 We started with this project with Junion Sao a couple of years ago, the idea was to prove some results and also to celebrate who persons joining the elegant club of over 70s gentlemen. So what you will see next, it's somehow updated version of that of the paper. So let's you know, do we move this? Okay, this is how my talk will be structured. And let's start with a few, let's say, notations. We have here proper smooth family such that all the fibers are killer. And as usual, we note KX here, the canonical boundary of the global family. And let's assume that we have a line bound alone on the total space on X. And for each positive K, we consider this shift KX plus L twisted with this ugly beast, namely, so the sections of this shift are holomorphic sections, modular t to the to the power k plus one case is deeper zero, but it's a multiple of these t to the power k plus one if you want presented in down to earth terms. And then of course, we have a projection from fk plus one to fk, which is simply the one you imagine very natural. And then the main question that I would like to discuss next is, so let's say that we have a section here in this sorry. Hi, we lost you, we lost you at your introduction of the fk she's I see. All right, well, for no more interruptions internet here has been, you know, intermittent. Okay, so we have this shift fk. So like the holomorphic sections are in k plus n modular t to the power k plus one. And then together with the natural projection between two successive such sheaves. And then the question that I would like to discuss now in the what follows is, I said, under which conditions, for example, positivity or whatever on a section here in fk can be lifted can be extended one step farther. Now, so let's see what would be the main results. The main result is as follows. So now we assume that the bundle L is simply a multiple of the canonical bundle kx. And we do not buy L straight, if I may call it so, the restriction of L to x. And here we have, we have a metric given by the right multiple of the section here restricted to the central fiber to the zero zero infinitesimal neighborhood, the center fiber. And then another part of the data is the hypothesis is that there is some extension of of s as a smooth section of k plus L. Such that so the D bar is multiple of tk plus one so this is of course not a hypothesis the hypothesis is that we can find such as k such that this L to hypothesis is satisfied for any epsilon positive. Okay, so this is by no means guaranteed given that here fire has has singularities but we assume that we can find such an extension as k. And then the conclusion is that we can extend our s one step farther so s is in the image of pi k. Okay, so module of this L to hypothesis which shouldn't be there as some of you know, but module of this hypothesis we can do that we can extend s one step farther. So yes, for some some more general result involving so abstract line boundary and not a multiple of the canonical. So I invite you to check to have a look on the version of the web. So to put this result in in a context, let's recall the following. Of course, it's good to start with the classic. What's our technology and so on. So let's say that we have a killer family and the line bound along x together with a section of k plus L over the central fiber. Excuse me. So here the hypothesis are that the curvature is any positive but on the total space. And together with the natural hypothesis and then the conclusion is that we can extend you to the whole space. So if you want here, we use defined for k equals to zero in the previous slide, and then we can extend up to k equals to infinity, so to speak, the whole family. Of course, the big difference is that we have this positivity assumption over the whole family. Another another result, which is somehow in the same spirit was established by Zawa Dema'i Matsumura in 2016. And it says the following. So we have here the still a killer family as before together with the line boundary and the section of our shift fk so holomorphic modular t to the k plus one. Again, the curvature is any positive on the global space. And and then another hypothesis that you would meet an extension UK such that this d bar UK divided by t to the power k plus one is n one form which is which satisfies this head to hypothesis on the whole family. And then again, you extends from k to infinity to the whole space. Okay, so so it's somehow in the same spirit. The big big important difference is at this level that our case we don't have any any privileged metric semi positively cut on the whole space. So what would be the motivation for this for this infinitesimal extension if you want, of course, comes from this important conjecture of you don't see who saying that pluricamical sections defined over the central fiber family stand to the whole x. And it is somehow let's say agreed among the people who work on this problem that is not just the problem is not the problem just in itself but rather what we have to understand what are the new tools we have to develop in order to solve it. In any case, what is known is that this was solved in you by you don't see in 2002 for for projective feminism. It's an embedding in the space. And it's one of the very important tools in the final generation of chemical rings. So good famous BCHM. And just to tell you why projective well projective it's needed because that the proof are used in an essential manner this linear series of systems multiple of kx plus a where a is an ample line bond and so on cannot do it without this ample line boundary and the tricky properties of of this linear systems. But okay, a must be there and it's not in the case. Still in the in the setup of of the main theorem that I stated before the for the killer feminists, the particular case of marketing was was known. So he was using host duty in the non reduced non reduced setting. We somehow limit the singularity second admit for this section we want to extend. Okay, so as I'm a very, very optimistic person and I think that you can follow everything very quickly online just like that, I will discuss the proof of this result. Just I will, the idea is to overview somehow the main steps came. So we have a hand this section as k which is this multiple of lambda k. And we are interested in the restriction lambda k divided by dt so this I hope the notation is very clear. So lambda k is an n plus one one form so of course it's divisible by dt. So we take the restriction to the to the central fiber x and then given this equation here. What we know about lambda k is that is deeper closed on the central fiber. And then the conclusion. So what we want to achieve is equivalent to the fact that lambda k is the bar exact. Okay, so but here we only require this on the central fiber. Okay, that's it's an important difference with respect to the usual setting when we try to solve the debugation on the space to the space x. And then so in trying to do so showing