Kähler manifolds and transcendental techniques in algebraic geometry

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Kähler manifolds and transcendental techniques in algebraic geometry
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Our goal is to survey some of the main advances which took place recently in the study of the geometry of projective or compact Kähler manifolds: very efficient new transcendental techniques, a better understanding of the geometric structure of cones of positive cohomology classes and of the deformation theory of Kähler manifolds, new results around the invariance of plurigenera and in the minimal model program.
Keywords projective variety Kähler manifold Hodge theory positive current Monge–Ampère equation Lelong number Chern connection curvature Bochner–Kodaira technique Kodaira embedding theorem Kähler cone ample divisor nef divisor pseudo-effective cone Neron–Severi group L2 estimates vanishing theorem Ohsawa–Takegoshi extension theorem pluricanonical ring invariance of plurigenera
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How will they thank you very much for me while considering the Booker From this I try to make more efforts to finish initiate a reasonable amount of time so I concerning an open-source software I will tell you a bit more agenda the lecture But in recent initiative but let us concentrate now on Geometridae long so the subject As A lot of progress in the last 4 or 5 years and I will try to outline on some of the maid Results that have been made In this Area you as you probably know protective under bright varieties and marginally Genoa Varieties Compaq can overwrite especially are Very important geometric objects of course They are used Alan To study of debt The structure of putting on brass of polynomials and so on but they have also it or whatever interaction with the law physics for instance according to some Modem series of physics on space-time could be 10 Dimensional space among which for Here All the placed on the remaining dimensions could be what To by Damita's called Kennerdell manifold namely A comeback And manifold which has As you roll Richie Cabot I saw a rich editor is given by Ken Einstein conditions so you see a gain during these equations have been already emphasized the lot Since the beginning of this conference With a fundamental where better I haven't done on the year the Richie Phillips alone So what I was starts bond if you look at it Torre concepts You can still get On a soda Noel goal is to study geometric topology chronological properties of Compaq and manifolds so what is this apologized to experts that I will give Most basic conditions so you start with Complex And dimensional manifold so it's a matter that is equipped locally coordinates with change coordinates and then Ala On these you consider Different forms Guide to chew you of disease which is a sort of You J. of J Cable But index notation DJ is just JAY went such from Rutledge Jerry wedge product He media forms and UK Boris 1 OK and then you can do costs Funchal exterior calculus as usual but we are interested in more general don't Currents So current is essentially the same A disclosure form except At the coefficients T. 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not body Using a holiday from Serie so you actually proved wife and 2 techniques and to college methods that . 8 feet shoe is given by The big harmonic forms With respect to price parade Associated to any get metric of course it is fundamental in this Setting that manifold is a compact get off our these these leads to 1 of the Millennium problems and they need much conjecture We It is still unsold So it's a very very basic questions of Andhra by free the question is to correct rise uh What All are The comedy classes all That it take under bright psycodes along with puke efficiency and the projector Saying that they should be exactly that Classes which are rational So the rationale condition is essential because of course 80 and a bright side has that deprivation on some hollow You have to take the intersection between the complex structure which provide I think you be the Which tells you about The fact that the varieties of close themselves so you have to pull information and you should take the intersection of these are the intersection here This is actually a vector space but this is is essentially a rational elected solar you could take about proficient than you would get re-elected as an end On the hearts the composition of course Can intersect the such space in a random way so this doesn't Pulido mating behavior some also a hot and tourists In many ways they very very arithmetic question Although the conductor is essentially for protective varieties I recently jet was came out in 2001 with account of them both For the kettle so you cannot Could ask the same question for him varieties but it turns out to be wrong So she exhibited a four-dimensional complex forests which has a nontrivial last in this Here Hodge last died due to damage it dimension which is in no way Related to a year and to write All analytical drink so don't even know coherent chief given which gives a thumbs up and classes This class so this is a very strong Don't or examples The Hodge conjecture Indicative of not explain much about The Based on I met construction of all the Also studied by heat to go out and accuses Iraq an lettuces namely a study of emission young connections but