1/5 Stable Pairs and Gopakumar-Vafa Invariants
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Transcript: English(auto-generated)
00:15
I want to thank, obviously, the organizers for the chance to lecture here, although I wish the original plan,
00:21
when this was set up a couple of years ago, was going to come with my family, and it was going to be this two weeks, and France, and all that. Obviously, this is slightly less than what I was hoping for. So let me begin by giving some motivation about what I want to talk about, which has to do.
00:50
So for motivation, let's just start off with the case of a smooth curve of genus G,
01:02
and then I can look at its Hilbert scheme of points, of endpoints, which since C is smooth, is just the same as taking the symmetric power of the curve.
01:24
And so there's a very nice formula going back to, well, probably in this version it's older, but on the level of, say, cohomology, goes back to an old paper of McDonald, which says that if I take the Euler characteristics
01:43
of these Hilbert schemes, and I sum over n, well, this just has a very nice formula, where I just take 1 minus Q to the 2G minus 2.
02:03
I can then make more complicated in kind of a dumb way by writing it as 1 minus Q to the 2G over 1 minus Q squared. And now the numerator of the right-hand side itself has kind of a geometric meaning. If I take the Jacobian of my curve,
02:24
and I take its cohomology, well, we know what that is. It's just the exterior algebra of H1 of my curve. And so if I take the Poincare polynomial here,
02:45
the Jacobian, where I'll just introduce a minus sign, that's exactly the numerator of this expression.
03:12
And so now what I have is I have some kind of slightly silly identity, where the left-hand side, I'm kind of summing up over all n, these Euler
03:21
characteristics of the Hilbert schemes. On the right-hand side, I just have a single space, just the Jacobian. But now I'm doing something a little bit more refined. I'm taking the actual cohomology instead of just the Euler characteristics. And the Q kind of has a different meaning depending on which side I'm on. On the left-hand side, the Q is just
03:40
indexing which moduli space I'm working with. On the right-hand side, the Q is this kind of cohomological variable. It's helping me keep track of my cohomological degree of the Jacobian. And so in some sense, one of the things I want to try to explain in these lectures is kind of an onsatz due to many people.
04:02
So I'll do the attributions later on when I actually get to it. But the iteration that I'll be talking about is kind of joint with the Yukonobu-Toda, which basically proposes a way of extending this to kind
04:25
of much more singular curve. So for instance, curves in the kind of setting of Calabi-Yau threefold. And one of the things that I kind of want to get to hopefully is that, OK, so generally, it's a conjecture.
04:41
And you can believe it or not, but in the cases where we can prove it already kind of gives you examples where something like this holds for extremely singular curves. And the technique of proof is kind of nice, because in some sense, what the technique of proof really does is it reduces it to the case of the smooth case where it's originally just McDonald's formula. So there'll be some kind of chain of logic where kind of the final step will just be applying
05:02
the original identity. So let's see. So let me give kind of, you know, I'll start talking about this properly in maybe the third
05:20
lecture. So let me just say now kind of what I'm hoping to kind of cover in these five lectures, which is in the kind of first two lectures. I want to give some kind of overview about Donaldson-Thomas
05:44
theory, some version of what you saw in Richard's lectures last week. But I'm going to focus really again on the setting of Calabi-Yau threefold. And in particular, the kind of perspective
06:00
that you get when you get in the Calabi-Yau three setting is that instead of doing things intersection theoretically, there's kind of an alternate approach where you work in this kind of constructible world. So constructible functions, and then that'll be today. And then eventually tomorrow, constructible sheets.
06:23
And then kind of in the remaining lectures, I want to then kind of talk about this kind of picture that I just sketched out, which is this notion of an approach to thinking about what are called Gokumar-Basson variants, which
06:41
are you should just think of this as the analog of the right-hand side of this McDonald's formula, some analog of what to put in the numerator in general, where we'll be using the kind of technology developed in the first couple of lectures to pursue that. And then in kind of the last lecture, I want to kind of talk about some related conjectures
07:02
to this story, in particular, a conjecture of Toda in the last couple of years, which is related. These kinds of topics are related to kind of defining right-hand side of this McDonald's formula.
