 Okay, so I want to start by talking about this. Yeah, that's cute. I'll explain what that is when you talk. But this is really about deformation of burdens. I'm going to say something very general that I can't say much about and then I'll keep me restricted to a case that I can say something about. So we'll start with complex manifold B. Another complex manifold B will just be some polymorphic subversion. Eventually, I'm just going to require a type of dropper, but partially non-dropper case. Shoot, one has done something like that. Current student has extended some of that work in some more depth of directions. But I won't speak about that. I will assume that this is profit for spaces. And I want to give them some kind of structure. Okay, so to do that, I'm going to basically describe first what the student sections are. And then I'm going to tell you how to compute the B-bar. The student sections, I take some, I can send this locally to some co-working vector field on the base. And then I can take a lift that I'll call a section. And the view is defined by, like, oh, it is anti-paper. Student sections, for instance, is this thing. So in the proper case, this is a smooth vector bubble. This structure is the bar, and it's still not a polymorphic bubble. Declare for yourself a set of smooth sections. Those are the size of the... No, this does not, because, oh, I forgot to say something. Yes, yeah. But I want it to be sort of polymorphic. Polymorphic on the fibers. So, yeah, that's crucial here. We want the property that when you restrict these to any given fiber, it should be polymorphic. The B-bar operator, this does not depend on the choice of the lift. Any two lifts will differ by, because if there are one vector field, it will be it. So this B-bar operator was defined by both. As smooth sections go, you sort of declare for yourself that you have a collection of sections that you call the smooth sections, and they satisfy some axiomatic things in this sort of general theory. And you get a family of Hilbert spaces, or field of Hilbert spaces, that would be called a smooth... So, it's confusing. I mean, there is a bit of control about the B-bar operator. I mean, this, I don't really know how... I mean, there is a notion of polymorphic section as a direct image, or... Yes, but... But these things will be directed at this, okay? Okay, you can do that. Basically, that's what BLS means. There's an electric circuit. So it's these smooth Hilbert fields, and the attachment of a complex structure. So I just kind of stole those ideas from the people, declared it for myself, and then I named it after them, and that was okay. Yeah, okay. This is not... With this structure, which I haven't been very specific about, this is not a BLS... So the problem is that these fibres don't have to be... That means that there will be some bad fibres, and concessions passing through every single, say, a dense subset of information. We don't have such a pattern passing through every single, or some large subset of these fibres, and that's a problem for us. If we consider this to be okay, we still try to prove some theory about it. So let me state... You should have no mean. It's also possible to know about M-positivity, of, say, in words, what it is. So you can compute the curvature of this H, and then you can use it to define a quadratic form on the tensor product of tangent vectors, one-zero tangent vectors with the vector on E. You can look at those tensors, and they have a rank. You can think of as linear maps from, say, the cotangent bundle to the fibre of E, and you can look at the rank of those things. If the tensor has rank one, then it's an indie composable. If you have a positive one, that's the same as Griffith's positive one. And then if you have M-positivity for all M, I mean the rank can only be as big as the minimum of the fibre dimension, the bigger M which redundancy, but that's called not... So if positivity of this H implies the local triviality of this capital, unless M is maximum. In that case, the Salatakegoshi theorem, as it was observed by Kou, the Salatakegoshi theorem tells you that any local section has an extension to some neighbourhood. But in general, I don't know if there is such an extension. If E is a line bundle, then M-positivity is just one positivity, and also not a positive one. That was the situation. So I'm just trying to read it. If E has dimension one, then it's what you're meant to do. But then at the point is that then we're just talking about ordinary visibility. Yes. Right, except that it should have no meaning because what is the curvature of a non-vector line? Yes. So that's some of what I want to try to explain. I would like to define the curvature of such a thing and give it some kind of structure. And so, as I said before, you don't have these extension of any element of the fiber to the neighbourhood to give you a section along which you could try to compute something like the churn connection before the curvature. And so, but on the other hand, what I'm going to try to convince you of is that you don't really need to extend it to a neighbourhood. You just need to extend it homomorphically to second order infinitesimally at the point. And that would be enough to give a well-defined notion of curvature that I have to sell it and I have to be willing to buy it. I don't have any other... So the trick is to embed this picture inside some ambient animal's paper. This is what he wanted to initially do, maybe, and he sort of did it into another case, a case in which the homomorphic family you start with is sort of trivially trivial in the sense that you cook it up to be trivial in all respects. But we'll have a non-trivial thing that will be some kind of sub-level. You can compare the differences and it's the ancient idea of Gauss in Romanian geometries and imported into complex geometry by Griffiths is that the difference of those two things is the square of some kind of tensor called the second fundamental form. And you want to estimate that second fundamental form in the... in this very trivial family case, particularly positive who can use the Skoda version of R-minor sphere to estimate that second fundamental form. You can really discuss this in the other