 Okay, so first about the question and about the stability conditions on MGN bar. I was correct. There is a paper, relatively recent one by YP Lee and a collaborator whose name is CHOU. And it's exactly about various our constraints for varying the has it weights. So they consider some sets of you have if you have, because you have to consider all modular spaces at once you have to consider which has that space you're going to allow for all the points at once and he has a notion for that and then he writes down versus our operators. So they exactly write down versus our operators in that context. And as I said that conceptually, in some sense, all of these things differ from MGN by some gena zero behavior. So there are methods to calculate descendants from one from the other, but to put them all together inverse our constraints is done in this paper other people have studied the differences between the cotangent lines on these spaces and Alex say I've been someone some years ago. Okay, so I hope you that's enough information to find the paper. If not, you can ask me. So the last lecture ended at this discussion of she's counting in dimension three, and the geometry of the Hilbert scheme. We went through the Hilbert scheme as a modular space of ideal sheaves that was here. And I noted we really consider all the bad points of the Hilbert scheme. Eventually you learn how to like these points because you spend most of your time working with what is considered the worst points in the Hilbert scheme. And the work of Richard where this obstruction theory is constructed and shown to give a virtual class. And then there was the observation that the dimensions of the virtual classes for the ideal sheaves and stable maps are the same in the sense that makes any sense at all. And where we exactly stopped last time was this transition to club your three fold so I'll start here. So, for club your three folds, the one of the nice things from our point of view here is both these modular spaces. The stable maps and the ideal sheaves have always have virtual dimension zero. So the first question that you could ask here although this is not the how the subject develop but nevertheless is, is there a relationship between the curve counting by stable maps and curve counting by these ideal sheaves. And so that's this question is their relationship and let's look for how when that might be. So the first the reason that you might think there's relationship is that they both virtually count curves. That's true. But there are some differences. And the differences, for example, in this Hilbert scheme you're counting all these little fat bubbles that are running around, but all, all the things you count are sub schemes for maps for stable maps. There's no zero dimensional nonsense. But on the other hand, there are multiple covers. So, somehow the modular spaces look a little bit different. It's hard to believe that there would be some equality on the level of modular spaces, but still it's not it's useful to consider the simplest hope. I mean in general, it's good to think about the simplest hope first, because sometimes it's even true. So the simplest hope is that if I take some class that doesn't break so an indie composable class like the class of a line on the Quintic, for example, to line on the Quintic. So it's a class that I can't break into smaller curve classes. That way I don't have to worry about multiple covers. So if I take this indie composable class I might hope that the Gromov-Wittnain variant which has genus G maps to this class is the same as the ideal sheaf count, where I somehow try to try to guess the other characteristic that this smooth genus G curve has and that I don't have to guess that that's going to be one minus G. So one could hope for something like this. And so we can try to see how this could be true, or whether this is true, and the way that this subject has developed is that there's an example which is the first example to study more or less for all questions. So that is the example of a P1 with normal bundle minus one minus one. So I take this P1 and the Klavier 3-fold, and it has normal bundle minus one minus one. This is somehow, so it's a smooth P1. And one imagines this is somehow generic rational curve, the Klavier 3-fold. So this is in some sense kind of like a thought experiment. We just imagine that our Klavier 3-fold has one. Of course there exists Klavier 3-folds where these exist. So I want to consider what this curve contributes to this Grumov-Witton invariant. And there are techniques about how to do those calculations, and this calculation is a kind of old calculation, maybe even a little bit before 2000. But anyway, it gives the answer, it gives the contribution to the genus G Grumov-Witton invariant of the ambient Klavier 3-fold in the class of this P1, this rigid P1. It's not a completely trivial thing. This, this, although it looks like it's only a genus zero curve, it contributes to the Grumov-Witton invariant in all genera. And that's because you can map higher genus curves to this P1 by contracting higher genus bubble so you can map, you can map a higher genus curve to this P1 by taking this is the domain of the curve is being a P1 and then the curve could have some tails of hygienists that are mapped to points. But nevertheless, in the stable maps. This, this innocent rigid rational curve actually contributes to the Grumov-Witton theory in all genera. And moreover, those contributions can be calculated exactly and it's given by this trigonometric series. And we put it in this generating function where I keep track of the genus and what the u variable for the genus this is the genus variable. So it's the only thing moving in the sum of the curve class is fixed. The Calabi I was fixed. It's the only thing moving as a genus and I put a genus variable. And if you do this, you get this very nice formula, the trigonometric series. And you could ask how are these integrals computed. And that was there was a time when there was a lot of work on how to do those things and the idea here is you can move to actually modulate some maps to P1. That's not surprising because we're talking about those maps where the curve gets to the quintic by first going through this nice P1 so you can reduce the entire calculation to P1. And then there's new techniques there. There's localization, the virtual class, and then how to deal with what are called these hodge integrals. And then on top of that a bunch of tricks and integration is just like in high school you're never guaranteed to be able to do an integral. And often there's some tricks involved. So this is this is some kind of direction which I'm not going to really investigate in this lecture. But that's kind of fun and you know what at that time, we were doing a lot of these integrals and it really felt like first year calculus in the sense that there's all these integrals, and you have to learn all the tricks and in the end you get trigonometric functions. Okay, so, so anyway that gives the left hand side that gives if we want to compare in this very simplest example it gives the answer for this side that's kind of good news gives the answer exactly for the side. So if we had an answer for the ideal sheaf the DT side if we had an answer, then we could try to see whether they're equal. Okay, so then we have to go study this DT calculation and this is done in these papers MNOP one and two. So that's Davesh Malik, Nikita Nekrasov, Andrea Kunkov and myself and we go to papers. And that was a long time ago by now. I don't know the dates 2000 plus something maybe 2005 something like that. Six, I don't know. But we do the calculations on the DT side. And that uses in some sense some of the same ideas. So there's localization but now on the