 Okay, so I want to pick up where I left off yesterday. And so just as a reminder of what we were doing yesterday, the setup was, you know, in some generality we started off with some kind of collabialed refold and then associated it some moduli of objects, sheaves, complexes, whatever on the collabialed refold and then we, you know, produce essentially a constructable function on this modulite space. The z-value constructable function, which has the property that, you know, if you're in the proper situation and you have these kinds of virtual endurance in the sense of intersection theory that this constructable function recovers that information. If you kind of, you have some notion of integrating a constructable function where you add up the Euler characteristics of the strata weighted by the value and this kind of recovers this virtual class invariant that was defined in Richard's lectures. And so what I wanted to kind of talk about first today is give kind of, you know, an indication of why this is kind of such a powerful tool and there'll be something that I'm gonna want anyways for later on in this course. And so the kind of modulite space I'm gonna take is what's called the stable pairs modulite space or the PT modulite space sometimes for a ponder ponder in common. And so the data here is, it's two pieces of data and this is kind of a modulite space that's meant to reflect some kind of curve counting invariant. So first you have a sheaf E, which is one dimensional and pure it has no zero dimensional subchef and then you have a section of the sheaf with the stability property, which I forgot to write that the copernel is zero dimensional. So this has one of these, you know, symmetric obstruction theories or whatever from yesterday. And so we can take it's the virtual invariant and we could sum them up in a generating function. It's not hard to see that this ends up being a Laurent series in the sense that for and sufficiently negative these modulite spaces are empty. And the theorem is that this generating function is actually first of all a rash of the expansion of a rational function in Q and it has this kind of Q goes to one over Q symmetry. And so this was proven in this generality by Kota and Bridgeland. And so what I wanna kind of first do is kind of sketch the proof in the kind of simpler case when beta the curve class which is the, you know the support of the one dimensional sheaf is irreducible. So that means, you know if you like all the curves appear in this class or beta is not of the form beta one plus beta two with the beta I effect it. So all the curves that kind of appear which have support beta are in fact, integral curves. And in the proof in this case this is actually one of the first this is kind of originally done by Ponder and Ponday and Thomas. And what, you know the reason I wanna give this proof is really just to kind of indicate kind of the power of disability to kind of work constructively instead of working intersection theoretically. This property of being a rational function is supposed to be true when you do curve counting on any threefold using the stable pairs modular space. But we don't really know how to prove it in that generality. Okay. And so I don't wanna get too much into the details I'll just state what kind of where the constructability ends up being useful and where in particular how you kind of see both this rationality and this symmetry. And so the kind of idea is that, you know to prove this kind of, you know rationality and symmetry it's actually a NOS. So for instance if there were a Laurent polynomial and the symmetry would just say that the, you know the Q to the N coefficients is the same as the Q to the negative N coefficient. This statement is a little weaker than that because you can have a rational function with poles. And so instead what you end up showing is so let's call this virtual degree for the given modular space I'll call it PT beta comma N. And so if I compare the Q to the N coefficient with the Q to the minus N coefficient it's enough to show that this is basically of this form. Some constant times negative one to the N minus one times. This is just some statement about power series. If the constant were zero then you would get an honest to God Laurent polynomial. And the idea is that this statement is something you can check kind of strata by strata on the modular space. More precisely, if I look at this pair's modular space for N and the pair's modular space for negative N I have a forgetful map where I can just forget the section and this will go to kind of M beta N which is, you know, some of the space of one dimensional sheaves, you know with these discrete invariance. And similarly I have a forgetful map here and then I have a natural isomorphism between these two spaces which sends a sheaf E what I'll call, you know E dual which is this sheaf here. So this is again some kind of pure one dimensional sheaf with the same support. And the way you should think about it is that for instance if the support were like a smooth curve and E was like the push forward of a line bundle then this dual would be the push forward of the dual line bundle tensor the canonical bundle of the curve. And so now to prove this kind of identity where I wanna compare the invariant for N with the