 Ac yn dwi'n ddim yn ymddangos y CKV konjectori? Mae'n ddigonol yng Nwysgrifennu Arfer. Dwi'n rysyn yn ddegonol. Mae'n ddigonol yn ddegonol. Mae'r ddigonol yng Nghymru yn ddigonol. Ac yn ddigonol yn ddigonol. A'r ddigonol yn ddigonol yn ddegonol yng Nwysgrifennu Arfer. Mae'n gweld glasfen, mae'r clywbeth yn gyfwyrd am y cyfwyrdd gwymau, Gopacum yma,付 yn ystod fod i'r cyfwyrdd. A ddechrau'n dyn nhw'n ddweud yn yr unig o'r cyfrifau C3 ac yn cyfrifio C3. Ond yn ychydig yn cyfrifio CKV, ac mae'n ddefnyddio yn cael ei gynnig o ddiwedd ymlaen theory so I'll give you a review of that and then explain the main tool which is degeneration and relative theory and the huge tool which is something called the M and O P congecture. Okay so grumホff witton theory is non-linear and so it's very hard. In particular you need two fields medallist to sort it out, just for a point, so the theory of mapping curves to a point. y front. Maen nhw eu maen nhw'n gweld yma Lmaxime Dyma yma i chi dweud yng ngyffreddがodd yn genus 0, mai hynny'n rhaid gyda'r linea. A gweld ymweld, mae'n mynd i gyd gan bod y mynd i'w gweld a wneud yma, mae'n gweld ei gwybod yma. Bwyd yn cyd-xiaeth. Yn thatd, gan gweld yma yma, ond hefyd yng Nghymru ym Mhwysgwyr Grymu'r Fwythyn, ym 3 ffordd. Felly mae'r ffordd yn ystod, mae'r gweld ffordd yn ym MnoP ac mae'n dweud o'r ffordd o'r ffordd a'r ffordd yn cael Llarbiau 3. Llarbiau 3 ffordd o'r ffordd o'r unionau o'r ffordd ym Llarbiau o'r Pandor Pandoron Pyxdyn. Yn ymwysgwyr, mae'n dweud o'r ffordd o'r ffordd o'r ffordd o'r ffordd o'r Fwythyn o'r Fwydd C3 three faults. It's still very difficult. You still have to compute the stable pair theory, which is somewhat more linear, but it's still very difficult. And you've still got to compute all these multiple covers and so on, but that's what we're doing. That's what we're doing this talk. So in particular case of K3 services in genus zero, there was this Yau Zaslo conjecture in 1995, which we've heard about before, involving modular forms. And so it's proved for primitive classes, so classes which aren't multiples of other classes, by Beauville and Brian Cohn and Lung, who's here. And then in all classes, but in a very roundabout way by invoking mirror symmetry in genus zero. So relating it to variations of hodge structure on the other side. And then for the full grom of Whitton theory of K3 in all genera, there's this Katz-Clemwaffa conjecture, which I'll explain later. And again, it's been proved in primitive classes. But today we're going to deal with all classes and all multiple covers. OK, so I give some brief review of grom of Whitton theory. So I'm not going to do anything symplectic. So I'll always be working with a smooth projective variety, some homology class. Then there's a modular space of stable maps. Maxime's modular space of stable maps. So you take holomorphic or algebraic maps of curves into your variety. And the curve is rather nice. It's almost smooth. The worst case is that it can be nodal. But the map can be hideous. But there's one condition, one stability condition, that the automorphisms are finite. Automorphisms of the curve which commute with the map. So if the map contracts a P1 or something, then you'd get lots of automorphisms which would commute with the map. So you wouldn't allow that unless the P1 had lots of special points. So there's a modular space of these things. It's a delimodford stack because there's some these finite automorphisms coming from here. And then the space of deformations minus the space of obstructions has a constant dimension given by a Riemann-Roch formula. This is called the virtual dimension. It's given by this formula. And so this fact, this says roughly that the space is cut out from by a bunch of equations in a bunch of unknowns. And the difference between the two is constant. So you might imagine that you could perturb those equations and get some kind of cycle of actual dimension this. So in general the modular space has too high a dimension. But you can get a cycle of actual dimension this done by Leontian and Baron van Tecky. Again proposed by Maxime. And this is called this virtual modulise cycle. And this is what replaces the modulise space. So the modulise space has the wrong dimension. But you use this cycle and you integrate against it. And this gives you gromofwiton variants and their rational numbers because of these automorphisms. So there you go. That's gromofwiton theory in a slide. So this example is conics in p2. So you might have some conics x squared equals ty squared in p2 degenerating to a double line. And in stable map land the degeneration is rather different. So you might take the embedding of p1 as this conic here. And then in the limit you can't take this scheme here because that's not a stable map. That's not a nodal curve. So what you end up with, you find the limit of the equations, these parameterised curves, is the double cover of this line. So the thin line, x equals 0, gets double covered. The branch points of the double covering are the intersections of these nearby conics with this line. I think it lets the score x squared equals ty squared equals x squared. Oh, sorry, yes. Whoops. OK. Put some other conic here degenerating to this. Thank you. Need your advisor to sort it out. OK. Here's another example of cubics. So the example's in black. These are smooth elliptic curves. Imagine cubics in p2 degenerating to a rational cubic curve. So a cuspital elliptic curve. So one way of getting that is you could imagine a smooth family of elliptic curves, blow it up at a point and you get a smooth family of elliptic curves except the central fibre is an elliptic curve with a p1 attached. And now that elliptic curve is negative in this surface so you can contract it. And when you do, you don't just contract this curve, you also contract an infinitesimal direction up the p1. And so what you end up with is not a p1, but a p1 with a cusp. And so here's a family of curves. And again, you could take the embedding of this elliptic curve as a stable map and then you could ask what's the limit in the moduli space of stable maps, which is supposed to be a projective delimodford stack, so it must have a limit. And the limit is this guy up here. So in the limit you get this cuspital curve, you can't just embed a cuspital curve, that's not a stable map because you've got worse than nodal singularities. But the limit is this guy, so it's this map here. This is a nodal curve and the blowdown map, this curve might have, this curve C bar has automorphisms, but the one which fixed this point are finite. So you get finite automorphisms of this curve fixing this point. And so those are the automorphisms of this map, so that's actually a finite group. But yeah, the limit stable map is from this nodal curve here to this. And you can see now you get, this moduli space can be horrible, much bigger, because now, because you're contracting this curve, you can imagine you can now vary this curve and you get the whole moduli space of elliptic curves with one marked point. You could attach to this P1 and also map those to this. So you end up with this huge orbifold, much bigger moduli space, big mess. And in grammar-footing