DT space, which is which is the Hilbert scheme of points. Sorry Hilbert scheme of curves, and of course there's lots of points. And the hodge and enroll are replaced now by box counting ideas. And Andrea Kunkov, also with Russia ticking to develop some box counting techniques which are very useful in this context. And then of course there's also lots of tricks there. In the papers there's various tricks about how to handle this. As I said that you're, I mean integration in some sense is always some kind of summation and you're not guaranteed to get to be able to get a close form answer, unless you're working on a nice problem. So anyway, it turns out this problem is a nice problem. And now we fix this curve class just as we did before. And now we look at all the ideal sheets. This is the DT invariant. And the moving thing now is the Euler characteristic. Can we give it, also give it its own variable and call it Q. And I should have remarked on this. It's a small remark here is that this is the genus and a genus of a connected curve starts at zero and goes up to infinity. So this is a sum for zero, one, two, three, so on. But this is a stranger thing because in DT theory we fix some curve class, and we're supposed to sum over all of the possible Euler characteristics. So if you remember that from yesterday that was a while ago yesterday. So I just remind you what's going on here. Oh yeah so here. This is the modular space of curves of a view is the Hilbert scheme, whose class is beta and who's holomorphic Euler characteristic is and I must write that somewhere. Yeah here, the red scheme of curves and and is the holomorphic Euler characteristic of this quotient curve of the curve itself and beta is the class of the curve. But it's not the case that this starts at zero or starts anywhere really in particular. You know if you have a genus G curve it's holomorphic Euler characteristic is typically negative it's one minus G. So you don't really know where this sum starts. You have to sum over all possible holomorphic Euler characteristics. You have to sum over all possible holomorphic Euler characteristics and you don't know so to speak when it starts. Of course in any particular case you might investigate like in this case, it's not so hard to see that this number starts at one. But in general you don't know and normally one writes this sum as this N and Z because it really could be negative. But the thing that's true by geometry is that it's zero for sufficiently negative so it's always a Laurent series. Anyway one has to execute the sum and by a different I said a different kind of set toolbox of tricks you can calculate this exactly. And this is the answer and you get something that. Well you get something very simple here. That's a rather simple function of q and you get something out you get something that is more complicated, which you get the McMahon function the McMahon function is this infinite product here. And this McMahon function is related to box counting and it's related to lots of things it's related to box counting actually McMahon apparently found this funger had the idea to start thinking about this function, because he was stacking cannonballs in the British So it's kind of some kind of military spin off actually this. And the another thing that it's not so surprising if you think about it is not only does this McMahon function occur, but it occurs to the oily characteristics to a power which is oily characteristic of the clubby a three fold. In particular, the answer here has to do with the whole geometry of the clubby a three fold. And here had no clubby a three fold in it at all because, as I said that the whole problem is can be reduced to studying the geometry of that rational curve things far away and the clubby out don't matter. But for the DT calculation they do matter and it's obvious why they matter because if you take ideal sheaths, even if you're interested in ideal sheaths that are have to do with this curve, you can have these little fat points running around everywhere. That's the nature of the answer. And then if you look at these formulas. So that now we have both sides exactly solved. And if you look at these formulas. In very particular cases you could hope for you get some things like this, but in general if you try to match these invariance by this series to these invariance by this series, you will not find any matches at all. The conclusion about a simple literal hope fails. The modular spaces and don't match there. They're two different that way. But a more sophisticated hope actually has as much more encouraging and that says that okay, let's actually just see what the answers are so we can start with the DT answer, which we've calculated. And this this crazy McMahon function is obviously not. We're not going to see that again so let's just divide by divide out by it so I've, I've taken it. I divide out by the McMahon function by multiplying it with the negative so it's gone now and I get this little series. I mean I guess I get this little rational function. That's a pretty nice little rational function. And then if I have the idea to make the substitution. Here is q is equal to the minus IU. And if I'm going to make any connection between these two series. At some point I'm going to have to confront the fact this is in you. And this is in queue and so at some point I have to confront in some sense change of variables. And so one has this idea of having this change of variables. And if you plug this in it's a small high school exercise to plug this change of variables into this function. And then you get some kind of series. And then you have to use the formula for sign. And if you do this patiently you find that it just exactly correct. So I say that again so if I take this DT series, I take out the McMahon part. I get a rest I get this rational function. And I substitute this rational function in exactly using this substitution and this this substitution does turn rational functions into trigonometric functions. And if you find out what exact trigonometric functions turns to and I said you can do this very patiently. I did it here probably it's even correct. You exactly get the grandma Witten series that you end in this fair for this formula that Carol Faber and I found. So that's very promising. And if you're very optimistic you can just just you can make a sweeping general conjecture based on just that example. And that's what we did that's his grandma Witten DT correspondence in this MNOP in the first paper of MNOP. The conjecture said qualitatively says that the relationship found for this minus one minus one curve over P1 this local geometry. Rigid rational curve holds in general that's the conceptual way to say it. And we had more examples in this minus one minus one curve. There were a lot of things going on at that time and there's this topological vertex which allowed to do, allowed a calculation of tort collobials and you could see that there that's like a box counting. And the vertex was developed by Aga Nage, Aga Nage, Klam, Marino and Waffa, and there's box counting methods that were developed by Andre Kuncov and Rashatikin. So there were a lot of tools there and we could do more examples than this one and we found this relationship was true there. But nevertheless, that's the, that's somehow the origin of this conjecture. And now I want to write it down precisely. So it but in some sense this is already the full content of it. But let's try to write it precisely in full generality. So let X be a collobia threefold, it can be anyone you want. So by this, I mean non-singular project of collobia threefold. And then this has a Gromov-Witton series, a Gromov-Witton potential, F. And it's a sum over all the genera, over all the curve classes, not