so let's start uh hello everyone so the speaker for this session is Professor PE David with uh from the University of Goldenberg uh who is also the recipient of the frontiers of science award uh today he will tell us the story about Duality between the stud effective and the mob projective yes please yes thank you very much uh and let me start so let me start by thanking the organizers for uh yeah inviting me and giving me this opportunity to speak uh and I I should also warn you that there will be a quiz in the end so you have to be very attentive throughout the whole talk okay uh okay um okay so uh the setting is this uh so uh we're going to consider a compact element x uh okay and then of given that there are sort of two uh natural uh Vector spaces to consider so we have the vector space of one one real horology classes um and then the vector space of uh n minus one n minus one chology classes okay so these are two finite dimension of vector spaces um okay and in those Vector spaces there are uh some um some convex cones which encode some very important um information about this manifold okay uh so I I save some time here by already drawing but uh so in this h11 space so first we have this uh red cone so that is called the C cone so and it consists of all classes that contain a c form okay and it's easy to see that this will be a convex cone and it's going to be open um okay uh so then uh the bar here it means that we take the closure so this will be a closed convex cone it's called the Nee okay so uh and a class is nef if it's in the ne okay so that's the first or the two first cones here uh okay but then there's this uh somewhat bigger uh cone um that did I oh I forgot to write it here okay uh okay okay so that cone um uh it consists of cles of not necessarily containing caliform but that contain some uh closed positive current okay um so of course a CER form is also a close positive current uh but there are much more of these so this would be a bigger code okay and this is called the this is called the pseudo effective cone or thee uh okay um and then we go to the N minus one n minus one space here so then we start looking at uh this thing that I've denoted by n here uh so now this looks very similar like this so we look at all the it consists of all the classes which contain a Clos positive nus one nus one current then okay and that is similarly called a super effective code uh okay and then there's this fourth cone uh M and its description is a little bit more complicated um so this ised as the closed convex cone by cles of this form so we we look at some modification of X U um okay and and there we pick uh a number of C forms so smooth CER forms uh so on X uh on x uh okay but then we can push forward that down to to X and then uh this will be a closed positive current okay but it might not be smooth okay um okay so we look at so so we look at the thing which is generated by all such uh things okay and that is called the movable code uh and the reason why that is called the movable cone is because uh if we imagine uh if we imagine that these uh CER uh if we imagine that these C forms uh uh are curvature forms with respect to some line bundle okay and then uh then uh I mean this line bundle will have to be of an ample uh and imagine that the line bundle has sections okay so then then uh you will have these sort of hypersurfaces okay and then so the and so this um CLA will be the same as sort of the class of these sort of hypersurfaces that you push down so this will correspond to the class on X of uh of uh sort of the intersection of these Maybe I didn't explain it properly but this will correspond then to a a curve on on X and that curve will be able to move because it's I mean these things will move upstairs anyway that that's the reason why it's called the mo uh okay uh and now so these two Vector spaces are dual uh to each other uh by this pointa so I mean so if you have a two cles I mean just take some smooth representatives and you can integrate um okay and then it's it's uh sort of natural question sort of how these cones sort of relate to each other given this Duality okay so the first theorem I want to mention now is that of Dem and pound uh which says that uh the nef cone and the pseudo effective cone uh are Jewel uh with respect to this pairing uh uh which sort of more concretely means that uh a class Alpha is nef uh if and only if uh for all pseudo uh effective n minus one n minus one uh claes uh this uh integral is uh non negative okay so that's what it means okay so uh if we look at it so actually so the uh the only if uh part here is obvious because uh if Alpha is NE okay so if say Alpha is uh uh is actually Ker uh then you can sort of you take K representative and then you take a close positive current here and then you integrate and then sort of by definition this will be negative okay so that's sort of immediate so the hard part uh is the is the If part to sort of just knowing so if if Alpha is a class and you just know that s of paired to any such beta this is bigger thanal to zero how do we know that Alpha is nef so this is the hard part uh but then that that follows from this deep result of of deay and porn about the numerical characterization of the ker code as far as I understand uh okay uh so a little bit later uh uh there was another result by bom de pound and pel uh which said that uh so if uh if X is projective then uh we have another Duality statement uh so and I will explain this uh very shortly so then e NS and MNS uh they are dual and now so in the projective case uh you can sort of instead of looking at these full uh these big Vector spaces you can look at the Subspace uh sort of gener generated uh by uh classes corresponding to sub varieties uh okay uh and okay and and those are called neuron sever spes and if you intersect these neur spes with these cones then you get these of algebraic versions um okay so where um so uh this is simply uh this intersection so if you're and so if you're doing algebraic