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Derived Torelli theorem for K3 surfaces

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Derived Torelli theorem for K3 surfaces
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Classical Torelli theorems are in their very formulation restricted to complex algebraic varieties. In this talk I will discuss a Torelli-type theorem for K3 surfaces in positive characteristic using the derived category of coherent sheaves as a substitute for the integral structure on Betti cohomology. I will also discuss various arithmetic consequences. This is joint work with Max Lieblich.
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when you're watching thinking organizers it's really a great pleasure to be here and to speak out and then I wanna as revisited that some old questions about to rally Theron said that are of course very influential papers especially with the Super singular case and I want to try to do rethink alone how we might freeze to really theorems and positive characteristics of uh I'll talk about a joint work with the lead Blake in time and just 2 against ideas for memory begin by reviewing some a basic to really clearance of the complex numbers and and also I have in mind literally theories for Monday's talk about catering services for this lecture but lessee we haven't the 3 service but in the end so as well would you consider you 1st considered that the letters which is the singular colleges z qualifications which has an inner product and so the sum of 2 bits of data there's less and which is so
that minute part and then there's the hot filtration the algebraic part hostel for the K 3 of course it just looks yeah well there's really just 1 piece of news I it really you just need to know this which is a really 0 in its candidates from from that and in a product you can get the the if 1 of them and so I the rally In standard formulation and says the following so I had to play 3 services critics and isomorphic if and only if there exists an analyzer morphism planned acts so that means including the In a product that a preserving the filtration but he donated to the vessel none of this is a delicate question so as people were hurt in the Galleria writing some of the answers officers celebrated remark that and this is a new idea U.S. because I 1 of the process .period impossible the characteristics so if it is in general he needed to take care cluster killer class so it I do it's a signal signal it's clear requests To Kill a class it been signal lifts to analyze a morphism that a products until the X to Y this is what you're after he can handle we have to ask if it's unique let not saving a spot of lunch on my feet on the original holiday so I was online I don't to want that's the question is is it and is injected now that's a questionable died I wouldn't say something on the on and so am I right enjoyed in general you can arrange this then after can arrange this by reflecting but the crust -minus 2 curves and synthesis that somehow very classical story and and so I mean in the very formulation it's not clear what you should do and positive characteristic .period because was integral and my fellow and of course there will this proved they had to rally "quotation mark pursued the singular cave trees but fan which is really the only result I know woods the users commodity as a sort of going around the arise more visible cohomology respecting some structure the Nunavut from this Moffat writes to think so and some of them the rough idea the the people of the reminds us of eventually goes virtually all rights to the rest of the year after that otherwise would have been a little models of the events in the last 5 years uh In my most of his foreign the century right under media so in the you're just relies on minus 1 the dominant legislatively because but did so and so the rough idea here is to try to make up formulation even posit characteristic will replace landed next year by the dried category come on category of coherent James when and the hot filtration that will get replaced by someone filtration can you put on them something coming at the drive category while you can go to the growth the group 10 secure which by Ramon Rocky is and that each job grew tense review have a filtration and which is filtration and it's great it's over gets replaced by the code mention filtration now this is a very In loose sort of analogy as but I'll explain it to rally care for K 3 is in this context and introduce consequences and so on but in general I don't know to what extent and could be that this is a much stronger can result heart condition that you could have truly theories of the derived sense and when it might hold the results are going give me new approval complex now in Baghdad this is Alex ladies against the argument for the proof which will reduce to the complex case from foot so let me give a precise statement and so it came within celebrated closed and characteristic not to and for some technical reasons and so and so precariously K 3 servants that means and so this was so when this came up on the previous talk about me it on the boards X and Y recalled for replied partners we have an equivalent sum triangulated categories between the bounded derived