invariant of negative N I can study it using the fact that the invariant is somehow defined now constructively. I can try to study this difference, you know fiber by fiber. So let's call this map price of N and I'll call this map price of negative N using this identification of the two bases. And so it turns out, well, if I pick a point by fix a sheaf the fiber in one projection is the projectivization of global sections of the sheet. You see the support is irreducible. So this business about the co-kernel being nonzero just means that the section is nonzero. And similarly, if I look at the fiber with respect to the other projection well given by the projectivization of this H zero which if you kind of mess around with duality ends up being naturally identified with H one of the original sheet. So the idea is that both of these are kind of you know you know the fibers are always projected spaces but the dimension of the projective space jumps around but since I'm working constructively it doesn't matter. I can just kind of assume I have a projective bundle. I wanna kind of compare the difference while I'm just taking an Euler characteristic. And so the difference in the Euler characteristics I'm just going to be taking the integral of my bearing function just over each of these fibers. Now there's one thing I need which is that I need to know something about the value of the bearing function here. And the key fact that they prove is that basically that the bearing function on the kind of PT space is just pulled back from the bearing function on the base. So it turns out that here I'm gonna use some fact that the bearing function is constant on fiber. So it's to assign bearing function downstairs. And so as a result, you know once I kind of sort out the signs I get exactly the statement I want. The difference between the you know the Euler characteristics of this projective space minus the Euler characteristics of this projective space is exactly H0 minus H1, which is an Euler characteristic. And so then when I now kind of integrate over the base when I kind of add up over all the strata of M I get exactly this kind of identity. And so really this is kind of the why this is such a useful idea you kind of just focus fiber by fiber and you can actually then turn that into an argument about the global invariance. So let me just say a couple of remarks about this. It was maybe not clear from what I said I was using the fact that the Perf class was irreducible you know, in a bunch of different ways. That's why this kind of argument is so clean. In general, you have to kind of be much more kind of clever. So you need this kind of much more complicated all algebra technology of the kind that you know is hopefully the subject of Veronica's lectures. Second. Yes. So there's a question about the proof. So is it obvious that the function pulls back? No, it's not. This is something, it's not hard actually, but this requires some proof. It's like a one page argument. And in fact, I mean, what's going on in the argument there are a few different ways of thinking about it but ultimately, you know, it's something that they proved using, you know the fact that the Baron function behaves well with respect to smooth maps and so on. And the fact that the Baron function in some senses really is, you know, it can be detected at the level of the schemes without keeping track of all the obstruction theory and stuff like that. But this really requires an argument. Okay, so maybe the second remark is that, you know this rationality is much more general. As I said before, this should always hold. Once you kind of put some, you know homology classes to cut the virtual dimension down to zero, but we don't really, you know right now that we don't really we can't access it in that kind of generality. Although there's some kind of recent announced work of Dominic Joyce that might kind of change that situation. The third remark has to do with this, you know question about the Baron function. So here I was really, you know the Baron function was really a long for the ride. What was important was that I was kind of could kind of study this question constructively. And really then, and then I needed to check something about the Baron function. But basically I needed to check that the Baron function was kind of, you know constant on fighters. And so in particular, this entire argument would work if you replaced the Baron function with something that was constant. So it would work if you somehow for, you know replaced this way of function that you're waiting everything by by something constant or maybe, you know a sign or something like that. And so in particular, this rationality I mean, this was actually originally what you can approve this rationality of this series is actually something about the actual or the topological or the characters is also not just the virtual invariance. Tomorrow I will kind of state a stronger rationality, rationality conjecture. Meaning a more kind of constrained version about what these rational functions are which only holds. So unlike this kind of weaker statement which this only holds and it's still open even in the clobby outcase. So I'll talk about this tomorrow. Actually, Andre, could you do me a favor? Could you repost, you know, the link if you're, I