geometry then it's very natural to look at these sort of cones instead okay so they Pro this dualis statement between the Su effective and the movable cone but in the algebraic setting okay uh but in the same paper uh they formulate uh they sort of formulate the conjecture that on any compact manifold uh we should have that the sort of the full s effective and movable cones uh are du okay so somehow completing uh completing this picture here okay uh any questions what yes yes yeah yeah because yes with respect to this okay yes uh uh okay and then so my result was then that uh well if at least if we assume that uh our manifold is projective uh then uh these cones are du okay yes sorry yeah but they will be automatically also or I mean but you understand the I mean if if you know the closure you can understand the interior as well aha okay yeah I mean this was yeah so might have a yeah they might have a a polygonal part as well yeah okay yeah um okay uh yes so how do you see this well uh uh so as I said here so there's like one one aspect which is sort of easy here so uh um so if uh if Alpha is s effective uh and beta uh is movable uh then you can see that uh this pairing is non- negative and that's because uh you can you can take close positive representative here right and then you can think of you can take a see so so you can take a close of this thing and the point is that you can pull back a closed positive one one current by simply pulling back the the function and and the pullback will still be positive so you will have a so you can pair then you pair that with these smooth C forms and then that will automatically POS okay so so that's that's uh sort of the obvious part and that's I guess the reason why you might think this is I mean might think this to be true um okay so so what that means is that uh the pseudo effective cone will be at least included in the Dual of the movable okay uh so we know one inclusion oh H yeah uh uh yeah you have to use your imagination sorry um okay uh okay so how do you how would you sort of prove the I mean the the equality then uh well so it turns out that sort of the key notion here is that of volume uh so uh if you have a sud effective uh class uh then uh you def you define the volume of that as well you can Define it in in in in sort of different ways here but I I will do it in the way which sort of serves sort of my later purposes the best so I Define it as the supremum of these integrals so uh I take uh I take a closed positive current T in Alpha and then I want to take the top power and integrate but you cannot really do that if you have a singular current so I need to sort of integrate away from the sort of the singular part of t Okay uh so and then this will be some number and I take the suprem and and here so I'm taking over all closed positive T uh uh in Alpha okay um and then if well so if if if the class Isn't So effective I just let the volume be zero so uh um okay so so this was um this was introduced by bom I believed in his I believe in his thesis and um and and he shows sort of two uh important things about this function so first uh so if if the Clause happens to be uh the first CH Clause of a line bundle uh then uh this volume that is defined here it it will be the same as the volume of the line bundle which was introduced earlier and and which is defined in the following way so you take the Le soup as K TS to Infinity of the dimension uh yeah the the dimension of the space of sections of L the power Kid K to the N divided by n factorial okay uh so this is a yeah uh okay this was done earlier in this El break setting um okay and sort of another uh key aspect is that uh you can see so a class will lie in the interior of the S defected cone so and you say that then you say that Alpha is Big so this uh is sort of true if and only if uh this volume is strictly positive uh okay and and and this is important because this gives sort of a way uh to check whether you're sort of in the interior of this cone and so of you need to do that when when when trying to sort of show this equality here you will need to be able to tell that okay so this uh this means that sort of the volume can be of used trying to establish yes it only depends on the numerical equivalence this uh I mean but that's um uh that's a theorem that it only depends on the uh numerical sorry no but then it will be zero yeah yeah so but it's a it's a theorem showing that it only depends on the numerical yeah okay um uh okay so so the volume uh will sort of help us to uh to try to show this uh sort of of equality uh but sort of this is is not enough you need more detailed information about this function um and then we have this uh conjecture due to De um which says that for all um pairs of um nef classes uh we have the following inequality so the volume uh of the difference is bigger than or equal to uh the yeah the integral of the alpha to the N minus n uh Alpha nus One uh beta um okay and uh and this is known as the um this is known as the transcendental polymorphic Morse uh inequality um yeah uh okay and and and the point here is that so in this um uh in this paper uh uh they show that if you assume this in equality uh then uh it implies The Duality that we want okay so I would like I mean I I would have liked to sort of explain the argument but it's a little bit too involved to have time to do but it's it's a nice argument but it sort of uh it sort of crucially uses them the volume um and the facts these two okay uh okay so ah yeah and okay so and I also want to say so uh so as you remember they Prov this duality in this algebraic case and that they did exactly in this way using the fact that if if you assume uh that sort of these classes are algebraic then it's not very difficult to show this inequality okay because in the algebraic case then you have a different description of the volume and then you can use sort of more standard chological methods and yeah okay so that's sort of how their proof went so so really the only thing that