 is to kind of be a little looser today and maybe not even use the full hour. I just kind of I'll pick up where I kind of, you know, ended things last time with a couple of questions and maybe, you know, talk a little bit about some, you know, other kind of directions that this circle of ideas goes. So we'll see how long. So let me just say where I left things off yesterday, which is that I was giving some examples of, you know, what I guess on the one hand, we have this kind of, you know, these, what I was calling these Gopukumar boffins variants, although to really be worthy of the name, they kind of have to satisfy the conjecture, but, you know, kind of some proposed on thoughts for defining these things, which again really involves kind of, you know, one dimensional sheets, along with a kind of modulite one dimensional piece, and then it's kind of, you know, perverse sheets that lives on the modulite space. And then kind of studying the how it behaves with respect to the math of the chow variety, you kind of cook up some numbers. And then the kind of, you know, the main kind of, you know, conjecture would be how does it relate to the kind of more traditional perp type of theories on X and the one that I was interested in with this kind of stable pair theory. So, you know, yesterday what I kind of spent most of the lecture talking about was what I do is the kind of, the most general piece of evidence that we have, which is that if you have, you know, basically an integral curve, it needs to be stronger, the push forward needs to be integral, but I'm just going to be a little sloppy about it here, inside of the total space of the conical bundle. And then, I mean, the statement that's kind of non-trivial in the sense that it requires both understanding some perverse alteration and this kind of mysterious chief on the modular space. But then, nevertheless, using just the fact that kind of formal properties of these constructions, you're able to prove it essentially by reducing it to kind of a much simpler case, the case of locally planar curves. And so, I mean, let me just mention one kind of question I have. And I mean, in some sense, that kind of two-step, that's kind of two-step reduction, you know, is one of the main things I wanted to explain, because that kind of technique shows up in a lot of different contexts. And so, and it's something that I personally found very useful in the last few years. Let me just kind of mention two pieces of speculation in this direction, which are not, I mean, I hesitate to almost write anything down because they're not somehow, these are kind of half-co-oped ideas. But one kind of questions I have is that, so the way I kind of explain this procedure in my lecture yesterday is that I used, I use this kind of perverse continuation idea where you kind of show some perverse sheaves at full support. And so then, if you want to prove something about them, you can prove them over the generic point where it's just a local system, all the curves are smooth, and you're in this kind of very classical situation. And so, I kind of used this kind of support theorem in the first step. One thing that's happened, you know, so that work of Ngo is pretty old, that one thing that's happened kind of in the intervening years is that there's been kind of a, you know, major advance in how we kind of think about a lot of those older questions, which is this work of Grochaneg, Wies and Siegler, where they basically give a much more kind of, you know, flexible way of proving many of the results, maybe not quite as strong as what Ngo does, but for all intents and purposes, for all applications, their techniques are just as good, and they're much more flexible. They have much less stringent hypotheses on the families in question, essentially via kind of some, you know, some version of piatic integration, although, you know, kind of, you know, some maybe more sophisticated way than was done in the 90s. And so, I mean, this is extremely beautiful work, and, you know, in his talk earlier, he alluded to a little bit in his talk earlier this week. And so, you know, one question that I had is, you know, how does that interact with this kind of circle of ideas? So, you know, you can imagine that there's kind of a more systematic way of thinking about some of the stuff that I was doing yesterday, where you kind of, you know, use their approach to things instead of this kind of more heavily perverse sheath approach. I'm leaving it open, because I don't really have a good feel for what the answer should look like. Although I think it is reasonable to expect that there is some kind of fruitful interaction there. The other question I have, which is, again, also kind of not even half-baked, is that, you know, so, you know, the way this argument worked for, you know, you have imagined gamma, some kind of very complicated space-curve singularity, and then you kind of push it down onto the surface where it becomes planar, and then you kind of use this kind of, you know, theory there. When you study, let's say, reduced planar singularities, so for locally planar singularities, the moduli spaces that show up, if you look at, for instance, the compactified Jacobian of the curve with locally