categories from coherent she gives and as soon and as I said I'm so there's 1 point which is widow come back to later but if you take the grant and a group of the triangulated category at and you have to choose this as amorphous office and the right way but do you have a nice offices here with the child groups and energy from the fear and I'm discuss that I could pass to numerical equivalents but anyway this month to important so I view this as being filtered In recent years and in some instances and went to and the Standard and simple way and then the pheromone this year an X and Y the case 3 services b 4 figs and there exists and equivalents triangulated category is the effects of the and preserving the filtration connection then exercise wanted to 1 of it is good this analogy between during the drive category as an integral structure and plus the hot filtration and life from the government 1 of the things that is supposed be carried you know I guess there was always a lose-lose along the yes yes yes think and he said that within the old noted that many of the 1980 there are those who tried category cottages and survival of these questions of the high quality of the water to the use of this I don't know the answer to this is a very interesting question which is inside and yes I do sketched the argument because it involves the deformation theory and that sort of a lot of the classical the tools of the deformation theory K 3 services but the proof does not in any way all how the producers related to the where you might want to preserve his only to the whole question is how long can you want also the canonical Sufism ample origins and ample then you know already that that need only the filter of I what interesting here of from here the remarks I found is that and I'll explain exactly what the examples are but here a few major equivalent to the Hawaii is not enough this is a very classical complied many people consider you need you can't just as for equivalent of triangulated category says is not enough to imply that X and Y or isomorphic arm but actually this result as I'll explain and I will tell you exactly to what extent the officials come back at the the wives of the season now and that follows the from having the equivalent of trying to such Salemme review said the son of Cho the ingredients that came out of the conference was talk but but may reveal that 1 of the few basics former cried transforms their so that's the right through it y smooth protective were carried the object of the dried category of the product and get to found during a time when sending K 2 for me to go start even in the midst of the standard transformed pullback tenser and push down and interference of Morrell it is really the sort of King intricate things off the ground is that if Helen equivalence it was the equivalent and trying to categories and infant care in there is of the form .period paid FOR Sunday In his and this this tells you not only "quotation mark gives you some geometry also tells you the candidates and how you produce as came of inference fast talk about what you do is you do I mean how could you have a complex on the product the most natural thing is to take quite a bit larger space and she's on X and Peter B. The universal family and here's the thing good yes all the derived categories will be the right categories hinges on smooth projector varieties so there's no issue about finance and regular their founded so what exactly the model spaces and yet so I wanted to write down the extra work is a catering service and so then and fix normalization so further complicates you have the more I actor witches the churn character and at at times the taught classes next square and I view this as being an element of the plan for a catering is defined integrally in general you would have to approve the terms of the number people of working for you the sooner than
planned just and has a very simple description it's a range C 1 of the men the rank last among new squared over Turin Online assumed to have a formula on Monday also fakes of polarization and so and class and then you can consider the staggering finally sold the securities the city stable she's on bags With book I victory the specter given I can see featured no and then you make use modular space this and so the Minnesota for good he is as Francois mentioned as you would have a have they reasons they receive she stated goal in which means that this thing here is that GM durable over some algebraic space and in fact this is a trivial she and overcame free surfaces which denoted Roman font like that and so I mean where the universal she Schaefer lives on the product of X with the stack but since it's a trivial durable and you can descend the universal she's too and so put universal chief media on banks across the globe took a free services and she found and induces an equivalence fighting you know it's you In 2 and so this gives a supplier of the former Kai partners for a K 3 surface it's just another the London stage the fear which is that every and for a local partner meaning case resurfaced with equivalent derived category the events and it is this form In doesn't mean it is clear that these those along the given by this doesn't seem all that is another equivalent now so I'll explain Idaho this follows from the the other theorem which where already that Ecuador's president I'm not saying anything about their work I don't know how to relate the equivalent to I don't want to lift the equivalent of that 2 direct category to and isomorphic In either case I would like this 1 of the following and the promises is a on page some the median age so I should say oversee this is somehow classical it's in the middle names here you put there as so but but I want sketch but in the spirit of these miners to Paris and how I view this in relation to the the the filtered resolved the the we eat so that by the Foreign opposition that In so latest opponent to hear case resurfaces and then flies From documents to equivalents triangulated category this drive categories and then there exists and choices the notice I'm H the it's hard to see the Winsford in here then after treatment posing 5 of the moderate equivalents I want to hear hands and equivalence the Kim page B he so arising in that way he can you can arrange 4 the equivalents To the filter this start I was something which is not a filtered equivalents but then you have these basic steps you could do a unified on equivalents the ranks and you can do this a game of replacing expire modest pace of shades of gray and so once you do that then you can make and this is just playing games with the children variants of the under the chat rooms and so then the corollary is that this result and so the that implies this the spirit here and those who want to add that the result that 2 filtered air if ever filtered equivalence than isomorphic than from this
