Thank you. Here's where we got to yesterday, we showed that the obstruction to extending a first order deformation of a sheaf, to second order was the cut product, the unedited product of the first order deformation with itself, and Dennis and Georg sent me some homework, many pieces of homework actually, but this is the one I was just going to mention, which was for higher order, is there such an easy description as a cut product of the obstruction class, so maybe I leave it as another exercise, but the statement is the following, so if you have something defined over an, so that's speck of cx over x, the n plus 1, so a flat family over there, and you want to extend it to the next order, the obstruction is this cut product, here again in the same group, it always comes down to the same obstruction space which only relies on the original sheaf on just x, but the data to form the obstruction comes from E n, from the sheaf E n, okay, so it does have this nice similar, very similar cut product description, okay I'll leave that up, okay so moving on, I wanted to talk about Kuranishi theory, so I'm not going to prove this, I'm just going to state it, and then I'm going to go into a more global version of it called perfect obstruction theories, so if you have a modular space of stable sheaves, I'm not dealing with semi-stables at the moment, then you can describe at what, any scheme, you can describe it locally inside, locally analytically, or formally inside it's the risky tangent space, and Kuranishi described it better, so he described how it's cut out inside it's, he gave explicit equations for how it's cut out inside it's the risky tangent space, by this Kuranishi map, so what we've been doing is working to finite order here, you can work over more general thickenings, more general ideals, I've been doing it over these curvilinear thickenings, these very simple thickenings, but the general story is not much harder, and then this does something actually a bit more, it tells you locally analytically what the modular space looks like, instead of just formally, which is what this would give you, okay, and so it says there's a, there's this map k for Kuranishi, which is the cut product, it's just e goes to e cup e that we saw before, plus higher order terms, sort of cubic and so forth, so it's a map from the risky tangent space, so you're in a, you're going to see your modular space has been cut out inside this vector space by the zeros of this nonlinear map to the obstruction space, alright, so the zeros give you the moduli space, they include the origin, that's, that corresponds to the point e in the moduli space, and k is entirely nonlinear at the origin, its derivative vanishes, it really starts with the quadratic term and then higher order, okay, so you, the tangent space really is the full vector space at the origin, okay, any questions about that, is that statement clear? Yes, I have a question, yeah, just, you said you described the m inside its tangent space, or is it the total tangent bubble? No, sorry, I just locally, a small piece of m, a little analytic neighborhood of the, right, so let me set it up precisely, so take your moduli space, take a point in the moduli space, and now I'm describing a little analytic neighborhood of this point, via an analytic neighborhood of the origin in this vector space, okay, okay, so what you see is the moduli space, its dimension is obviously less equal to the dimension of the vector space it sits in, this is a risky tangent space, and it's greater equal, you know, the number of, the dimension of the space it sits in minus the number of equations cutting it out, okay, so this is sort of unknown, the number of unknowns minus number of equations, and then this mysterious comment in grey, you'll understand a bit better in a few slides time, so maybe I'll talk about that later, and then this quantity here, so this, the dimensions of these x groups, which is called little x1 minus x2, is, when that's a constant, for instance when the higher x's vanish, then it'll be given by Riemann-Roch formula, so when that's a constant it's called the virtual or expected dimension of m, so in good situations there is no x2, there's no obstruction space, and your moduli space is smooth, it looks like it's a risky tangent space, analytically, locally, analytically, in that case the moduli space has this dimension, and in general you imagine that the moduli space is cut out by this many equations, in this many unknowns, so that if those equations were transverse you would expect this to be the dimension of the moduli space, of course they're not transverse, this k really is non-linear at the origin, but this is what you, there's good reasons to expect this to be the dimension of the moduli space, it's what you would expect it to be, if you were allowed to perturb k, which you're not, then once you made it transverse to the origin in the obstruction space, then you would get a smooth moduli space of this dimension, so that's called the virtual dimension. Okay so we're going to fantasize for a while, and then later we'll come back to actually proving that, although this fantasy isn't quite true, it's, this is near as damn it true, so it's close enough, okay so this is the kind of thing I fantasize about, so we're going to pretend we have a global Karinishi model, so the whole moduli space is cut out of a smooth ambient space by a section of a vector bundle, such that you know the dimension of the ambient space, so the number of unknowns minus the number of equations is the virtual dimension, and you know I haven't really defined, I've only given you, I haven't really defined the virtual dimension yet, I've given you an idea of it, but we come back to that. Okay this is how I want you to think about moduli spaces, this is, I mean for the purposes of the rest of this talk it's extremely important that you should think that moduli spaces always look like this, so the Karinishi model is the local version of this, and this is also, therefore