planar singularities, these are, you know, very closely related to what are called affine springer fibers in type A. Which came up in Eugene's talk last week, and then also Frank Chen's talk yesterday. And so one question that I've always wondered is, what is there, is there, you know, a role for these, you know, space-curve singularities? Is there some kind of analogous, you know, role for them to play in geometric representation theory, where you wouldn't take the comology of these kinds of spaces. You would take the kind of comology with, you know, values in this kind of sheaf that we've added on top of it, like this. This is, you know, the contribution of this curve gamma to this moduli space, I may be able to write it. And I mean, I don't even, in this case, I don't really even have an idea for what you could ask for. But I think this is a direction that I think is worth exploring. And actually, Dori Bejleri, at some point, he had some speculation along the lines, although I don't know how precise it was. Okay. So what I'd like to talk about today, next, is maybe what are the, so these, this kind, this is kind of, you know, in some sense, the most general evidence we have for this kind of connection. And, you know, obviously still a pretty restricted situation, but there's some generality to it in which the singularities of gamma are very bad. And so what I'd like to first do in today's lecture is just talk about some kind of non-reduced example. You see the, this kind of relationship between these two theories in the, in the integral case is kind of the simplest. Basically, you know, you look at the PT theory in this class gamma, and you look at the, this, I'm going to go from Vafa endurance of this class gamma, and then they just match up with some denominators thrown in. But if you have a non-reduced or a reducible cycle, then they're going to be all these corrections that come from effective sub-cycles. And so, and sometimes the, the relation is much more complicated in that thing. And so I think it's kind of important to have examples where, where it's true there. And we don't have so many, but here, there are kind of two main sorts of examples where, where you can produce examples where it's true. And so, okay, so let me just say that here. So the relation now, let's say the contribution of the stable pairs and variance for this class gamma in terms of mg gamma prime. And this is in particular where that kind of exponential and stuff really show up. And so the first source of examples where it works out is by taking flops. So a flop is a situation where you have a colabi out three fold, and you have a vibrational equivalence with another colabi out three fold, which I'll call, you know, X cross, which is, you know, a different, you have some contraction of X and you have some kind of different resolution of the contraction Y and where the exceptional loci are curves and, you know, X and X primer isomorphic and co-dimension one. So basically, you know, there's going to be some kind of curves here that kind of get contracted and then, you know, you blow them up again and get some new curves and X prime. And so there's all, you know, this in birational geometry, this is kind of a well studied situation, in particular one thing that you get, which is the theorem of Bridgeland is that the, well, okay, so there's, first of all, the, let's call this birational map, maybe not see, you can identify the, you know, H2 on these two three folds, although effective curves may or may not go to effective curves on the other side. And then there's a theorem of Bridgeland that actually gives you a derived equivalence between the derived category of coherent sheaves on these two three folds. And what that means that you can take, you know, a coherent sheaves, like our one, our modulite one-dimensional sheaves or a modulite space of stable bears, and you can apply this derived equivalence and then you'll get a family of, now not necessarily sheaves on the other side, but maybe more complicated complexes. So assume you're in a situation where you have a kind of a one cycle, say an effective one cycle on X. You have to think a little bit about what this means, but it makes sense to talk about what its push forward is to the other side. It'll be a cycle in the class corresponding to this identification between H2 of X and H2 of X cross, but you can actually kind of refine that to get an honest to God one cycle. Assume this is also effective. It's not always true, but it often is, for instance, if you start off with an irreversible curve that isn't contained in the exceptional locus of X, you'll get some effective curve on, effective one cycle on X prime, but typically it'll now be reducible and have multiplicities and so on. And so the theorem you can over-approved is that if you calculate the contributions of the Gopal-Khomorov-Avach invariance on each side, they just match up on the nose. When I say local, that means I'm just taking the contribution just at this one point of the chow rite, local invariance I defined yesterday. I'm going to put a little asterisk here. The star here is because as always, there are these, to define these invariance, there was this choice of orientation I