not time here to tell you more OK so Now I will Give us Some hints about results that are related to the good Iife betting furor So let me recall this fundamental theorem which goes back to the early work of but art in the fifties 53 54 which actually led into the field medal He Seoul of Purim say is that The following 3 conditions I Quebec They make around random comeback complex and dimensional metaphor It did equivalent That banks It is protected and right Can be better In complex Based Sekhar aren't Eds But prices have mission for more inclined bundle which has positive cavity So this is a differential geometric condition namely positive covered so of course we are In the complex case still cover cherish be meant And A convex Convex condition in the complex case and complex he again is 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On wheat polished duly host was called the class OK and now dissent paper In 2004 actually Proof was already in the works 2001 of we got completes a contractor isolation of the kettle couldn't enough so it it is The originals urine Baker And on maturity comeback can manifold And now we are going to define P O P stands for positive mainly the new really positive facets of what means Medicaid positive so you pick and not maturity And that it did Some variety of dimension He I met You we climb that When you integrate the form On Such Varieties aren't you get Positive number will be evaluation of the comedy class And and her bright on an elliptic psycho should be people that's OK because they finds Of course which has no reason to be convex the Gary by but the buttons But Not Convex Regional and the here and is that The can of course is 1 of the chambers of said And of course Open conditions The cab code is just 1 of the connected components of that set up a Nevada keep gage of more Actually you have only 1 OK is exactly he so you have A very precise characterization of the chemical even tho it is Continental It is characterized by a bright objects your project so in some sense it is a step toward the Hodge conjecture Because the Hodge conjecture Tells you that you can relate to all Continental object to enter bright wants and he has leased year-end tells you will die Although you look at all the Continental classes they can be computed in terms of purely bright objects Of course I don't claim this is by far from thing that will lead to the proof of the hot conductor but at least it is 1 of the 1st results we have at hand That It gives you a strong hand that You have Really on a coincidence between Continental object under bright This can be viewed as a continental version that world Bright Newman tributaries known as and emotional picture on the other hand if you pick a genetic complex borrows situation it is easy to describe Because It is well known On the generally complex Torres You have no and that it eagle-eyed your bright subsets at Old except finer points Frank said on the Taurus itself so denying Torres Has absolutely no income notes part And therefore only conditions for For years It is when you integrate on makes itself Since orgies Is trivial It's just given by constant forms They said It is just The forms Whichever Positive that down positive that dominant thought permission for me is of course an even number of Agon value Of negative agonized and therefore you have several chambers in that case deposited the timbre but also owns 2 McGerty baggage values And minus 2 was it wants this nice another tempering P and of course the chemical is just a positive this is what we decades an example where is is different from Pete OK so what's the proof of the truth
The game also uses the Richie equation Can and yellow durum so you make a class which is necessary cable are And you want to compute cave art And you assume that But you have the required conditions here that is the voodoo Mr. keep positive and now you can use the and was here and to solve the Mojave equation Here and fire press It's honor guard rest I bought Epson on again It is something so He Function that you should sold for And here It is the right hand side yellows Theorem tells you that you can pick at That's sit on at random so any right-hand side here It is Acceptable provider There The total Is what you expect of course this is a necessary condition that you get all of this is the same as the interval of this because of Stokes theorem gives no contribution interval and therefore you should have condition but you can't draw The board you exactly what you so The meeting of Dallas era is that you can put the Camacho where you like on and putting the temperature away like is the same as putting the volume where you not so you can put the value here and very last year Small here or conversely you can put the body exactly where you want on the manifold OK and then You take profit of this by concentrating will you actually Along a given Analytics a variety of successive variety and then you can concentrate devote whom belong that's a variety and go to the limit at actually you may have new and that it thinks of varieties so I follow your fathers But then you take The diagonal Trixie take acts by X and then you are sure to have a big sobriety thing the diagonal so you do this with a diagonal Are you have to take of course the sum of the projections here Of the castle on each checked And then you concentrate the body on the diagonal of the product I'm doing this and then report