proposition you get that every and for a workout partners monasteries the pieces of his having I could do it on the other side to about hour the I want wide to be a major release of a free composed to went to work so ,comma say which is more subtle and which is the only finally many were forgiven came forgiven attempts to only finally In the 1st time foreigners and to lower the effects of supra similar from cousin characteristic and I didn't know non-trivial the so and so even tho I think this condition about preserving the the Dakota mentioned filtration very strong and in the case of cave resurfaces you can recover a known the what's known about for a workout partners of K 3 surfaces from that 1 sought to rally and it it would also run and discuss the that's the lady of the proves that all /slash deformation theory but K threes and caterers with the former Kai transforms them and so on and so am I. let me give some of the deodorant or see the the the filter to deride the Kremlin's gives a nice morphism follows very easily from the classical and throwing them wiser but the point is that if you have X and Y it is and then you get this fight P which is given by pullback tenser and push down so that passes to any college series and and so 5 induces In an isomorphism the this what French slumped to devoted age children I think when normalization is slightly different but here's now think original get confused about what think you had a shift by 1 of the the digitalization this this the Kyllonen and I'm age tilted the side this is Soroush Ford is not going to define it in this way and so you so you can look at it as a morphism on these bodies will cry lattices and the filter condition told you that it doesn't it induces an ice water Vermont true for filtered sometimes you get a nice more margins on an and on inch Cho and preserving the tradition the consultation and and so were done by the usual to His words and some sense is a much stronger for imposing much stronger condition than the usual to relieve congestion and let me just remind over here we should have said this before in this case the so if you think about it if I have a point here are a few the kids needs a lot of the skyscrapers chief point is in this modernize basic corresponding Shiva over here it is the Schaefer corresponding to the point z so the ranks in general get get messed up right so here when you have this kind equivalents and it's not going to preserve the filtration I want you do birds are done all these things get scrambled out of the wanted reserve the filtration and in the usual to releases said OK so am have 1 enduring characteristic for now we're in a bit of a delicate deformation theory problem of 3 surfaces and we want to think about it so how to deformed the form and its blind he was the triple consisting of 2 K 3 services amounted to interact and hot and went ahead you do that and welcome the consent there's very little strange stock which is start the perfect complexes the on its crossed 1 story on the contents of which are simple I'm an operative universally global with his peers some object of the job categories of simple means that the the other automorphism functor but this Jeep just yet looks like a simple sheaths and this universally global means that the negative ads from Honduras the 2 0 0 4 I listened to you or post on condition that it imposes conditions then this is an old however may Blakey said which is that this is an outrage that cancel some of the geometric object to
work with our Francis Bacon I'm separated them what kind thing was allowing generally you don't put the simple conditions on the simple condition actually gives us a reasonable space here so the simple conclusion implies that you have the to stay in the it's Xinji endurance all renowned rates space the which is removed 1 DAX preceded the follows also given it was kind of put the equivalents 5 .period so we provide the rest of the events to do what I have we had this gene and products and you can do with some asymmetrically as being the World War II hero and here's explores why accumulators of family of should of complexes on acts as parent tries by what it used to be and and what circulated point you have to verify things like new business to compute the text of a complete why he so for these conditions half European at 0 and the negative effects and this is happening in the drive category of but because this is a former colleague this is the same as XII I do have what and Kenwyne New computer you use you can bring it to the triangulated category on the other side of city's conditions easy to verify when you have a affordable cried equivalents and so what you end up with is a map From what I have to stand which is new give us new peace so if you give you this map and improvements in people are and the conditions the the fight he isn't violence as you pull of this fall if the people of this village and is equivalent to this new Peabody is and open the incident and so you you know some not translated the condition to have a formal cry equivalents to so-and-so map from the St. Louis algebraic space from them right so what can you do with this Barcelona proposition that while it was considered another defamation functor of X so we have a catering service we have its defamation functor which we know and everything about it then so they those 2 listings 1 of the Brazil .period what is going on here is not really the Roman fonts that nothing is separated enunciating a global geometry and liftings 2 officials of the defamation founder of X and a you can don't put it over here the opposition there's lots of Marxism the nice morphism if a fund terrorists a program presentable so In this of defamation founders not such a big deal but it such that but and for every