have to make. So the assumption is that you can kind of take orientations on both sides, probably his Kalavi-I orientation. And so again, what he's doing is he's taking a one-dimensional sheath, moving it over to the other side and then kind of analyzing what you get on the other side. And similarly, this is an older result of, you know, also you've Nobu and John Calabrese. The stable pairs invariance on x and x prime can be related. This is a little bit more complicated relationship, but again, there's some kind of very explicit relationship between stable pairs on one side and stable pairs on the other side. And again, the technique of proof is always the same. You have a sheath or a complex of sheath now, and you apply this derived equivalence to get something on the other side, and you just see how far away that is from a stable pair on the other side. So I won't write down this relation explicitly just because it's a little complicated. But the upshot is again, subject to these conjectures, this basically gives you that if you know the kind of correct relationship on one side of your flop, it implies it for the other as well. You can just follow what's supposed to be expected behavior of these Gopal-Kumarov invariance under this more complicated relation. And what's nice is that this, you know, kind of flopped one cycle, gamma prime, actually can look very different from gamma. So this gives examples, you know, gamma could be one of these examples that we've already proven the theorem for in the integral. And then gamma prime will look kind of, you know, we'll have different singularities, we'll have lots of components, we'll have multiplicities, reduced, non-planar. So this is kind of the first main example we have of kind of non-reduced one cycles where things work out. The second example is actually classical, but it still gives, I think, a kind of an interesting check. And this has to do with Higgs bundles. And in fact, most of what I want to say for the rest of today's lecture is really about this case of Higgs bundles. So, okay, so this is kind of a more classical modular space. And so let me just kind of go over what the definition is. You haven't seen it before. So the starting point for Higgs bundles is you start off with a smooth, productive curve. The modular space of Higgs bundles on this curve, rank R, and then I'll set chi equal to d is I take two pieces of data, it's going to be a vector bundle. So e is locally free of rank R. I fix its Euler characters to be d. And then c is a twisted endomorphism of e. So it's a map from e to e twisted by the canonical bundle of my curve. So kind of fiber-wise, it's, it looks like an endomorphism of fiber, but globally it's kind of twisted by this canonical bundle. And okay, there's some kind of stability condition to get this to work. And so this is some, you know, this is really just a smooth modular space. It's a smooth quasi-projective variety. You could also consider more general kinds of twists. So the ones that kind of are better understood is when I kind of further twist by an affected advisor on my curve, all the same definitions, except now my twisted endomorphism goes to, I twist by the canonical bundle with a little bit of extra. And this twist makes a surprising amount of difference in the theory. So both of these spaces, this kind of regular Higgs and this kind of twisted variation on it, carry what are called a Hitchin maps. So these are maps, just the affine spaces. So in the case of the Higgs space, the original Higgs space, the way this is kind of traditionally thought about is that you have this twisted endomorphism, and then you just kind of take the twisted characteristic polynomial. So at every kind of point of my curve, I could take the characteristic polynomial, the endomorphism of the fiber, and that'll give me, you know, the coefficients of that. And then as I kind of study how it behaves on the twist, I end up with an element of, this is basically taking the characteristic polynomial of Higgs field. And if I do it with poles, I mean, I can do the same thing if I have this kind of, you know, instead of K, I do K-twisted by D, I just take global sections of K of D to the I. And so in either of these situations, these are, you know, proper maps. And generically, the fibers of this map are, you know, basically Jacobians of some curve that covers C. And so this is very similar to the situation I had yesterday. This kind of generically looks like some kind of family of abelian varieties, but then the fibers get very badly singular. And so it's similar to what I was doing yesterday when I was considering versatile families of planar curves. So what does this have to do with kind of colabi, so these are, you know, whatever, these have been studied forever. These are some kind of nice classical modular problems. So what do they have to do with this kind of colabi-ow story? Well, the idea is that you can kind of embed these problems into the kind of colabi-ow modular problems we've been looking at. In kind of a dumb way, it's kind of amazing to me that this ends up being a useful way of thinking about it. So first, if I, let me do just do the right case of, I guess I'll do both at the same time. So first I