OK, so first about the question about the stability conditions. On MGN-BAR, I was correct.

There is a paper, a relatively recent one, by Y.P. Lee and a collaborator whose name is CHOU. And it's exactly about VIRISAR constraints for varying the HASSET weights. So they consider some sets of, if you have,

because you have to consider all modulized spaces at once, you have to consider which HASSET states you are going to allow for all the points at once. And they have a notion for that. And then he writes down VIRISAR operators. He exactly writes down VIRISAR operators. So they exactly write down VIRISAR operators in that context. And as I said, that conceptually, in some sense,

all of these things differ from MGN by some genus 0 behavior. So there are methods to calculate descendants from one from the other. But to put them all together in VIRISAR constraints is done in this paper. Other people have studied the differences between the cotangent lines on these spaces. In fact, there's a paper by Alex Seyev and someone some years ago.

OK, so I hope that's enough information to find the paper. If not, you can ask me. So the last lecture ended at this discussion of sheaf counting in dimension 3 and the geometry of the Hilbert scheme.

And we went through the Hilbert scheme as a moduli space of ideal sheaves that was here. And I noted we really consider all the bad points in 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 the 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 for this transition to Clavier threefold. So I'll start here. So for Clavier threefold, one of the nice things from our point of view here is both these moduli spaces. The stable maps and the ideal sheaves

have always had virtual dimension 0. And the very first question that you could ask here, although this is not how the subject developed, but nevertheless, 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 there a relationship? And let's look for how that might be.

So the reason that you might think there's a relationship is that they both virtually count curves. That's true, but there are some differences. The differences, for example, in this Hilbert scheme, you're counting all these little fat bubbles that are running around,

but 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 moduli spaces look a little bit different. So it's hard to believe that there would be some equality

on the level of moduli spaces, but still 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 indecomposable class, like the class of a line on the quintic, for example,

it's a 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 indecomposable class, I might hope that the Gromov-Witten invariant, which has genus G maps to this class, is the same as the ideal sheaf count,

where I somehow try to guess the Euler characteristic that this smooth genus G curve has, and I don't have to guess that, that's gonna 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 the study, more or less for all questions. And that is the example of a P1 with normal bundle minus one, minus one.

So I take this P1 in the Calabia threefold, and 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, a Calabia threefold. So this is in some sense kind of like a thought experiment. We just imagine that our Calabia threefold has one. Of course there exists Calabia threefolds

where these exist. And I want to consider what this curve contributes to this Gromov-Witten 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 us the contribution to the genus Gromov-Witten invariant of the ambient Calabia threefold in the class of this P1, this rigid P1. And it's not a complete trivial thing. Although it looks like it's only a genus zero curve, it contributes to the Gromov-Witten invariant in all genera.

And that's because you can map higher genus curves to this P1 by contracting higher genus bubbles. So 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 high genus that are mapped to points.

So it looks silly, but nevertheless, in the stable maps, this innocent rigid rational curve actually contributes to the Gromov-Witten 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 we keep track of the genus and put the U variable for the genus. This is the genus variable. Genus, though it's the only thing moving in the sum. The curve class is fixed. The Calabia was fixed. It's the only thing moving is the 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 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 everything to actually modulate space of 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 we can produce 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 some kind of direction which I'm not going to really investigate in this lecture. But that's kind of fun. And in a way 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 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 Devesh, Malik, Nikita Nekrasov, Andrei Okonkov and myself. And we wrote two 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 that there's localization, but now on the DT space, which is the Hilbert scheme of points,

sorry, Hilbert scheme of curves. And of course there's lots of points and the Hajj and Enroll are replaced now by box counting ideas. And Andrei Okonkov also with Reshetikin developed some box counting techniques which are very useful in this context. And then of course, there's also lots of tricks there. If you look at those papers,

there's various tricks about how to handle this. As I said, I mean, integration in some sense is always some kind of summation and you're not guaranteed to be able to get a closed form answer unless you're working on a nice problem. So anyway, it turns out this problem is a nice problem. And now we fixed this curve class just as we did before.