this is always true locally, and it's not quite true globally, but it's very close to being true globally, and it's extremely important to think of moduli spaces like this, because that's how the whole philosophy of virtual cycles goes. The philosophy of virtual cycles is that M will have the wrong dimension in general, it'll be singular, because this S won't be transverse to the zero section of E, but you want to do something like what a topologist would do, which would be to perturb S to make it transverse to the zero section of E, and that would give you a new M which was kind of smaller of the correct dimension, and that would have a fundamental cycle, and that's what you're after, that's going to be the virtual fundamental cycle. So we'll do even better than that, because we have more tools in algebraic geometry, and we'll get a virtual cycle of the correct dimension, and it won't just, it won't come from perturbing things, it'll actually come, it'll actually be a cycle inside of M, so it won't move off M, as would happen with perturbations. It'll be something better, but yeah, it's very important to have this picture in mind. Is it true, Siriski, locally? Is it true? Yes it is, yeah, we'll come to that, yeah. Now, what's going to be true globally is more or less that this picture holds to first order about M. It's not going to hold completely, you're not going to be able to construct the entire A or an E, but you will be able to construct the information of A, a first order neighborhood of A around M, okay? So to first order about M, this information is just sort of packaged in taking the derivative of this picture, okay? So that's this. So you have the tangent space to A, it contains the tangent space to M as the kernel of the derivative of the section, okay? So remember the zeros of the section describe M, and then you look to first order and you see whether S, as you go in a direction along A, you see, to see whether you're still moving along M, you see whether the section is still zero, so you take its derivative, all right? And in algebraic geometry, the derivative of course doesn't make sense, it would involve picking a connection or something, but on the zero section, which is where we are, where S is zero on M, the derivative does make sense, so maybe that's an exercise, all right? And then the co-kernel of this derivative is what we call the obstruction space, so that was the thing that was sort of like the X2 on the previous lecture, or some family version of X2 varying as you move over the moduli space. So this is the thing which is involved in the implicit function theorem. The implicit function theorem says everything, you know, the moduli space looks to sort of linear or the risky tangent space information like the zeros of dS, and if dS is onto, so S is transverse to the zero section, then there's no obstruction space, and that linear description is completely accurate, the moduli space looks locally analytically like its tangent space, okay? The linear theory tells you everything, that's when dS is transverse, and when it's not the implicit, so that's the maybe the inverse function theorem, the implicit function theorem is when dS is not transverse, there is some co-kernel, and then what it says is this is a good model of the moduli space, but on top of it what you have to do is you have to cut out further, so this is the linear space like the X1 on the previous slide, then you have to cut out further by a non-linear piece which takes values in here. At each of the we have a question on the Q&A, should X2 be curly, shouldn't it be only the global section? It's got a pi down below, so it's the relative X2, so it's the X2's down the fibers, so it is global down the fibers, it's global on X2 on, yeah, there's base change issues, but if you take that sheaf that I've written there and you restrict it to a point of M that is the global X2 of the previous slide, okay? So it turns out this is the thing that will globalize, this sequence, this fantasy is just that, it doesn't really exist, but this sequence will exist and it's called a perfect obstruction theory, and there's two things to notice about it, I'm going to call the two terms in the middle, E0 and E1, and they're locally free, okay? So that's that's an artifact of the, that's the remnants, that's all we can sort of remember of the fact that A is smooth, so that's why E0 is locally free and E is a vector bundle and that's why E1 is locally free, okay? And so this this sort of resolution of the tangent and obstruction spaces of M, this is a locally free resolution of them, it's called a perfect obstruction theory, okay? And we call just some notation, we call E0 to E1 the virtual tangent bundle, hopefully that makes some sense to you, and it's dual, so we use this standard notation that when you dualize a complex you move the subscripts to the top and make them negative, is called the virtual cotangent bundle, okay? So this isn't precise yet but this is the rough idea, this will be the thing which does globalize, this is the infinitesimal part of the fantasy will make global sense, and that's called this perfect obstruction theory. Yeah, should we think of the function k in the previous example as a local expression of this section S? Yeah, yeah that's right, you know I've cut down in in that expression I've kind of cut down A and E or TA and E to be minimal at the point E, you remember that you know at the point E that we worked on on the previous slide this ds would be zero, so I've kind of removed some but up to quasi-isomorphism it's the previous, yeah it is the previous slide but it's somehow I've taken these two guys to be on the previous slide I would have taken these two guys to be a minimal possible dimension but it doesn't matter, you can always add on a little you know some kind of smooth direction to A and some extra piece of E which is cutting out the origin in that extra smooth direction and it just changes this up to quasi it doesn't change this up to quasi-isomorphism