 So today I'm going to talk about the course formula for Haya Direct Image. So naturally I'm not an expert about that. So now we have a lot of experts here in Boo, and Su, and Pro. But yeah, anyway I will explain. And if you have any questions, you can ask me. But it may be better to ask them. Anyway, so let me introduce you first. Let me give the setting. We have a family of contact telemanipers that is given by the subjective property of the submersion. So every fiber is contact telemanipers. And we have a permission vector boundary on the total space. Then the direct image is given like this. So whose fiber is actually the converse group of the fiber is that is by the law of motion that is actually just an e value of pq4 closed e value of pq4. Of course that this property is not called in general, but if we give this assumption, then we have that identity. So under this assumption, then this Haya Direct Image is a local relationship. So we first give a fiber-wise telephone on the total space. The goal of this talk is compute the curvature formula for this Haya Direct Image. Of course in terms of some intrinsic quantities, so for example C omega that is shielded curvature, and AS is a logental table of the total space class, and Xeta HE is a curvature of the vector boundary, and ATX are explained. So first, let me introduce the horizontal lift. So in this talk I always assume that the target space S is a unique disk in the complex plane, and the general case is easily done, because the curvature is a local thing. So first, if we have an alternate vector field, we have partial S, then we find the horizontal lift V with respect to this fiber-wise telephone by the two conditions. First one is the horizontal lift is orthogonal to the fiber direction, and the second one is if we push down the lift, then that is the original coordinate vector field. And the second one is Jewish curvature. Jewish curvature is nothing but the length of the horizontal lift. Here omega is not telephone, but the fiber-wise telephone, we don't know that it's positive or not, but if that is correct, then that is positive. And AS is given by the diva of V, that is actually a good representative of the Gouda-Responsor class. If the boundary is relatively line-bundled, then if we give the metric of the line-bundled by a line-side metric, then that representative actually harmony. And the last one is the last one defined by just plugging V to the set-up, and this is actually a component of Gouda-Responsor class of the pair, but if you just consider the deformation of vector-bundled, then that V is nothing but the coordinate vector field, and that is exactly the class of the deformation of vector-bundled. But here we deformed the manifold, yeah. Can I ask a question? So when you talk about the fiber-wise scalar form. A fiber just decodes the one-on-form, which is possible on each byte. So it's an auto-spawn on absolute total space. Yes. What we should assume? Yeah. Okay. Yeah. So yes. So this, of course, if the family, the family of manifold is a trivial, then that is exactly same as the Gouda-Responsor class of the vector-bundled. But here we actually deformed the manifold and vector-bundled, both of them. So there is no notion of Gouda-Responsor class, but we can define the Gouda-Responsor class by some Athea-bundled, but I don't want to go there. Of course, I'm not an expert. And yes. But that is not so related with today's talk. So anyway, that is kind of some important quantity. So let me briefly report the history of the curvature of this setting. So 1986, first the CU compute the curvature of veripetism metric in the borderline space of the canonical polarized manifold. And here, she introduced how to use re-derivative to compute the curvature formula. And 2009, Berns-Baud proved a very important theory. Maybe I think this is one of his favorite paper. I'm not sure, but yeah, maybe. So she proved that if... So it is positive that that direct image of relative canonical line-bundled twisted by L is Nakano positive. And in the next paper, he also compute the very explicit curvature formula. On the other hand, Morgan and Tatayama used that Booz method to show that if E is a vector bundle, which is Nakano positive, then this higher direct image is also Nakano positive. On the other hand, Geo proved... Geo computed the curvature formula for this type of image. And here the one thing I want to emphasize is here T and here N minus T. First, we start with TQ, but here she assumed that T plus 3 would end. And Philip Neumann replaced this tensor power of the canonical line-bundled by the arbitrarily polarized polarization L. And recently Booz and Su and Bihai also computed curvature formula for this higher direct image. And also they considered some other cases. And also they proved some extension when the vibration is not submerged. So let's go to that higher direct image. So higher direct image, we first have to see the compact structure of that thing. So that is locally free-shift. So we can think that is a vector bundle. And a compact structure is actually given by indicating what is the atomic section. So here the atomic section is nothing but... Because this is... Yes, the element here is a class that is represented by some diva closed form. And so here the higher direct image, the frantic section of that image is diva... And this is the diva operator in the focus space. And the representative is actually diva closed, not peak form, the generative form, but this... So this shape variable, 0, 2 form. So if a class is represented by such kind of form, then that is frantic section. And we can give natural elementary by that here h is the harmonic projection. So we only consider the ellipse norm of harmony. So to compute the curvature, so we have to choose nice representative, which is given by the first part. So if we are given a local polynomial section, then that is... There is a good representative which is diva closed form, but that is not peak form, of course. That is that shape variable, 0, 2 form. So that is diva closed. And also whenever we restrict that representative on the fiber, that is harmony. And via global isomorphism, you can also think about that psi as peak form on the total space. But in this case, the first condition is crazy by this one. So psi is not diva closed, but if we rest the pullback of this, yes, that is diva closed. And also psi is also harmonic on each fiber. So using this good representative, we can say that that ellipse norm is actually the ellipse norm of this via psi. So again, what do you think of psi? Psi versus 0, 2 form with valetian omega p, right? Yes. Psi is just p24. So you first take a template of H and diva, you