07:27
Are there any questions before I get started properly? So again, if there are things that show up in the chat, I'm not really probably going to be able to see them so easily, I think. So hopefully, Andre can read them aloud in his dulcet
07:44
tones. OK, so today, I want to start talking really in some generality about numerical Donaldson-Thomas
08:08
theory. And so again, the setting for today, I'll be focusing on kind of the geometric setting,
08:22
and specifically in the case where we start off with some kind of Calabi-Au threefold. Does not necessarily have to be a projective, so just some smooth algebraic threefold with a nowhere vanishing algebraic reform.
08:43
And the kind of basic moduli space that you associate here is going to be some moduli space M of x, which is going to be some moduli of stable coherent
09:04
sheaves, the moduli space of stable coherent sheaves, or more generally complexes of sheaves on x with a fixed discrete invariance during classes
09:32
indexed by some vector v in the cohomology. And so the kind of classic version of this story
09:44
is where you just take, maybe you fix the polarization, and then the moduli space you look at is just,
10:02
let's say, Giesecher stable sheaves on x to avoid kind of stacky issues or issues with the obstructions. You often will kind of fix an isomorphism
10:23
of the determinant of E. So here I'm giving some kind of examples.
10:42
For curve counting purposes, it's usually better to work with some kind of variation on these spaces. So the old version that we kind of, the subject kind of started off with was to work with the Hilbert scheme of curves,
11:05
or rather Hilbert scheme of one-dimensional sub-schemes on x. So here maybe I'm going to fix an element of H2.
11:24
And then I also fix an Euler characteristic. And so this would parametrize sub-schemes, one-dimensional sub-schemes, where the support, the kind of one-dimensional piece
11:43
is class beta, and then the Euler characteristic of the structure sheaf is n. And then one example that we'll see if I have time to talk about maybe in some is the special case when beta is zero,
12:00
and then you're just considering the Hilbert scheme of points. Now in these examples, the way you kind of put it into this framework of moduli space of stable sheaves is that instead of thinking about the sub-scheme and then the kind of surjection from the structure sheaf, you just remember the ideal sheaf. So you think of this.
12:28
You look at the ideal sheaf of your sub-scheme, which is a rank one sheaf on x with the trivialization
12:42
of the determinant. And this turns out to be equivalent to looking at the Hilbert scheme. So the way you put it in this framework of moduli space of sheaves is by forgetting about your sub-scheme and instead remembering just the ideal sheaf that cuts it out. The version that's cleaner, and maybe actually Rahul
13:04
already has spoken about this in his lectures, or if not, he will soon, I'm sure, is a variation, again, for curve counting purposes of the Hilbert scheme construction where you work with what are called stable pairs.
13:21
And so this theory was developed by Padre, Ponda, and Thomas. And so here, the moduli space, again, you kind of fix a curve class.
13:40
You fix an integer. And then the data here in this moduli space is there's two pieces of data here. First is a sheaf, e, which is one-dimensional support
14:09
and is pure. So pure meaning it has no kind of zero-dimensional sub-sheafs.
14:20
And then the second piece of data is just a section of the sheaf. And then the stability condition is just the statement that the co-kernel of this section is zero-dimensional.
14:43
So what does an object of this space look like? So the simplest kind of example to think about, if you haven't seen this before. Oh, I should just say what the discrete invariants are. So again, the support of the sheaf, just the cycle theoretic support
15:03
is gonna be in class beta. And then the Euler coordinates is n. So the way to think about what this space looks like, the first approximation is, let's say here is x.
15:22
And then imagine the support of e is let's say a smooth curve inside of x.
15:42
And so you could last, what are all the stable pairs with fixed support like this? Well, the simplest is I just take the surjection, O x to O c. So that's an example of a stable pair. But what I could also do is I could also take,
16:02
if I give you some line bundle on this curve, I could take e to be its push forward, which is now a coherent sheaf on x with one-dimensional pure support. And then if I just any non-zero section of L will then produce a stable pair on x.
16:22
So given non-zero section here, it defines for me a section of e and the co-kernel of this,
16:44
of the corresponding section on e will exactly just be the zero locus of the section on the curve. So the way you can think about this is you have some line bundle on this curve and now I have some section.
17:00
And so the same curve will contribute to many different stable pairs. Just you take any line bundle with a section or equivalently, any collection of points on my curve that will be a corresponding stable pair on x. So the contribution of c
17:24
to all of these stable pair spaces ends up looking a lot like just taking the different symmetric powers of my curve.