so yeah there's some homotopy hidden in this that I'm not going into I don't think my answer helped there but anyway okay so in some sense this is the key slide or this is the main idea and now we try and make it a bit more precise okay so because it's algebraic geometry we dualize everything so instead of taking tta and e or e lower zero and e lower e1 we take their duels okay so this is e upper minus one and e upper zero okay and we map them the correct way to express that resolution is to consider this map to this object here okay so this is um I here is the ideal sheaf of them inside A that should be a curly A this is the cotangent bundle of A and if you imagine a nice situation this is really the conormal bundle sat inside the cotangent bundle to A and so in nice situations D here would be an injection and this complex would be quasi isomorphic to its co-kernel which would be the cotangent bundle of M and in general this is a slight replacement for the cotangent bundle of M called the truncated cotangent complex of M denoted here so maybe I can bring that up okay so um in in good cases where M inside A is smooth or cut out by a regular sequence this D is an injection and this this object here is really just the cotangent bundle of M this LM but in more complicated situations where M has singularities maybe it's non-reduced um maybe you know calculate this in a few cases so take speck of the dual numbers inside the affine line and you'll see that in those cases this is not an injection there's a little bit of fluff over here as its kernel um and that's called the um oh there's something homology what's that called andre quillen something or other uh anyway the fit there's minus one the homology of the cotangent complex is um is some it it's some intrinsic thing associated to the singularities of M which tell you something about their singularities okay but this truncated cotangent complex is a very natural thing to consider it's independent it's very easy to show it's independent uh of the choice of embedding you know hint if you have two embeddings do the usual thing just take their product and compare it's it's rather easy to prove this and what it what it gives you is it gives you you know this is the data of it's two cormology sheaves and some extension data of how they're glued together so the cormology sheaves are the zeroth cormology sheaf which is the co-kernel of this map is just the cotangent cotangent sheaf of M okay so the the co-kernel of this map is always the cotangent sheaf of M by the exact sequence of kehler differentials that's in hartshawn and the the minus one cormology so the kernel here is some kind of up to dualizing it's some kind of intrinsic obstruction sheaf so we've we've started to see obstruction spaces you can always make an obstruction space bigger you could always add pieces on to it it would still be a perfectly good obstruction space but you can't always make it smaller there's some minimal size of obstruction space which is this intrinsic obstruction space which you have to have on any modular space which is a reflection of its singularities on any space and so that's the dual of this um that of this uh kernel here what is your comment on this kehler differential you say i didn't follow that um so if you look in hartshawn at the exact exact sequence of kehler differentials there's there's two of them i don't remember which one um one of them says there's there's a short exact sequence there's nothing it doesn't say anything about that it doesn't give you a zero here but it starts here it says ideal model ideal squared goes to the kehler differentials or cotangent bundle of a restricted to m and then its co-kernel is the kehler differentials or cotangent sheaf of m okay so it's the three term exact sequence that this one goes to this one goes to omega of m goes to zero okay but this is just a repackaging of the exact sequence on the previous page in fancier language designed to intimidate you and precisely the definition of a perfect obstruction theory it can be described duly in in terms on the previous page but it complicates base change and you have to do it it's complicated to write down um but li and tian did it this dual way is much slicker to write down but it's a bit harder to follow it's due to baron and fantaki and the precise definition is from any space for me all my space is going to be quasi-projective or projective so um it simplifies their definition a tiny bit it is a choice of two term complex of vector bundles up to quasi isomorphism with a map to the cotangent complex which is an isomorphism on zero-th coromology sheaves and a suggestion on minus one-th coromology sheaves okay well that's basically saying is this this virtual cotangent bundle here is zero-th coromology is really the cotangent sheaf of m um as it is in this diagram here and in in this suggestion on h minus one says that the obstruction space associated to this complex here well really it's dual when you go e zero goes to e lower one the obstruction space there contains the the intrinsic obstruction space which has which any scheme is associated to any scheme so it says that the obstruction the intrinsic obstructions embed into the obstruction space given to you by this complex okay so this is a this is a tricky definition or well i mean it's a very easy definition but uh it's tricky to unpack but hopefully i've convinced you that my fantasy on the previous page gives you one of these okay and there there it is and what i'm going to show on the next page is that one of these gives the infinitesimal version of my fantasy so um so it really is usually this is a very usable easy definition to understand it you should just understand it as being the the linearization of the fantasy on the previous slide all right the linearization around it any questions yeah the map from the dot to m does this have to be a more more really come from a moreism of chain complexes or no yeah it's really just a map in the drive category okay