And now we look at all the ideal sheets. So this is the DT invariant and invariant and the moving thing now is the Euler characteristic. Can we 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 fixed 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. Where's the, oh yeah, so here, this is the moduli space of curves that I view as the Hilbert scheme, whose class is beta and whose holomorphic Euler characteristic is N.

I must write that somewhere. Yeah, here. Hilbert scheme of curves and N 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, its holomorphic Euler characteristic is typically negative. It's one minus G. So you don't really know where the sum starts. You have to sum over all possible holomorphic Euler characteristics. In the sum here, 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 it. 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 the sum as this N and Z because it really could be negative. But the thing that's true by geometry is that it's a 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 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 function,

or had the idea to start thinking about this function because he was stacking cannonballs in the British army. So it's kind of a, some kind of military spinoff actually. And 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 Euler characteristic,

to a power, which is Euler characteristic of the Calabia threefold. In particular, the answer here has to do with the whole geometry of the Calabia threefold. The answer here had no Calabia threefold 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 in the Calabia don't matter. But for the DT calculation, they do matter. And it's obvious why they matter, because if you take ideal sheaves, even if you're interested in ideal sheaves that have to do with this curve, you can have these little fat points running around everywhere. So 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 invariants by this series, to these invariants by this series,

you will not find any matches at all. So the conclusion about a simple literal hope fails. The modular spaces don't match. They're too different that way. But a more sophisticated hope actually is 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. So we started the DT answer. And this crazy McMahon function is obviously not, we're not going to see that again. So let's just divide out by it. So 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'm going to get, sorry, 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, the substitution here is Q is equal to E 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 U and this is in Q. 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 a series and then you have to use the formula for sine.

And if you do this patiently, you find that it's just exactly correct. So I say that again. So if I take this DT series, I take out the McMahon part, I get this rational function and I substitute this rational function in exactly using this substitution. And this substitution does turn rational functions into trigonometric functions.

And if you find out what exact trigonometric functions turns to, as I said, you can do this very patiently. I did it here, probably it's even correct. You exactly get the Gromov-Witten series, you end in this formula that Carol Faubert and I found. So that's very promising. And if you're very optimistic,

you can make a sweeping general conjecture based on just that example. And that's what we did. That's this Gromov-Witten DT correspondence in this MNOP in the first paper of MNOP. And the conjecture said qualitatively says that the relationship found for this minus one, minus one curve over P one, 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 toric collabiales. And you could see that that's related to box counting. And the vertex was developed by Agonagich,

Klam, Marino and Vafa. And there's box counting methods that were developed by Andrey Kunkov and Rashidikin. 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 somehow the origin of this conjecture. And now I want to write it down precisely.

So, 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 collabial threefold. It can be any one you want. So by this, I mean, non-singular projective collabial threefold, and then this has a Gromov-Witten series,

a Gromov-Witten 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 sum over all genera, all non-constant maps, I put the Gromov-Witten 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 novel 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 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-Witten 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-Witten theory. And that's, of course, just taking the exponential.

So this is the connected version of Gromov-Witten theory. Take the exponential, get the disconnected version. And if I do that, I get a disconnected Gromov-Witten theory. That's a series in U for every curve class beta. It's just calculating the disconnected Gromov-Witten invariance in that curve class beta. On the DT side, we sum over all beta,

including zero now, because we'll remove it by hand. And 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 in DT. Yeah, so this is the, geometrically, it's a little different. In Gromov-Witten theory, we can just throw out the constant maps, because the modulate space is fine without them,