don't know if you're online but you know there's a link to that the backup. Could you repost that? So I tried using it but I don't see it. Oh, is it not working? Oh, hold on. Oh, is it not working for you? I just see initial state. You get a link, you share it, can you access it? The host one or the participle one? Well, the one that you sent to everyone. In the chat. The Google Drive? The Google Drive. So I just see the, oh, I see it now. Yeah, it's working now. Yes. Oh, okay, all right, good. Could you just repost the link? Actually, I'm not sure if it. It will, yeah, okay. Okay, so what I'd like to talk about for the rest of today is, okay, so in general now let's go back to this kind of generality where we have X, we have some modular space, and then we have this kind of constructible function on it. And so we would like to, what I want to kind of explain is how to kind of promote this story to some kind of comology theory associated by modular space. With the property that the, you know, the virtual inheritance will just be the, you know, alternating sum of the betting numbers. And so what we'll actually do is we'll actually kind of, you know, construct an object here, which I'll call the, you know, the DT on M. And, you know, what this will be is, well, you should think of this as a, this will end up being a perverse sheaf on M, but that, you know, to first approximation, you could first think of this as a, like a constructible sheave for a complex of constructible sheaves or something like that. And then what this will have the property is that if I, for instance, take the stock wise or the characteristic at some point, this will recover the value of the Baron function at that point. And then if I take, you know, the global comology and I take the only characteristic, that'll be like integrating the Baron function, which in particular gives me the virtual number. I just want to make one point which is always a little confusing is that, you know, on the one hand that went, you know, when we do these modular problems, we're doing modular coherence sheaves on X. This object here is really a constructible sheave, you know, with respect to the, so I'm thinking of the, you know, taking the analytic topology and so on. So what is the idea behind this construction? Now it has to do with something that I said yesterday, which was a kind of, you know, one of the examples of where, you know, I said what the Baron function was, which is where I had a smooth variety with a function on it. And then the kind of mod by space I was looking at was the critical locus of that, which is just the zero locus of the differential of that. And then in this setting, you know, the Baron function at a point was, you know, something like negative one to the dimension of V, one minus the topological Euler characteristic of the Milner fiber at this point. And so the idea is that if I want to kind of promote this down to instead of just a number, it is some kind of, you know, homology, what I can do is instead of just taking the Euler characteristic of this Milner fiber, I could just take the homology of the Milner fiber and, you know, and then do that as P varies around M to get a sheaf or a complex of sheaves. So there is a standard way of doing this. Using this notion of what are called a vanishing cycles. Vanishing cycles, you know, in a word is kind of measured, if I think of this as a family of varieties over A1, vanishing cycles measure is the, you know, difference in homology between the singular fiber and the smooth general fiber. This way that kind of set that up, I have V over A1, which is my function. Unless I'm interested in the fiber over a zero, which I'm going to assume is singular. First of all, the complement of zero and at the universal cover, corresponding cartesian diagram. And let's call, you know, these inclusions, let's call this one I, and let's call this inclusion all the way here from the universal, you know, from the pullback of the universal cover all the way back to my original variety. I'll call that, let's say J tilde. If I can do the following, I'm going to produce a sheaf on the central fiber of the zero or really a complex of sheaves where I do the following. I can take the constant sheaf on V and then I can, you know, pull it back and then push it forward back to V and then restrict this to the central fiber. So this is what's called, I'll call this, so IQ. And the way you should think about this is that if I look at the stock of this at a point, this is like this, the, the comology of this millner fiber at that point. But now if I want to do, you know, the analog of this, you know, one minus and so on, I want to kind of, this is like taking reduced comology. So to kind of do that construction, I have a, just from a junction, I get a map. So this is all happening in this kind of derived category of constructable sheaves. And then I can just take the cone. And so again, up to shifts and so on, this is really the analog of taking, you know, this reduced millner fiber to get the bearing function. And again, what this is, you're supposed to think about this is measuring the kind of the difference in topology between the nearby fiber and the kind of clothe, the singular closed fiber, five rovers zero. And so once I throw on a shift, these two