element .period ruled comments that's something that someone can refer uh K 3 and I have a former THC in 1 arm and off sorry your right the money further assumed by funded him myself foreign and so want to get it when you're in this situation so you can after after changing your choice changing the choice of key so you always have to do this kind of the basic operations can range that 1 5 P 1 0 0 years contends it was there and I'm assuming this is filled tunnels GM of putting a lot of conditions but what that it can arrange that ended the takes the ample :colon the next to compulsory liners of gamble ,comma 1 the United see why there's really not much control over the Stuyvesant and general equivalents make it filtered and ended the war operations to have make further so so there's not much control 1 should do this and the proposition is that you have got reflection is for like that and I view it by very rough analogy with the classical theory as making reflections and an and things like that you know so then you have an isomorphism defamation from tourists Delta the formation of To the deformation why such that very well in the car group and so will make a three-day so the deformation founder of X with the line bundles of some from parents and and you can ask what it does so Delta out of this the formations of the pair the Mets isomorphic lead to the deformations of the fired and so on and because of the way said about this is this a class alignment bottle and so in other words you can you can use a former client transformed to identify the dead defamation spaces in a way that matches up the alignment of stars are explained by phone maybe I should say there is the basic problem we encounter which I think will have time to mention in the servicing of the case and you run into the so so in the classical literature within your paper as you consider deformations of excellent Elbogen could also consider deformations of X with a bundle on bottles and I get more and more complicated than the waterline bundles you consider this is huge and important problem I think because 1st a number of situations where you might try to deform the hiring K free that you might want to deform the case 3 plus 12 courses if it superseded you can deform the whole and let us but hold the answer would like to 1 of the big letters to the In a bid to card of the generic tried for working and right summonses is gonna works so the construction Delta that yeah so well I think it's basically this picture so you take I would say I have a an local W-algebra them and I have a a all and so then and I'd get lifting of arms the DAX ended and the the I get when the DAX added and then here I have the creates of story with and and then here's my wife which here this is the new people or and this is and so will open its subsets
lift uniquely so
ministers the latest and most of information because this device Delta you don't want to the note that it doesn't matter I just have an open sunset and I have a noble think New :colon thickening of X could be as bad as you will know have the votes in the sense that the the local you find Russian army at the stopover there is also a lot of us and as a seller .period which is special to the K 3 which is that I need to know this is flat over over the local environment -dash so much on I was eventually being allowed under the rug but there the main subtle point is and why couldn't just the DX and it was indeed the flatness of this over the fastest way to and I don't believe so that's a lot of work experience so that his clubs yeah most of these have to sit down and think about it but it but eminence and important point that it doesn't actually mean you have to verify that it has some reasonable properties but not enough to be a projector variety the militants opened subsets market and that in fact that there's a little bit more here which is you can lift the peace you see there's an annoying thing which is that the Roman find the units of something like the course modular space and you really want to lift the former Kai transformed to and so in fact it takes a little more work can list the penis and it which is related to me that I mean this is not new have to keep going over in the car and this a 1 because of the stakes from more work but not dwell on that OK and so now as hard perceived as news this year because we want live to characteristics are you better preserves some ample line bundles can algebra rise and get the whole thing so now and so conclude so what you I'm getting a that's makes characteristics the yard the EU and that deformations ex-wife please Over the lifting its ban on but then you know that the only for he the I and II guess absolute certainty that I have at least 1 amount to a makes characters your so and little or no mention a related pointed out right and so on and so then as generic fibers and actually admitted it there's a lot of traffic but this deformation actually if I start out with a filtered equivalent satisfying those conditions this defamation is also going to be a filtered equivalent and this is the time of the year and is in the year of the Rooster this this announced this thing this season have to calculate on the I don't know I donated to be unique and it has a deal with this that the soldier and whether there could be a chronological Minotaur asserted that well do you use the well OK so I thought I'd do I choose my there's an ample line on acts and then I look at the threshold of the dead deformations