theory you have to do integration over such so-called degenerate contributions where you have a P1 with an elliptic curve attached and that elliptic curve is allowed to vary through the moduli of elliptic curves and you have to take integral over that moduli space. And that gives you the grammar-footing theory sort of contributing to this point here. But so that's one reason why you get rational numbers. This is another reason. This double cover here counts as a half. But there's this BPS formalism, which tells you there must be some underlying integers. So you can see in this case there's a half here in my class two times a line in the class for conic. There's this half counting this double cover here and then there's all the other double covers. But there's an underlying integer which is the one counting the embedded line in half the homology class. And the BPS formalism tells you that that's kind of universal. Once you know the invariance, there are some integer invariance in some smaller homology classes. And once you know those there's universal formulae which tell you where all the rational numbers come from. They tell you about all these multiple covers. So this is all conjectural due to Gopakumavaffa. What you do for the purposes of this talk, it's just numerology. So you rewrite these rational numbers in terms of equivalent numbers by this formula. And these are equivalent data. So given these gromafwitniun invariance, you get these BPS numbers. Given a set of BPS numbers, you get gromafwitniun invariance. And you can see roughly why is because the power expansion of this sign here to first order when D equals 1 gives you this term here. So this is some upper triangular linear. There's some infinite matrix which is upper triangular relating these numbers to these numbers. And it's got ones on the diagonal. So it's completely invertible transformation. And conjecturally, these numbers are integers now. So this encodes all the multiple covers. So that needs in particular that all the denominators in the gromafwitniun things come from the knowing numbers? Yeah, well, I would ask you that. Is that correct? OK. The powers of sign are a century or higher. OK, yes. Conjecturally. So in particular, they are restricted. Yes. The denominators can't be very big. That's right. So here X is a free fold where X is anything. Well, X is more or less anything, actually. Conjecturally, one can also do this. Once you put appropriate insertions in and so on and get numbers, then Rahul, Pandra, Pandra, ITM talk conjectures that this also holds. Yeah. But the physics derivation only applies to Calabria free fold. The Gopacum of Aphra only conjecture this for Calabria free fold. So Rahul has an extension in it. It's more or less. Well, let me be careful. He has an extension for arbitrary free folds and you get versions for surfaces and so on from that. Maybe. Yeah, you're right. No, in higher dimensions. He has other papers conjecturing. There should be some integer structure, but it's not entirely clear what it should be. In fact, it was another source of integer structure. You write from integer to some integer numbers which are caseurating invariants by given number. You can consider this stack of the stable maps through Calabria. It's a horrible minimum number stack. You can see that H is a integer number. Yes. It gives some kind of formula. It's based on trunas or other triangle metrics. But nobody knows whether it's a good position or not. I don't know. And it will actually give another approach to write to Gopacum of Aphra here because the universe of Gopacum of Aphra and the relation between Gopacum of Aphra and given that it's integer is imposing some call integer here. And there are other now approaches that Ian, Ellen, Parker have some integers. These stable pairs we'll talk about later are integers. Anyway, it's clear that underlying grammar for written invariants there are some integers, but some things approve, some things are not. You're not supposed to understand this formula. You're just supposed to understand the philosophy that underlying these rational numbers are some integers. For instance, in an irreducible class, what you find is to the first order these BPS numbers are the grammar for written invariants. There's no room for multiple covers or degenerate contributions. And then to next order, once I double the class in genus 0, let's say, I get this definition here. So this says I count my curves in class 2 beta and then I need to subtract off a contribution from those in class beta, which have been multiply covered. And I subtract off the contribution given in genus 0 by this Aspinmore-Morrison multiple cover formula. This is the prediction. And that is indeed what this formula tells you. And we'll see something about these numbers later on. But these are congetrally integers. And indeed they are in all cases. They'll be computed. Is that right? OK. But I mean, just in this case, you mean? No, no. You mean there's recent work of Inel and Parker? OK. Right. So any questions? That's grammar, foot and theory. Done? I don't know why, then. Can I ask a specific question? So there is also some denominators because of the genus, like in genus 1, because of the modular space itself being a stack? Or is that not? No, I'm just asking something stupid. No, I think. I just don't understand. Say again. So if you look at genus 1, there might also be some denominators from constant. Constant, mapping to constant. Yeah, so integrals over modular space of curves. Yeah. And they're all in there. Yeah. And what's the intuition for how to extract the integers that lead? What's the? Ten-side. Yes. Yes. Beta, non-zero. I'm sorry. But those are all in there in the sense, like in that cubic example I had where you have these degenerate contributions. So any time you have a, imagine you have a P1, then I can glue on any of those curves you said to that P1 and contract. So I get integrals over the modular space of curves. The modular space of curves in that way. And they're all encoded in here. And this conjecture is that they all go away once I do this. So I guess the question is, in the genus 0 case, you kind of explained to us naively, why expect there are integers around? Right. But in the higher genus case, what's the intuition? The intuition. I mean, one intuition is Rahul computed all these integrals over the modular space of curves and Faber. And they got all these formulae. I mean, I have no intuition. Yeah. Gromwff-Witton theory is really a silly... I should be careful. I mean, to try and understand the numbers is really a silly game, in my opinion. One should not try and understand the numbers. I mean, maybe the generating functions, that's a different matter. I think going backwards we would not have started with Gromwff-Witton theory if we'd chosen to try and count curves in a variety or something. Okay, so now we do K3 surfaces. So the virtual dimension of this modular space is genus minus one. And the reason for the minus one is that I can deform my K3 surface so that my class, my homology class, is no longer of type 11. So then I can't possibly have any curves, so the answer must be zero. Okay, so all