operations, I could, instead of taking the constant shape, I could have taken any sheaf on V. And these two operations define functors from, you know, sheaves on V to sheaves on V zero. This lives on kind of, you know, on V zero. And in fact, kind of the surprising thing is that they preserve kind of this abelian category of perverse sheaves. So I have V sub F, which goes from perverse sheaves on V, sheaves on V zero, and also size of F. But this is the one that I'm going to be interested in. Okay, there's kind of a lot going on here. So I really, I mean, what I really want to encourage you to do is just to think of this as this original definition where I took the reduced Euler characters and replace it instead with reduced comal. And then imagine doing that in families to get an actual complex of sheaves. And so in particular, the specific object that we'll be interested in is where I take in the constant sheaf on V. Well, the constant sheaf on V isn't quite perverse. You have to shift it. So instead I'll take this object here, which is a perverse sheaf on the zero fiber that's supported on the critical of this. And in particular, it has the property, I'll just call this, you know, piece of M. It has the property that it's stock wise Euler characteristic is exactly this parent function. And so the goal in which I'm gonna sketch is that we basically want to kind of take these kinds of things and glue them together. But to do that, I have to argue why this is even a good local model. And so actually, I mean, there are a few ways of thinking about this. So again, so now let's go back to our situation where I have X and the corresponding model to my space M of X. And what I'd like to argue is that at least locally on M, this, you know, my space looks like something where I'm taking the critical locus of a function. And so this is actually easier to think about if you're willing to work analytically or, you know, kind of formal locally. So for instance, in the kind of gauge theory world where let's say M of X is like a moduli of like vector bundles, the way you could try to model what your, you know, moduli space looks like is you could take a, you know, a C infinity bundle. And then what you're interested in doing is putting some kind of holomorphic structure on it. So you're kind of looking for, you know, integrable D bar connection on sections of E. And so if you, you know, write any such kind of, you know, zero one operator in terms of some kind of base one plus some kind of correction where this correction is like a zero one form on the endomorphisms. Then it turns out that you can write down the integrability condition purely in terms of a critical locus condition down what's called the holomorphic turn sign and functional, which is some explicit integral over X. X is Calabiow. So you have, let's say, you know, holomorphic three form and then you kind of, you just kind of cook up some zero three form to kind of compensate it. So this is actually what was first written down as far as I know in Richard's thesis. All right, whatever. This is, you know, this is some kind of construction that you knew that picks out exactly the condition that you want. And this is modeled off of an analogous condition in for real three manifold. But this is not the only way of thinking about this actually. For instance, the approach that I like the best is using deformation theory. And this works much more generally if I give you some point in my modular space. So some object on X, then if I look at the formal completion of my modular space at this point, well, anytime you have a scheme and you take the formal completion, you can embed it inside of the formal completion of the tangent space at this point, which in this case is just some X to one. And then you can actually write down explicitly a power series on this tangent space that whose critical locus cuts this out. So the idea is that if I look at the X to algebra of my object, it carries a product, the Yoneta product, but actually carries operations, basically massive products. Induced by the fact that the way I get this is I take the homology of some differential graded algebra. So you have these higher operations go from, once you write them down, they can go from symmetric power of X one to X two. And then you can just write down a formal function on X one, which is just the sum. I might screw up the denominators, but I think this is right. And here, what's going on is that here, this pairing that I'm using is exactly the serduality pairing. And so what you then show is that this formal completion is just literally the critical locus of this, of this formal power series. So there are a few ways of thinking about this. I should say both of them, the Kalavia condition showed up in some pivotal way. Here it showed up because I was taking a holomorphic three form. Here it was showing up because I was using the fact that I have a pairing, a natural pairing between X one and X two, which in some precise sense is compatible with these operations. Do you mind giving us a bit more details about what just happened? Because I'm a bit confused as to what F is here. Oh, right. So, okay, yeah, maybe I shouldn't have gone into this. What I was trying to do is I was trying to kind of give some idea of why the modular spaces