can see that. Oh, okay. So that... And no VS or... No. No, okay. Yeah. So, yeah, of course in Bu and Mihae and Su's paper, they kind of studied what is the complex structure in the hyodartic image. So they defined the connection explicitly. But for computing the curvature, actually we don't need the very explicit curvature connection. So of course, we want to know that, but here we just go through the curvature directly. So first, we take that good representative and this is the metric. So that is a harmonic relationship. So we have a frame. We have a local Hamilton frame. And that is psi 2 e k. And if we take an inner product, then we have a metric. Then now we... So here we can assume that this is a normal frame at some people point, which we want to compute the curvature. And the curvature is just some second derivative of this electric tensor. So because we have... We take a normal coordinate, a normal frame. So this is exactly the same as the curvature. So in this, of course, simplicity, I will denote this psi k and psi f by just psi. So they... Of course, we have to compute when they are different. But yeah, you can't imagine what computation goes on. And usually, general formula comes from the polarization. So let me denote that this psi and here the curvature, the overall s direction and psi direction is r. And the first derivative is even like this. So if we take a d d s, then that goes into the interval. But we have to take a derivative. Here v is a horizontal lift of coordinate vector field. Then by life's rule, so you can decompose this by two columns. Here the redevelopment of the vector of undervalued pq foam is defined like this. Here d is the channel connection. And to write this term as an inner product, we have to change the order of this derivative at the hot star operator. So let me just recall the definition of the redevelopment. So the redevelopment is defined by this paragraph's magic formula. And since the connection is composed into two parts, so the redevelopment does not preserve the binary. So we start with pq foam, but we have pq foam. And also if we take this one, then that is not pq, but t minus 1 comma q plus 1. So we have two columns. And also if v phi is a zero-one vector, then we also have two columns. One is pq and one is t plus 1 comma q minus 1. So I denote the binary preserved term is just prime. And if the binary changes, then I double prime. Sorry, v is a vector field here? Yeah, vector field here. Yeah, one zero vector field. Only consider the horizontal lift of the coordinate vector field. That is enough to compute the coverage formula. Of course, you can define the redevelopment for any vector. Then, yeah, so we have those kind of good identities. Yes, maybe I have to say that, yeah, this talk is not very, maybe very intuitive because all identities comes from some tensor calculation with respect to some covariant derivative. But yeah, anyway, we have those nice formula. And if we use the third one, before using the third one, so now we are here. So if we use the lattice decomposition theorem, then we can decompose this into two columns. So here it is entirely part and it is entirely non-trivial part. So here the values are different and these finishes. And we still have this one, this is this one, but here this part is actually divided by that. So this slide is divided only to these dimensions. So the first column only suffocates. And how about the second derivative? So yeah, maybe I'm running out of time, so I just skip this slide. And yeah, you can use a lattice decomposition and using some more concentration. Anyway, altogether you have this form. So here this is the concentration of two derivative and here we have every sine prime, double prime, every bar prime and double prime. There is some different signs. And just this proposition, we can replace the double prime part by the cut product of the lattice plus, the lattice plus and this side product is just a contraction with the vector field part and vector part. And so it is quite well known that the commutator of the derivatives is given by the derivative of the commutator of vector fields and the curvature of the vector wonder. And here that the commutator of two vectors actually does not contain the derivative along the horizontal direction. So it only has the vertical direction. So if we replace this one and those two does not depend on the horizontal variation. But still we have two commutes, this one and this one. They contain the derivative along the horizontal direction. So we have to compute that two terms. So first I want to compute the every sine prime and that is the norm of that every sine is that one. But here every sine is actually to the harmonic space from the fact that that psi is a numeral frame. So we can take that green operator and that's Laplacian. And this is every sine prime a diva star that is vanishing. So we have this one and finally we get this one. And here we can use this sine fk so basically we want to get rid of all differentiation along the horizontal direction. So here every operator that is defined on fiber is operator and not defined on the whole total space that is defined on fiber. So if we plug in this from here and now we get rid of all horizontal direction, derivative of horizontal direction. Likewise we can also compute that every bar psi prime and here we also have those identities. If we plug in this there then we don't have some derivative along horizontal direction. So now we have kind of general curvature formula but that is not so enlightening and also it does not have some particular application so I just hold to some special cases. So first case is before going there I also want to introduce this formula. So if you compute we compute that derivative that is even more like this and here w is actually gradient 1 0 vector field of the geodesic curvature. And actually I don't know how to make this formula simpler. But anyway so that those two terms actually that is also appeared in the twisted Bokno-Odaira Nakano formula but I don't know how to manage that since it's simpler. So this term appears in the twisted Bokno-Odaira Nakano formula. So for proving Ozawa-Temoshi extension. Yes. And in particular its side is the star close that formula becomes much simpler and also its side is deep close then that this that becomes this one. So we only have this Bokso-Odaira and some minus of these points. And let's move to the untwisted case first. So maybe this theorem is proved by Griffiths a long time ago. So untwisted case the D-Bara Nakawajan and D-Lakawajan are same. So we actually have this one. We have this one. Because if we start with that one and we come here and we can change this