Gromwff-Witton theory is zero. And we can go. However, the reason for that is the obstruction theory contains this trivial piece which tells you whether the curve class has any zero-two part. And that's what makes the virtual cycle zero. And you can remove this trivial piece and you get something called a reduced obstruction theory. And this has virtual dimension g in genus g. And you get reduced Gromwff-Witton invariance. And these aren't defamation invariance, obviously, but they're defamation invariance so long as I move my K3 surface inside the locus, inside the modular space of K3s. Where my class remains of type 11. So this is called Noeta-Lefshed's locus. And it's a divisor. So it's the divisor in the moduli space of K3s where the holomorphic two-form pairs to zero against this curve class. This is a good point to consider the Gromwff-Witton. Next slide. It can't be simplecticated on this and how it may depend on that. It's simplecticated on the same. So these are not symplectic invariance now. This is algebraic geometry now. But you will see on the next page why it can also be interpreted as symplectically. OK, so you should think of, I don't know, elliptic K3 or something. So you have a one... The elliptic fibas have genus one. And indeed you have a one-dimensional moduli space of elliptic curves. And you can destroy that by deforming the K3 to no longer be elliptic so that the curve class, the fibre class is no longer of type 11. But while you're in this Noeta-Lefshed's locus you can expect it to be one-dimensional. And that's correct. And indeed in there you have 24 singular fibas and they give you genus zero curves normalising the nodal curves. And they indeed appear in dimension zero and you can count them and get 24. And that's one of these reduced Gromwff-Witton invariance. OK, so there's a three-fold point of view and this is what answers the two previous questions. So what you can do is take a disk in the moduli space of K3. I'm not really telling you... but anyway. Intersecting this Noeta-Lefshed's divisor. So this is some kind of moduli space of K3 and here's this divisor where my class is of type 11 so it might contain a curve. And you take this disk intersected transversally and that corresponds obviously to a K3 fibration. So these are all K3s. All the points of this disk correspond to K3s so I get this K3 fibration over the disk. With the K3 I care about at the origin. And really what you're doing is considering curves in this guy. In the fibres of this guy. And ordinary Gromwff-Witton theory. And that changes the obstruction theory. So you can see the problem. If you were just dealing with a point here you could move it off this divisor and you get zero. But if you consider a disk then of course you can't move it off the divisor. It always intersects. OK, so all curves in this fibre class actually lie in the central fibre. But you get a different obstruction theory and you find the virtual dimensions now zero because you're in Calabi Island. And so the invariance about this torque is concerned with these invariants. You integrate one over that moduli space with that virtual cycle of virtual dimension zero. And you can express it in terms of s itself. What you do is you take that reduced class I was talking about before and you integrate a certain hodge class over it or lambda class or the top-chain class of the hodge bundle. So any questions? This talk is about this number. So these are the Gromwff-Witton invariants we think of as the K3 surface. But really they're the contribution actually I might say it, there we go. They're the contribution of how this K3 surface contributes to the fibre-wise Gromwff-Witton theory of three-folds. So these are really kind of three-fold invariants but these are the invariants we're interested in. Right, yes. You can relate it to the Gromwff-Witton theory of compact three-folds which might have many curves where my curve class becomes of type 11. I'll do that later. But for now it's a non-compact three-fold. It's something like the twister three-fold of the K3 but it's not really because everything's algebraic in this talk but it's some kind of approximation to the twister three-fold. Right. Okay, so what's the Katz-Clemwaffa formula for these invariants? I rewrite them in BPS form by those universal formulae. So these are now rational numbers which we conjecture integers and then two miracles happen. Not only do they turn out to be integers but also they depend only on the square of the curve class. Not on how divisible it is. So it's clear that these Gromwff-Witton invariants we've defined by deformation invariants and Tirelli and so forth they only depend on the square of the curve class and its divisibility. They're the only invariants, really, of a homology class in a K3 surface. So they depend on how divisible the class is how many times it's a multiple of another class. But these BPS invariants do not. This is some kind of incredible thing and shows you that BPS invariants are the right things to consider somehow. So if you're looking at curves in two times some class you subtract off some contribution as told to you by Gopakumar and Vaffa for the multiple covers of curves in half that class and then you want to compute what's left. Incredibly you can just forget about this homology class and go to a completely different homology class which should have no relation. The only relation is that square is the same and that homology class can be primitive it's not a multiple and you compute there and you get the right answer. It's really incredible. It's really compact and so forth there will be no multiple covers. Whereas in your guy you've got some non-compact modular space because you started with a compact modular space with loads of multiple covers and you took out some contribution from the multiple covers. So that's part of the conjecture and then the other part of the conjecture is it just computes them all in terms of modular forms. So when you take genus to zero here this becomes the Yazaslo formula this becomes slightly more familiar Yes, when you take Z to one what you get here is all that survives is the genus zero part and you get the Yazaslo formula. So when you take Z to one this becomes 1-Q to the end all to the 24 this famous modular form. So this is the kind of thing you'd want for general varieties you'd want to know you get special functions modular forms or something so that if you just compute a few classes that determines all of them. K3 surfaces. Any questions? There is no explanation of that non-humerical explanation. The modular form has a kind of geometric conscience. There is no geometric directing. I mean this also comes up in the if you look at the coromology of K3 surfaces and their Hilbert schemes this also comes up and then there's a vaguely geometric explanations to do with partitions and Hilbert schemes clearly have something to do with partitions. I think it's not satisfactory. That's correct. In particular in this talk this is just going to be computed by brute force. Particularly at these here's a table of numbers. These are the Yazaslo numbers along the top. These are really incredible. The fact that you get zeros down