base of X L and then against paired up here and then there are the only thing I have not serve discusses why you have to get this an actual piece of because you only have a map to the Roman font of elemental cost money as they left a trivialized and during Over the last 2 2 2 to get their actual peak so then generic farmers admit filters former Cleland's and so on and after replaced made by another extension to get the generic Firas isomorphic and then you go back down because using from an amount you cavers were and then use characteristic 0 result the result and then go back to and to close fight so there's a lot of technical points here but I think the key ideas this kind of he's sorry I had to make it completely not so loosely you can the government has hit the hole now it's not something I just follows from the view come from the from the extent Haga and prize-winning making remarked so that that gives that if you never filtered equivalents in the isomorphic the finances alternative the findings result on the I think this over similar cases may be the most interesting because How could you prove that In fact but what if you if you have a super singular case training than any formal counterpart has to be done itself this this argument that it is sketched out to not itself very well to that but the the the problem you encounter sufferers have a singular military trade I saw lots of people delight series know that if you have your answer very which is rank at least 3 in characteristic 0 then you don't have natural for a part but there's another game becomes deformed the King plus they rank was arraigned 3 subgroup inside the SBA it's a little odd to do but that's a basic deformation theory problem you encountered and so it
and so in this case should you can calculate using the 1st is the deformation theory of the polar ice skate said you do still have a characteristic 0 point of the the defamation space from a mail put it as a question in general and I don't know the General I don't know how to describe no information summarized this way deformations and and so on fixed groups on authors paper there's discussion of a fair and you can get a quadratic senior another maker quadratic extension In the vertical and so on and so on but in general what is the deformations faces of something like that look like I don't know In the current to flow that might not have any characteristic 0 . the for over there were no takers who presented the case to this season of the year and the year so of those users yes there is value in history the Union of Local the full license there "quotation mark there no on I think this basis could be motivation to classes 0 so that the point is that you do some trigger the determinant uh of the P and related that to the obstruction class the so so there is a serious argument there are many it's not going to have to do you're right there is an obstruction in general now but if you calculate what that obstructionism is related to the determinant of peace and so if you do it set it up right do you can arrange for the obstruction you don't go to you mean you can because it's a one-dimensional obstructions face and and European Union you have to relate what that obstructionist or something happened the 1st is a nontrivial appointed the OK it's U 2 or 3 minutes although some may be just mentioned sort of and a couple more questions here so I don't know how to do this in general and the other .period is which was raised several times is the way this is set up this is no and U.S. morphism that will construct the isomorphism From the events and I really don't know how to just take the derived
equivalents and then construct the map students that I don't know where to to do it but it's tempting to think about the child group of 0 cycles and soared over you if it's filtered to get a map on 0 cycles but that's what looks like issued to see the morphism but I don't know how to do that and you that and Agassi other questionable right it is I don't know in general and whether filtered equivalents and much stronger condition right equivalence is much stronger than from classical to remove seems to promise a new programs in the hope of 450 about where we think you are few on the other the staff of the colony generating utility the you know the the problem is that some of the way resented the way I understand it I don't know I mean in the in the in the hottest to rally situation yours I know a lot about the ice amorphous mn relationship to the wouldn't have wanted during college write-offs of transformations that they understand and even if international presence and to testicular a prosecutor plants and so on so but here I think it's a total mess and what but what is the relationship between the eyes from office and the the prevalence of triangular pattern of for and the it was only but right now so so give and take this saying how could use event's you know maybe you could impose on condition that says that there's an actual I think it's kind the question is please it was obvious the only thing today and what about the former what was then what is the only thing anywhere in the world you actually Over the past year during war news of what is going on 1 of the workers in Realty actions including the death of whatever ICI graphics yes yes ISO 1 have and what happens under this the Delta well what is going on in here I don't know because I don't understand some of the general by the rights of women somehow to visit the region of the these a woman years with the law In the last year or so that would be great the only answer then I have to see what is termed classes still hasn't votes over to the other side of things yeah the I see idea Francois suggested I I think this is another promising may work to prove it using Arthur's there on form to the Super Single referring to there were questions mm