zero. We don't want constant maps. So these are non-constant maps. There's a whole discussion about this at the separate discussion. It is enlightening to think about the constant maps, but I don't want to have that as a distraction at the moment. So I sum over all genera, all non-constant maps. I put the Gromov-Witton invariant. And then I have a variable for the genus. That's the same variable we had in this example. And then I need some way to keep track of the classes. And so this is normally put in some kind of nova covering or anything, but we can be kind of naive about it. It's a variable that keeps track of the curve class. And this is, this prime here was some terminology that was used. I'm not sure it's good terminology, but anyway, it's to remind you that we've taken out the constant maps. And then I can take the exponential of it. Exponential means that I start with connected theory, then it's a disconnected theory. Taking the exponential is quite important because it didn't show up in this example. So when I said that the relationship found here holds in general, it's maybe a slight lie that that's the only thing here. But one of the things that doesn't show up in this example is that all the curves in this example are irreducible. In Gromov-Witton theory, the way with for connected curves, the support would always be connected, but in the sheaf theory, it can be any support. So there should be some disconnected version of Gromov-Witton theory. And that's, of course, just taking the exponential. So this is the connected version of Gromov-Witton theory, take the exponential, get the disconnected version. And if I do that, I get a disconnected Gromov-Witton theory, that's a series in U for every curve class beta. It's just calculating the disconnected Gromov-Witton variance in that curve class beta. On the dt side, we sum over all beta, including zero now, because we'll remove it by hand. We sum over all n, and then there's the dt invariance, and now the variable q, and the curve variable is the same. And we can write it this way, where that's the contribution of the curve class beta part. And the first conjecture has to do with the constants, the constants and dt and dt. So this is the, geometrically, it's a little different. In Gromov-Witton theory, we can just throw out the constant maps, because the modular space is fine without them. Because there is a connected modular space, a connected map. In dt theory, we can't just throw them out. We'd be very happy to do that, and we're going to do that later. But that's a different theory of stable pairs. But if you have a Hilbert scheme, the Hilbert scheme's general points looks like this, right? I drew this fuzzy thing. And you could say, I don't like these guys. I don't want them. I want a Hilbert scheme that just looks like curves. And, you know, you can't do that with the Hilbert scheme. It's not a legal move. It will lead you to some problems. We have a question in the chat. Is there some intuition for why exponentiating gives a disconnected Gromov-Witton theory? What's the reason for that? If there is some intuition. Yeah, that's just what it does. Like, for example, if you want to count, if I take a Gromov-Witton variant one in a line class, when I exponentiate that, what happens is that, yeah, I mean, so the exponential turns connected to disconnected and kind of in big generality. And can I try to, I mean, I can explain one aspect of that, but it's good to think about it. In some sense, it's very simple compensatorics. But suppose I have some geometry where I have one line, and that's my connected invariant. When I take the exponential of this series, this one line will be a one times this variable, which keeps track of the curve class to the line. And when I take the exponential of that, what's that going to do? That's going to somehow contribute. What does exponential do? It'll contribute some, that particular term, when I raise it to the exponential, it'll have some series where I take m times the line, and that one will stay here, and it'll be divisible by m factorial. All right, this is what will happen in the exponential side. And what is the meaning of this? It means that I can look at a map that's not just one line, but m lines mapping to my one line. And then this map has an automorphism of m factorial, and that's why it's here. So this is just one little piece of why this works. But the reason why exponentiating gives you disconnected is more or less just a matter of the combinatorics of what the exponential map does. So I invite you to think about that. Or maybe that can be discussed in the problem session also. But anyway, when I'm here, in the DT side, you don't have the option of removing these fat points by hand, by geometry. Actually, you do, but that's a different space. But in the Hilbert scheme, you don't have that option. And so in order to confront it, we have to calculate those in every case. The easiest conjecture at MNOP, there's three conjectures. The very first one says that you take any clubio three-fold. And look at the degree zero DT theory. That's a DT theory of the Hilbert scheme of points of that clubio three-fold. That's some kind of new thing. It's some virtual class on the Hilbert scheme of points of clubio three-fold. And the first conjecture is the answer for it. It's this McMahon series to the order characteristic of X. And this was the first one that was proven. There's now there's many proofs of it. There's a proof by Jun Lee, there's a proof by Baron Fantechi, and there's a proof also using co-orders and methods that I gave with Mark Levine. So that's the most modest of the conjectures, but it's required. And then the next thing for the DT theories, we want to remove these points. And so we remove it from the level of generating functions by dividing the DT generating function by the DT zero one. So that's the idea is just to remove the constant contributions on the level of generating functions. And then I have a DT series that's kind of a reduced DT series without the, I interpreted it as a DT series without the constant, without the fat points. And now the next conjecture, which is kind of important also is that the DT invariance then are always the Laurent expansion of a rational function q. That's exactly what we saw in this example. If we go back to the example, we take this, this is the raw DT series, then we remove the point contributions that's getting rid of this. And you get a rational function in Q. So the conjecture is this always happens every single time that once you remove the point contributions, the, this DT series is the Laurent expansion of rational function q. And moreover, it's a very special rational function q it's invariant under q goes to one over q. This was maybe not something you checked when you saw this but perhaps you should have you look at this function and you substitute q goes to one over q. You find amazing at the same function. It's a very special kind of rational function that satisfies this functional equation. And so we conjecture that happens every single time. And that's conjecture to and this turns out this conjecture is also accessible and it's accessible by methods that when we formally the conjecture weren't there but came into the subject very soon which was using ideas of wall crossing and it's proven by Bridgeland and Tota that's kind of extremely nice proof. And it shows that it proves that this is a Laurent expansion rational function for clubby at three folds. So maybe I should just right here, because you can talk about this more in general for clubby at threes. It proves, so their work proves that it is a lot of rational function more of it satisfies this functional equation. So those are those two conjectures are proven, and then the