Mihai Paun

we started with this project with Junior and saw a couple of years ago the idea was to prove some results and also to celebrate uh 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 how 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 a proper smooth family such that all the fibers are Keller and as usual we not KX here the canonical boundary of the global family and then let's assume that we have a line bounded on on the total space on x and for each positive okay we consider this shift X Plus L Twisted with this ugly beast memory so so the the sections of this shift are holomorphic sections module or t to the to the power K plus one okay so deeper is not zero but it's a multiple of this t to the power K plus one okay you are presented in a big down to earth terms and then of course we have a projection from SK plus 1 to FK which is simply the one you will imagine very natural and then the the main question that I would like to discuss next is um so let's say that we have a section here in this sorry we had a pause I mean I we lost you we lost you you uh your introduction of the FK sheaves I see all right well let's hope for for no more interruptions internet here has been uh you know intermittent okay so uh so we we have this this shift FK so like the holomorphic sections or in K plus n modular t to the power K plus one and then um together with the natural projection between two successive such shifts and then the question that I would like to discuss now in the what follows is um and under which uh conditions for example positivity or whatever on L uh a section here in FK can be lifted can be extended One Step farther now uh so let's see what would be the the main result the main result is as follows so now we assume that the the bundle L is simply a multiple of the canonical boundary KX and uh we did not buy l straight if I may call it so the Restriction of L to X and here we have uh we have a metric given by the right multiple of the the section here restricted to the to the central fiber if you want to the zero zero infinitesimal neighborhood the central fiber and then another part of the data is of the hypothesis is that there is some extension of of S as smooth a 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 L2 hypothesis is satisfied for any Epsilon positive okay so this is by no no means guaranteed the event that here fire has uh has singularities but we assume that we can find such an extension Escape and then the conclusion is that we can extend uh our s One Step farther so as is in the image of Pi K okay so module of these two hypothesis which shouldn't be there as some of you know but modular this hypothesis the we can do the we can extend S one step further so yes for some some more General result involving abstract line bundle L 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 um in a context let's recall the following uh of course it's good to start with a classic and so on so let's say that we have a Keller family and the line bundle on X together with a section of K plus L over the central fiber excuse me so now here the hypothesis are that the curvature is semi-positive but on the total space and together with the natural hypothesis and then the conclusion is that we can extend U to the whole Space so if you want here um we use Define for k equals to zero in the previous slide and then we can extend up to k equals to Infinity so to speak over the whole family of course the the big difference is that we have this positivity assumption over the whole of the whole family another another result which is somehow In The Same Spirit was established by matsumura in 2016. and it says the following so we have here the uh still a killer seven years before together with the line bundle and a section of our shift FK so holomorphic module or t to the K plus one uh again the curvature is semi-positive on the global space and and then um another hypothesis that you admits an extension UK such that this uh d by UK divided by T to the power K plus 1 is the N1 form which is which satisfies this head to uh hypothesis on the whole family and then again U extends from K to Infinity to the to the whole Space okay so uh so it's somehow in the in the same spirit the the big big important difference is at this level that uh our case we don't have any any privileged metric uh semi-positively curve on the whole Space so um what would be the motivation for this for this infinitesimal extension if you want of course it comes from this uh important conjecture of saying that plural chemical section is defined over the central fiber of the family extends to the whole X and um it is somehow let's say agreed among the people who work on this problem that is not just a problem it's not a 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 Newton by Anton Co in 2002 for for projected feminism and it's an embeddings and it's uh one of the very important tools in the final generation of chemical Rings so-called famous bchm and just to tell you why projective well projective it's it's needed because the proof or use in our essential manner this linear series of uh you know systems multiple of KX plus a where a is an amp aligned bundle so I cannot do it without this ample line boundary and the three Key properties of of this linear systems but okay any must be there and it's not in the current case still in the in the in the setup of uh of the main theorem that I stated before the for the feminist the the particular case of Mark Levin was was known so he was using horse theory in the non-reduced non-reduced setting we somehow limit the the singularities you can admit for this section we want to extend okay so as um uh I'm a very very optimistic person and I think that you can follow everything very quickly online just like that I will uh um uh discuss the truth of this of this result just I will the idea is to overview somehow the main steps okay so we have at hand this section as K which is this multiple of number okay and we are interested in the the jurisdiction Lambda divided by DTS or this I hope the notation is very clear so Lambda K is a n n plus one one form so of course is divisible by DT so we take the Restriction to the to the essential fiber X and then uh given this equation here uh what we know about Lambda K is that these people are closed on the