because there is a connected modulate space to connect to maps. In DT theory, we can't just throw them out. We'd be very happy to do that. And then 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 schemes general points looks like this, right? I mean, 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 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. Yeah. Is there some solution for why exponentiating gives a disconnected Gromov-Witten 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-Witten 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 combinatorics. 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. So the easiest conjecture at MNLP, there's three conjectures. The very first one says that you take any Calabi-Yau

threefold and you look at the degree zero DT theory. That's a DT theory of the Hilbert scheme of points of that Calabi-Yau threefold. That's some kind of new thing. It's some virtual class on the Hilbert scheme of points of Calabi-Yau threefold. And the first conjecture is the answer for it. It's this McMahon series to the Euler 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 Junlei, there's a proof by Baron-Fanteki, and there's a proof also using cobordism 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 theory is we want to remove these points.

And so we'd 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. Then I make this, 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 invariants 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 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. When you look at this function and you substitute Q goes to one over Q, you find amazing and get 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 two. And this turns out this conjecture is also accessible and it's accessible by methods that when we formulated the conjecture weren't there, but came into the subject very soon,

which was using ideas of wall crossing and is proven by Bridgeland and Toda. So that's kind of extremely nice proof. And it proves that this is a Laurent expansion of rational function for Columbia three-fold. So maybe I should just write here because you can talk about this more in general for Columbia threes. It proves, so their work proves that it is a Laurent expansion of rational function

more of it satisfies this functional equation. So those two conjectures are proven. And then the conjecture, the most complicated conjecture is the one that's the relationship with the Kromovitin 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 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 state in the conjectural discussion, and it's actually proven now, since this is a rational function, such as substitution is always legal, 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 Gromovitin-DT correspondence. And in this level, and the way I explained it, it's a very clean statement. And it's a really, if you think about this, 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 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 Klabia torque geometries, this was proven in the first pass.

That's the other O here is Oblomkov, I'll write the names later. This is for torque geometries, it was proven for complete intersections, I proved an argument with Aaron Pickston. And also this argument, Aaron Pickston basically shows that if you have a Klabia threefold with a good sequence of degenerations to torque varieties,

we can also prove it in that context. So it's proven for a lot of familiar Klabia threefolds, but I don't know how, if you want to say, is it proof 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, is the microphone on? Yeah, yeah, could I, is there some geometric significance of this substitution or is it just the thing that works to match the mark? I mean, you know, when we were playing with it, it wasn't really something I have a question

because it is, 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 Hadjina 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 going to, if you 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 examples. We were calculating things all the time there and one new, for example, that the Hadjina girls that come up when you substitute this way, you get rational functions and the box counting also gave rational functions. I mean, maybe there's a question of the sign. The sign is not really, you know, for the qualitative stuff,

the sign is not really that relevant. In some sense, the sign has to do with some ways that, I mean, what are the pluses and minuses in 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 there are 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. You can find them on, well, the same links that Andrei sent before.

Okay, so are we back in business now? Can you see? Yes, yes. Okay, so that was the basics of this Gromov-Witten DT correspondence. And as I said, there's a lot of ways to go with that, but I'm going in the descendant, in the direction of the descendants. And so somehow that was the link

that's put all of these lectures together. So we're gonna go to descendants. I hope you like descendants because we've had a lot, we'll have more. So descendants for curves and sheaves. So we've discussed descendants for the modular space of stable maps. And I wanna revisit that construction in a slightly more abstract way. I don't know if it's more abstract or not,

but from a different perspective, from the perspective of correspondence. So I wanted to define this symbol, this tau K gamma that we saw inside the brackets in the Gromov-Witten series, the Gromov-Witten invariants. I want to define this symbol now

as a particular cohomology class on a particular modular space. And specifically a cohomology class exactly at this grading in this modular space. So there's no genus index. So, I mean, you'd consider all the, and there's no beta index. So this fellow is a cohomology class in 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 cohomology 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 genus G curves, I can take the curves 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 in this way, this top space,

the one-pointed space provides a correspondence between the target and the modular space. And when I have this correspondence, then I can use it to move a cohomology class from X to the modular space. And that's what I do. I pull it back and then I push it forward.