conjecture. The most complicated conjecture is the one that's the relationship with criminal theory. And it says step three says, once you've done that once you know this is a rational function. Then I can do this substitution. The substitution on the level of power series is very slightly illegal because it starts with one or minus one on the left hand side this starts with minus one. And if you substitute power series when with a minus one, then you're confronted with issues like convergence and things like that and analytic continuation and we don't want to do that and we don't have to. Since this is a rational function. Since by this by this state in the conjectural discussion and it's actually proven now, since this is a rational function. So this substitution is always legally you can always substitute into rational function, and then you get a series in you. And then the claim is that that's exactly the series. So that's the grommel wouldn't DT correspondence. And in this level and the way I explained it it's very clean statement. And it's a really if you think about this if you have if you have not thought about these things before. It's entirely real it's in some sense and can be entirely grounded in a careful understanding of this one one small piece of geometry. Okay, did I want to say something more. Oh yeah what's the status of this, I would say it's open. I mean many cases we don't know but it's proven also in very many cases so all the clubby our torque geometries. This was proven in the first pass. So the other Oh here is a Blomkov all right the names later this is for torque geometries it was proven for complete intersections I prove an argument with with Aaron Pickston. And also this argument Aaron picks and basically shows that if you have a clubby a three fold with a good sequence of degenerations to torque varieties, we can also prove it in that context. It's proven for a lot of familiar clubby a three folds but I don't know how if you want to say is approved for every single one. This is not the case. All right, so that's the now I finished yesterday's lecture, and I'm going to go on to today's lecture. So maybe it's a good time to ask a question if you want to. Yeah, yeah, could I, is there some geometric significance of this substitution, or is it just the thing that works to match them up. I mean you know when we were playing with it it wasn't really some have a question because it is a. I mean it was kind of obvious when we were playing with these things. That it's the, it is the thing that changes the box counting problems into the Hodgner girls and it was kind of understood that I mean, yeah I don't know how to actually answer that question maybe the way if you're if you're going to. I just want to pose the question in the following way which I encourage is that how would you find this substitution without doing any calculation. I don't really fully know how to answer that. So I don't know how to do it by pure thought, but on the other hand there were lots of example we were calculating things all the time there and one. The Hodgner girls that come up when you substitute this way you get rational functions and the box counting also gives rational functions. I mean maybe there's a question of the sign the sign is not really. You know, for the qualitative stuff the signs not really that relevant. In some sense the sign has to do with some ways that I mean what are the pluses and minuses and the DT theory, for example. But I think that if I'm honest I'll tell you that I don't know how to justify this substitution without doing any examples. I mean people try to make up some stories and you know maybe they're good stories but I don't know how to say anything in a completely rigorous way. But on the other hand the example is pretty simple. All right so I have to find now the next file. I think we're on D now. So this is part D. So we can find them on, well the same links that Andre sent before. Okay, so are we back in business now can you see. Yes, yes, okay so. Okay, so that's the that was the basics of this ground with DT correspondence, and as I said there's a lot of ways to go with that but I'm going in the descendant direction the descendants and somehow that was the, the link that's put all of these lectures together so we're going to go to descendants. I hope you like descendants because we've had a lot of more. So descendants for curves and sheaves. So we've discussed descendants for the modular space of stable maps. And I want to revisit that construction in a slightly more abstract way. I don't know if it's more abstract or not but it's slightly from a different perspective from the perspective of correspondence. So I want to describe to define this. The symbol this tau k gamma that we saw inside the brackets in the government series. The government in variance I want to define the symbol now as a particular comology class on a particular multi space and specifically a comology class exactly at this grading in this marginalized space. This index so I mean it's you'd consider all the and there's no beta index so this fellow is a comology class and all of these modular spaces at once. So k will be as before the power of the cotangent line. And this delta tells you where the comology class gamma started in X, and the idea is to use this correspondence so this is a basic strategy that if I have the modular space of genius G curves, I can take numbers with one mark point which is like the universal curve over the left hand corner. And that has an evaluation map that's evaluating the map at that mark point X, and it has a projection map to the underlying curve. Well the underlying map without the mark point. So this, in this way, this top space, the one pointed space provides a correspondence between the target and the modular space. When I have this correspondence, then I can, I can use it to move a comology class from X to the modular space. And that's what I do I pull it back, and then I push it forward. But in, while it's you know in the, in the transit in this transit stage, I am allowed also to apply some comology classes here and I apply the cotangent line to the Cape power. And this is the, this is the definition of this descendant now realize as a certain comology class and it comes by using this correspondence to move the comology class that you start with gamma. You move it first up here and then you multiply with the descendant with cotangent line and then you push it down. And if you do this, if you keep track of the bookkeeping you'll find it lives here. And the nice thing about looking at it this way is that formally the same things happen. The same constructions can be pursued for the modular space of sheaves. So what does the typical modular space of sheaf it has some. That's the modular space I'm calling it I because we're going to eventually use it for the ideal sheaves or anyway. And then over the modular space, there's the modular space cross X there's a universal sheaf. And then I have two projections on to X into the modular space. And if I start with some class, comology class and X and if I want to move it to the modular space, then I can use this correspondence. I can pull the class back to the product. And then while I'm in transit there I can apply churn characters and you can pick whatever characters that classes or however you want to organize it but. A straightforward thing to do is take the churn characters of this universal sheaf. And then push it down and then I've, if the module if the target variety, or the variety in question is dimension R. I'll shift the churn character by a little bit to make sure that I have the same dimension rule that I used for Gromovian theory, you don't have to do this. There's now a descendant in sheaf theory