17:44
And so in particular, this will be kind of the left-hand side of this kind of McDonald inequality in general. And so again, how do we wanna put this in this kind of setting of DT theory? So again, the setting of DT theory as I wrote it is that we're gonna wanna consider moduli space of sheaves or complexes of sheaves on x.
18:07
And so here I have this kind of two-term complex. I just think of the corresponding object in the drive category. So I take this two-term complex.
18:23
I think of this as an element in the rise category of coherent sheaves on x. And this is how I'm gonna think about this moduli space. It's a moduli space of a certain kinds of two-term complexes.
18:45
Again, so one thing that's kind of instructive to do is if x is projected, then this space of pairs is also projected. And so kind of a fun thing to try to do is just to understand what certain limits look like
19:00
in this space. So for instance, maybe it's an exercise. This is a local question. So it doesn't really matter what the ambient threefold is but imagine I have two lines that are kind of colliding,
19:21
two skew lines that are kind of about to collide in some limit. And then we know from Hartron or something that the limit in the kind of Hilbert scheme of one-dimensional sub schemes is you get two intersecting now coplanar lines and a little fat point there. But this kind of limit isn't allowed
19:40
in the stable pair space because the support of the sheaf, because the sheaf isn't pure. And so instead what you can try to work out is what the limit of this kind of thing in the stable pairs moduli space is. Where the support of the sheaf
20:00
is still going to be these two lines, but now there's going to be what used to be a kind of a fat point now gets replaced with some kind of non-zero co-kernel. So these are kind of the examples
20:21
of the kinds of moduli spaces, geometric moduli spaces that one can look at. But actually everything I'll be talking about today, it doesn't really have to be a geometric setting. So they're kind of the kind of main example of a non-geometric examples to look at
20:41
or come from looking at representations of quivers with potential. So everything I'll be saying today and tomorrow kind of makes sense in that setting. In particular, there's a lot of the material and Marcus Reinike's lectures, I think will be relevant.
21:05
So these are the examples I want to look at. And so as always, we're interested in some kind of virtual structure on these moduli spaces.
21:22
And so in the context of Richard's talks last week, the kind of initial piece of data that you want to understand is something about the deformation theory. So you have a deformation obstruction theory for understanding these moduli problems,
21:44
which because I'm just working with moduli spaces of sheaves or maybe complexes of sheaves, they're given by X groups. So the deformation space, the tangent space is given by the self X to one
22:02
of whatever your sheave for complex is. If I fix the determinant, then we usually do some kind of traceless thing. And then the obstruction space is then given by next two.
22:31
And so again, my understanding is Richard kind of talked in more detail about how these show up for a moduli of sheaves. And so already the first nice thing that happens in the Calabi-Yau three case,
22:43
so this is off course, so far everything is completely general, which is you get to apply Sare duality. If I take the dual of the obstruction space,
23:10
I get X to one with this twist by the canonical bundle of X, but because I'm in the Calabi-Yau three setting, this is just trivial.
23:21
So I get exactly the deformation space. And so particularly it means that the virtual dimension of my moduli space was just the difference in dimensions is zero. And so if X is proper,
23:45
or at least if my moduli space is proper, that's really all I need. I have this virtual class, which is a zero cycle. And under the properness hypothesis,
24:00
I can take its degree, which will give me a number. I'll call this kind of the virtual number associated to my moduli space. And it's just some integer. And the procedure for doing this, again, this is something that Richard sketched out
24:21
is the way you produce this virtual class from all this data is basically by using some techniques from intersection theory. So that's kind of the world where these constructions live most naturally.
24:43
Okay, so what do I wanna explain? First then is, again, in the Calabi-Yau setting, another way of thinking about what these numbers are. So this is this notion of what we now call Baron functions.
25:08
So the setting here is that let's say I have some moduli space M and I've equipped it with this kind of
25:25
perfect obstruction theory, meaning I have some two term complex that calculates the deformations and the obstructions to my moduli problem.
25:41
We say that E is symmetric. I have a quasi-isomorphism between E and it's shifted dual.