can take a surface, if I start off with a curve, and I can associate to this curve a non-compact surface, which is just the total space of the canonical bundle or the total space of the canonical bundle twisted by D. And the first, so you can think of this, this is kind of like a non-compact, you know, K3 surface, non-compact trivial, and this is kind of like a non-compact final surface. And then if you just think about what does it mean to give you, let's say, a one-dimensional sheaf on this surface, well, if the support of the sheaf is proper over the curve, that's the same if I push it forward, I'm giving you a vector bundle on the curve with one of these twisted endomorphisms. The one-dimensional sheaves on these surface really correspond to either these modular spaces of Higgs bundles. And then what is this Hitchin map, so the, if I look at one-dimensional sheaves on the surface, the sense of just being the same as the corresponding Higgs space. And then this Hitchin map ends up just being the map to the linear system of the surface, so the chow variety of the surface, which just remembers really the support of this one-dimensional sheave, ends up being exactly this kind of base of the Hitchin map. But actually, I don't really want to work with surface, I want to work with Claudia III fold. So I'm going to do kind of, you know, the same thing that I've been doing always to take a surface and turn into a Claudia III fold, which is that I'll take the total space of the canonical bundle of the surface, this is going to be my x, which in the first case is just, you know, the total space of this rank two bundle on my curve on both cases. And so now if I take my kind of curve class beta on this non-compact Claudia III fold to be, you know, r times the zero section, then this moduli space that I've been interested in, one-dimensional sheaves, all the characters one, well, this just ends up being kind of, let's say, the twisted Higgs moduli space, let's say for D, it's one on my curve, 1D is greater than zero. And in the case without the poles, I just get the regular Higgs space on my curve. But then cross a copy of A1, that's because in this case, in the K3 case, when I've done the total bundle, total space of the canonical bundle of the surface, that's just taking S cross A1. So there's always going to be this trivial A1 factor. This is my moduli space on my Claudia III fold. And up to this factor of A1, it exactly recovers the kind of classical moduli spaces that people have studied. And then what I was saying earlier, the kind of thing that we've been looking at is we've been looking at this moduli space of one-dimensional sheaves on the Claudia III fold, and I'm mapping it to the chauvinite of the Claudia III fold. And in this case, again, this is just ends up being exactly the Hitchin map, maybe with this extra copy of A1 floating around. But again, what's nice about this geometry is that from the perspective of what I'm interested in, this includes one-cycles that are non-reduced, reducible, and so on. The support of the sheaf in this case, depending on which point of the chauvinite I'm looking at, it can be non-reduced. The most extreme example is when my Higgs field, endomorphism, is nilpotent. And then the corresponding support is just R, literally R times the zero section. So it's kind of non-reduced with multiplicity R. And then what are the these kind of Gopel-Komar-Vauffen variants in this case? Well, all right, so again, what are we doing? So in this case, M beta is smooth because it's just this Higgs moduli space. And then, you know, after the shift, what I'm interested in is I take the kind of perverse homology sheaves when I push forward to the chauvaryte, namely the Hitchin base, and I take their Euler characteristics as sheaves on this base. And so concretely, one way of thinking about that is that I'm taking that kind of perverse, kind of extra perverse filtration, or grading, even on the homology of M beta, which is just this Higgs moduli space. And then I'm just forgetting, because I'm taking this Euler characteristics, I forget the homological degree, and just remember this kind of the perverse degree. I'm remembering this kind of which perverse homology sheaves it's coming from, but I don't necessarily care what the what the homological degree was. And what's nice is that this has been, you know, studied a lot. So in D equals zero, meaning the case of, you know, traditional Higgs bundles, this object is exactly the subject of what's called the P equals W conjecture, non-Abelian Hodge theory. This is kind of the big subject, and it's super interesting, but I'll just give the part that's relevant for us, which is that, you know, so again, D equals zero, if I'm just doing Higgs bundles where I'm twisting by the canonical bundle, and nothing more. What this says is that if I looked at the moduli space of Higgs bundles, there is a certain diffeomorphism, so some kind of C infinity diffeomorphism, with the character variety of the curve, which is, you know, this involves maps from the, you know, topological fundamental group of the curve, maybe with the puncture, maybe a modular conjugation, and with the property that the kind of the loop around the puncture gets