so what's the relationship between perfect obstruction theory and this fantasy global coronetian model okay so as i said the global coronetian model will always give you a perfect obstruction theory that's the top of this slide here conversely a perfect obstruction theory always gives you local zariski local coronetian charts with to which you know compatible with the perfect obstruction theory so uh let's see that so you can see it as follows um so uh locally any scheme is cut out of a smooth scheme okay so you just locally pick generators for your local ring uh with some ideal eye and now answering your previous question locally any morphism in the drive category is a is genuinely a map of complexes can be made it's homotopic to a genuine map of complexes because these locally free sheaves are just projective modules okay locally a locally free sheaf is a projective object in the category of modules so you can always lift okay so we have a genuine map of complexes from my my e dot to this uh representative of the uh truncated cotangent complex and now i can extend my bundle that i'm my e1 to a bundle locally over over a because i'm just working locally zariski locally my bundle on m always extends to a locally and then i can lift my map here uh i missed i missed something out i can assume that the map from e minus one to i mod i squared is a suggestion because i could just add trivial piece onto e minus one and the same trivial piece onto e zero um and it wouldn't change this complex up to quasi isomorphism so anyway i can assume this diagonal map's a suggestion without loss of generality uh that's sort of defined on m and now i can again because it's a projective module because it's locally free and i'm only working locally so because it's free i can pick a lift to a map here and then again by shrinking if necessary i can assume this is a suggestion because it's a suggestion on m okay and so what does that mean i found a bundle with a map to the ideal sheet with a suggestion to the ideal sheet you compose it and think of it as a a map to o that's just a suggestion of e oh sorry that's just a section of e and it's a section of e which cuts out m because it subjects onto this ideal sheet okay so the the section is is generating the ideal sheet it's cut it's giving you all the equations which cut after m how do you extend the the to the entirety uh well it's just you know i'm only working locally so you may as well assume that e minus one is just a trivial bundle because it is a locally trivial so and then this is compatible with the previous slide you know now you can take this local fantasy description of your local piece of m and you can produce a perfect obstruction theory from it and it's the one you started with all right so zariski locally these two notions are the same and then there's a weak categorical homotopy infinity category nonsense way of glow gluing these local coronetian structures and they glue extremely weakly they don't really glue in any geometric sense um over open sets to define something called a coronetian structure on on m and so you find these coronetian structures on m suitably defined are the same thing as perfect obstruction so the upshot is that this the the model is a fantasy but it exists is risky locally we've just seen that and then you can patch them in some week if you if you if you're sufficiently infinity homotopy enough um then you can imagine a world in which you can patch them and if you don't like that instead it exists globally to first order about m and that's what a perfect obstruction theory is that it's just the derivative of that fantasy model all right so i i don't know if i helped at all but um if you prefer that is what a perfect obstruction theory is there's the definition it's only you know one sentence um but it's to me it was always a an intimidating sentence and this helped um for you it may not do all right so um it turns out to be a very useful concept moduli spaces which have this structure of a perfect obstruction theory okay but i i always think of it as meaning that roughly speaking the moduli space is cut out in a smooth ambient space by a section of a vector bundle of the right kind of rank we will come to the virtual dimension in a minute of course you can always do that right you any quasi projective scheme is embeddable in projective space and then you can always pick some vector bundle of huge rank which has a section which which cuts it out um but when you do that if you take an enormous rank vector bundle you're essentially sending the virtual dimension negative you're cutting out by too many equations when you perturb you'll get the empty set we don't want that okay so now here's a here's a case where um there's a natural perfect obstruction theory so this is a moduli of sheaves on some smooth projective x and this is just a globalized version of the previous lecture all these x ones and x twos that we discussed deformations and obstructions this is just a the global version of it all done in a family over m so suppose you've got a universal sheaf on x times m so over any point of it so it's flat over m and then over any point of m when you restrict you get a sheaf over x and that sheaf over x is the one corresponding to that point of it okay it's a tautology so it's impossible to describe at the board then uh there's something called the tier class which i'm not going to go into um of e it tells you roughly how twisted up he is it defines so ignore these truncations here they're just to confuse you um what what what this says the way you should read this uh equation here this arrow is better to take its derived dual so that check there means derived dual so r home to the structure sheaf if you take the derived dual of this what it says it says you should take the the cot the tangent complex of m and it should map to the complex r home ee so that's just the complex which whose cormologies are the x one and the x two that we've been discussing so this is just the map from the tangent