here really incredible. Whenever you get a curve you get all these higher genus curves mapping to it they don't contribute to BPS numbers they're all there in gromofwitnyn merits but the BPS numbers have already removed those. All you get is lower genus curves mapping to your curves and how do you get that? It's because you're only picking up the curves which are singular. When your curve becomes singular then you can get lower genus curves by normalising and so on. This number is kind of familiar. This is just the signed Euler characteristic of the linear system. You take your curves and the projective space they live in the sections of the line bundle and take its Euler characteristic and that's where this comes from. These numbers are familiar but then all these are pretty amazing. How are we going to do this? We're going to degenerate and as I said pandropandropix improved this MNOP conjecture which I haven't told you what it is yet for projective three folds which can be degenerated to unions of toric varieties. We're going to use that but that's a very global thing and we want to use some degeneration arguments to get a local MNOP conjecture local to S so for this kind of twister space. Then the multiple covers once we've used MNOP conjecture to convert everything into this theory of stable pairs which I also haven't told you about yet the multiple covers become scheme theoretic thickenings of stable pairs and one needs to calculate these and that's not trivial but we'll do that using some other tricks there'll be a vanishing theorem and a localisation calculation OK So because MNOP conjecture again is the fact that everything depends on the square of eta is that the conjecture? Well the conjecture is two parts is that and it's this formula so I have to tell you what stable pair is so this is more like so the slogan is gromafwit and theory counts parameterised curves and this is unparameterised curves this is just their images they're embedded sub-schemes more or less but you'll see why you can't get away with just embedded sub-schemes a bit of a mess so you use these things called stable pairs so a stable pair formally is a pair so a coherent sheaf with one-dimensional support and a section which is stable so it's the stability conditions that the sheaf is pure has torsion free on it's support and that the section has zero-dimensional co-kernel OK so if you don't like that a stable pair is a curve a pure curve no embedded points no points knocking about a pure curve it could be non-reduced it could be a thickened curve but not just at points it has to be non-reduced everywhere if it is non-reduced it can be disconnected and a zero-dimensional sub-scheme of that curve or a divisor on that curve OK that's what a stable pair is so for instance if you just have a curve in a variety you get a stable pair you take the structure sheaf of the curve that's your sheaf that's the torsion sheaf supported in dimension one and the section one or you might have a curve with a divisor on it a Cartier divisor on it then you get a line bundle on the curve with the section vanishing on that divisor that is also a stable pair so that's the curve with the divisor where you get very divisors it gets a bit more complicated so a simple example is where you have the structure sheaf of two curves let's say intersecting in a point and you take the obvious section one that has co-kernel wherever these curves intersect so you take the trivial line bundle on both curves where you intersect that's kind of got rank two in some sense so the section only hits rank one sub-sheaf and you get co-kernel at that point so what you find is you get points at the intersection of C1 and C2 so that's what a stable pair is they form a projected moduli space that was proved by Le Potier and we fixed some numerical invariance so before it was the genus and the curve class here the genus is replaced by this holomorphic Euler characteristic of the sheaf and the curve class is the same and so this holomorphic Euler characteristic here is if the curve is let's say reduced then it's one minus the arithmetic genus of the curve but then plus the number of points and this is a defamation invariant quantity I'll show you in a minute until now X is just a projected variety but it's just about to become a three-fold smooth projected variety so an example is a surface then what you find is it's precisely this moduli space of stable pairs really is curves with points on them so you take, I'll tell you how to read this you take the Hilbert scheme of curves in S in class beta so if the surface has no fundamental group this is just a projective space of sections of a line bundle and you take the universal curve over that and then you take the relative Hilbert scheme of points on the fibres so this is really just curves with zero-dimensional sub-schemes of the curve but for three-folds which is what we're going to be dealing with in this talk and this KKV formula is crucially a theory about three-folds it can be written just in terms of K3 surfaces in Gromwch Whittenland because all these multiple covers live on the K3 surface but in stable pair land you'll really need to thicken out of the K3 surface into a three-fold so it's really a three-fold theory so I'm really interested in three-folds here so for three-folds the genus can jump which is why you can't just take the genus to be your your data up here but it gets traded with a number of points so this is the standard example you have two curves in three-space this can't happen in surface, there's not enough room and you move them until they cross and at that point you've lost a point because they crossed here and that point sort of bubbles off and becomes the point in the stable pair and then in a further family you can move that point around the curve smooth the curve so in this picture you can see the genus has increased by one but the number of free points has also increased by one and this quantity remains the same and in flat families that's always the case and this is the reason you can't just take sub-schemes you can't just take curves as sub-schemes because these points start coming off and you get these curves plus points so in stable pairs at least the point is confined to live on the curve and these are the moduli spaces we deal with so any questions about stable pairs? this is just curves with points on them so this result here says that they're very simple for surfaces and we use that later on for three-folds they're a bit more complicated they can't be described quite in this way because of this phenomenon now for three-folds now I'm going to work on a three-fold I've formed this complex here in the derived category you don't have to follow any of this and its deformations minus substructions have the same properties as before they have constant virtual dimensions the dimension here remains the same it's given by Riemann-Roch formula and it's the same one as before because I'm in dimension three this isn't true for surfaces so there's also a statement here that for three-folds deformations of this complex are the same thing as deformations of the stable pair that's certainly not true for surfaces for