defined by integrating one forms over curves so the topic of interest for Griffiths were precisely the one forms of the N forms in higher dimensions but for us we don't really start with N forms they are just something that has to be there in order to be able to define the norms so we come to the same situation almost but from different starting points but anyway the case of when L is trivial L equal to 0 if you want that is a theorem of Griffiths ok so what does this mean then to compare it to Precova well if you a metric on a vector bundle I call it h now there are many h's here it's not a holomorphic function anymore this is locally is given by a matrix h and the curvature of such a thing is a matrix value differential form which you can write like this so this is positive definite matrix you take the d of that and you take h inverse which exists and you take d bar of the results and then if you count like if you write like a beginner you say that this is equal to minus d d bar this is the d d bar of the logarithm of h because here is d h inverse d h would be d of the logarithm of h and then I have d bar and d in the wrong order so I get a minus sign there so intuitively it's something like d d bar minus d d bar of the logarithm of h and but of course logarithm of h does not exist this is just intuitive so it says anyway that minus log h is plural subharmonic but minus log h in the preq of a case it corresponds precisely to e to the minus phi tilde because it's somehow the norm is the fiber integral of the u and before we had in the preq of a case we had that e to the minus phi tilde was equal to integral 1 e to the minus phi tx dx and this depends on t so it's like formally similar that the norm 1 squared doesn't matter so much but just for philosophy should be 1 squared so this is in the preq of a case you think of that as l2 norms in the complex case you look at the whole norm which is then a matrix and the conclusion then looks pretty similar to preq so from this it does not follow that if you take any section of e you can take the norm of that with respect to our metric you can take the logarithm of the norm get something scalar here this is not necessarily plural subharmonic so this is not the conclusion that you can draw unless the rank of e happens to be 1 which is a very particular case but it in order to get the precise convexity statement if you want you should look at the dual instead so if you have if xi sub t it's a holomorphic section of the dual bundle then the dual bundle inherits a metric from the bundle itself and you can look at the log without the minus sign there and then this is plural subharmonic so that's the that's the more concrete statement that you get just knowing that such a gadget is positive in some sense is not so useful but the useful consequences are that all such objects all such functions are plural subharmonic that's it and actually that's equivalent if you know that all such functions are plural subharmonic as soon as you take a section of the dual bundle you get something plural subharmonic then the curvature is positive so that's an equivalent statement of the theorem ok so that was the most complicated generalization because it's about manifolds and line bundles and such things the more elementary version which is still a little bit more tricky it's the second version here so I look at non-proper vibration so now I just take I can still use the same figure here so now this is it's called beautiful D instead and this here is C N and this is C M so like this so I have a domain in the product space C N plus M and I look at the slices of those things so this is if I take a point T in here I get the slice DT like that and I get for each T I get a Bergman space now you don't need holomorphic forms you can just look at the holomorphic functions the slice this is supposed to be straight holomorphic functions on the slice that are square integral against a plural subharmonic function so phi is plural subharmonic and I look at those things so we get something similar for any T you get a Hilbert space which is now generally of infinite dimension it's not finite dimension anymore that's why this because if they are just bounded domains in C and they can have infinite dimensions but it doesn't matter you can still get a family of such Hilbert spaces a bundle of Hilbert spaces let me wait with that we get the bundle of Hilbert spaces but it is not really in the technical sense of vector bundle because it's not locally trivial so one has to be a little bit careful there we have to say that if I take another slice here there is no natural way so this is DT prime there is no natural way to identify the Hilbert space here with the Hilbert space there in general so it's just a bundle of Hilbert spaces but it is not a Hilbert bundle but still we can think of it as intuitively a vector bundle but we have to be a little bit generous with the definitions and now we take for any compactly supported measure so if we have a measure which is supported on here mu is supported there on a compact subset of the fiber we can define its norm it's not the usual norm of a measure you take the supremum of the integral of h with respect to the measure over DT where you take the supremum over all holomorphic functions of the Hilbert space of Hilbert norm less than one Hilbert norm less than one so you think of that mu defines an element in the dual space of ET and this is the norm in the dual space integrating with respect to a measure defines an element in the dual space ja and then we have this conclusion it's a little bit technical because I cannot just say that it has something just as positive curvature but we formulate it like this instead it's simpler let's see ja, so we have all this situation we have pseudo convex domain in cn plus 1, we have all the slices we have something pleurisobarmonic in there phi, and we have a family of measures for any fiber we have a measure so here is another compact set and another measure here lies there and we assume that those integrals if I take a function h which is holomorphic in the big space and I integrate it over the fiber with respect to this measure I get a function of t and I assume that this is always holomorphic if h is holomorphic so that takes a little bit of time to digest but intuitively that means that mu t define holomorphic sections of the dual bundle this is a condition that the map t to mu t is holomorphic for any t I get an element in the dual bundle and it depends holomorphically on t in this sense then the conclusion is that the logarithm of those norms here is pleurisobarmonic so that will take some time to get used to unfortunately we will not really need to do that but we look at, that's just what I wrote that it's a holomorphic section the dual bundle so let's