that are showing up are actually critical loci, at least locally. And so if you're kind of comfortable in the gauge theory world, you can see it for vector bundles just by explicitly writing down this functional. If I took the derivative of this functional with respect to A, this is some incident dimensional thing. So you have to kind of deal with that. Then you get exactly the integrability condition, which is something like, D bar A plus A commutator A is equal to zero. So that kind of falls out exactly here. What's less clear is why this kind of formal thing also works, but I just wanted to indicate it that there is a way of making sense of this purely algebraically without thinking about gauge theory. But so maybe it's better best not to get into this. But I just want to indicate that there is a picture that lets you kind of talk about in kind of much more generality how to see this critical structure. So thinking about this, we have one more question. Is this some sort of A infinity deformation theory? Yeah, that's exactly what the, I mean, this higher operations is, I'm thinking of this as an A infinity algebra. And then the ser duality is giving me the structure of what's called a cyclic A infinity algebra. And so then there's some kind of general theory about understanding your moduli problem as in terms of exactly the critical, in terms of the critical of this thing. So there's some, let's see, a reference for this. There's, you know, basically there's some nice notes of conceivage and Soigelman from like the 2000s that kind of discussed this. But there are other places as well. I mean, in general, I mean, okay, this is kind of a tangent. In general, even if I don't have a colloquial, if I want to understand what are the equations cutting out this formal completion inside of the tangent space, you can always just looking for the zero locus of these, you know, of these formal functions, which go from X one to X two. And then what's special in the colloquial case is that you can take all of those functions and group them together as coming from derivatives of this one power series. Okay, well, okay, maybe that was a bad thing to try to emphasize because these kinds of results are nice, but they don't really, they work analytic locally or formal locally. And what ends up, what you really need is something a much more stronger result, which is due to kind of a Pante of Taun, Vakyeh and Vesosi, and then Chris Braav of Vittoria Bussi and Dominic Joyce, which says that again, so if I have my moduli problem, my moduli space and I take some point on it, and I'm assuming I'm the kind of the scheme setting, so M of X is a scheme, there exists as a risky open chart for my point, which can be written as the critical locus of a function on a smooth scheme. So these kinds of, you know, deformation theory arguments or gate theory arguments only give you kind of, you know, very, you know, small, maybe, you know, assuming you could show convergence, you'd get really just like an analytic open, where this is true, what's kind of striking about this result is that you really get kind of a risky open set where it's true. And this proof is really much harder. You see the idea is these guys, the first paper introduced this notion of what are called minus one shifted symplectic structures, which is really a notion that's kind of properly in the world of derived geometry. And then the second group kind of proved that there was essentially a local structure theorem, a risky local structure theorem for whatever these symplectic structures are. What I like about this result, by the way, other than the fact that it's just like a cool result, is that it really kind of, in a strong way, uses a derived geometry in the sense that, you know, a lot of the other constructions in the subject, like, you know, all the stuff about like obstruction theories and virtual classes are kind of, you know, dodges to get around Avig to actually work with, you know, derived schemes directly. And this result really, I don't, there's no, as far as I know, anywhere, there's no real way around it. And so any kind of construction that builds on it, which is, you know, what I'll be doing for the rest of the lecture, ultimately really kind of relies on that kind of formalism in an essential way. So let me just say, we've seen already some example of, so for the most part, this theorem tells you that you have these Zyrsky local kind of critical charts where you have this critical of this description. It doesn't really give you some very good idea of how to find them or how to find the function and so on. But in some examples, you know, you can actually see explicitly, we've seen some examples already. So for instance, there was an example from yesterday where I had this kind of, you know, three, three hyper surface inside of P2 cross P2. And then I wrote, I just get some explicit modular space and I described it as a critical locus of a function. Another example that I like a lot, where again, you can actually write down a global critical locus description, is the case of the Hilbert scheme of points on C3. Here, the ambient space that you work with, so you can think of the Hilbert scheme of points on C3 is giving you, you know, three commuting matrices on C to the N, along with, you know, cyclic