if I have a modular if I have a modular space of sheaves that's, you know, really respectable modular space so it has a universal sheaf over the product with this with the space in question. Whenever I have that I can use this construction to get descendants in the theory of sheaves and I get some total logical comology classes in the modular space of sheaves, and this is not a new idea it's an old idea. So examples like the really classical example is when X is a non singular project of curves and when this R is equal to one that's the classical example. If I pick a line bundle. This is just to you don't have to do this but if I pick a line bundle, then there's this space the modular space of rank two stable bundles with fixed determinant L. So that's a space that's been studied a lot. And if I pick the term the degree of L to be one then there'll be no semi stable so this is a very nice modular space and I smooth modular space, and I can define descendants, exactly using the strategy that I take that modular space rank two bundles. I can use the universal sheaf, I can pull back comology use the turn character that universal sheaf and move them down. And, and a theorem that's relevant to this is that actually the full comology the modular space of rank two bundles of fixed determinant is generated by such descendant classes, and you have to use, in this case, X the curve the curve has also odd in this case we have to use this descendants of the odd comology here too. And this is a results by Mumford Kerr ones are gay and also a discuss a there's also a related investigation of the relations and also you can change to to higher rank and there's a whole chapter of algebraic geometry related to this from our point of view it's it's it shows that this descendant construction is pretty useful construction. It gives everything in that case, I mean you don't expect every time to get gives everything that gives everything. In the example of the surfaces we considered that was exactly a parallel construction that was used. I mean it's used also to define the Donaldson invariance often it's called the slant product there. And when I explain various computations for the quote scheme. So, as I put our churn characters of this total logical sheaf and this is very close to descendants in fact, if you look at that. If you look at the dimension of this, this is churn cut, these were churn classes after push forward by growth in the ground rock they're related to churn classes before push forward. So if you use growth in the ground rock you can exactly change this into the descendants of this lecture. Anyway, so my, all I'm arguing here is that this descendant construction for modular spaces sheaves is some kind of general universal construction that has already been used many times and more or less is very useful. And that's exactly parallel to our descendant from the point of view of correspondences the descendants in the modular curves. Okay, so, so now we go back to our three folds. And, and the idea here is that we're already happy with the Gromov Witten DT correspondence for Columbia three folds, but we want to promote that whole theory to descendants. And if we do that of course we can't just be satisfied with clubby aspects clubby I have dimension zero so when we started with insertions we'd like to have some curve consider some three fold X with some curve class beta such that the virtual dimension which is positive so we can consider these but anyway, the Gromov Witten theory allows us to create some descendant integrals using the descendants I've described in this lecture. I warn you these are very slightly different from the descendants that from the previous lecture with the brackets because those were defined using the markings and there's a little difference here. And since we're I'm only discussing this matter here a bit formally I'm not going to worry about that difference. The Gromov Witten theory has some descendant theory and the DT theory of ideal sheaves also have the descendants. And with the ideal sheaves it's perfect the Hilbert scheme as a universal sheaf over it, and I can just off the shelf use the definition for descendants for sheaf theory. And I get the notion of descendants there. So, the question here is that, since we somehow know what to do for the Gromov Witten DT correspondence. In the clubby outcase where there's no descendants it's just somehow an integral is one on both sides. The clubby outcase we're just integrating one here on the modular space, and also one on the ideal sheaves. That's exactly what to do here that's the Gromov Witten DT correspondence and, as I said it's proven sometimes not proven other times but we're pretty confident that that thing holds. That's really stable ground. But now the question is, can we lift this to deal with all the Senate insertions. And if you believe that it's kind of like believing that correspondence whatever the reason that correspondence, whatever reason that that correspondence held that reason should somehow propagate these kind of parallel correspondence constructions. And that's a pretty serious leap and it didn't have to doesn't have to be true. I think that's fair to say. So that here's the question can we extend this Gromov Witten DT correspondence of MNLP to descendants and this was already considered some first steps of this was taken in the second paper MNLP to we wrote two papers where there's some ideas about this. Okay, so. So now, in order to make progress on this one can continue with ideal sheaves, but it turns out it's not the best idea that and the symptom of that the early symptom of that was in the case. The clubby outcase which, as I said we're supposed to be completely happy with now, at least on the conceptual level. There was something that was little unpleasant on the DT side which was that there were these fat points running around. And we had to remove them by hand, not in the geometry but level of the generating functions. And it turns out that when we consider these descendants, and maybe I could say it was a little bit lucky we were able to do that. So we'll start considering these descendants. We'll see that it becomes harder and harder to do it. And at the end it poses some serious difficulties. So what happens now as we switch horses a bit that in the world of sheaves, there's many many different modular spaces. And this can be viewed in some sense as stability conditions choices of stability conditions that's probably the best way to think about it. I'm going to switch now and I talked about the ideal sheaves first for for two reasons one is because it's actually how the subject started. And the second reason is that most algebra at least people on the algebraic side algebraic geometry side have an idea of the Hilbert scheme that's kind of part of the standard repertoire of knowledge and algebraic geometry. So that's what that's the reason I started but in fact, nowadays in the discussion of the theory almost all discussion is, is not about the ideal sheaves it's about the multi space of stable pairs. If we're discussing these sheaves supported on curves and three folds and and the stable pairs more or less repair the shortcoming of the ideal sheaves which is to say, they give a geometric way to excise these points to get rid of them. That's somehow the one line sentence about why it's good. But after doing that it turns out they're better behaved and basically every way, and the development of both the calculation side and theoretical side has more or less all switched to stable pairs because they're just better. But of course they have, you have to pay a little bit to start. The definition is not as well known as ideal sheaves. So I have to tell you what it is. The X be a non singular projective