26:03
So you should think of E as being kind of supported in degrees negative one and zero and then E dual is gonna be shifted supporting degrees zero and one. And then I shift it back. So it's again, supporting degrees negative one and zero. Well, I have some isomorphism like this
26:21
with the symmetry condition. So such that this isomorphism itself has some kind of self-duality property. So the easiest way to get a kind of data like this is that for instance, if I have, let's say if F is a vector bundle with a symmetric bilinear form alpha,
26:49
then you can produce an example of a complex with this kind of symmetry just by taking a map from F dual to F. And then you have using this bilinear form you can produce a map like this
27:07
which exactly has this kind of symmetry. And so an example of a kind of a, so the baby example of a moduli space
27:22
with a two-term perfect obstruction theory with this kind of symmetry is where your obstruction theory is kind of dumb. So let's say M is smooth and then your obstruction theory basically consists
27:40
of the map from the tangent bundle of M to the cotangent bundle of M which is just the zero map. So this is like, I have an obstruction space but those obstructions are all unrealized because the space itself in fact happens to be smooth.
28:01
And in this case, if you calculate what the virtual class is, so it should be zero dimension, it's a zero dimensional virtual class cause the deformation destruction in dimension. It's just going to be, it'll just given me the Euler class
28:20
of the cotangent bundle, which is my obstruction bundle. And if I take its degree, I just get up to assign the topological Euler characteristic for them.
28:44
And so the first observation with everything here I should say I'll be saying is due to Pi Barand. Except for the name, he didn't of course name it after himself but at some point it caught on. So in the Calabi-Au3 setting,
29:00
so all the examples that I said before, again just the same spirituality calculation I did before tells you something a little stronger. It tells you that the obstruction theories are always symmetric.
29:33
And so let me give the kind of key local example of one of these symmetric obstruction theories kind of use again next time.
29:44
Which is an imagine I have some ambient smooth space V which is just affine space and I have some function on it. And then the kind of space that I'm looking at
30:02
my actual moduli space is just the zero locus of all the partial derivative, just the critical locus of this function F.
30:21
So because it's some, it's a space cut out by a bunch of equations and in particular has a nice two term obstruction theory. And then you can just write down what the obstruction theory is in this case. And it's basically determined by taking
30:46
the Hessian matrix for F. You take this kind of symmetric matrix given by taking the partial, the second partial derivatives. And this defines exactly that kind of a symmetric obstruction theory.
31:03
And this will be kind of the main example for us. So this kind of baby case is the case where the function was zero and it's kind of dumb, but in general, it's more interesting. So this is the framework for us.
31:21
We have a moduli space, we have this kind of two term obstruction theory and it has the symmetry property. So definition, a function, if I give you any kind of complex scheme,
31:41
a function from the complex points to the integers, it's constructible. If the set of points where the function has some value, so new inverse of A, this is a constructible set.
32:11
I have some variety M, there's gonna be some open set where it has some value zero and then maybe there's some locally closed set where it has value one and then maybe some other stratum
32:20
where it has value negative one. And then given one of these constructible functions, I can kind of use some version of integrating it, some discrete version of integrating it where what I'm gonna do is I'm gonna sum,
32:41
I'm gonna look at all the kind of strata where the function has some value and I'm just going to add up those strata weighted by the, add up the Euler characteristics of the strata weighted by the value of the function there.
33:04
For instance, if I just had the constant function one, I would just be getting the topological Euler characteristic of M, but of course in general, I'll get something else. And then in particular, I can look at, for instance, just the abelian group of constructible functions,
33:22
Z value of constructible functions in M. And one way of thinking about this is if I just look at characteristic functions, this is a basis index by irreducible sub-varieties.
33:43
Do you only assume finitely many non empty fibers? Yeah, yes, that's right. I'm sorry, M is gonna be just a finite type thing. So that's right, only finitely many. And so the key theorem that kind of kicks off for me,
34:02
at least this is the whole direction of the subject is that if I give you M with the symmetric obstruction theory, in particular, any moduli space of sheaves or whatever on a colobiad threefold,
34:23
there exists, associated to M and E, there exists a constructible function on M such that if M is proper,
34:45
and a virtual number of my moduli space, meaning in the sense of taking the degree of a virtual class, is the same as what you get by integrating this constructible function.