vector modulo conjugation. And so that, the commuting condition kind of imposes some strong singularities. So the ambient smooth space is I'm just going to take three matrices on C to the N, you know, a cyclic vector V inside of CN, and then that's it. And then I mod out by the action of GLN. So the cyclic vector just means that if I take X, Y and Z and I apply to V, eventually it spans all of C to the N. So this is a smooth space. And then the way I think of the Hilbert scheme inside of this smooth space is this is the critical locus of an explicit function, which is just a trace of X times the commutator of Y and Z. If you just explicitly calculate what the critical locus condition is, I just take the derivatives and set it to zero, it exactly picks out the condition that the matrices commute. And this is a really nice example because, you know, it's explicit enough that you can imagine you can actually try to calculate millner fibers and so on. But of course, as N is big, it also gets kind of unwieldy. So I won't maybe go into too much detail about here, but you know, what I said already is that M of X is basically covered, you know, what we call these critical charts, which again, just some description open sets, which are described as critical loci. And then, you know, not surprisingly, there's also some kind of compatibility on overlaps. And this, you know, one way of thinking about this compatibility is what's known as the notion of what's called a D critical scheme, which is basically some notion that was developed by Dominic Joyce to avoid having to talk about these shifted symplectic structures all the time. Okay, so what we would like to do then is the following. So on each of these critical charts, I have a, this perverse sheath that I've constructed on you. So if the chart is given by this data of U, V and F, where U is the actual open set of M of X, I've produced this object, which is, is vanishing cycles, essentially the constant sheath. And you would like to glue these together, but there's a problem with doing this. And this problem is kind of, there's an obstruction to doing this gluing, which you can think of already in the kind of the simplest case, which is that suppose V, you know, V is, so let's say some smooth variety. And I can think of V as a critical locus on itself by just taking the zero function. And if I run through this construction, so the critical locus of S in this case is just V itself. And then this kind of, you know, vanishing cycles object that I'm gonna produce is just a constant sheath, again, with this shift. But there's another way of getting V as a critical locus, which is that what I can do is I can take, let's say I'll call L to some Z mod two local system on V. So soon V has non-trivial local systems. And then I can take the corresponding, you know, two torsion line bundle. I can write down a function on the total space, F tilde, which is basically just fiber-wise, is, you know, T going to T squared. So using the fact that the L squared is trivial. And so it's not hard to see that if I take the critical locus of this thing on this bigger space, it's again just equal to V. But now if I calculate the vanishing cycles, I don't get the constant sheath anymore. I get this kind of rank one local system associated to L. So depending on how I described, you know, V is a critical locus. Yeah, I either got the constant sheath or I got this kind of Z mod two local system. And so this is gonna cause some kind of problem when I try to kind of match things on overlaps to kind of produce a global object. So to solve this, you actually need a little bit of extra structure, which is something that already came up a little in Richard's talks. And it came up essentially for similar reasons. So on my moduli space, I have this two-term complex, this kind of obstruction theory, which is if you like the dual of my complex of deformations and destructions. And what we call the virtual canonical bundle is just the determinant of this thing, which is a line bundle on my moduli space. And so the extra structure that we need is what's called an orientation. It's just a choice of square root of this. And it turns out once I pick this choice globally, then I can kind of solve this problem about bloom. For the theorem, which is due to, see Braves, Boussi, DuPont, Joyce, and Centroid. Because if I take my moduli space and choose an orientation, choose the square root, then you can kind of solve this gluing problem. You'll call this DT sheath, which is now some perverse sheath that lives globally on your moduli space. And this choice really makes a difference. So first of all, there's a question about why does the square root even exist? And then if choices will differ by like two torsion line bundles on M, how does that affect the answer? And then the first thing to point out is it really does affect the answer. So different choices will kind of change this DT sheath by tensoring with this kind of local system, which seems like a mild thing. But when I kind of, for instance, take homology, it'll totally just change to what the homology is. What is its orientation in the kind of the critical case? So for instance, if M is just globally a critical locus, then, well, so the obstruction theory in this case, might remember it looks like this