space of m to x one ee that's this is just a fancier way of doing of writing that down in two ways one globalizing it so not just doing it at a point and in the second thing we're doing is we write we're not taking cormology we're writing this thing down on complexes before you take cormology but when you take cormology of this map then um or at least the dual of this map then on zero cormology you're getting that the tangent space of m at a point maps to x one and you're seeing that the intrinsic obstruction space of m embeds in x two um okay and there's this truncation where we we remove the complex we truncate the complex so it really only has x one and x two okay this is an obstruction theory for m so that's a theorem uh i haven't really told you what an obstruction theory for m is but what it is is it's the same as the definition of perfect obstruction theory without the requirement that this complex here can be written as a two-term complex of vector bundles okay so it is also a perfect obstruction theory so this this guy on the left can be written as a two-term complex of vector bundles uh whenever it can be written as a two-term complex of vector bundles very good ah okay so so modulo this issue of whether this complex can be written as a two-term complex of vector bundles this is a perfect obstruction theory Okay, so when can it be written in this in this way when that that occurs for instance if all the higher x's vanish So when all In general you're going to have x ones x twos x threes x fours and so on If if your big x is of low dimension, maybe you only have x ones and x twos If you're on a curve, maybe you only have x ones And then things are better The case i'm interested in Is calabi our three folds and in that case you get the vanishing of these higher x's by Serduality so the higher the for instance x three is dual to homes And we already saw that when e is stable these homes vanish At least the trace free homes. Sorry. There's zero here means trace free So the homes from e to e are just multiples of the identity So when we when we take the trace free ones, you get nothing And similarly x four is dual to x minus one which vanishes when ease of sheath And so on so when when all these higher x vanish point wise Then you can prove By some kind of reverse of base change that this complex is really a two-term complex of vector bundles Okay So what we end up with is one of these perfect obstruction theories For moduli of sheaths on calabi our three-folds. So you have your x one and your x two defining deformations and obstructions You essentially have no x threes and no homes So because you only have these two guys They are the cohomology of a complex of vector bundles the the difference in dimension Is a constant in fact? It's zero This topological constant this virtual dimension is zero in the clubby our three-fold case Because the x one and the x two are said dual to each other. So they've always got the same dimension And the fact that they always have the same dimension means that they can be resolved in this way That they are the cohomology of a two-term complex of vector bundles May you come to take a trace free home right? Yeah No, but it's still the case You don't really it's still the case that you can use the full x one and the full x two So I'm just trying to state things more simply. So this full x one and full x two still form They still are the cohomology of a two-term complex of vector bundles Even though it is true x zero and x three don't quite vanish, but they're just they're very constant, right? They're just They're just the trivial line bundle. So that doesn't really affect the argument I was just trying to simplify things I don't want to get into the difference between x and trace free x But you you can use this I don't yeah, you don't need that I mean, you you'll probably get invariance which vanish, but you can use these Okay, so um, so that's the obstruction theory if I've lost you just think about that fantasy model Just pretend it's true Now what I want to get out of this Is I want to get the correct cycle Um for the moduli space So at the moment the moduli space is it's got virtual dimension zero if I'm in the kalabiya 34 case Um, but it's got actual dimension way bigger than that. So moduli of sheaves on kalabiyaus Satisfies some kind of murphy's law. They're as bad as you like. They're as singular as you like um Actually, that's not quite true because they're critical schemes, but um, they're They look dreadful and they're of too high a dimension in general So, um You don't want to work with their fundamental cycle to start integrating over or trying to define invariance You want to find some virtual cycle some cycle of the correct virtual dimension And so let's let's go back to the fantasy if we're in the fantasy case Then we can perturb the section So that it was transverse to the zero section And then um, it zeros would now have the correct virtual dimension So you'll get a homology class of the correct virtual dimension in a Unfortunately not in m because when you perturb the zeros of s will move off m slightly Okay, and and it would you know what? This homology class will be it would be the Poincare dual of the Euler class of e Now we can do better because we're algebraic geometers um So we can get the class actually to line m And to be a chow class so really an algebraic object By using Fulton McPherson intersection theory So we can localize the Euler class of e to the zeros of this section even when it's not transverse Um, and then it'll turn out that the the procedure which I'm about to remind you at the Fulton McPherson intersection theory It it won't actually require the full fantasy model to hold it'll only require the the That it holds locally that you have these Kuranishi charts locally which you do And that'll be enough to do the Fulton McPherson intersection theory. So that's kind of baron and fatakis insight So let me explain that so Sorry, okay, so so here's a picture in the fantasy model