35:05
And so what it means is that, this intersection theoretic quantity, it means that you can kind of study it using ideas from kind of constructible geometry or micro local geometry. So let me just say a couple of remarks about this. I'm gonna say something about why this is true
35:21
in a second, but let me just kind of say what's kind of so interesting about this. It allows you to do a couple of things that you couldn't really make sense of intersection theoretically. So for instance, this virtual cycle, so it's really a cycle class.
35:41
And so if I give you some subset of M, it doesn't really make sense to talk about, what is the contribution of Z to the virtual class? Cause you can't really localize. There's not a clean way of localizing
36:01
this kind of zero cycle class to all the different, along some stratification events. On the other hand, if I give you a constructible function, it's very easy to do it, because I can just restrict my constructible function to Z and I can integrate it there.
36:39
Second, the right-hand side makes sense
36:41
even if M is not proper. So usually the left-hand side, unless you're in some kind of equivariant setting, like in Richard's lecture, if I have a non-compact modulized space,
37:00
it doesn't make sense to take the degree of a zero cycle class on it, but you can always just integrate this constructible function. On the other hand, the left-hand side, of course, because it's defined intersection theoretically is a deformation at least in the proper situation.
37:29
I mean, which is the only time it makes sense. And something like the right-hand side, if I just take, for instance, the actual topological Euler characteristic, that is as M varies in a flat family, that's certainly not going to be a deformation invariant.
37:41
And so there's something going on with this kind of specific choice of constructible function that's kind of correcting for the failure of the Euler characteristic deformation. This is really not at all obvious from how it's defined.
38:12
And so this ends up being an extremely useful theorem. Let me just say, maybe I won't write this down. One way this gets used a lot is when you study how these invariants change
38:21
under a change of stability and wall crossing and so on, which is that when you kind of cross some kind of a wall and your stability condition changes, your moduli space usually changes maybe by some kind of flip or flop or something like that. And so understanding how the cycle class transforms might be kind of delicate, but this kind of weighted Euler characteristic,
38:43
if there's some open part where the two moduli spaces are just the same, then you can just throw it out because the contribution to this kind of integral is going to be the same. And you can just focus on the kind of the actual strata where the stability is changing. So this ends up being kind of an extremely powerful tool for those kinds of analysis.
39:13
So let me sketch the proof of this result. And it goes into how this kind of virtual class
39:26
is defined, which again, I think, I believe Richard covered in his first couple of lectures. Well, the idea is that if you have N embedded in some kind of smooth space, let's say,
39:43
then the way you get this virtual cycle is that you have M sitting inside V and then there's some kind of a vector bundle over V and then there's some kind of cone
40:02
with multiplicity sitting inside of S. This is some kind of conical cycle, not a cycle class, an honest to God cycle inside of this vector bundle. And then when I intersect it with the zero section, I get exactly this virtual class.
40:23
But this is true just in general. What Cai showed in his paper is that if you now add the condition that the obstruction theory is a symmetric
40:40
and like in the Collabia situation, you can actually refine this picture so that this vector bundle S is actually the total space of the cotangent bundle of V and this conical cycle inside of the cotangent bundle
41:07
is not just a cone, it's a Lagrangian cone. This cotangent bundle has a natural symplectic form. And when I say this cone is Lagrangian, I mean that the smooth locus
41:22
of every irreducible component of this cycle is Lagrangian in the usual sense. The symplectic form restricts to a zero and it has middle dimension.
41:41
So why is that so special? Well, so there's an isomorphism between on the one hand constructible functions on V
42:02
and this free building group of conical Lagrangian cycles, which is known as the characteristic cycle map. It takes a constructible function here and sends it to what's called
42:21
the characteristic cycle of this function. And this is defined in some sense via some kind of Morse theory type construction. The fact that there is an isomorphism like this shouldn't be kind of super surprising. Each of these spaces has a basis that's indexed by these irreducible sub-varieties. So I already talked about how
42:41
irreducible sub-varieties just by taking the characteristic function defines the basis here. Similarly, if I give you an irreducible sub-variety, I can take the smooth part and I could take its conormal bundle, which is the Lagrangian side of here, and I can take its closure. And so that gives me a natural basis here, but that identification is not what's used to define this isomorphism.