two-term complex coming from the Hessian of F. If I take the determinant of this, I get the canonical bundle of V restricted to M squared. And so then the kind of the natural orientation, if you have a global critical locus, is just to take KV restricted to M as kind of the natural choice. But of course, it depends on my description as a global critical locus. But for instance, in the examples like the Hilbert Schemer, something where I gave you some kind of nice function that cuts it out, then in particular, it gives me also a nice orientation that cuts it out. Another example where there's a natural choice, this is kind of one case where there's a natural choice. Another example where there's a natural choice is when my Calabi A3 fold is the total space of the canonical bundle on the surface. So again, this is the kind of geometry that showed up in Richard's talks last week on the Waffa-Witton theory. In that case, if I consider for instance, she's on X that are kind of proper over S, I have a map where I kind of take this she's on X and I just take its push forward to S and I get a point in the modular stack of she's on S. E maps to pi lower star E. And let's just assume I'm in a situation where this push forward is coherent interest. Then there exists a natural orientation on M of X that you can think of just by looking at how to think about the obstruction theory on M of X. So this R-HOM is what basically the determinant of this is going to calculate my virtual canonical bundle. And I can trap this in terms of two complexes that are built out of S. So on one hand I can, this is built out of R-HOM FF and then the kind of other piece as the triangle is R-HOM of F, F tensor KS with a shift here. And so then, well, if I combine serduality with just taking the determinant of this is going to be the product of the terms of these two things. I combined that with serduality, I get that the canonical bundle on M of X is naturally the virtual canonical bundle on M of S. Square. And so again, this gives me a natural choice of square root. But in general, you know, a priori it's not clear that they even exist let alone that it's kind of a nice choice. And so the kind of two theorems that kind of help with this is first of all, there's a very soft theorem. But if I had more time I would kind of explain the proof of this due to Necker-Soff and Okunkoff which is that orientations always exist. And their argument is in kind of very great generality. Anytime you have some kind of, you know moduli of objects in a colabi out three category or something like that. But they don't tell you how to pick it. And so then the more recent theorem from a couple of years ago of Dominic Joyce and Marcus Upmeyer is that, okay. So I'm not actually sure about with a precise hypothesis. So maybe it's important for them that it's a projected colabi out three fold. So maybe this might not be exactly correct. They kind of provide that kind of a canonical choice. And in particular what's great about what they've done is that it's a choice that's kind of compatible with kind of varying the moduli space under things like extensions and direct sums and so on. But their proof is heavily gauge theoretic. I have to confess, I mean at some point I was meeting with them and they were explaining to me and I, you know, if I understand correctly which maybe this is wrong that's like what they do is they take the colabi out three fold and they cross it with a circle and then they kind of do some analysis about the corresponding real seven manifold and some kind of special holonomy for those things. So at least from my point of view I don't really understand I don't have a good feel for what's going on in this there in this section. So I'm almost out of time. So now we can just, I can just state the upshot of all of this though all this kind of formalism. This is pretty much all I'll need is that, you know, we started off with X we took the M of X we maybe have to choose the square root maybe the choice has been made for us or maybe we are in a situation where we have a natural choice and that produces this kind of DT sheath that lives that some perverse sheath that lives on my modular space. And then if I want to, for instance kind of get a comology associated to my modular space I can just take the hypercomology of this perverse sheath. And so what properties does this satisfy? Well, first, I mean the first one is the thing that I said at the beginning which is that if I take the stock wise Euler characteristic I get exactly the Baron function. That's really cooked into how we kind of pick this choice of feet of glue. If I instead take, you know the global comology and I take the Euler characteristic there well, this is the integral of the Baron function which in particular is my virtual invariant. What else? Well, we get some properties that just come from the fact that, you know vanishing cycles has some nice properties. So for instance this DT sheath is closed undertaking duals, Verdi-A-duals which concretely means that if I take the kind of comology it's naturally dual to the kind of compactly supported comology. So for instance, if M is proper you basically get Poincare duality. What else? I mean, this won't be relevant for me but this is extremely interesting and useful in general is