look at a simple example of this theorem which is easier to digest so I first recall the definition of the Bergman kernel so if h is a Bergman space of square integrable functions the Bergman kernel is the integral kernel for the orthogonal projection so there are square integrable functions on some domain with respect to some measure you can look at and they are holomorphic but you can look at the space of all l2 functions and the holomorphic ones will in general be a closed subspace and you have an orthogonal projection from all functions to the holomorphic ones and that is given by an integral kernel and that kernel is the Bergman kernel this is there the most important object is the restriction of the Bergman kernel to the diagonal and then you can define it uh more easily by saying that it is the supremum of the value of u at c squared divided by the norm over all u in the that are holomorphic that are in the Hilbert space so if you want to with my definition before write it here you can say that you can say that k cc you can interpret this as the norm of a particular measure namely the Dirac measure at the point c squared that's the norm of this operating on holomorphic functions the norm of the Dirac measure the norm of the Dirac measure as a measure is one but here it just operates on holomorphic functions you compare it to the Hilbert norm of the holomorphic function then it is the Bergman kernel that's what it means what this means here so that plays a big role in complex analysis okay and now the theorem is this what is the theorem says that if you have this setting you take the Bergman kernel for each t you have a Bergman kernel and the conclusion is that this function here the logarithm of the Bergman kernel is subharmonic as a function of both t and c both t and c it's very classical in the theory of Bergman kernels that it's plural subharmonic with respect to c it's very elementary but important but here the theorem the content of the theorem says that it has some subharmonicity property as a function of t also that's the point of the theorem and this follows from the theorem because these Dirac measures here they qualify as the measures mu in the theorem so in this simplest case here you will get that the logarithm of the Bergman kernel is and this was proved before by Yamaguchi and Maitani Maitani and Yamaguchi maybe in the case of one complex variable and no weight function so that's their theorem in that case hmm and as a consequence we get this eh yeah before I do that I can just say that if we are in the real setting in Rn we can look at the Bergman kernel in the same way for the space of constants and if you think of it a little bit you will find that the Bergman kernel is actually nothing but it's a constant and it's one over the volume of the set so the Brunminkowski theorem says that the real variable Bergman kernel is log concave and here we have that the complex Bergman kernel is log Plurus Brunminkowski and therefore it looks very similar and then a special case of this is when you have some sort of symmetry so we assume that the sets and phi are symmetric under one dimension of circle action so you multiply all the variables by e to the i theta the domain should not change and the function should not change and we also assume that zero lies in all the fibers we have a zero section in here and then the conclusion is that the logarithm of those integrals is Plurus abharmonic and then it looks really a lot like Brickoba and it's just no holomorphic functions left anymore but just integrals of subharmonic so the complete Brickoba does not work but if you have this symmetry property then it does work and this is just because in that case when you have all this the Bergman kernel at the origin is precisely this so it's just a particular case of the theorem about the Bergman kernel and one can go one step further and we have the following theorem so we have more symmetry so we have a Plurus abharmonic function in our entire space and if you assume it does not depend on the imaginary part of c so it's just the function of t and x if x is the real part and then we define this phi tilde like this and then the conclusion is that this is Plurus abharmonic in t and this is really an honest generalization of Brickoba's theorem and therefore Bruminkowski then if phi is also also only depends on the real part of t then it's precisely the same thing as Brickoba's theorem because Plurus abharmonic function is convex if it does not depend on the imaginary part and from this follows for instance Schismann's minimum principle which has been used quite a lot in Maryland I think by Thomas and by Janir Schismann's minimum principle it says that the infimum over all x of a function phi like this is Plurus abharmonic so this is a bit strange this is a useful theorem in many situations as is well known here but normally infima of Plurus abharmonic things are not Plurus abharmonic infima or convex functions are not in general convex maxima of convex functions again it goes in the opposite direction but it still holds that this theorem still holds if you have this invariance property all the assumptions infimum of this thing is Plurus abharmonic so this is an important ingredient I think it's fair to say in the work of Thomas and Janir on geodesics in the space of Kepler matrix for instance and why does it follow from this thing here because all you need to do is that you need to multiply phi here by m very large number and then you take m throats which you can also do and then you let m go to infinity and then this integral here will be precisely this thing here it's a consequence of the fact that LP norms tend to infinity norms when P goes to infinity so it's just a minimum principle ok, and then as the very last thing I just give another statement due to Dario Cordero which is more maybe of an honest application which says that so this is from analysis it says that you have two you have Cn you have Cn and that's a that's a Banach space if you want to you can give it different norms you can give it one norm zero norm and another norm of one norm they are Banach norms now so they are just one homogeneous thing and then you can use the Ristorin theorem Swedish theorem and find intermediate norms here by the method of complex interpolation and you can look at the unit balls with respect to those norms they are the BT and you can look at their volumes and then this is a concave function of this so I don't think that was known before so this uses again this business about one of the theorems I had before consequence because you can view this as some sort of Bergman kernel the volume of BT and okay so that was what I had planned to say for today and then the next time I will give some applications to complex analysis and complex geometry yes