three fold, and the discrete invariance are going to be the same as for the ideal sheaves I pick a, I pick a curve class and I pick another characteristic and and this P for pairs P and X beta is the modular space of stable pairs. This modular space has two things that's why it's a pair. It has a sheaf and it has a section. F is a pure sheaf of dimension one. So what this means is it's a sheaf on X. So here's X, there's a nice picture in the next slide by the way, as a warm up here is X, and this F is a sheaf on it. So that means it could have support up to dimension three, because X is dimension three but we asked for has pure, it's a pure sheaf of dimension one. Dimension one means that it's support is dimension one pure means that it's support is pure of dimension one or more even fancier means that every sub sheaf has supportive dimension one. Maybe that's a better way to say it. So that's what f is, and what is s s is just simply a section of f but it's not any section of f it's a section with co kernel dimension zero. That means it's a section that it can't the section can't be the zero section. It's a section that's a co kernel dimension zero so that's it and so in some sense the definition is not so bad. It's a multi space of stable pairs is these sheaves the sheaves of pure dimension one and pick a section and the section can't be stupid basically. The picture of it is this that's the picture. This is a picture that basically everybody has in their head when they think about stable pairs and here's X. Now this the stable pair, the f is this black sheaf that I've drawn, and I've kind of tried to draw its fibers. Its support is this green curve and the picture you should have in your mind that kind of the ideal picture, the ideal element of this modular space is this the support is a nice smooth curve. Of course doesn't have to be but that's the ideal picture, and the sheaf isn't is a pure sheaf on a smooth curve so it's nicest the nicest picture for that is it's just a line bundle. The f is a line bundle on your smooth curve, and you have to pick a section. And the section can't be zero so that's my section it's like zero it can't be identical zero so zero, a couple points. And you know an algebraic geometry. If you have a section of a line bundle on a smooth curve, the line bundle, both the section and the line bundle up to some C star actually just a turn by the zeros. The data of the stable pair is incredibly simple to think about it's this green curve with the divisor. But those are all the only the ideal elements of it I mean the best behaved elements of it. The modular space has degenerations you can have pure sheaves that are more complicated they're not line bundles on their support. These sections could be zero with the singularities of those sheaves so it can be more complicated. So the construction of this space can be you can just look at. There's a book by Le Portier who studied before. We were thinking about DT theory, etc that he studied constructions of modular spaces of sheaves with sections, etc and stability conditions. And he found out that precisely this definition of a stable pair is his theory already covers the construction of it. You can look at his book. This the references are in some paper in some papers I wrote with Richard Thomas. So the construction is already there it's a it's more or less. So, one doesn't have to develop a new theory to construct it they've already been constructed. So it is a scheme, like the Hilbert scheme and somehow, you know, the having this section takes out the automorphisms rigidifies it. So it's pretty nice space. That's a this guy says this is scheme to find modular space it's a scheme. It's very much like the Hilbert scheme. This module space of pairs and the Hilbert schemes are kind of cousins. So the, the, the somehow first classical example is when access p3. That's a three fold and inside this modular space of pairs is the classical locus. This is just my terminology the classical locus and that parameterizes these ideal objects I said, and those ideal objects are non singular useful curves of degree D is kind of space curves together with a line bundle and a section. And there has to be some. And then there's somehow some linear relation that tells you what the homomorphic or the characteristic is but roughly speaking, to be in this classical locus of stable pairs on p3. It's just the data of a line bundle on a smooth space curve with a section. So as I said you can also think about is just a smooth space curve with some with some points that points determine the line bundle and the section. So the, that's kind of nice and meaning that somehow the idea of what the bulk of the spaces is simple from the point of view of geometry. The interesting part about the space of course is what else is there in the space and it's always the case with these modular spaces that, while one likes to imagine the general object, which is incredibly nice. In fact, all of the study and the analysis in any case is always on the most degenerate our objects. That's just how life works. Okay, so, although I'm skipping all the analysis of the generate objects, I'm telling you that actually all the, all the thought is about those. So, the first thing we need to get started is we need an obstruction theory. And this one is significantly more subtle than the Hilbert scheme. And the reason is that in order to get the right deformation theory, one has to view this pair. Well, you can view this pair as a map from OX to F because I have a sheaf and I have a section that is a map from OX to F, but you have to view this. This is a complex in the drive category. The deformation theory that the deformation obstruction theory that we place on this stable pair space is as deformation of objects in the drive category. And so one has to prove this stuff makes sense and this is explained in that first paper with Richard, I have the title down somewhere. So that's, that's pretty technical, subtle discussion, but it turns out that you can define a deformation obstruction theory on the space of stable pairs, where the deformation space is given by traceless X1, where of the stable pair with itself, except viewed as an object in the drive category, and the obstruction space is X2 and the higher X vanished by before. So in some sense it's very similar, but things are, it's very similar to the Hilbert scheme, but things are slightly more complicated and one has to take slightly different perspectives. Okay, but after that's all done, you have a nice virtual fundamental class on the space of the exact same dimension that one we've already seen twice before. And yeah, here's the paper, the first paper is the counting curves via stable pairs with Richard. Okay, and then what about descendants, what we have these descendants also this universal the stable, the space of stable pairs as a universal sheaf which is the universal sheaf in the stable pair also as a universal section. The universal sheaf, and then we have maps. We have the same correspondence before, and we can define the sentence exactly before it turns out it's smarter to do something else. It doesn't change much but it's smarter to not take the just the churn class of F, but that take the churn class of, well, because the complex stable pair and not only as universal sheaf but as universal complex with the universal section. And it's a little smarter to take the churn character, not of the sheaf but of the complex. And why I say it's only a little bit smarter is because this is after all the trivial bundle so it's not going to change much but it'll change a little bit in one place. And that will help you. So we define this churn character, the, we define the descendants and we call it instead of the tau CHK. So