But while it's in the transit stage, I am allowed also to apply some cohomology classes here and I apply the cotangent line to the kth power. So this is the definition of this descendant now realized as a certain cohomology class.

And it comes by using this correspondence to move the cohomology classes that you start with gamma. You move it first up here, then you multiply with the cotangent line, then you push it down. And if you do this, if you keep track of the bookkeeping, you'll find it lives here. Okay, 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 a typical modular space of sheaf? It has some, that's the modular space. I'm calling it I because we're gonna 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 onto X into the modular space. And if I start with some class, cohomology 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. I mean, you can pick whatever characteristic 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 Gromovitin theory. You don't have to do this. But this is now a descendant in sheaf theory. If I have a modularized space of sheaves, that's a really respectable modularized space.

So it has a universal sheaf over the product 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 cohomology 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 projective curves. And when this R is equal to one, that's the classical example. So 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 modularized 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 degree of L to be one, then there'll be no semi-stables. This is a very nice modularized space, a nice smooth modularized space. And I can define descendants exactly using the strategy that I take that modularized space of rank two bundles, as some universal sheaf. I can pull back cohomology, use the turn character of that universal sheaf

and move them down. And a theorem that's relevant to this is that actually the full cohomology of the modularized space of rank two bundles of fixed determinant is generated by such descendant classes. And you have to use, in this case, X is a curve. The curve has also odd cohomology. So we have to use this descendants

of the odd cohomology here too. And this is results by Mumford, Kurwan, Zaghi, and also there's also a related investigation of the relations. And also, you can change two to higher rank. And there's a whole chapter of algebraic geometry related to this. From our point of view, it shows

that this descendant construction is pretty useful construction. It gives everything in that case. You don't expect every time it gives everything, but it 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, the insertions I put are churn characters of this tautological 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 Grothendieck and Rauch. They're related to churn classes before push forward.

So if you use Grothendieck and Rauch, you can exactly change this into the descendants of this lecture. Anyway, so all I'm arguing here is that this descendant construction for modular space 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. OK, so now we go back to our three folds. And the idea here is that we're already

happy with the Gromov-Witten DT correspondence for Calabi-Yau 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 Calabi-Yau's because Calabi-Yau's have dimension 0. So when we start with insertions, we'd like to have some curve. We want to consider some three-fold x with some curved 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 from the previous lecture with the brackets because those were defined using the markings, and there's a little difference here.

And since I'm only discussing this matter here a bit formally, I'm not going to worry about that difference. But anyway, there's a Gromov-Witten theory, has some descendant theory, and the DT theory of ideal sheaves also have the descendants. And with ideal sheaves, it's perfect. The Hilbert scheme has 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 Calabi-Yau case where there's no descendants, it's just somehow an integral

is 1 on both sides, the Calabi-Yau case for just integrating 1 here on the moduli space and also a 1 on the ideal sheaves. So we know 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 descendant 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 to these kind of parallel correspondence

constructions. And that's a pretty serious leap, and it doesn't have to be true. I think that's fair to say. So 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 2. We wrote two papers where there's some ideas about this. And OK, 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.

And the symptom of that, the early symptom of that was in the case, the Calabiao case, which as I said, we're supposed to be completely happy with now, at least on the conceptual level. There was something that was a 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 the level of the generating functions. And it turns out that when we consider these descendants, maybe I could say it was a little bit lucky we were able to do that. But when we 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 is 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. So we're going to switch now. And I talked about the ideal sheaves first 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 in algebraic geometry.

So that's the reason I started. But in fact, nowadays in the discussion of the theory, almost all discussion is not about the ideal sheaves. It's about the modular space of stable pairs. If we're discussing these sheaves supported on curves and threefold.