part by this one. And we both hear D-Lakawajan saying that we can communicate each other. So again, this one. So the result is some orthogonal some projection to the space of orthogonal to harmonic space. So we have this one. So previously our formula is even like this. So this part is actually also not part of this one. So they just the harmonic part only harmonic part remains. So we have this one and the second one can be computed as same way. The second one is finalized flat vector boundary case if the set is Nakano semi positive or Nakano semi negative. And also if the curvature is flat on fibers and also there exists a fiber-wise kind of form on the total space then you can compute not higher than image but the primitive software that is given by just a primitive representative. So in that case we can generalize the previous formula in this case. This formula is first proved by Buen, Mihai, and Su. And also actually they used actually Buen developed his very nice language which analyzed the representative of PQ4, the space of PQ4 but here I just used a tensor calculation. It looks almost parallel and anyway we can have this one. So let's go to the second one. So let's go to the line boundary case and if E is a holiday line boundary and then the permanent line boundary which is positive on the fiber on each fiber then one can take the fiber-wise kind of form by that curvature form. So in this case that eta s that is some plus of the vector boundary repair that actually vanishes because we take that set up by omega. So omega is defined. So the free is defined by oh that is nothing but degree is straight from the fiber that is 0. And by Boknok data and a kind of formula we have the difference of two Laplacian. So the difference of two Laplacian is some constant times I get it again. So actually I try to compute the curvature formula the general curvature formula is wonderful but the second one is not very easy so I just assume that this size it must not lose then we have this formula. And if P plus Q equal n then those terms these two terms and also these combinations that is go back to the who through the formula or philoderma's formula and once again I have to mention that here those three terms have plus sign except the harmonic part of this one the harmonic part actually has some minus sign here and those three terms are negative sign but only this part the harmonic part of this one is some opposite sign. So anyway we have so if that size give us a key star group we have this formula and of course this is not the curvature formula for this form or image so if we want to find the application maybe we just go to the simpler case so that direct image of relative along the P form then the curvature formula is given like this so here n is n minus p then n is positive but still we have some negative here so the negative always come from the D oxide and we can also consider the negatively curved line boundary case that in this case we have to assume that the sign is D closed and the part of the formula is given like this of course here also people's click on n this formula moves back to the known formula and here the difference comes from this one you assume either that size D closed in this case yes so we only we can only compute the D closed direction but then are you assuming this for one particular fiber for the particular fiber where you compute the curvature or do you assume one particular one particular yeah one particular yeah yes so of course you can assume that this size is D closed only one particular five then it's enough okay yeah but then it doesn't matter so much actually I didn't know very much about that really yeah you can explain me later yeah that you say that the center is quite small but it doesn't harm anybody it's a because I mean it's going to be representative of well so here yeah this yeah the size D closed condition is automatically satisfied the case of Q's direct image of relative technical line under so here we have a negativity actually I don't know the negativity of that this like image shape is already known but actually I'm not so here we have three those more negative forms but here we have also some more symptoms but here we put these two inside that two types of size is actually the actual size of the size so we can find the condition of these which guarantee the sum of these two forms that we have but I don't know the negativity of that the direct image shape is already known or not I don't know so what was the assumption on L? it is negatively curved on each fiber so of course for setting the positivity so we have a notion that L is positive on the outer space that guarantees this C omega is positive I mean so here we A is that omega by the curvature of the line under so if the line under is so of course that is positive on each fiber so we don't know the geodesic curvature of omega is positive we don't know but if L is negative on the total space then that guarantees omega is positive then C omega is positive so here if L is negative then so if L is negative then this is positive and this is positive so those terms are negative and we can find the condition for the sum of these two terms are negative in that case that the direct image is negative and also we can think about the Hermit Einstein Vecta Wonder the family of Hermit Einstein Vecta Wonder in this case so the the family of calomonic fluid is actually treated so we just consider a three-year family and we consider a Vecta Wonder on the color space and if we assume that the H is Hermit Einstein on each biker then we can compute the curvature of this ion in this and this maybe if p equals 0 then then the result is already known by Georg and the final case is final case is yeah maybe this is Takayama and Morgan's case we want to compute the curvature formula of that q-style hyodineic image and show that the positivity there so here if we assume the kind of positivity then we have first this one and we also have the variation of this form but so we so compute more we have to compute we have to analyze this form some mist form but yeah so far I have no idea how to compute that and the problem is that so this form is not harmonic even though if that form is harmonic then if we compute a three color then that is 0 but yeah that doesn't have to jetter and yeah so actually I have no idea maybe I have fast su or bu this one and so anyway yes about this hyodineic image shape there is a CRM by Brua and Takayama and it is Nakano positive and this hyodineic image shape is Nakano positive and also Wang and Bu and Bu Ji Hai su also prove that I want to recover from so my formula but yeah I am not successful yet so yeah that's all thank you very much