Inc On 1 of the sectors Our gives you was think positive Because Essentially You have to the begin along the diagonal because of put all the volume on the diagonal And then you project on 1 factor so you will reported the diagonal 1 factor And therefore you you have stick positivity because a diagonal Both projects On subjective to each factor in this way you actually produce the can car so this is the proof of these fundamental that Which is that shattered Last In the Of all the chemical here just made the bomb but If the volume is 20 positive Our then Actually It cannot guarantee each shed Lien Chan of effective so what this means you may think it is that possibility To codes and patched together but If the Dutch Get on the mound of each year Now I can already be at a point Where the body was 0 Because otherwise you can't 20 model Croatia so I think this is another very very used food Khan sequence of the Richie Croatia which is 1 when was but it has been was also in similar ways for other recently OK I will maybe keep this The A essential step in the proof of the computation of Kate because you proceed now by induction So The did a lot of X which would be the also on every under bright variety so you do we section on dementia you would you make the same argument on every Analytics a variety so you produce It can afford cannot current on each sobriety and by using induction then you conclude the made you're OK and then Long once you have approved this thing get back Gary so I another general statement which characterizes the can Classes without taking necessary Connected component which is independent Of selecting the timbre Jedi Just a rephrasing actually will not insist that you can get in his way also the long-delayed on so it doesn't have if an audience Every can afford the intersection On that As this will get a forum on any And that it take variety is known they get money journalists Exempt Exactly characterizes The Open best Do all of that is just to think that you can here that generator body Add and edit cyclists meant applied in ecology achieved by afford so it gives you and not very precise geometric picture of the dual of the chemical and they do will And point accountability is just generator
1 of these Uh which products OK so not 2 of the fury of the formations Of course it is very important that an entity Dumitru to study on the basis of you Yemen and bike object but you want to know what all the objects that look similar to the 1 you diploma delivered a bit of course this is the same as When you have a pretty normal change in the coefficients And then you look at the zeros and then if you move in a bid to coefficients 0 said forms itself and you want to study the geometric forms Variety OK so it does not their formation is just a matter where as the Barnett based on you assume that you have smooth virus fixed he poked fun I'm on the cheek by meant and assumed that the fibers are Compaq complex manifold internally will assume that they are compact can you protect the basic questions raised by could I in the sixties It Following 1 It Is it possible To obtain And combat Can manifold has limit of auditing variety Always mused That You should realize The given manifold xxx as sent on fiber Which is They met No you by fibers Eckstein you which our project Care so Real numbers are limits of rationalist and some ways of the Continental Can a manifold should be of measure bright amount this is the expectation Well this is true in many cases there are many examples but I doubt whether very recently found for killed This by producing a very clever Example Which is not that difficult after all this just a modification of the forest Any mention floor so you can buy can manifold which does not have A little bit died of any buttock may force account via the formation doesn't have the correct apology God she even produce something That even after you make surgery on even after you roll up anything you want like this it is not even a good With the peak died after you so there's nothing you can do to get this OK so this is a negative results But on the positive side How could they Spencer Proved that results We know that can Properties table so if you did form Can a metaphor Man is still Taylor but this is emetic topology So it's just Small bowl on you'd like to know whether This is true more underwriting so the projector is as the you Baker defamation here The family of medical depending on some space A news show that 1 of the fibers get and the questions For which Monitors In advantages Based on the Fibers can book the project is that There should be more most everywhere Tho All parameters I'll decide Accountable Union of and that it accepts that the quality base shoulders as measures it will said the cabin union of could be mentioned 2 varieties in the real sense um For those lands which I'm not get out It should be a by normal together so eager for duty this is the expectation OK and then we got something like this Which is the 1st step So we True At least that The can of course that the debate saying 2 Chief those would be studied Formation of the chemical so we essentially proved that it cannot cope this is constant outside accountable you And then we are Almost at the point that we can prove the conjecture but still I don't think it this down because they still 2 points to check what might be done in the near future The main thing is that the can of are essentially stable by what it called the ghost many connection so you can of connection given by By variation of pushed cut And then using this connection You can't show That the Caddo code essentially is constant with But constable according to connection pay no Oconto another Which is the study the Croat so There is a mechanical which I explained long-term so you have now of the dual columns So you have the cones of all But Crows currents degree and this 1 and minus 1 of them in the duel Say you have to be effective currents Yeah the theater bright spot of the you've achieved goal yet the movable curves So in this dumb energy the movable All the Carol But you can obtained by taking a complete intersections of type thing sections