instance for surfaces you could imagine this could be OX goes to OC and then this complex just becomes O minus C it just becomes the line bundle associated with the curve and linearly equivalent curves will give it the same complex but for three-folds this pair determines the sorry, this complex determines the pair and so you can do the deformation theory of this complex and find a virtual cycle of this dimension okay so this is the stable pair's virtual cycle and so you get integer invariance now there's no automorphisms anymore and so I'm interested in the Calabière case and you get these integer invariance by just integrating one over this virtual cycle taking the length of the virtual cycle alright so if you didn't like any of that you just need to know there are invariance coming from these stable pairs so we count stable pairs on a Calabière three-fold any questions? okay so now we come to the hard bit so we need to degenerate to make our geometry simpler so I'll describe the general picture due to John Lee this is up in gromofwiton theory initially but there is a similar theory in stable pairs I'll concentrate on stable pairs so what you want to do is you have deformation invariance of gromofwiton invariance and stable pair invariance as you deform the variety the invariance don't change but you'd like to deform the variety to something singular in particular some normal crossing thing and then you'd like to be able to compute there and get the same answer as you get for the smooth guy so morally you think of curves on this central fibre here as being curves on one piece times curves on the other piece but fibre product over that they must meet in the middle they have the same boundary on the divisor in the middle alright that's roughly true and that works fine so long as your curves have a boundary value on the divisor so long as your curve intersects the divisor in a finite number of points you can make sense of this both at the level of modulised places and virtual cycles but what you don't want is the curve to fall into the divisor so if the intersection with the divisor is a whole curve then you're in trouble oops so what you don't want is this picture where you start with a smooth X you degenerate it to a union of two pieces and you follow your curve along and suddenly it degenerates to something with a component lying in the divisor so if that happens what you do is you blow up the total space of your family here and you get a picture like this this is the exceptional divisor I'm assuming I guess that D has trivial normal bundle here in general this is some P1 bundle over D and you get a new picture like this ok so we're not just going to count curves in X1 and X2 we're also going to count curves on objects like this and then again this curve here might move over here and drop into a divisor and then you've got to bubble again it's like stretching the neck in Donaldson theory and I should say there's a C star which acts on this space scaling the P1 fibres and you want to divide by that so if your curve is here or you move it by the C star action it lies over here you want to consider those the same same thing so you want to form a modular space of such things so Jun Lee did this for grom of Whitten theory and then with his student he did this in stable pair theory so what they showed was there were modular spaces of so-called stable pairs relative to a divisor so I'm just doing one piece now just X1 with its divisor D so here's X1 and here are all the bubbles attached to it and I'm trying to get out a curve here which has a well-defined boundary value along D I can intersect with D and he produced such a modular space so in this example we've got four bubbles there's a C star to the four acting and you consider any stable pair two stable pairs equivalent if they differ by this action and they form such a modular space of such things where no curves lie in D they all intersect D in a finite number of points and this is really a compact modular space so every time the curve tries to lie in D you bubble again and there's a complicated art in stack it's generations with these stabiliser groups and they work relative to the stack it's all very ingenious and it works and you end up with a formula like this so the virtual cycle there's a virtual cycle for this modular space of these kinds of objects that I've drawn and it has a map to the Hilbert scheme of D given by intersecting with this final boundary divisor and then you get this wonderful formula which is as a form I said before you intersect in the Hilbert scheme of D you take the boundary values of curves on X1 the boundary values of curves on X2 you intersect so you see where they match up and that gives you a new virtual cycle which is defamation equivalent to the virtual cycle on the original smooth guy all right is that clear I guess in the total space so if I have a one parameter degeneration of a smooth three fold into this guy then I get a family of modules to this guy then I get a family of modular spaces and family of virtual cycles okay so this is what we're going to use so we start with our twister three fold so we're interested in stable pairs for instance they're all going to lie on the central fibre but possibly thickened out of it and you know we can't compute that because these dam case three surfaces keep changing they're moving across the base and we degenerate so we replace this by its defamation to the normal cone so if I take this guy I times it by C I blow up S in the central fibre and I end up with a new central fibre which looks like this okay so I've degenerated this guy to this guy where I've just bubbled off a P1 times S so what I'm doing is I'm doing I'm forming the relative modulised space of curves in T relative to S so the curves can't lie in S because then they wouldn't have a boundary value in S they're supposed to intersect S in a finite number of points and of course here that finite number of points is zero because my curve class dotted with S is zero because it's a fibre class so what I want to do I have to bubble and bubble again and so forth and what I find is now my curves in this relative geometry of John Lee must all lie in this piece they can't lie in the twister three fold because if they did they would have to lie in this crease here and that's illegal in John Lee's theory they always intersect these creases transversally so they must all lie here but now we've made some progress because here S is sat inside with it's really trivial, it's really a the trivial family of S is C times S and we can use localisation and so on so what we end up with is I didn't explain this very well but John Lee's theory tells you that instead of computing here what you can do is compute here so you end up computing curves which look like this so we've essentially degenerated the three fold to its normal cone of S and we end up replacing the geometry of just the twister three fold by this kind of geometry where you're computing curves in these bubbles modulo the C star actions and they all lie in the fibres and you're in much better shape now because everything locally just looks like S times C and now you can use localisation in