Or a local version of it So a is this horizontal axis The section of e is the the graph of it is this red thing e is the vertical fibers And I'm interested in taking the zeros of s so I'm interested in intersecting the graph of s with the zero section of e so with a here And then that's what the fat Um red blobs are that's the moduli space So the moduli space is the zeros of s and I've drawn this one fatter than this one because this one's like a You know, it's got some non trivial scheme theoretic thickening Now what Fulton and McPherson tell you to do is they say replace this graph By the graph of not s but t times s So make the graph more and more vertical And take the limit as t goes to infinity and the graph becomes something called a cone So it becomes c star invariant on the the c star action on the fibers of the vector bundle And this is called the code the normal cone of m inside a All right, so when when um, let's say m is smooth Maybe of the incorrect dimension, but when m is smooth Then this cone is really a a copy of the normal bundle of m inside a But made vertical of that using the derivative of the section So embedding it in using the section All right, so this is this this cone here Is the limit of these graphs But it can be described just purely algebraically in this way As I say it is just you should think of it as the normal bundle of m in a In good cases, it's just the normal bundle of m in a and it embeds an e Using the section s And this has a huge advantage replacing the graph by the cone Has the huge advantage that now I have no data left away from m everything all my data is now concentrated over m So I don't need a anymore once I have the cone Which is good for us Sorry Oh, yeah, that's not good Okay, well, you know who to blame Yeah, sorry that that should be to the power of i and to the power of i plus one I'll correct it and send it for the the notes on the web Yeah, so this should be the ideal to the i over the ideal to the i plus one Okay, so this cone is supported over m And so the big insight of better end of fateki is that even though My current ishi charts don't really glue in any geometric sense only in some categorical sense um The fact that the global perfect obstruction theory glues so the vector bundles e glue and so on And the tan the tangent bundle to a that was e zero The fact that they glue is enough to show that the cone actually glues All right So if this is your picture on an open set when you pass to another open set You can't glue these a's but you can glue these cones and these ease from one open set to the next And now what we're going to do is instead of intersecting the graph with the zero section We're going to intersect this cone with the zero section And that will be the virtual cycle. So the definition of the virtual cycle is that you intersect the cone with the zero section of e one And this intersection is called a geysin map and it's defined again by fulton mcpherson It's in fulton's book Um, and what it is defined one way of defining it is as the inverse of the tom isomorphism So there's an algebraic version of the tom isomorphism Which is if you go from left to right here You take cycles on m and you pull them up To get non-compact sort of borrel more cycles or chow cycles on e Then that's an isomorphism So all cycles in e can be written as the pullback of a cycle on m So it's it's rationally equivalent to some some vertical cycle Okay, and then the the geysin map is just the inverse of that So it says take any cycle on e like this cone And write it as a vertical cycle a pullback of a cycle on m and that cycle on m is your intersection So, how these rationally equivalents come? So so these oh, I beg your pardon. I pressed the wrong button These chow groups are these algebraic versions of homology groups They're defined via algebraic cycles up to rational equivalents So instead of taking cycles up to co-boundary Cycles up to boundaries you take algebraic cycles up to rational equivalents Okay, so um just to get a feeling for what this map is I give you some examples of how you do computations in folton mcpherson theory So these are nice. I encourage you to do them if you've never you know battled with folton's book And I encourage you to read folton's book. It's a masterpiece Um, so suppose you have a line bundle for instance With a meromorphic section Then by taking the limit of these graphs You can uh in two different ways you can take the limit as the graph becomes vertical Or you can take the limit as the graph becomes horizontal And because they're equal in chow because they're rationally equivalent um You can show that this vertical cycle the line The entire line bundles over the zeros of the section Is equal to this vertical cycle the entire line bundles over the infinity of the section Up to the zero section, which is not vertical So what that says is In chow the way you make the zero section of l into a vertical cycle the pullback of something in the base Is you write it as You know the pullback of the zeros of s minus the pullback of the infinity and the poles of s Now so therefore when you invert that and you intersect with the zero section What that says is that the intersection of the zero section with itself in a line bundle Is given by the zeros of s minus the poles of s. Okay, which I hope you familiar with already So that's just the Euler class of l Another exercise is suppose you have a sub bundle and you have a section of e Which lies in the sub bundle so it can't possibly be transverse to the zero section of e But it could be transverse to the zero section of e primed Then what that will mean is that the cone Is really e primed But that's not vertical. It sounds vertical. You know in english. That's vertical and hopefully in french It's vertical but in mass. It's not vertical. It's not the pullback Of some cycle on m because it's not all of e. It's just e primed And so you have to make that into the pullback of some cycle on m and that defines its geasing map and When you