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 in 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, 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. So let x be a non-singular projective threefold. And the discrete invariants are going to be the same as for the ideal sheaves. I pick a curve class, and I pick an Euler characteristic n. And this p for pairs, pn 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. And this f is a pure sheaf of dimension 1. So what this means is it's a sheaf on x. So here's x. There's a nice picture in the next slide, but anyway. 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 3 because x is dimension 3. But it's a pure sheaf of dimension 1. So dimension 1 means that its support is dimension 1. Pure means that its support is pure of dimension 1.

Or even fancier, it means that every sub-sheaf has support of dimension 1. 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 of dimension 0. That means that the section can't be the 0 section.

It's a section that's a co-kernel of dimension 0. So that's it. And so in some sense, the definition is not so bad. It's a modular space of stable pairs. It's these sheaves. The sheaves are pure dimension 1. And you pick a section, and the section can't be stupid, basically. The picture of it is this.

That's the picture. This is the picture that basically everybody has in their head when they think about stable pairs. Here's x. Now, the stable pair, the f is this black sheaf that I've drawn. And I've tried to draw its fibers. Its support is this green curve. And the picture you should have in your mind, the ideal picture, the ideal element of this modular space

is the support is a nice, smooth curve. Of course, it doesn't have to be, but that's the ideal picture. And the sheaf is a pure sheaf on a smooth curve. So the nicest picture for that is it's just a line bundle. You're taking f as a line bundle on your smooth curve,

and you have to pick a section. And the section can't be 0, so that's my section. It's like 0. It can't be identically 0, so it's 0 at a couple of points. And in 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 are actually just determined by the zeros. So the data of the stable pair is incredibly simple to think about. It's this green curve with the divisor. But those are all 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. And the sections could be 0 at 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 Poitier who studied before we were thinking about DT theory, et cetera. He studied constructions of moduli spaces of sheaves with sections, et cetera, and stability conditions. And it turns out that precisely this definition

of a stable pair, his theory already covers the construction of it. So you can look at his book. The references are in some papers I wrote with Richard Thomas. So the construction is already there. It's more or less, so one doesn't have to develop a new theory to construct it.

They've already been constructed. And it is a scheme, like the Hilbert scheme. And somehow having this section takes out the automorphisms, rigidifies it. So it's a pretty nice space. So this guy should say this is a scheme.

It's a fine moduli space. It's a scheme. It's very much like the Hilbert scheme. This moduli space of pairs and the Hilbert schemes are kind of cousins. So somehow first classical example is when x is p3.

That's a threefold. And inside this moduli space of pairs is the classical locus. This is just my terminology, the classical locus. And that parametrizes these ideal objects I said. And those ideal objects are non-singular irreducible curves

of degree d. These 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 holomorphic Euler 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 it as just a smooth space curve with some points. Because that points determine the line bundle and the section. So that's kind of nice, meaning that somehow the idea of what the bulk of the space is

is simple from the point of view of geometry. The interesting part about this space, of course, is what else is there in the space. And it's always the case with these moduli 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 objects. That's just how life works. OK, so although I'm skipping all the analysis of degenerate objects, I'm telling you that actually 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 as a complex in the derived category. And the deformation theory, the deformation obstruction

theory that we place on this stable pair space is as deformation of objects in the derived category. And so one has to prove that this stuff makes sense. And this is explained in that first paper with Richard. I have the title down somewhere. So that's a 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 derived category. And the obstruction space is x2 and on the higher x

vanish by the floor. 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. OK, 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 here's the paper. The first paper is counting curves via stable pairs with Richard. OK, and then what about descendants? Well, we have these descendants also. This model space of stable pairs has a universal sheaf, which is the universal sheaf

in the stable pair. It also has 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 just the turn class of f,

but take the turn class of, well, because the complex, the stable pair not only has a universal sheaf, but has a universal complex with a universal section. And it's a little smarter to take the turn 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 turn character. We define the descendants, and we call it, instead of the tau chk. So that's the descendant insertion, and it's given by the turn character of this complex.