the C direction so the first thing that's happened is we've degenerated to a modulai space which really depends only on S all these curves here they don't see the rest of the twister three fold they're far away from it so it really only depends on S that wasn't clear to start with it wasn't clear at all that my definition of my definition of stable pairs on this twister three fold appeared to depend on the twister three fold so even though the stable pairs do thicken and they stick out into the twister three fold this invariant only depends on S and we have to compute it so we're computing curves of this form and what you find is this is what guys in this area call rubber because of all these C star actions so it's effectively we're computing curves on P1 times S but relative to this boundary divisor and this boundary divisor and we only allow bubbles on this side and blah blah blah but anyway the fact that really the twister three fold is attached over here changes the deformation theory and what you find analysing all that and I won't go into the details you find the following relation so maybe I just say why we're doing this part of the problem of stable pair theory is it's inherently a disconnected theory your curves don't need to be connected and Gromwff Whitten theory curves are always connected so when you take this log this is the connected stable pair theory so this generating function here once you take log it's a generating function of new numbers which you can call connected stable pair theory if you wish and we express that in terms of the invariance of this so-called rubber space so this is reduced invariance on this space made entirely out of S okay so we now have these connected stable pair invariance defined by this the generating function the coefficients of this generating function and we prove we've now we've now done two things we only depend on the k3 surface and we've computed them we've shown in some sense how to compute them we've related to this geometry here which is rather trivial and we can use localisation on and we'll do that in a slides time okay but the first thing we do is we've proved a so-called pairs not to less chess correspondence because I want to use a global X not just this twister guy because the global X I'll have a so-called MNOP conjecture for relating it to Gromwff Whitten theory so the idea of this you don't this is a little bit hard to pass and the reason for that is in a global k3 fibration you can't just talk about classes on one k3 surface because there might be monodromies so you have to sum over all classes which push forward to the same class on the three fold so it all becomes a bit of a mess explaining it but here's the explanation so what this says is connected curves counting on a global projective k3 fibration come from the curves locally this sort of a twister theory the curves in one fibre times by the number of fibres for which my class becomes of type 11 the shape of this formula is very simple so it says in general you have a k3 fibration how do I get curves in my fibre classes I only get them when the k3 fibres lie in the null to left shift's locus that's this contribution and when they do I get this contribution coming from the twister three fold okay and there's a similar gromofwitten null to left shift's correspondence proof of stable maps some time ago this is a simpler situation because everything's connected and everything lies on the k3 surface there's no thickening out so there's a very similar situation so we have we can relate stable pairs on a three fold to these local contributions we can relate gromofwitten theory on a global k3 fibre three fold to these local contributions and then we're going to apply both to a specific example so this is a k3 fibration it's global, it's projective and it can be degenerated to unions of toric varieties and we choose this one because these correspondences are invertible this formula here determines this once you know the null to left shift's numbers it determines these numbers in terms of these numbers so once I know the global three fold invariance I get these local three fold invariance twister invariance local to the k3 surface so if I can relate pairs to gromofwitten theory so if I can relate this guy to the pairs invariance of the global X then I'll be able to relate these local guides so I'll get a local MNLP conjecture so we just choose our three fold very carefully so that I can invert this formula and I can deduce from the fact that gromofwitten and stable pairs were related for this global guy I can deduce that gromofwitten and stable pairs are related for this local twister geometry so the way we do that is this MNLP conjecture I'll do this quickly the MNLP conjecture is just a relationship between stable pairs and gromofwitten theory and again you're not supposed to understand this formula it's a preposterous formula but again you turn connected gromofwitten theory into a disconnected theory by taking this exponential you then get the stable pair theory but only after this crazy change of variables but all you're supposed to understand from this formula is that the two sets of invariance contain the same information and also there are some integers underlying gromofwitten theory and why does this change of variables make sense is because part of the conjecture is that this right hand side is a loron series of a rational function so here's an example and this rational function is invariant under q go to one over q not the loron series but the rational function if you put q inverse in here you'll see multiply top of m by q squared you'll see you get the same function so it makes sense to make this substitution this formula looks formally very strange because before you wrote the sum with the gromofwitten variance was the sum with the pps right I'm not doing that now yes I agree and then that formula in a special case was a Jacobi formula which is a product it looked like something you would take the log of you take logs of products you don't like to mention it and just formally it's very weird but you really are once taking the log you want to mention it these things are much simpler in some sense there's also a bps formula of this I know you have a matrix in general I can't answer your question you want to take the log that doesn't matterabse when you do both the same series you taken the log in those same series and that looks very weird but the story here is that here you get kind of one dimensional keep it's kind of have two or something around here in case you you do not Mae hwn i oedaf ei fod wedi cinnwys'n bwyd yn gwneud defnyddio. Mae hwn i oedaf am gyfyniadau ond i hyn yn fwy am leuysau'n gartfeytag yng Nghaerfod. Mae hwn i'n defnyddio arno. Mae dweud y gwerth rydym yn credu, fel bod eich hwn yn eto! Mae hwn i atweinyddio arno. Yn nhw mae'n gwneud y ffordd yma. Mae hwn i'r ffordd y galedad. Mae hwn i'r ffordd y galedad a phrys agor'r pwn i'n eu trefod. Mae'r gŵr frying i'rhaeg, gorydd. Felly un o hun gener ein sefail ar gyfer un i tri最 fasten wedi' nhwysnt Mae rows i dd Arist playback'on i'r tŷ Mae'r wneuddwad fy Llywodraeth i bŵie Dwi yma yn llefod mae lle ac mae'r dd putra Mae'r