sent to a certain group of units. This is what's called the twisted character variety. This is a diffeomorphism, but of course it's not an algebraic map, because the right-hand side is an affine variety, some kind of, you know, kind of exotic construction, but since you have a diffeomorphism, you can identify the singular comologies of this one-handed Higgs space, gets identified canonically with the comology of this character variety. This is some kind of, you know, known thing. The conjecture, which is due to Taldo, Housel, and Migliorini, is that, okay, on the left-hand side, I have this kind of extra information coming from perverses on this chow variety, which is just the hitch and base. And on the right-hand side, I have, well, this is an affine variety, so if I look at the comology, it has a non-trivial weight filtration, and the conjecture is that these are also identified. There's a, I feel like, scale the grading by two or something like that, but I'm going to skip that. So why is that relevant for us? Well, this is the thing we're interested in if I forget the comology degree. I just remember the perverse degree. That's the thing that we want for these Boko Komarov invariants. If I believe this conjecture, then I can compute it instead by looking at the comology of this character variety, where I only remember the weights. And this is something that you can actually compute. So this is done through a Hausal and Rodriguez Viega. I'm just going to use initials for the second time, which is that if you're only interested in the kind of the weight polynomial of the character variety, you can do it. This is a thing that you compute by a point counting. It's a cool story because it's really related to, you know, basically you replace this group GLR with GLR over a finite field and something you can compute using the representation theory of GLR over a finite field. But you can do it. And then what was observed later on by flying the Akinescu and Pan is that this matches, when you kind of go through this entire procedure, this matches on the nose what you would expect from the stable pairs theory of X. X in this case being this kind of local curve, the total space of K plus and the numerical level is not hard to compute. So this is kind of an extremely long chain of reasoning and it's kind of built on this conjecture. But it means that every time, every case where this conjecture is proven, you then get the corresponding connection with the between the Gopal-Koroboff variance and the stable pairs theory. So the upshot here is that basically anytime you have the P equals W conjecture, it implies the kind of Bromow-Witton PT relation, sorry, the Gopal-Koroboff PT relation. And so in particular, we know this conjecture. For instance, we know this when the rank is two. This was done in this original paper. The conjecture was formulated. We also know it, so this is R equals two, but genus can be arbitrary. And we also know when the genus is two and the rank is arbitrary. And so in particular, in these cases, everything here is ironclad, and then you really do get kind of provable cases. What is special in genus two? Sorry, say that again? So what kind of feature of genus two you use to prove? I'm sorry, could you try that one more time? You mean why can we only prove it in genus two? Yes. Oh, yeah. Well, yeah, that's kind of a complicated story. I mean, the way the proof goes in this paper is that we prove the theorem by embedding a genus two curve inside of its Jacobian and then studying one-dimensional sheaves on this abelian surface. And so what's special about genus two is that there's kind of, you know, basically the comology of the hitch in space is, you know, surjected onto by the comology of this compact hypercaler variety. So there's a theorem that you get for higher genus, but you only kind of get it on some piece of the comology of the hitch in space. So there's a version of this theorem that works for arbitrary genus, but it's not strong enough for this application. We only get a certain subalgebra of the comology for which the injector holds. And then what's going on in the rank two case is the rank two case, even though the genus is arbitrary, you actually have a complete presentation of the comology of these, of these the hitch in space. And so you can really kind of write down all the generators or relations and see where they go and match them up and what piece of the filtration you want. So these two proofs are very different in the sense that this first one is kind of, uses a lot more information about the comology. The right hand side uses kind of, fact that you kind of, there's a compact geometry that kind of governs the story without any loss of information. What are the kind of next cases I think to think about? So my feeling with this whole, this relation is it's still kind of in the stage of every example kind of shed some light on what's going on. For me, what I think is kind of the most accessible one actually, I mean, the one that's kind of, I think really low hanging through that we didn't pursue, but not for strong reasons was that when we, when you study local surfaces, you know, we only restricted ourselves to the situation where the push forward of