that everything here can be kind of decorated with hodge structures. So the kind of formal way of doing it is that this fee lists to this category of what are called the mixed hodge modules. But concretely what that means in practice is that then when I take global sections this carries a natural mixed hodge structure. This is extremely useful for calculation. And so finally, let me just say the non-property which is that one of the things I mentioned in class yesterday was that, well, the virtual class by, you know, because it is defined by intersection theory has a property that is deformation invariance. If you have a family of x, x's and you're in the proper situation then the virtual degree is going to be independent of where you are in the family. And if you translate that to a statement the bearing function is something kind of very non-obvious because the actual family of modular spaces well, maybe it's proper but it doesn't have to be flat or anything like that. So we know this kind of, you know, so if n is proper then by virtue of this kind of index formula from yesterday, this weighted Euler characteristic is a deformation invariant as x varies. But that's not going to be true for this kind of a comological thing. You can, you know, a simple example to see how things can go wrong is where you consider the following family. These aren't really modular spaces just a family of things that have this kind of, you know that are, you know, the kind of critical structure which is you just take a, you're just going to take a Riemann surface m of E, C, C and then I'm going to just take a omega to be a one form on it, one more for one form. And then I'll define a family of spaces which is just the zero locus of t times omega. So when t is non-zero, I just get a bunch of points corresponding to the zeros of my, of omega. And in that case, you can just see it while the, you know, everything is isolated. So the, the, the comology is all supported in degree zero and you just get, you know, here to the two G minus two in degree zero. But t equals zero, I get the entire curve. The vanishing comology in that case, the fee in that case is just the constant chief up to a shift. And then now when I take the comology, it lives in different degrees, you know, I get the q, q to the two G and q. And of course, these have the same Euler characteristics, but they're just different. And so in general, you know, that makes this problem a little bit subtle if I'm working with, for instance, if I'm interested in like, you know, the Quintic three fold, it's a, not at all, you know, it doesn't really make sense to talk about in general, these kinds of, you know, these kinds of comological invariance for a modular space sheets on a Quintic three fold. You have to tell me which Quintic three fold. Like in general, there's no reason to expect the answer to be insensitive to it. All right, let me stop here. We have questions. We have one from the chat. I can try asking it myself, but I'm not sure of you. So the question is, if the, actually think that the question is if the orientation data on the stack of compactly supported complexes is, if it comes from the global one of choice, Upmire. Oh yeah, that's a good question. So this is like, I mean, so I actually don't know, can I see this Q and A maybe? It's answered, it's in the answer tab. Oh, I see, there's a long discussion here. The short answer, so, okay, hold on, if you could expand the stack where you know, this is from Boyko, is that correct? I mean, he would know better than me, but make sure I try to understand this question. I was wondering, oh, I see, did the canonical Hilbert scheme orientation data come from the global one of Joyce Upmire? I have no idea. I mean, again, I think he's in a better position to answer this question than I am, but what I'll say is that, you know, again, it's a gauge theory construction. So if you have like a torsion chief, the very first step is you're going to kind of, you know, let's say you're interested in like, you know, something on C3, well, you're going to embed C3 as I understand their argument inside of like something compact, or maybe, you know, maybe you'll do some kind of boundary and framing or something like that, but then you're gonna resolve your torsion chief by vector bundles. And so then you lose a lot of this structure. Like, so I think even for this, even for a hill, it's not clear to me that their construction will agree with this kind of what I was calling the canonical one, which is maybe not the right choice of work. But so, you know, so a priori, it's different setting because they're working in a compact setting. But for instance, you know, you could, in an analytic neighborhood or something, you can, you know, identify these two modular spaces. And then it's not clear to me that those two orientations agree. But again, I'm not really the person to ask there. One more question in the Q and A. In the local case, there isn't a canonical one. Oh, it's an answer to what you said. In the local case, there isn't a canonical one. You could prove that there is one depending on a choice of compactification. Okay. All right. So a question is an answer, I guess. Yeah. Okay. All right. Any other questions? No, then that's fine. Thank you. Congratulations again. Thank you.