fderdd yma yn y symbolion Mae'r dd800 dreffan yw hypighabweith wi'n sefais ar gyfer le differ Kyw a triangular Dyna gyda 1 perseverance a mwy o chyfrion Fe rydym ni ch diagon allan to do what you said. I don't know. The formulae turned out to be not so bad. I'm not, I'm not good with formulae. Yes, I really mean this. OK, so this is actually proved for projective complete guys which can be degenerated to unions of toric varieties. So it's proved for this guy and for this guy I can invert those correspondences relating counting curves in the K3 fibres plus the counts of which K3 fibres my classes of type 1 1, these in order to left shift numbers, I can invert that and so I can deduce from there proof of the MNOP conjecture, this local MNOP conjecture local to us. So we get this kind of MNOP conjecture for this twist of threefold. So we're kind of, we've done all the MNOP stuff now. This Gromov-Witton invariance that we want to compute has been reduced entirely to computation of stable pair invariance. So now all you have to do is compute the stable pair invariance and as I said remember they thicken, they stick out of the K3. They're really threefold invariance. But we reduce those as well to this, these computations on this so called rubber, this K3 times P1 with this group acting and so on. And you can further degenerate and localize on this, this rubber and you can imagine now you can reduce everything to computations on S times C localize with respect to the C star action on C and that's correct. Okay, so you reduce everything to a computation on here. All right. So we still have to do that. We have all the multiple covers correspond to thickenings in this direction but all the stable pairs are going to be correspond, are going to be situated at S times the origin because I localize and one just has to compute. Now so we can forget everything that went before. I just need to do a stable pair calculation. The KKV formula is now giving me a conjecture about what I get when I compute stable pairs on this guy. And now there's several advantages of stable pairs over Gromwell-Fwyrton theory. And one of them here is that it has something called a symmetric obstruction theory. So half of you know what that is and half of you don't. It doesn't really matter but roughly speaking what it says is vector fields on the moduli space are due to obstructions. And we used that before when we took this reduced theory. So you can take the obvious vector field where you translate along the C direction and that's due to the obstruction that we removed before to get the reduced obstruction theory under this serduality that comes up. And now there's another vector field and that's going to give us a vanishing theorem. That's going to show just as the previous vector field gave us a substruction which showed the ordinary invariance vanish and you have to pass the reduced invariance. Similarly this extra vector field is going to give us a new vanishing theorem, a new obstruction which shows that the virtual cycle vanishes in most cases. So it's the following. What you do is you take a vector field on this moduli space and it sort of flows outwards from the fixed points, the C star fixed points. Okay so what are the C star fixed points? You have some stable pair which I'll consider as just a curve. It's supported on the central fibres but scheme theoretically it sticks out in the C direction. And now I'm going to produce a vector field on this moduli space. It's not along this one. It takes you out of this one. And what it does is I flow kind of the last part of my thickening here. So you imagine an n fold, transverse to the curve, you're imagining an n fold thickened point and breaking off the last point to give you an n minus one fold thickened point and a separate point. Okay so one can write this down globally on the moduli space but this is the basic idea. This gives you a vector field on this moduli space. No no no sorry the flow does this and now I differentiate it and I get a vector field. There's a flow just local to here. So there's a flow on this sorry there's a flow taking this into this. So it is as follows. I take my thickened curve and I break off this point. Okay and now I differentiate that at time equals zero and I get a vector field. So on here I get a vector field with sort of values in vectors in here. So I get a vector field which points into here but just along here. I don't get it on the whole moduli space. I just get a vector field along here which points in this direction. Okay. Does that make approximate sense? So this gives me a new vector field on here. All right. Not along here, on here. I don't know how to say this. Does everyone understand? There's a vector field on here but it's not tangent to this. It's tangent to this. All right. But that vector field is the same as the translation vector field that I mentioned on the last slide. That's just a weird property of deformation theory. Clearly the flows are not the same. If I just translate this guy along the C direction that's not the same as this flow. But to first order when you work out the deformation theory they're the same so long as they do the same all that matters to first order is what they do to the kind of centre of mass of this curve, of this thickening. You can see that breaking off one point and moving it along the C axis is the same. What that does to the centre of mass is the same as just moving the whole thing at 1 over D times the speed along the C axis. Okay. So you just have to believe me. These are the same vector field. When you say poignet, can you move the curve? Sorry. I'm thinking locally transverse the curve. I've just got a thickened point and I'm breaking the last point off. What that does to the centre of mass is move the whole, it's the same as moving the whole thing at 1 over D times the speed. This gives a vanishing result because what you find is, locally these two vector fields are the same. But if you move along the curve, if the curve has different thickenings, this D changes along the curve. Also the curve has points and if those points have different thickenings, then that D changes again. So what you find is these two vector fields are linearly dependent if and only if that D is uniform is just constant over the whole curve. So what you find is that these two vector fields are linearly dependent only on the pairs which are uniformly thickened. So this is a uniformly thickened pair. You have a curve with some points on it and you just times it by C and then intersect with the thickening of the central fibre. That's uniformly thickened. For that guy, this vector field gives you nothing new. It's just linearly dependent on the previous vector fields. It's just a multiple of the previous vector field. But for something that's not uniformly thickened, this vector field is completely different because here it looks like a multiple and here it looks like a multiple but it's different multiples. So it's really, globally it's linearly dependent. So what you find for that reason is when you have a new vector field which is over here in the non-uniformly thickened guy, you have a new trivial obstruction in the virtual cycle vanishes. So