is missing in the sort of transcendental setting uh is this uh inequality um okay and okay so what I showed was uh I showed that uh the inequality uh sort of at least holds when uh the manifold is projected okay see uh okay um okay so I will I will I will sketch the proof and then I'll see how much time I have left afterwards if if any yeah okay uh okay so how how would you prove that kind of inequality well so the first is some um some observation that uh it's sort of enough uh uh to show a somewhat weaker inequality uh like this uh uh where um so you can also assume that uh Alpha is not only nef but ker so it of contains a c form uh similarly for beta uh but then crucially you because because X is projective we here we can also assume that uh this beta is integral so it's also the CL uh of the current of integration along a hypersurface um okay and then we're also allowed to assume that uh the difference lies in the interior uh okay uh and and the key thing here is of course uh that we have this um that we have this hypers surface uh okay uh okay so what do we do with the surface well so then we can write so this current of integration it it lies in the same same CL as Omega beta okay so we can write it as Omega beta Plus ddcg for some function um okay and now G will have this kind of logarithmic Singularity along y so that may indicate it uh like that uh and uh for reasons that will be apparent in a bit uh so so G is singular but if I take some maybe large value R then I can sort of regularize G so I sort of do like that so so what this means that so gr uh is a regularized maximum of G uh and minus r so and now this is a smooth uh function uh and another important thing here is that it's going to be sort of psh with respect to this uh Omega beta and this simply means that if we take omega beta plus ddcg then this is a positive uh form uh yes yeah uh okay so now now comes the the sort of the key thing here uh so we let F of R be this kind of supremum so we take the supremum of all five uh bounded from by by gr well all F which are psh with respect to uh Omega Alpha so this is an envelope this is an envelope construction which is very classical in PL potential uh Theory and a key thing with these envelopes is that they are themselves then sort ofab harmonic uh with respect to this sort of uh reference okay so uh and it might well it it might look like this so this is fire um okay and and and two sort of important things to note here is that first uh if we take uh if we take this current that we get using the envelope uh and then we take the top power so this is not necessarily smooth but it's sort of smooth enough so that you can actually do this so and you can so you can integrate over the whole thing and this will be the same thing as we if we just integrated sort of that thing so so this is just uh the integral of alpha to the N okay and okay so and the another important thing here is that if we okay so this thing it's be a positive measure um okay but actually and and this is one thing which is extremely useful about these envelopes is that you can say very much about uh these measures in fact uh it's going to be so it's going to be supported on the set where the envelope is equal to this obstacle uh okay so you see here so so here there's going to be nothing okay so it's going to be supported where these two agree and and then you can sort of naively there I mean f r is equal to J Jr so naively you would say well so it should be you can just uh change it here okay so but and and the good thing is that it actually works but it's not it's not as easy as it looks because you have to in order to do that you have to know something about the regularity of f r and that is sort of a subtle issue but this was actually shown by uh by man uh some years ago and then uh in this setting and then there's now a more General result by DSA and Trapani showing that this is this is this is actually true in in a great generality um okay so this is an absolute sort of key property uh of these envelopes and now as as when you know when we know this then we can do a very simple estimate uh because so what you can do is that you can just add this Omega beta I mean it's a CER form it's just going to be much bigger yes it's it's it's very simple and then uh this is then I mean and this is sort of positive everywhere so you can sort of you can also forget about that thing if you want it's I mean the the inequality goes in the right direction okay and then uh now this is some this is a of positive form and so now we can of use the binomial theorem uh so we can write this as uh um okay so and now okay so there's just one sort of idea left to to to use here okay so uh so this works for any R okay but and now we let R tend to Infinity uh okay and then it's easy to see here that the p r will decrease to something that we call F Infinity this you see here it's it's this is going to be bounded from above but by G sort of by construction uh and the way these P sub armonic functions work is that this is also going to be sort of ponic with respect to Omega Alpha okay uh and then okay so then we can look then we can look at this current now you notice here that so f Infinity is less than or equal to G so it means that this current will have this kind of Singularity along y okay so it will have like the long number at least one along Y and then we know that so that means that we can actually remove uh this current and it's still going to be positive so here we get the positive current and we see here that uh this thing lies in Alpha uh minus beta which is good um okay now now now we're ready here uh okay so now I want to estimate the volume of Al minus beta and S by definition here the volume is the supremum of sort of all such integrals uh so it's going to be bigger than or equal to uh this integral um okay yes uh now okay so this is this is the singular set of this uh current okay so the singular set it will sort of contain or it might contain uh y okay it might also contain