central fiber and then the the conclusion so what we want to achieve is equivalent to the fact that uh Lambda K is D Bar exact okay so but here we only require this on the central fiber okay that's it that's a deep important difference with this time to the usual setting when we try to solve the D by equation on the space to the SpaceX and then uh so in trying to do so showing that Lambda K is B 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 um I mean so the the on the central fiber and point wise in the complement of this set s equals to zero so D Prime is the the with you corresponding to this to this L and the singular singular connection and um so this is certainly a good news because it looks like what it should right if we have a look at the Hodge Theory this is Alpha and beta would be smooth we will be done but that's of course not the case we only have coefficients here uh 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 total space we want to get rid of this uh multiple T2 to the power K plus one so the most natural idea would be to uh simply take the derivative of that with respect to T now 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 um those are very very well known so we construct a d derivative in order to do that so the the component 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 C Infinity but still the formula still makes sense and in the end we will restrict on the central fiber so in this case uh honestly an automorphic section the function and then another another object here is that is a vector field a smooth Vector field on the whole family which is D over DT plus something else okay so this is the existence of such object it's standard and then we Define the lead derivative acting on forms of this type we simply contract with the vector field here x i and take D Prime okay so in doing so of course uh we get some map within those spaces but it's not exactly correct because we get some poles right because of the Prime here is singular nevertheless so we can we can he was delivered 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 on the left hand side let's see so we said that we take the contraction with PSI and then we Theta D Prime so when you commute PSI and V bar you get this term here and then um uh Computing D Prime and D Bar then you will get something like the curvature form so the the the formula will look I mean you'll replace this last by this expression here so somehow given given the the type of the connection that we choose the curvature term here has uh 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 um guy here in the image of D Prime and moreover this guy is singular right because now we would like to do this uh to repeat this process in order to get rid of t to the k and then the the important remark here because we have a new enemy here D Prime of this form so the the remark is that when we take a D Bar oxide so PSI was D over DT plus something so the D Bar kills the over DT so the the contraction is still multiple of DT and then delete the derivative of D Prime of such object has the same shape now this is just a differential geometry uh formula rather long but nothing spectacular is happening here so the point is now that um we can do this that we can repeat this process right so you can iterate this process so of the K plus 1 derivatives if we are at this point and the shape of Alpha and beta of course it's uh if you have the Curiosity of doing this already after two derivatives it gets very complicated but we know that we know that uh the coefficients would be metamorphic forms of the type indicated in the claim okay so let's say that we have this uh this equality here at this point now the the point is that this um uh 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 uh can be somehow used in order to improve Alpha and beta here to make to to replace the ugly ones that we have here beautiful ones which is uh which by which we mean the four I mean the following so we take a low resolution of the of the devices so the device is maybe 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 y e plus F in one second in one click so somehow the this equation uh so we take the pullback of of the integral on X hat and then due to some arithmetic uh conditions here with the m and the vanishing order of s what you can what you can do is to transform Lambda K in a form with values in E plus n where e is has this shape and L has this metric with coefficients strictly between zero and one okay 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 Lambda K divided by the let's say the section Associated to this it's in the image of D Bar plus d Prime as here but now the the novelty is that we can assume that those Alpha and beta have logarithmic poles that is a huge gain that we have uh out of this simple number causing condition and after this L2 okay so so we transform this equation which is rather unfriendly into something with log poles which okay and then uh so now let's look at this this equation a little bit closer um so uh we can assume Alpha equals to zero to start with simply by moving this term on the other on the other hand on the other side and observing that se times Alpha is is regular because Alpha has uh log pulse and then somehow the heart of the matter would be to establish the establish the following general statement so here Omega C uh would be a metric with let's say chronic singularities but coefficients are standard like one minus one over m well one minus one over some integer which is divisible enough so that the coefficients that we have in the curvature of L are so when multiply with this here we will become integers okay so we need this this to work we expected this sort of metric and then uh we have a hermitian line boundary on X which which looks like that the curvature is has some divisorial path and we allow some some smooth semi-positive one one but in the curvature and then we also consider this time a NQ form for Q at least one a form with values in L plus e so uh and we assume that we can find forms better on and better too with this sort of log pulse in E plus the singularities that this metric has such that we have uh the quotient Lambda divided by SE is in the image of D Prime of beta 1 plus this smooth part of the curvature wedge beta 2. so uh so um and you will