the one cycle, so the local surface is the one, the total space of the canonical moment surface. So we kind of restricted ourselves to the situation where this was integral. But I think with, you know, with a little bit of, you know, elbow grease, it should be possible just to handle kind of, you know, reducible instead, reduced and reducible should be okay. The statement is more complicated because now you have corrections coming from these kind of irreducible components, but still, you know, the situation is sufficiently nice in the sense that, you know, the theorem of myself and Ewan and Migler and me and Chenday has already been extended in the locally planar case to this setting with the right corrections. But more interesting, the local surface case is, you know, non-reduced. And in fact, already, you know, kind of, you know, the simplest example where I think, you know, this is still open is, you know, when maybe the modular spaces are kind of still smooth. So for instance, if I take the total space, the local surface for like a del petso surface or something like that, in this case, the modular space is smooth. And I think understanding what's going on there is very interesting. You know, I think accessible as well, although, you know, maybe not as easy. What's interesting in this case is that if I look at a stable one-dimensional sheath on this threefold, it's scheme theoretically supported on the surface. But on the stable pair side, the one-dimensional sheath that I'm taking sections of will be thickened off of the surface, and that kind of leads to some contributions. And you can see that in the conjecture. And an example that Jim Bryan mentioned to me is something that he's kind of interested in, I should mention, this is Jim Bryan, is still the local curve geometry, so like what I was doing in the Hitchin case, but maybe not where you kind of twist it so the spaces are all smooth. So for instance, if I have a curve, and I could take a theta characteristic of the curve, a choice of square root of the canonical bundle, and he's been interested in kind of basically understanding what's going on in this kind of geometry. And then the last case, which I mean, the last kind of specific example that I think would kind of shed a light on it on what's going on a little bit more is the following, which is again, I'm going to take, it'll be a local curve geometry, but I'm not going to have assumed that the normal bundle is split. So let's take n, which is a kind of a generic rank two bundle that's determinant, the canonical bundle of the curve. And then I just take the total space of this rank two bundle. And so this is a little bit like what you would, this is kind of what you would get if you, you know, when I was talking about, on Wednesday, I was talking about how one way to think about these conjectures that you imagine you're in an ideal situation, where all your curves and your colabi out threefold or maybe one smooth and isolated and rigid in some sense. And you could try to imagine what it would contribute. And, you know, one of the issues is that, so this is kind of an example of what you would get if you thought about it that way. But what's interesting about it, and you can see in this case, is that if you consider, let's do, for instance, the simplest case that I already don't know how to do, which is beta is two times the class of the curve. And so set theoretically, if I look at the moduli space here, what I get is I just get a rank two stable bundles, I get all this, in particular, that none of the sheaves that show up are thickened in any way. And so you might think, well, all right, this is great. I, you know, we know we've known for, you know, millennia with the comology of the moduli rank two stable bundles, so I should just be able to compute the variance and see what happens. But what happens is that actually, this is not a scheme theoretical quality. So m beta actually has some kind of non reduced structure, which is a little mysterious, in the sense that, you know, I, you can kind of see what the, where, you know, the locus that it's supported on the non reduced structure, but you know, I don't have a good feel for what, what it looks like. And I mean, the, why you have this non reduced structure is just something you can see. If, let's say, if, if F is a rank two stable bundle, you know, the tangent space to the moduli space of stable bundles on the curve is something like, you know, x one of on the curve of F comma F. And there's a map from here to x one on the three fold of, let's push forward, but there's some, you know, long exact sequence and there's a non trivial co kernel here that can happen. Generically, it's zero, but then non generically, there's some piece here, which is something like, um, I want to say f, that's your m. And this group can be non zero. And in particular, this, this group can be strictly larger than this one. It's kind of like some kind of real nether question on this moduli space that affects the co-mology. And so, okay, so I, I, in terms of studying this conjecture, this is, I think, the kind of main example of the thing about next. But let me kind of conclude with just another direction, which I actually was planning on