do that what you find is That you get the Euler class of the bit of the bundle that you haven't used yet So e over e primed. So the obstruction bundle you get the Euler class of the obstruction bundle So that's a great exercise to do So that's the situation in Virtual cycles where you have a smooth moduli space but of the wrong dimension So it's of too high a dimension So what that means is your obstruction sheath becomes a nice obstruction bundle. It's locally free Over your moduli space and you take its Euler class and that gives you the virtual cycle And the sort of the c in furnace diversion of that would be that you would write e as a direct sum you would split the extension so you'd write it as e primed plus what's the plus the obstruction bundle And your section would be of the form s comma zero So you've not used this obstruction bundle and now you perturb it to some s comma s primed And that would give you Once your s primed is now transverse to the zeros of this obstruction bundle What that would give you is that the virtual cycle Is the zeros of this s s primed or the zeros of s takes you to m And now on top of that you want to impose the s primed is zero And so what you end up with is on m you end up with the zeros of s primed Which is precisely the Euler class of the obstruction bundle Okay, but this Fulton McPherson in section theory gives you a way to do that Without these perturbations So These virtual cycles are extremely hard to compute. It took many years before people worked out ways to compute these things There's four main techniques So one is very occasionally the fantasy model really holds globally so An example of that is when you compute um genus zero grom of witton theory of the quintic You can see it the the ambient space is the space of genus zero stable maps to projective space p4 that turns out to be smooth and then it turns out there's natural equations You can impose on that to cut it down to the stable maps which lie in the quintic And that gives you a global currency chart and you can compute and that was one of the very first computations in enumerative geometry In in virtual enumerative geometry, but it's extremely rare there's another fantasy model which i'm going to talk about in um Two lectures time Where you have something called virtual degeneracy loci And there again you can compute instead of by taking an Euler class by taking certain other churn classes Um, so that's rather new I guess The older there's um degeneration standard in algebraic geometry degenerate these virtual cycles have a deformation in variance a homotopy in variance And so you can degenerate your situation to maybe an easier one or um So sometimes you degenerate to a union of pieces which um, maybe toric or have have some symmetry It can be extremely difficult to work out How to put everything back together when you smooth again But modulo that issue on these toric pieces the way you can compute is using torus localization So this was um worked out by graber and pander pander how to How to compute by localization in a virtual setting? So when you have a symmetry group you can exploit that to do computations And then there's another type of localization called co-section localization Which again, I'll briefly discuss in two lectures time And in fact, we're going to use three To define baffle-witton invariance. We haven't got there yet and then we're going to use two and four to help compute them And so I just end here by um very quickly Yeah, I probably yeah, um So I've more or less said this anyway. Let's suppose x is a smooth Projective colabi i3 fold you'll see the relevance to baffle-witton theory tomorrow So what that means is that the canonical bundle of x is trivial And suppose you have a stable sheaf on x Then as I've mentioned before the higher x groups vanish modulo this trace issue Um the x2 is actually dual to x1 So in particular the virtual dimension is zero and um what that means is that the the obstruction theory given by the x to v that we've been talking about the defamation obstruction theory is actually perfect of virtual dimension zero and so If there are no strictly semi stable sheaves if all the sheaves are stable Uh, then the moduli space is projective and compact And there's a virtual cycle of dimension zero And uh, you just integrate one over it or you take the length of that cycle and that gives you an integer If there are strictly semi stable sheaves, there's a Joyce songs generalized et invariant. Um, which we'll discuss a tiny bit later on But that's much more complicated to define use this whole algebras and fancy technology Um, and that's a rational number, but it it strictly generalizes this so when in the stable case it reduces to this integer here So that that's what's called dt invariance And I just briefly for completeness tell you the mnop conjecture Which is that when you take rank one dt invariance So when your sheaves are of rank one, then they're really up to tensoring by a line bundle their ideal sheaves And their ideal sheaves are curves and points in x So what you're doing the dt invariant here is essentially counting curves in x But it's really counting one-dimensional sub schemes. So it's thinking of the curves as being cut out by equations There's another way to think of holomorphic curves, which is as instead of these Unparameterized curves being cut out by equations You can think of them as parameterized curves the images of holomorphic maps And that's the subject of gromov-witten theory And the mnop conjecture is that the two sets of data are equivalent So the the rank one dt invariance are equal in a very this is a very complicated binary relation here To the gromov-witten theory of x. So these integers are equivalent to these rational numbers Been a very complicated way which I won't go into And pander pander and picks have now proved this conjecture for most colabi are three folds That can be degenerated or under various moves covers degeneration and so on Um, you can end if you can