see so if we ignore here Alpha which we said that we can do that's exactly the type of hypothesis we are here in two so we assume that this whole is pointwise the complement of this divisor and the conclusion is that Lambda is the bar exact which is exactly how which was exactly our goal so so this is uh somehow the the main result which makes uh things work at this point and of course if this slide is called DB bar lemis for obvious reasons right because um is uh somehow if you have a generalized version of the usual typical name and the host Theory okay if you have the novelty is that we allow the curvature of Health 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 the the tools that we use in order to to prove such a result of course since uh it's about digital Lemma no wonder that host the composition comes into the picture it's a little bit more General version that what we learn in school as horse decomposition but still given that we are in a very um somehow friendly situation having here singularities but which are so that the connecting light is are of the standard type are all default type so the only reason we need this is because of the singularities we have in L and still get a we need a complete analog of the house decomposition which we can do by simply mimicking the visual proof and another piece of of uh information if you want is that we have we are able to establish as a consequence of this version of host decomposition and we have the so-called the ramkodea decomposition for currents which are induced by a lot for not forms without points now um so that if we have a current on a manifold of a PQ type then you can write it as we can Define its projection on the space of harmonic forms and plus the LaPlace of the green operator of some other current okay so so the the horse decomposition holds if you provided that it's written and interpreted in the right way sense of current and if you want one of the the first uh very nice application that I know of that was by Judo noguchi who showed that actually holomorphic forms with blog posts are closed in his uh in his proof this was somehow the code era the composition of a current was uh that the first I think very nice is 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 um and we are able to show that we're able basically to use that uh the the familiar like the The Familiar Duality Duality argument to say that Lambda which the form of the right degree um is bounded by the quantity that you know so um yeah unfortunately I cannot I cannot go farther into the details of this but um 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 um for Q equals to 1 we have exactly what we are waiting for we're expecting that Lambda is um debug exact sign so let me um now uh present an application of this this and uh it's that statement is the following so we have a compacable manifold together with a simple number possible noise e and we have also a line bundle which here is called f which is in down with a metric non-singular well this exists after all in more similar Matrix such that the curvature is semi-positive so it's a semi-positive hermitian 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 uh intersection of the components of E this is not Vanishing identically okay and the conclusion is that the map this section induces in cohomology for HQ in this form so NQ forms with values in E plus f is injective for any Q positive so this is uh one of the [Music] um results in a whole industry called injectivity theorems and then um I put here some some of the contributors started with tanker Colonial server or maybe uh somehow at the beginning some many things happen in the meantime and in uh nowadays maybe the main contributor would be here like uh fujino Chan and Choi and matsumura somehow in an axis projective this was established by uh and he asked this question one of his work like um some time ago um uh in the meantime many particular cases were established but um somehow not in this degree of generality so let's see what this has to do with the DP bar Lemma so um I will only discuss a very particular case when this zero set of of the section which gives the map is a smooth hyper surface it's why and then things since we have this hypothesis that as restricted on any intersection is non-vanishing not identically Vanishing then of course the sum here would be would have a simple number Crossing and then so let's take a NQ form Alpha with values in this e plus F such that when we multiply it with s then it becomes D 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 naturally in the complement of the zero set of those right because I mean we didn't solve the problem right we 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 convex combination of the the smooth metric that we have this Phi F and a small fraction of the uh of the section s itself okay in this this defines a Pneumatic on F which of course looks at the first side like a crazy thing to do because we have here a perfectly smooth semi-positively Curve Metric and we replace it with something which is singular but the idea is that um by using this by using the singularities of this metric we can construct these forms Theta beta1 and beta2 such that this new form here has only log poles along e as opposed to this one which has spawns along e plus y and then the the other ones better and beta2 they have log calls along the B plus y so once again the the in order to do this the singular part of the of the metric is crucial okay so so somehow we do simple manipulations with this and uh Vector Fields smooth vectors tangent tangent to y in the tangent in the logarithmic tangent excuse me uh Associated to Y and then uh by by contraction and derivatives we can replace this with the distance the idea is we can eliminate the poles of the pulse of along y and now so if we combine this equality which holds again in the complement 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 the prime plus Theta f now this is again uh in the complement of this device but the the the as opposed to this equality it holds in the sense of currents on the whole on the manifold for x and anyway in any case we can apply the the um previous result and showing that which gives us precisely that this difference is uh is the deeper exact and so that's the case for Lambda