that's a descendant insertion is given by the churn character of this complex. Okay, so that was a little detour to get us up to the same level and stable pairs, as we had achieved in in the DT theory of ideal sheaves. And now now the question is why did we do it I tried I tried to give you some example I'm trying to give some motivation. It is the case since the stable pairs by definition or pure of dimension one. They're supported on curves there's none of this nonsense of these fat points running around. That's just not there. And you could say well how did we make that profit without any payment and the answer is of course there is some payment. You had to pick the section. The Hilbert scheme has just a sub curve, it doesn't have the sheaf with the section. So there that's the payment. You get rid of the running around fat points but you have a little more complicated structure on the curve. There is a place where they kind of overlap you could say the Hilbert scheme. What's on that curve is actually Oh, of that curve that's the sheaf and the section is one. And that's a that's the place where the two ideas overlap. Okay, maybe if you haven't thought about those things you can think a little bit. But anyway, that's the roughly the the transaction there that you get rid of the fat points, but you the price of that is to put the sheaf of the section. But the overall transaction is a profit, because everything's happening then on the curve. And one of the consequences of that I mean the first place where you see that's a real advantage is the study for descendants on stable pairs. And that's what the end of this lecture which we're getting close to and tomorrow is about. It's about this descendant theory of stable pairs. And there's a kind of a circle of ideas here. The first is to promote the court the GWDT correspondence to the case of the descendant theory of stable pairs. And there's various papers, some even quite recent. And all of that, as I said that we had on the in the girl wouldn't DT in the first past rationality of the series played a serious role. And so one of the one of the parts of this wheel is rationality of the descendant theory of stable pairs. I wanted to still say a little bit about that today, which I will. And then somehow the most interesting piece of this puzzle is the various our constraints, which is to say that if you believe in here this should be now a clear goal in the sense that if we believe that the descendant theory theory of the descendant groom of Witten theory, it corresponds to the descendant theory of stable pairs on a three fold. Since we already know there's various our constraints on the groom of Witten theory, there must be a way then to find these various our constraints on the stable pairs theory and that's correct. And so we can find that and even prove it in some cases. So there's a, I tried to write here some. Well, these three basic ideas and then some of the papers that are relevant. And then there's even some key here. One oh is Andre and two O's is a Blumkov and a conco. Okay. Maybe they're all two O's here. No, there's one other. And the last topic today before we it's it's the, so tomorrow will be about this various our constraints on the stable pairs side that's kind of in some sense, a lot of open questions in that direction, but I want to start with the rationality. And the rationality statement says that if I stay if I take my non-singular projective three fold, and I can define this descendant generating series so there's nothing that's happened here now that we haven't already seen. So this is the descendant generating series I should say for stable pairs just so it's clear for stable pairs. And it's defined by this bracket, the bracket tells you what the space is what the curve classes. And then what are the descendants insertions these churn characters given by those are given by those correspondences. The left hand so that the left hand side is defined by the quality the quality is we sum over all or the characteristics holomorphic or the characteristics in this way and then we take this integral of the descendant operators on the fundamental class of stable pairs. This is now completely well defined we've discussed every aspect of this definition I think. And that's a series in Q. Lots of them because you get to choose which come all the classes you put in here and you get to choose which numbers of the turn characters are lots of them you get to choose your turn cloud you get to choose this. And of course you also get to choose your space. It's important that this is the series is a Laurent series that doesn't go infinitely negative that's the very basic aspect of it. And that's because the modular spaces they're just physically empty for and less you don't need to know anything to show that these that you get zeros for very negative numbers are just empty. Well you need to know a little bit of classical geometry to prove that but it's empty. The first rationality conjecture, which is an incredibly clean statement. It says that this series as defined is always in every single case the Laurent expansion of rational function in Q. And if you want to see an example of it. There's a paper I wrote on descendants of stable pairs for this Donaldson volume. Can you see that. It's a little small. Can you see it better now. It's a little fuzzy. Yeah, you don't have to know too much about it but just you know this is a show there's some complexity here. So this is an example of degree two with a town nine. So maybe the notation has changed here but don't worry about this too much for p three it's a for twice a line it's kind of kind of complicated series. But if you see it it's rational. Technically this is this technically this computer program outputs a conjecture, but it's 100% certainly correct. That's not the issue. I'm just trying to be honest. We can discuss why that's true later but if you want. But if you look at it it's there's something kind of remarkable about it's not a random rational function. First of all the bottom has some roots, those roots are, you know, not complicated roots they're plus minus one in fact they're all roots of unity. And the top is a very far from random thing it's palindromic. If you see it. And 73 here and 73 in the minus 825 minus. So it's in, it has this entire palindromic symmetry. So that's kind of a striking thing. And that's of course formulated in the second part of the rationality conjecture. And that's related to that's by the way that's related to q goes to one over q that we saw for Gromov within the T correspondence. So the second part of the rationality conjecture which is formulated with Aaron is that this rational function is very special so if I look at this rational function. It only has poles that at roots of unity and zero actually this was something we would already goes back I think it's MNOP2, but the part that's precise was formulated precisely later is it satisfies this functional equation that if I take this, whatever rational I take it out here. And if I substitute one over q. It satisfies this functional equation and the terms of functional equation depend a little bit on what's being submitted here, the case. And also the degree that this size of the first the virtual dimension here. So those are the two rationality conjectures the first just as it's rational the second says, actually it's a very special type of rational function and you can look at this as a rational function. And there's there's ways to calculate it we could calculate this rigorously, but that takes longer. And it's not so clear what the point is, but Alexi so Alexi has the some computer programs that allow you to calculate the answer but not rigorously because the computers you know, they're not that