that lambda k is the bar exact. The first step would be to do the following to show that in fact this lambda k thanks to the property we have here can be written in the image of the operator d bar and d prime. I mean, so the on the central fiber and point wise in the complement of this set s equals to zero. So deep prime is the covering going on the river corresponding to this to this L and the singular connection. And so this is certainly a good news because it looks like what it should write if we have a look at the hodge theory, this alpha and beta would be smooth. We will be done but that's of course not the case. We only have coefficients here. Smooth divided by some powers of the section s and who's responsible for that you will see it in a moment here. So let's see how we go about this, this claim. Of course, we have this equation on the top total space. We want to get rid of this multiple t to the power k plus one. So the most natural idea would be to simply take the derivative of that respect to t. Of course, we are on a manifold. We cannot take derivatives just like that. So we have to construct some some intrinsic objects, which will do this for us. And this is those are very well known. So we construct the derivative in order to do that. So the covalent derivative that we will take is the one induced by the extension we have at hand. So it would be singular here. So again, fk is just same infinity but still the formula still makes sense. And in the end, we will restrict on the central fiber. So in this case, honestly, an automorphic function. And then another another object here is that is a vector field, a small vector field on the whole family, which is d over dt plus something else. The existence of such object is standard. And then we define the derivative acting on forms of this type. We simply contract with the vector field here XI and take the prime. Okay, so in doing so, of course, we get some map within those spaces, but it's not exactly correct because we get some poles, right? Because of prime here is singular. Nevertheless, so we can use the derivative for this equation. And what we get on the right hand side, it's not difficult to guess. It's simply this, right? So we take the derivative, this lowest index here with one degree. And then on the left hand side, let's see. So we said that we take the contraction with XI and then we take the d prime. So when you commute XI and d bar, you get this term here. And then commuting d prime and d bar, then you will get something like the curvature form. So the formula will look, I mean, you'll replace this by this expression here. So somehow given the type of the connection that we choose, the curvature term here has influence on the right hand side. If you want, it becomes this. So after one derivative, this equation transforms like that. So we lower the index here, which is what we wanted. But we have some additional guy here in the image of d prime. And moreover, this guy is singular, right? Because d prime is the case for d prime. Now, we would like to do this, to repeat this process in order to get rid of t to the k. And then the important remark here, because we have a new enemy here, d prime of this form. So the remark is that when we take d bar of XI, so XI was d over dt plus something, so the d bar kills d over dt. So the contraction is still multiple of dt. And then the derivative of d prime of such object has the same shape. Now, this is just a differential geometry formula rather long, but nothing spectacular is happening here. So the point is now that we can do this. We can repeat this process, right? We can iterate this process. So after k plus one derivatives, if we are at this point, and the shape of alpha and beta, of course, if you have the curiosity of doing this already after two derivatives, it gets very complicated, but we know that the coefficients would be meromorphic forms of the type indicated in the plane. So let's say that we have this equality here at this point. Now, the point is that this equation can be obtained without any L2 hypothesis, but the L2 hypothesis that we have at the beginning, which was exactly this one on the central fiber. Now, this one can be somehow used in order to improve alpha and beta here to replace the ugly ones that we have here, the beautiful ones, by which we mean the following. So we take a low resolution of the divisor. So the divisor is maybe a very singular hypersurface, and we make this divisorial and the simple normal crossing of the support. So we write the components like e plus f, you will see why plus one second in one click. So somehow this equation, so we take the pullback of the integral on x hat, and then due to some arithmetic conditions here with the m and the vanishing order of s, what you can do is to transform lambda k in a form with values in e plus l, where e has this shape, and l has this metric with coefficients strictly between zero and one. This makes the difference between these two type of components here. And then so if we change the rotation and alpha and beta and lambda and everything, so we can forget about everything else. Now our equation looks like that, lambda k divided by the section associated to this. It's in the image of d bar plus d prime as here, but now the novelty is that we can assume that those alpha and beta have logarithmic poles. That is a huge gain that we have out of this simple normal crossing condition and out of this l2. So we transform this equation which is rather unfriendly into something with log poles, which and then so now let's look at this equation a little bit closer. So we can assume alpha equals to zero to start with simply by moving this term on the other hand, on the other side, and observing that s e times alpha is regular because alpha has log poles. And then somehow the heart of the matter would be to establish the following general statement. So here omega c would be a metric with let's say conic singularities, but coefficients are standard like one minus one over m, where one minus one over some integer, which is divisible enough so that the coefficients that we have in the curvature of l, so when multiplied with this here would become integers. Okay, so we need this work expected this sort of metric. And then we have Hermitian line boundary on x, which looks like that. The curvature is has some divisorial part and we allow some some smooth semi positive one one part in the curvature. And then we also consider this time a nq form for q at least one from the values in n plus e. So, and we assume that we can find forms better one and better two with this sort of log poles in e plus the singularities that this metric has such that we have the quotient lambda divided by s e is the image of the prime of better one plus this smooth part of the curvature which better two. So, so, and you will see so if we know here alpha which you said that we can do that's exactly the type of hypothesis we are here in two. So we assume that this