See you take I based on the auditor varieties they completed sections they get down to occur And now you you take this in the real world Are you push forward I developed now You get a cup of which feels the variety and moves cover a whole variety of this called the movable And we get The cold end And And you can do this also with firearms so then you replace a height of infections by Cannibal and this is this year this year so you have here the 2 columns I explained the effective quota
Careful this is inviting 1 year and here yet said ability But here are your inviting me and I want a nice 1 year of devised here So these are the movable and these are the random effective maybe don't move the exceptional cover are OK and now it turns out Net the movable device that began buyers on new loans to the curves I don't yet NTT visors are billed to the movable and this is setting you resell time was not for the question for full of Where Entally I'm exaggerating it a bit because
The is the obvious intersection Monday that result Is that These are I explained Blood For the other day Of course we we can't prove it only had bright category So exaggerated limited and the picture because it option of this is still unknown Should be true but Some arguments are missing So this is the consul poplin Gilbertian So I have no time to our explained this because of worries amount of time but the proof realized a game Only on the intersection of currency Decomposition Which I explained I will skip this this is the intersection product defined by bookstore also like Chile On you can intersect in a positive way in and the Canon Inc Away on any compact manifold you have What I call Movable intersection fury means that you take you take psychosis you take intersection but you remove the thinks parts so that you always get something positive I. it turns out that this can be done a completely generally wait our even Passes and this is extremely useful and approved by explained much more You get A composition as well OK and then you have The amendments conjecture The relates The could AID mentioned a member of sections to The United but dimension of the 1st class So I This is it Somehow Dramatic The area it measures the People Of positivity of because that's the conjecture Saying that This You might getting married and should be equal to Niger bright barrier which is a good idea mention things Mrs. Popper state of abundance and is known to be true in some cases but a hot cases often This is the proof of the theorem which is Essentially a Comdex but I will skip the Gave up Currency sequence I will finish with this It is the game a solution of the long-standing conjecture Which characterizes the unity ruling Variety variety is called Unity rural if it is covered By And rational the you have The key 1 complex line which moves In the variety and covers everything tho the despite condition You knew would arrive and the here and prove is that Part the Kennedy told bundle is Still effective if and only if the variety is not so that the Not rude variety on exactly the ones which I said my positive Richie 2 or On the big do you would what are exactly the wines which have not said my positive Richie cover So This way you had very very clear genetic feature of the wood varieties in terms of cabbage OK there's been a aura Other progress On this area The body of people of stew for instance on variants of religion or using consonantal techniques and it is amazing That Rather it's results at the moment no enter bright troops Even those statements are completely under bright so I hope Given fair Account of some of the important techniques in the field stock you mathematics 5 1 minute I should tell you just 2 seconds Something about Open source and open publications so I could take you put in the Meeting to announce that We have Created the site which is called open science that tightly connected to the archive Brick Prince basis armed If I were I hope the goal is to have A completely Open way of publishing and mathematical papers so we are providing automatic Management All of us review with a referee Mathematical papers on the This would be a song called point where you could find them Hopefully The future old papers from archives John willing to put your papers or cut And then we will help People Organized Committees And we are already corroborating children also and of course This depends a lot On the On the willingness of people to collaborate more will be announced very soon so the is Open signs You'll J and But nobody does that I hope that you will Take a look See if it is can be useful for mathematics just to have a completely open and free access to the medical papers in the near future thank you very much