smart. And the last thing I want to say is that this is great for stable pairs and if you believe this, you should be convinced that if you want to think about descendants for chief theories you should work on stable pairs because they have this beautiful rationality property. In every case, at least conjecturally, but what about ideal sheaves, what happens to them and then here is the one of the reasons why we don't want to carry them along in the descendants study. And that's an interesting thing to do but maybe the right way to say it's harder to carry them along. And that's because they fail to be rational. So it's not true that every function in the world is rational. And if you do the same construction for the ideal sheaves and the Hilbert scheme, you define the same descendants and the same series. The same descendants series and I put an eye to show it's ideal sheaves now. And that's not rational because the first ones is McMahon series and that's not rational. But that's not the problem because we knew how to fix that. So what the idea the first idea was you take this descendants series in the DT theory and you divide out by McMahon because you know McMahon's an irrationality that we've already dealt with. So divide that out. So this is our this is basically the best hope and the DT on the ideal sheaves side. And the problem, the first problem is this is still not rational and q. And maybe this is not such a serious problem, but it makes it the subject harder to develop in that line and the whole source of this irrationality is those fat points. And if you want an interesting mathematical conjecture to prove there is that with Alexei and Andre we conjecture that while this thing can be irrational and it definitely can that's not. That's not an issue it can be irrational. This normalized series is a polynomial. So it has rationality in it, but it has all of these other functions. These are the this QDQ this iterated derivative of a special function f3 and that's the f3 which kind of looks like the Eisenstein series except Eisenstein series have odd powers here is even power. So this is so we this conjecture says precisely exactly how much more there is in these kind of functions and rational functions we don't know how to prove this conjecture. And maybe I stopped with this comment that for some years, we would call this the Frankenstein series because very close to Eisenstein series but doesn't have any good properties. But then people in the subject said that this was irreverent in some way so we don't do that anymore. So whatever this thing is whatever the name of it it's not the Frankenstein series. Okay. We have a couple of questions in the chat, but before I want to say that the summation should go over and I guess. Oh yeah sorry I screwed that up. There are questions in the chat. There's one which is, which actually pertains to what we've learned before. How do you understand the fact that in many concrete examples of Galavia threefolds we can in some sense interpolate from the stable pair moduli into the Hilbert scheme moduli by a wall crossing. Is there a conceptual explanation for this interpolation. Yeah you can I mean a conceptual I don't know but I mean you can formulate it as I would say more generally you can formulate it as a. I mean, you know, it has to do with what you. Yeah I don't know how to answer this question exactly. Yeah, in terms of stability conditions in the drive category, you can view the Hilbert scheme. It's the result of one of the stability conditions and you can review the stable pairs as a result of another stability condition and that wall crossing so I think that Richard wrote some paper, where he explains that very carefully. And these ideas. They also have different. I don't, I don't know how to say something somehow really much more about that but they are there view you can view them as different. What you view as a stability condition so in some sense. It's a question whether you ask for this section to be, you know how much weight you want to put on the section. And one of them you ask section to be surjective. And that that gives you the Hilbert scheme but if you loosen it to being surjective then you get the. Then you get the stable pairs. Because you know one of them is really just that if you write it like that one of them is all going to. Oh, see, let's say one of them is all going to F with a surjection and the other is going to F. And this is not surjective and when when it's not surjected then you have to pay some price for it. And that's why you get the other conditions in the stability. So that's the question we have maybe you can go for it says a rational function. Because the question is, what are the roots of the of the talent from a polynomial. Oh that I don't know I don't know what that is. I mean they're going to be, they don't have to doesn't have to factor over q I can tell you that. I don't know what the meaning is the roots and roots of unity we have some kind of control over which roots of unity occur, and roughly speaking the roots of unity that occur are related to how, how multiple the classes. I can this case for example you get only. You get minus one whose square is one and that that to has to do with the fact that this is a to here. But this that's that that stuff's conjectural. I don't know how to prove. Yeah, even this statement that this this functional equation is always true. Maybe I should have said that that this rationality conjecture has been proven in some context that it's going to be very important that we've proven it in all of the K all the toward cases and access torque. This rationality conjecture is proven. If it's one leg torque, then I think we have control of the functional equation. It's possible, but in the general toward case I don't think we have control of the functional equation. So this part is still a little bit mysterious. But it's really striking when you do the examples they turn out to be these beautiful palindromic polynomials and and I should say that I did say that in the case of the club yeah that there the functional equation is proven to goes to one over q. And how is that proven there why is it proven there it's because there's a different technique in club yeah which is not really being covered in these lectures at all which is this example by Kai to use some kind of weighted whole other characteristic to get the club yeah and variance and the one q goes to one over q is proven using some properties of that or the characters that weighted or the characteristic together with some ser duality that relates higher coefficients to lower coefficients. So there's a very nice geometric reason why I don't know what happened Alexis function. There's a very nice geometric reason in the club yeah case why we get these palindromic polynomials, and actually it turns out that whether it's palindromic or anti palindromic depends on the sign here but but those arguments don't work in general there those arguments only work in the club yeah case in the case like in p three, we just have no argument and in the full generality. I have one more question. The series that you would expect to be rational in the DT for case. Yeah, the people are working on that. I don't know who asked that question but my advice is to ask young Han, when people are working on those DT for calculations. And I think that it's the first hope would be some kind of rationality if you. You know, the DT for here we have one curve class and we have only one other parameter and DT for you have two other parameters. But so, I mean some of the people, I mean some of the people in my group who are there we're working on that young Han I think one on is also about it so I would address those questions to those two. Thank you very much.