holds pointwise in the complement of this divisor and the conclusion is that lambda is d by exact, which is exactly, which was exactly our goal. So, so this is somehow the main result which makes things work at this point. And of course, if this slide is called d v bar lemma is for obvious reasons right because some is somehow if you have the generalized version of the usual d v bar lemma in the Hodge theory. Here if you have the novelty is that we allow the curvature of L to be part of the story. And still if we have the right sign which means positive then we can draw the same conclusion. Now just I will just comment one second about the tools that we use in order to prove such a result. Of course, since it's about d v bar lemma no wonder that Hodge decomposition comes into the picture. It's a little bit more general version that what we learn in school as Hodge decomposition but still given that we are in a very somehow friendly situation having here singularities but which are so that the conic singularities are of the standard so type or before type. So the only reason we need this is because of the singularities we have in L and still get a need of a complete analog of the Hodge decomposition which we can do by simply limiting the usual proof. And another piece of information if you want is that we have we are able to establish as a consequence of this version of Hodge decomposition and we have the so-called Deram-Kodaira decomposition for currents which are induced by log forms with log points. Now so Deram and Kodaira prove that if we have a current on the manifold of a PQ type then you can write it as we can define his projection on the space of harmonic forms and plus the the plus of the green operator of some other current. So the Hodge decomposition holds if you provided that it's written and interpreted in the right way in the sense of currents. And if you want one of the the first very nice application that I know of that was by Judo Noguchi who showed that actually holomorphic forms with log poles are closed in his in his proof this wasn't somehow the Kodaira Deram decomposition for currents was that the first very nice spectacular if you want application of these techniques somehow we do something like that here. Okay so this we use it for this type of currents and then well a few things are happening and and we are able to show that we're able basically to use that the the the familial like the the familial duality duality argument to say that lambda which the form of the right degree bounded by the quantity that you know. So yeah unfortunately I cannot I cannot go further into the details of this but somehow if you remember these keywords I think that's rather that's good enough and in any case this solves the problem here so for q equals to one we have exactly what we are waiting for expecting that lambda is um um d bar exact. Fine so let me um now present an application of of this this d d bar lemma and it's the statement is the following so we have a compact manifold together with a simple normal possible divisor e and we have also a line bundle which here is called f which is endowed with a metric non-singular where this exists after all non-singular metrics such that the curvature is semi-positive so it's semi-positive emission semi-positive line bundle and we also have a holomorphic section of some multiple but it has the the property that when we restrict to any intersection of the components of e this is not vanishing identically okay and then the conclusion is that the map this section induces in cohomology for hq in these forms so nq forms with values in e plus f is injective for any q positive positive so this is uh one of the um results in a whole industry called injectivity theorems and then I put here some some some of the contributor started with Tankev, Kolay and Osava maybe uh somehow at the at the beginning some many things happen in the meantime and in nowadays maybe the main contributor would be here like Fujino, Chan and Choi and Matsumura somehow and and if x is projective this was established by Osama Fujino and he asked this question one of his work like some time ago um in the meantime many particular cases were established but somehow not in this degree of generality so let's see what this has to do with the db bar lemma so I will only discuss a very particular case when this zero set of of the section which gives the map is a smooth hypersurface it's y and then things since we have the this hypothesis that s restricted on any intersection is um uh non vanishing non non identically vanishing then of course the sum here would be we have a simple normal crossing and then so let's take an nq form alpha with values in this e plus f such that when we multiply it with s then it becomes the bar exact and then we will do something horrible we will divide by s and f e so of course this equation will still hold but uh naturally in the complement of the zero set of those right because I mean we didn't solve the problem right just reform reformulate this equality here and now so the idea is to somehow do something with this form in order to make it look like in the image of the d prime plus eventually something having to do with the curvature of f and then this is done as follows so let's say that we consider the following metric on on f like the this complex combination of the the smooth metric that we have uh this phi f and a small fraction of the uh of the section s itself okay this this defines a new metric on f which of course looks at the first slide like a crazy thing to do because we have here a perfectly smooth semi positively curved metric and we replace it with something which is singular but the idea is that by using this by using the singularities of this this metric we can construct these forms theta beta one and beta two such that this new form here has only log poles along e as opposed to this one which has poles along e plus y and then the the other ones beta one beta two they have log poles along e plus y so once again in order to do these the singular part of the of the metric is crucial okay so so somehow we do simple manipulations with this and vector fields smooth vector fields tangent tangent to y in the tangent in the logarithm tangent excuse me associated into y and then by by construction and derivatives we can replace this with this the idea is we can eliminate the poles of the poles of along y and now so if we combine this equality which holds again in the component of the sum with the the previous one it follows that we have this new expression lambda minus d bar of this form divided by s e is in the image of d prime plus theta f now this is again in the complement of this device but the the as opposed to this equality it holds in the sense of currents on the whole on the manifold on x and anyway in any case we can apply the previous result and showing that which gives us precisely that this difference is is the deeper exact and so that's the case for lambda