OK, so that was a little detour to get us up to the same level of stable pairs as we had achieved in the DT theory of ideal sheaves. And now the question is, why did we do it? I tried to give you some example.

I tried to give some motivation. It is the case, since the stable pairs, by definition, are 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 this sheaf with the section. So 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 O of that curve. That's the sheaf. And the section is 1. And that's the place where the two ideas overlap. OK, maybe if you haven't thought about those things, you can think a little bit.

But anyway, that's roughly the transaction there, that you get rid of the fat points, but 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 it'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 GWDT correspondence to the case of the descendant theory of stable pairs.

And there's various papers, some even quite recent. And in that, as I said, that we had in the Gromovitin DT in the first pass, rationality of the series played a serious role. And so 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 Virasara constraints, which is to say that if you believe, and here this should be now a clear goal in the sense that if we believe that the descendant Gromovitin theory

corresponds to the descendant theory of stable pairs on a threefold, since we already know there's Virasara constraints on the Gromovitin theory, there must be a way then to find these Virasara constraints on the stable pairs theory. And that's correct. And I will explain that and even prove it in some cases.

So 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. 1-0 is Andrei and 2-0 is Oblomkov and Okonkov. Maybe they're all 2-0s here. No, there's one 0 there.

All right, so the last topic today before we, so tomorrow will be about this Virasara constraints on the stable pair 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 take my non-singular projective threefold and I can define this descendant generating series, so there's nothing that's happened here now that we haven't already seen. And this is the descendant generating series, I should say four stable pairs just so it's clear. Four stable pairs.

And it's defined by this bracket. The bracket tells you what the space is, what the curve class is, and then what are the descendants insertions, these churn characters given by, those are given by those correspondences. And the left hand side is defined by the equality, the qualities we sum over

all Euler characteristics, holomorphic Euler 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. And we get lots of them

because you get to choose which cohomology classes you put in here. And you get to choose which numbers of the churn characters is lots of them. You get to choose your churn class. You get to choose this. And of course, you also get to choose your space. And it's important that this series is a Laurent series.

That doesn't go infinitely negative. That's the very basic aspect of it. And that's because the modularized spaces, they're just physically empty for n less. You don't need to know anything to show that you get zeros for very negative numbers. They're just empty. Well, you need to know a little bit of classical geometry to prove that, but it's empty. So then 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 a 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. Not really, no. Can you see it better now? It's a little fuzzy. You can see that it has 10 terms in the numerator. Yeah, you don't have to know too much about it. But this is to show there's some complexity here. So this is an example of degree 2 with a tau 9.

So maybe the notation has changed here, but don't worry about this too much. For P3, it's for twice a line. It's a kind of complicated series. But if you see it, it's rational. 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, if you want. But if you look at it, there's something kind of remarkable about it. It's not a random rational function. First of all, the bottom has some roots. Those roots are not complicated roots.

They're plus minus 1. In fact, they're all roots of unity. And the top is very far from random thing. It's palindromic, if you see it. There's a 73 here, and a 73, and a minus 825. So 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 Q goes to 1 over Q that we saw for Gromov-Witten DT 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 at roots of unity and 0. Actually, this was something that already goes back, I think, to MNLP2. But the part that was formulated precisely later is it satisfies this functional equation. That if I take this, whatever rational function I get here, and if I substitute 1 over Q, it satisfies this functional

equation. And the terms of the functional equation depend a little bit on what's being submitted here, the k's, and also the degree of this size of the virtual dimension here. So those are the two rationality conjectures. The first just says that, for example,

that's rational, and the second says, actually, it's a very special type of rational function. I mean, you can look at this thing. It's extremely special type of rational function. And there's ways to calculate it. We could calculate this rigorously, but that takes longer. It's not so clear what the point is. But Alexei has some computer programs that allow you to calculate the answer, but not