other things but you can show I will not say exactly how but you can of show that actually the singular set will at least be small so it's sort of in the I mean it will have a uh yeah it would have like zero volume okay so I take a pretty large set avoiding that thing and I call that U uh okay and this this current sort of does nothing on you so I can write it like this so this is going to be bigger than or equal to thegal over of the closure of U like that okay but but now f r converges to F infinity and decreases to F infinity and then there's a continuity uh I mean so so these measures they uh they are sort of continuous with respect to these sequences uh so this is going to be bigger than or equal to uh the limit of these integrals where I uh have R okay and uh and now I use this classic trick of so I can write the integral over U as um okay so I mean I can write it as the difference of the integral over the whole of x minus the integral over the complement okay it's pretty obvious uh uh okay but we know what the full integral is that's just uh okay so the full integral is as I said just uh this thing uh and and now I sort of use this now I use this uh uh this inequality and I integrate over the complement of U uh okay so I get then minus over K okay but now you see here so the in I mean the inequality I mean I can just it just get better by integral integrating of the whole of X I mean it just get better uh right and now uh sort of these things is nothing but uh uh these numbers that I had before uh and I said that I can actually make I I can sort of make the complement of you arbitrarily small sort of in this say in this sense so I can actually assume this to be NE negligible so I of this is going to be approximately zero and then uh you have it um okay so maybe I've already run over time I haven't kept track of the time uh okay okay so this is uh okay so this is the proof uh I haven't said anything about sort of the I mean applications or why this might be useful so I mean the thing is that it it is actually useful I don't have time to sort of go into it very deeply unfortunately uh but it has various consequences uh on I mean even if you so you don't even if you're sort of only really interested in ker uh forms or k classes this actually has consequences because um it tells you things about how Cal manifolds can deform so I just want to give you uh very quickly a a sample sort of application uh not of okay so it's an application not actually of this theorem but a slightly stronger um a slightly stronger uh statement but it's sort of very very much related okay so um so you can prove you can instance prove this thing which I don't know I I thought was interesting so if you have a uh say that you have a compact C manifold uh then for all positive Epsilon uh you can find uh some ker form uh in the same class as the given one um and ker embedding uh sort of into uh this so where uh so e so e uh is an ellipsoid uh in CN so and and this is just the standard ukian C form on CN okay and uh such that sort of the embedding captures almost all the volume of x so okay so you uh so here's X and it's sort of the the large part of X really looks like a standard ellipsoid okay so it's just an example that's sort of these kinds of things tells you sort of what kind of Kor metrics uh sort of you can find okay so I leave it at that [Applause] okay uh very nice talk so uh any questions or comments sorry it's related yeah it's very much related uh so because you you always have I mean you always have approximate risky decompositions but this is a uh this is like a quantitive version of it so you really need some very uh you need some strong control of how good uh sort of the approximation is and what does this ellipsoid mean in algebraic geometry oh then in okay the ellipsoid yeah so this uh okay so this um I mean this was just one example so this is say connected to uh if you want to bodies um yeah okay thank you so any more uh I mean yeah it follows okay so it follows from this that uh it follows from this that actually the movable cone is the same as another cone which has also been studied called the balanced cone or maybe it's a closure of the balanced cone um okay so and that has in a way a simpler definition because it's just uh it's just a smooth positive n minus one n minus one forms you need my theorem for that and I think actually that somehow uh you you kind of want you actually want the connect to to the modifications because what's interesting is well from my perspective so so really what's interesting is are these modifications you want to have some information about these modifications so um at least for sort of Def the kind of applications that that I'm uh I'm interested in so I think actually uh well I think that so so really so so what you're really interested in you're really interested in the behavior of this volume function that is sort of the key rather than The Duality statement um and and and the point is that sort of and you need you need you want information about the volume because this is connected to uh I mean sort of modifications of your of your manifold and you and you would have seen that maybe better if I had used the other definition of volume yeah so so there are e yeah so you can sort of Define you can Define the movable cone sort of more easily but actually it misses the point a little bit because actually the the connection to the modifications are s of what's most interesting about it [Laughter] I just it was a trick I just wanted you to be yeah attentive so any more questions or comments so maybe I have one so so so your X is a projecting manifold right so how about the DU phenomen in like singular case if you allow X to have Singularity and maybe my Singularity I'm sorry you're asking the wrong the wrong person uh uh so oh it's fine yeah yeah okay okay uh yeah so I think I mean is probably good person to ask so so any more question or comments so if there are no more questions Let's uh thank the speak again what