way and kind of recurve things because there's some kind of finite spirit. By and large, you've correctly got this what he would have liked to do in this case, but there's no obvious complex structure for the bundle of L2 spaces. I'm going to tell you there are many. Okay, so how do you construct some DLS field new sections, content some theta inside of the total space X and I want it to be horizontal. H is sitting inside H has homomorphic on the fibers. This one is not homomorphic on the fibers. I mean the sheath theoretic H0 sections of this sheath. So you've got one of these horizontal subspaces. That means you're choosing a horizontal direction in every fiber and then if you've got a tau a lift that I'll you know, see theta to tau so that's where the sections are going to be vector fields and then I can use the complex content of this on the E-bar operator. I'll have some E-bar theta and the way I would differentiate is to compute an E-bar question if you compute. So this is how you do non-sections and there's an obvious extension to differential forms. I'm not going to bother. I've asked whether or not there's 0 and that can be true if not only if this theta is 0. Initially I made an error when I first sent out this preprint. I thought it was always going to be 0 but it's not. It's no big deal. If you have a curvature bundle with a deltium metric you can still define that a churn connection and so it has some kind of curvature and the first formula I was going to put on the board the first formula I was going to put on the board was going to be the curvature of this L and this E-bar theta. So I don't want to put it up because I don't think it's going to be particularly helpful for that. So in this kind of trivial family setting when we computed this curvature the curvature operator was just a curvature of the vector on the E-flies and you restrict it to directions in the horizontal direction because it's a product structure you can lift the horizontal. In this setting it actually is not just that multiplier it also has contribution theory of the family X. There is a contribution from the federa spectrum. And also if this E-bar square is not zero then you don't get a one-one form in this setting you can always find a theta so that E-bar square theta is zero and that's because of Erisman's theorem. It's not Erisman's theorem itself but it's a version of Erisman's theorem proved by Oranishi at the MIE through publication in the book of Lausanne. And maybe also, yeah, so there's a way of finding these sort of complex sub-manifolds and just use the tangent spaces of those as the phagos and they're automatically linked. Review and everything becomes Yeah, and then everything we will fall into the classical. Yeah, following. It's griffons. That is so overly difficult but even if it's not positively griffons I still have a theory here. Okay, so anyway then as I said you want to get the sections that are even in terms of the polymorphic in the second order if there are sections instead of characters I will give this a horrible name. This is probably a sub-soft bumblebee. I emphasize that this is not a theory of spaces that do not fit together like a sheaf. B, what should this sigma-7 be? B, sigma-7 be? So instead of all the germs, the first thing I want is a restriction of a single file is HB but also this other weird looking condition B bar restrict that to the fiber of zero. It's like saying not only is B bar restricted to the fiber of zero but also the horizontal derivative of it. Maybe I don't care about it but the D bar restricted to the fiber of zero is this condition and this is the second order condition. And amazingly this is independent of this state. It's not terribly complicated calculation but it requires by sort of following a general theory of tournament. So for this value there is the next one. The fact that you call that D L theta and then I can just take the finalized Bergman projection of that and that would be the formula if you had a locally trivial because I'm never going to use it. It's just like an analogy. It suffers from the same problem that you want to restrict it to sections of H. It's not going to be able to hit but when you compute the curvature it's going to be deficient. This is independent of the state. And to form is the remainder of the form this does depend on. So for example it's easy to see that this is C infinity linear unprepared for it. That does not mean that it's given by a tensor. When you have infinite dimensional things that doesn't mean that it's computed by a tensor. So you need some extra information to make sure that it's sort of tensorial. If you restrict to this family of sections it is tensorial. That's surprising. Take this theta L theta and then you subtract the square of the second fundamental form. Plus here but it's because I'm treating these as differential forms so the sign change but really this is a negative operating in the way it's written. This is when this H is locally trivial it's the Gauss formula or Griffith's formula that this is exactly equal to the theta H. So I define this thing to be theta H and then I need to prove that it's positive and okay so what I can do is quite similar to both proof of his theorem except you have to modify things a little bit but you use the hodge theorem to find particularly good representative of these sections and the primitivity that you get. Well there's a lot of stuff that happens. The Kodakir Spencer contribution from the curvature of L also appears in this expression here but then you add to it another contribution from the Kodakir Spencer class. When you add all these things together the part from here cancels out and then you're left with this other thing you want to choose it in a good way using the hodge theorem to get positivity and then it's some kind of integration by parts argument. There is a kind of geometry of these spaces that's really different from the directive which I don't know yet but it could be used for but one of the things I would like to use it for is for example answering the question of local triviality not knowing it a priori I think I can do that but it's not finished yet so I won't play right now. Okay, sorry for going over the time. Thank you.