spending maybe half an hour talking about, but I'll just, I'll just say a few words about it. The kind of, the other direction in this circle of ideas has to do with what are called, you know, kind of kind of dependence questions. You know, in some version, this is conjectured in, you know, my work with Yukonobu and then kind of more systematically in the paper he wrote, the semi-stable case. And it has to do with some, it was a choice I made at the very beginning when I defined these invariants. So when we define this approach to Gopal-Kmarvaf invariants, what I did is I started off with some moduli space of one dimensional sheaves. And which one did I take? I took, you know, sheaves where the support was in class beta, and then I fixed the Euler characteristics to be one. And so you could ask why, what was special about Chi equals one? Why did we pick that? And the answer in some version is, you know, in some sense it's just laziness. You see, when Chi is equal to one, you don't have to pick a polarization to define stability. But if you're willing to make that kind of choice, then you could have picked any value for this, for Chi and, you know, you could still get a moduli space. So the simplest case is when, you know, you pick some integer k, and, you know, assume that in some appropriate sense it's relatively prime to the curve class. So you don't have to worry about semi-stables at first. And so now you get a moduli space and sigma beta k, where sigma here is some kind of stability condition that you have to use as well. Or maybe coming from, let's say, a polarization. And so just as I did before, you can define some invariance now depending on this integer, which is the value of k and then also the stability. So I mean, then the, so the first conjecture, which, you know, we made in our paper just to deal with this objection, was that, okay, in fact, none of this mattered. Whatever choices you made here, independent both of this stability, which is, and, and, and this, and this choice of Chi that you picked. And in fact, the independence of, of the stability condition, this actually was proven later on. Again, under, you know, under some assumption that these orientation issues work out. And if you want, I mean, you could even ask for something stronger. We never formally made this conjecture, which is that you have this math from your modular space to the child variety. And the child variety doesn't require any choice of stability or doesn't depend on what k is. It only depends on what beta is. And then you can just ask for this push forward of the, you know, this DT sheaf, which depends on sigma and k, just to be independent. This is a little surprising, depending on your point of view, this is either surprising or not surprising. So you see, why is this kind of, you know, not surprising? Something like this might be true. If I specialize all the way to Euler characteristics, I mean, I look, I just take the gena zero thing. So the gena zero statement, kind of, you know, n zero beta, say my k is independent. This is actually conjectured a long time ago. This is in fact a prerequisite. This is, you know, equivalent to this strong rationality conjecture that I stated. So this is something that people have been interested in for some time and, you know, basically it boils down to the independence of these numbers on all these choices. And so this is just some kind of, you know, souped up version of that. And, you know, whatever reason is making this numerical statement true is maybe also making this kind of sheaf theoretic statement true. So from the point of view of this kind of DT perspective, this is not, we did not view this as a big leap. But it is a little more surprising from the classical point of view. Well, the surprising is maybe just on the word, but it's a little, in the following sense, which is that, let's say I take a smooth curve of genus G. And then I take the modular space of stable bundles on this curve, or rank R and chi. And so assume, and again, let's assume I'm still in the co-prime case. So, you know, you could ask, how does the comology of this thing depend on the choice of D? And well, I can always twist by, like, a degree one line bundle. And that'll relate, you know, one modular space with the other modular space where the degree is shifted by R. But what Harter and Narha Siman showed was that basically other than this move, and other than, like, taking duals, that in general, these comologies are different. So the comology, like, for instance, you know, I think already when the rank is five, and then D is one, and D prime is two, these spaces are different. These comologies are different. And that's a little bit like what I'm asking here. I'm having a modular space of, you know, things supported on a one-dimensional scheme. And so you could ask, why, what's special about this Kulabi-Ao-3 situation that isn't kind of happening in the case of curve? I don't really know a great answer to that. But one thing that kind of gets explained from this perspective is that, so you see, okay, so this is a very classical question about stable bundles on a curve, and you don't get any kind of independent, kind of independent statement. However, if I look at