you go vanishing theorem. So none of these guys contribute. You can just throw them all away. Only these guys contribute. So this is a multiple cover formula in stable pairs. And this is really much simpler because the modulite space of these is just the modulite space of stable pairs on the original K3 surface because the thickening bit is trivial. There's no information at all. There's just one integer. So we reduce. So that's that statement there. So we reduce this to a calculation really on the thin K3 surface with no thickenings at all. And so this gives us multiple cover formula and it shows the stable pairs, satisfy this BPS formula and so on. And then finally we just need to do this computation to get the actual KKV formula. And to get the actual KKV formula, we just have to compute on this modulite space on the surface. But if you remember I told you that surfaces are much better than three folds, the modulite space is really the Hilbert scheme of points on curves. And so these calculations, Martijn Kul and I did a few years ago. And what you do is you express this is the last slide, the next slide is the last slide. You express this Hilbert scheme here as an incidence variety in this obvious smooth space. So in our case, the Hilbert scheme of curves in the K3 surface is just a projective space. You have your points on the curves. That's an incidence variety inside the space of all points on the surface times by this linear system. And the incidence variety is the obvious one that this curve here should contain these points. And so this is cut out by a section of a total logical vector bundle over this smooth space. So there's the vector bundle. You look at your section cutting out your curve and you see whether it vanishes on these points. And so that defines you a section of a vector bundle over this space. And that's what cuts out this stable pair theory. And this really works at the virtual level. So what you find is the reduced virtual cycle on here is really given by this data. It's really just the Euler class of this vector bundle. So you find the reduced virtual cycle is really just the Euler class of this vector bundle, at least after pushing forward to this big smooth ambient space. This is really a remarkable thing. You don't usually see moduli spaces cut out of compact ambient spaces by sections of vector bundles of the correct virtual dimension. But this is, unless you're on a convex variety doing genus 0 grwm of Whitton theory. But this is one such case. And so you can really compute, because it's just total logical integrals on Hilbert schemes. And what you find is this magic, the answer only depends on the square of the curve class. It doesn't depend on the divisibility. And this is where you find this other gopacum KKV conjecture, that the BPS numbers do not depend on the divisibility of the class. And you can really calculate it now. The moduli space is nice and holomorphic, symplect it. You're doing things on case-free surfaces. Everything's very simple. And this calculation was done a long time ago at Kawhai and Yoshiocha. And you get the KKV for that. I think people are telling me to stop. OK. Thank you very much. Can you add some variable to your formula? Because you are refined? Yes. Yes, you can. And you can almost prove it at the gopacum of vaffa level. So if you look at the original gopacum of vaffa prescription, where you take these moduli of torsion sheaves and a vibration to a base and so forth, you can really, that works beautifully in this case. And it not only gives all of these answers, but it gives these refined version as well. It's also a refined formula. Refined, homophobic. Because it's everything still. You can try and push the refinement over the other side. Yeah, I don't know. I don't know. Sorry, say that again. Then why is the first one the woodwork formula? I didn't hear that. What's the corresponding formula for the refined case? I don't know the names of these things, but it's the one that people know from the Poincaré polynomial of the cohomology of Hilbert schemes of points on K3 surfaces. So it's a very well-known thing. I can give you the formula later, and you'll recognise it, but I don't. Any more questions? Yes. Maybe it's the same question. Could you try to put that? Well, the formula that I spoke about, the one that I want. You want to go right back to the formula? Well, the one that, well, I mean, you don't have to show it. The one with the one over one minus two to the end of the 20 and so on. So that function is a name. The function of right is one, yeah. Up to a factor, if you, if on the left you put two G minus two in the X, instead of two G, then the right-hand side will be exactly one over a Jacobi. That's the power of Q and C, and it would be on the nose. That Jacobi form is a name, it's called Phi-10-1. It's the first cos form. The first cos form that there is, is that one, maybe a 10, an index one. But then it can be moved up from a Jacobi form to a function of tau and C, to a function of tau and C, and so on, so it's a zeal form, and it's the first zeal cos form, and it goes so on. That's the one that comes up in BPS state counting. So this is where you change this to like X times Y or something like that? No, no, no, not at all. There's a, no, this is the constant. This, you don't change anything. This is the leading term in the expansion, on how to that it's a Z. The same Z, you don't change anything. Z, Q, but that's a Q prime. And this is just the coefficient of one over a Q prime. But it's a whole of the wrong side. And that the wrongs here is, again, is one over a cos form. And that cos form is this, because the cos form of A-10. And that one has been studied a lot by people, including Waffa, because it's also BPS state counting for something of that kind. It's a 150-page paper. So this is done, I know it very well. So that's a function that contains a third variable, which is a Q prime. And so I'm also asking him, had that variable, that's very different from another multiplicative variable, sort of another elliptic variable, Z. It's another multiplicative variable, another Q. Okay, so I'll show you the four. I think it is almost certainly the same. I'll show you the formula later. Yeah. I have a general question about this stable pair. So you said that you're thinking about this as a deformation of a certain particular type of objects in the derived category. Yes. Can this be generalised? I mean, can we also deform other theories when you deform some more general objects in the derived category? Maybe some. Are you asking, is there a theory of counting objects in the derived, stable objects in the derived category? Yes. In some sense, can that be viewed as a special case of that? Yes. Yeah. So this is a special case of the invariance of Jan and Maxime and these generalized DT invariance. So, yeah. In effect, there's some risk in it because if we consider stable pairs, it's like reached a lot closer to some function, yeah? So this should be finished in a cycle. Consider converges, this should work. Consider one variable number. Absolutely, and that's what gives this refined. Yep. Yep. Good. Good. Thanks for watching. Thank you.