eventually write them in terms of toric pieces Then they proved it for you So this is really a theorem And then recently so halo phase batch and I Have shown that the dt theory in higher rank is actually governed by the rank one theory Okay, so the information of curve counting gives you the information of the whole dt theory So there's a paper on my web page. It's not it'll be on the archive in a few weeks Um on modulo a certain conjecture, which is proved for various colabi are three folds, but not all of them But in particular for instance for the quintic it's proved And so for the quintic where we know pander pander the m and ap conjecture and this bogamolov kizika conjecture Um, it's proved that the gromov-witten theory of the quintic Determines via, you know insane formulae that can't possibly get a handle on Yet, um, it determines the the dt theory Okay, so it turns out dt theory is just governed by curves Um on your colabi are three fold And um, so i'll stop here, but next time Um, we'll define waffa-witten invariants. They're extremely close to dt invariants But they're they're defined for surfaces not three folds, but associated to every surface There's a local three fold a local colabi are three fold um And uh That's the relationship. That's why i'm describing dt theory the two subjects are very closely related But we'll see that next time Any question or comment? Uh, yeah, so you mentioned that so about the bogamolov kizika conjecture Yes, we have like an idea of some class of various for each fold Not me. I'm not really an expert on that. Um, there probably are experts here Um, it isn't it's not been proved on many things It was a huge breakthrough when Chunyu Lee proved it for the quintic three fold recently Um, but it's the thing that you need to be true Or it's the best route to producing bridgeland stability conditions on colabi are three folds And so where it has been proved which is on Colabi are three folds which are closely related to abelian surfaces And the quintic and then a few other things like double covers of p3 and things like that um In those cases we now know that there exist bridgeland stability conditions on those three folds So that that's where it comes from. So it's a work of um By a makri toda stilari in different papers They make this conjecture. It's it's definitely not always true, but Morally, it's true. It's true enough in most cases for our purposes But yeah, it's something that's more or less Some version of it is expected to be true on any colabi are three fold But we're not there yet Yeah, it's not my field but yeah And so then the last implication you wrote is uh, expected to be true for like most Yeah, I absolutely expect this to be true for all colabi out. Yeah, but but it's not it's not proved yet. Yeah Yeah No idea Does it even make sense? I'm not even sure if if high rank dt theory makes sense on a local colabi out Maybe you'd need a framing or something Yeah, that's low. That's rank zero on the colabi out. Yeah Yeah, we'll talk about that next time. Yeah, I'm not sure You're setting me homework again. You two are going to be banned from these lectures I get uh, yeah, I'll think about it and I'll get back to you. Can you replace the one by by zero? Yes Yes, uh, yes Um More or less so so long as you allow the rank zero so long as you allow the the important thing is the um The sheaves which are supported in dimension two you need those Um, if you take the ones supported in dimension one Then it wouldn't be true perversely Um, so if you just think about curves Dt invariance on sheaves on on curves in your colabi are three fold That is not enough to recover the gromov-witten theory incredibly Um Because you that'll only recover the genus zero gromov-witten theory. Yeah There's kind of a bit of a paradox But if you use refined dt theory where instead of them these just numerical invariants You get sort of vector spaces and graded vector spaces and homology groups and so on then you can recover gromov-witten theory Yeah, that would take a bit of explanation. I probably shouldn't go into it Yeah, is there some reason we should morally expect that the rank one theory controls the higher rank or we just observe You know, I mean it's really as simple as saying that all rank R bundles Or sheaves can be written as iterated extensions of rank one sheaves and you sort of controlling What the contributions of those extensions are and checking that they're kind of essentially Given by universal formulae and churn classes and so on That's some kind of dumb way of describing the wall crossing formulae that we use That's one way of thinking about it. Yeah, it's just their extensions and rank one sheaves It's you know, maybe you should think of it a bit like Cyberg-witten theory governing donelson theory. Cyberg-witten theory is a rank one abelian theory Donaldson theory is a higher rank Non abelian theory and one governs the other But this wasn't predicted by physicists. So I don't know. I don't know if there's an argument like their argument that Donaldson theory is governed by cyberg-witten theory So you have a lunch to go to right? I feel like I'm delaying it. Is there a question? Yeah Uh, so on the modern space, you have this The abstraction series you have this easier and the minus one and also you have push forward of the universal sheet So if there is some kind of interaction between the three so some maps um They The push forward of the universal sheaf down to the modular space. Was that the question? So yeah over a point you're taking the the sections of the universe of the sheaf Rather than the x groups from the sheaf to itself Yeah, I mean I would consider that more to be that that object the push down of the chief I would consider that to be more like an insertion You would maybe take its churn character or something and integrate against modular space or Um, yeah, I mean it yeah It's relevant. Uh, it comes in in various in some places, but yeah, I don't have anything intelligence to say. Sorry Okay, thank you