Higgs bundles or twisted Higgs bundles, then in fact, it is true that the comologies of these Higgs spaces, with or without the D, are independent. And so, you know, this is something, let's see, I think this has proven a couple of different ways. I think maybe the first proof was basically by just calculating, finding a formula for these punk-ray polynomials. So this was essentially done, I think, by Schiffman, in the denontrivial case, maybe Schiffman, Moskvoi, Gorman is what I wrote down. So I think this is the final paper that proved this equality here for nontriviality. And so you might ask, why is it the case that Higgs bundles have this kind of kind of independent statement, while just regular stable bundles on a curve do not? And so from, you know, my perspective, the explanation is just what I said before. These kind of Higgs-modulized spaces are secretly a kind of colabi-out-3-modulite problem. But just stable bundles on a curve are not. There's not some embedding of the curve into a colabi-out-3 fold, such that the modulized space of one-dimensional sheaves is just stable bundles. But then you could push it forward, is that you could ask a little bit more. Here, I just focused on the case when beta and k were co-prime, because then this modulized space, I don't have to worry about semi-stable sheaves, and like there's some stack and some core space. You could ask if there's a way of pushing beyond that. And the answer is yes, there is. So now there exists semi-stable sheaves. You'll have some kind of stack of semi-stable sheaves with some kind of coarse-modulized space, which again, maps to the chow variety. And okay, maybe I won't get the intricacies of the construction here, but there is basically a way of modifying everything I said to handle this. So this is developed by Yukinobu. He basically produces a perverse sheave on this coarse-modulized space, which is now sometimes called the BPS sheave. And then you can kind of run the same story. And so already in these kinds of classical modulized spaces, if you specialize, you get kind of some interesting statements. So the kind of something that was proven recently by myself and Jin Liang, if you do twisted these twisted Higgs bundles, but now in this kind of non-coprime case. So this is some kind of singular coarse-modulized space. Well, it turns out that this, whatever this mystery sheaf is, it's somehow, I haven't told you how to define it. It ends up just being intersection comology. This is something that was proven by mineheart. And this is kind of what you get. And this is actually a theorem that you can prove. So in the intersection comology of this singular variety twist is independent of D. And in particular, it just equals the regular comology in the coprime case. And so this is, you know, in some sense, this is kind of a classical statement that you could have formulated in the 70s or something like that. And, you know, the reason to expect something like this is true, the only one that I can really see is by thinking of it as a kind of a specialization of these kinds of considerations. But the analogous statement for if I remove the twist, if I just consider regular Higgs bundles, it's still open. You know, now you don't need intersection comology. This Toto sheaf is going to be something a little bit more mysterious. And already in that case, we don't know how to prove what should be true there. All right. Okay. So I ended up using the full hour, which was not my intention. But let me stop here and thank you for your attention. Thank you. Any questions? Yeah. Yeah. So why does like the trick that you use to go with Yidunyang-Shen to prove this household radius conductor not work for, for example, using this kind of pendant, doing this kind of pendant? Oh, yeah, yeah, yeah. That's interesting. Right. So the that trick, which I didn't, you know, I meant to actually talk about that trick a little bit. But, you know, I mean, so what this trick is about, just let me just say a couple of words about this. So there's this kind of the same technique that I used in yesterday's lecture can also be used just to study questions about Higgs bundles. Higgs bundles are a smooth modular space. You wouldn't think it would be so helpful. But the idea is that you can embed them inside of these twisted Higgs bundles as a critical locus. And so this allows you to take statements that are easier to prove for the twisted case and turn them into stasis theorems for the untwisted case. So it's exactly the same kind of philosophy as that from yesterday's lecture. And so what happens for these kinds of questions? Well, in order for this to be true, it's important to work with stable Higgs bundles. The analogous statement is not true for semi-stable Higgs bundles. The critical locus is bigger. So you get some, you know, thing that's true, but it's like it's the comology of some bigger modulite space. So it's not actually the thing that you're interested in. And so then, you know, what you have to do is you have to somehow relate that bigger space with maybe the smaller modulite space that you actually care about. Does that answer your question? Yes, thank you. Any other questions? Well, thank you very much again. Thank you very much.