rigorously, because the computers, they're not that smart. OK, so then 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 sheaf 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 here is one of the reasons why we don't want to carry them along in the descendant study. Although 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 in the Hilbert scheme, you define the same descendants and the same series, the same descendant series. And I put an eye to show it's ideal sheaves now. Then of course, you know it's not rational, because the first one is McMahon series, and that's not rational. But that's not the problem, because we

knew how to fix that. So the first idea was you take this descendant 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 basically the best hope on the ideal sheave side. And the first problem is this is still not rational

in Q. Maybe this is not such a serious problem, but it makes 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 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 this QDQ, this iterated derivative of a special function F3, and that's the F3,

which kind of looks like the Eisenstein series, except the Eisenstein series had odd powers here. This is even power. So 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 it's very close to the Eisenstein series, but it 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 wanna say that the summation should go over NIS. Oh yeah, sorry, I screwed that up. So there are questions in the chat.

There's one which is, which I think pertains to what we did before. How do you understand the fact that in many concrete examples of Caladia threefold, we can, in some sense, interpolate from the stable pair modulate into the Hilbert scheme modulate by a wall crossing. Is there a conceptual explanation for this interpolation? Yeah, you can, I mean, 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, 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 as the result of one of the stability conditions, and you can view 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,

yeah, I don't know how to say something somehow really much more about that, but they are viewed, you can view them as different, as the results of what you view as a stability condition.

So in some sense, it's a question whether you ask for this section to be, how much weight do you wanna put on the section? And one of them, you asked the section to be surjective, and that gives you the Hilbert scheme, but if you loosen it to being surjective, then you get the stable pairs.

Because one of them is really just, if you write it like that, one of them is O going to O C, let's say one of them is O going to F with a surjection, the other is O going to F, and this is not surjective. And 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. Another question we have, maybe you can go for, say, the rational function. Because the question is, what are the roots of the palindromic polynomial?

Oh, that I don't know. I don't know what that is. I mean, they're gonna be, they don't have to, it doesn't have to factor over Q, I can tell you that. So the question is if there's a meaning behind the complex numbers you get or whatever. I don't know what the meaning is. The 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 multiple the classes. Like in this case, for example, you get only, well, you get minus one whose square is one. And that two has to do with the fact that this is a two here.

Okay. But that stuff's conjectural. I don't know how to prove. Yeah, even this statement that this functional equation is always true. Yeah, maybe I should have said that, that this rationality conjecture

has been proven in some contexts. It's gonna be very important that we've proven it in all the Toric cases. When X is Toric, this rationality conjecture is proven. If it's a one leg Toric, then I think we have control of the functional equation. It's possible. But in the general Toric 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 I should say that I did say that in the case of the Calabi-Yau, that there the functional equation is proven, the Q 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 Calabi-Yau, which is not really being covered in these lectures at all, which is this as developed by Chi to use some kind of weighted whole Euler characteristic to get the Calabi-Yau invariance. And the one Q goes to one over Q is proven using some properties

of that Euler characteristic, that weighted Euler characteristic, together with some Serre duality that relates higher coefficients to lower coefficients. So there's a very nice geometric reason why, well, I don't know what happened to Lexi's function. There's a very nice geometric reason in the Calabi-Yau 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 those arguments don't work in general. Those arguments only work in the Calabi-Yau case. In the case like in P3, we just have no argument in the full generality.

So I think we have time for one more question. Are there any theories that you would expect to be rational in the DT4 case? Oh yeah, the people are working on that. I don't know who asked that question, but my advice is to ask Young Han. I mean, people are working on those DT4 calculations.

And I think that it's, the first hope would be some kind of rationality if you, but you know, the DT4, here we have one curve class and we have only one other parameter. In DT4, 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

who are working on that is Young Han, I think Wunan is also thinking about it. So I would address those questions to those two. Okay, thank you very much.