43:01
It's a little bit more subtle than that. But what's great about this construction is that on each side, there is an evaluation map to the integers. On the left-hand side, when I take a constructible function, I can just integrate it. And on the right-hand side,
43:21
if I give you a conical cycle, I can take the degree of its intersection with the zero section. And the way this characteristic cycle construction goes is that this diagram from use.
43:44
This is what's called the index formula. I don't know, due to many people. So, you know, maybe Dubson.
44:01
So this is just a very general statement about constructible functions on V and Lagrangian cycles in the cotangent model that you can kind of set up an isomorphism, which makes this diagram commute. I'm not gonna, I won't actually, if I had more time, I would actually, I had an idea from sketching a proof
44:22
of this kind of index formula. There are a lot of proofs. The one I liked the most is in a paper of Schmidt and Valonin, where basically they just reduce it to the case of understanding kind of past to the real analytic world, and then you reduce to the case of understanding like a tiny ball.
44:41
And so this is great. You see the right-hand side is exactly what we want to define the virtual class, the degree of the virtual number. The left-hand side is the kind of thing that, you know, Beren's theorem is about. So to produce this kind of Beren function in the statement of this theorem, I'm gonna take the conical Lagrangian cycle that's associated my obstruction theory
45:01
and just move it over to the left. So this is now what we now call the Beren function is just it's whatever constructible function maps to this obstruction cone
45:24
under this characteristic cycle map. And so that's exactly how you kind of go from the intersection theoretic world over to this kind of constructible world.
45:48
So then this begs the question, what do we know about this Beren function? How do we, this is kind of a somewhat abstract statement. How do we kind of compute it? You know, any examples? And in general, it's quite hard. So, you know, I would say,
46:01
if I give you a kind of a random moduli problem and I give you some random point in the moduli space, you know, it's not so easy to kind of compute this thing. But some cases, we have some kind of statement. So for instance, the easiest case is when M is smooth.
46:24
M is smooth, you can again kind of put the stupid, you know, this is the baby example where I just have the zero section defines for me a symmetric construction theory.
46:42
And then the Beren function in this case is just a constant, just negative one to the dimension. And then Beren's theorem says exactly what I wrote before. If I integrate the Beren function, I'm getting negative one to the dimension times the topological Euler characteristic of M,
47:06
which is the degree of this kind of virtual cycle that we associated it before.
47:29
What about this local example I did up here? So here I kind of wrote down this key local example where I take the critical locus of a function.
47:49
M is the critical locus of a function S in N variables.
48:05
And so in this case, if I give you some point on M, the value of the Beren function, again, it's something pretty nice. This is related to what's called the Milner number,
48:23
maybe the reduced Milner number of my function at P. So it's some kind of notion in singularity theory, which let me just state what it is. So given a function and some point in the critical locus,
48:41
I can take the Milner fiber, which is just, I take a tiny, I take a ball of, a closed ball of some tiny radius
49:02
around my point P, and then I intersect it with the fiber. So let's assume that F of P is zero. Make my life easier. And then I just intersect it with a nearby fiber
49:22
of my function. So here, epsilon is much less than delta, much less than one. And so then what the Beren function is in this case is, I'm just taking, again, up to a sign, taking one minus the Euler characteristic
49:41
of this Milner fiber. So this isn't super explicit, but it's, again, something familiar from singularity theory. So there's a question in the chat. Do you assume that the singularity is isolated
50:01
and what if it's not? Oh yeah, I am not assuming the singularity is isolated. You can still, this definition makes sense in general, and it makes it a little harder to think about, but this definition is still not. I'm just- So on the rotational thing, you wrote X is in V, do you mean X is in AM? Oh, sorry, yeah, V was AM, that's right, yeah.
50:32
So let me give a more complicated example where you get to kind of see this.
51:00
So let's say I want to,
51:01
so I'm going to take the following kind of Calabi out threefold. I'm going to take a three, three hyper surface inside of P two cross P two. Which if I kind of, this is a Calabi out threefold complete intersection. I project onto P two and I get an elliptically fibered Calabi out threefold.
51:26
I'm going to pick it, you know, to pick the defining equations such that, maybe some elliptic.
51:42
So that one of the singular fibers of this vibration is sitting inside of P two, just given by, you know, X squared times Y. So one of the fibers, it's this reducible, non-reduced cubic inside of P two.
52:20
This fiber looks like, you know, maybe two C one plus C two. And so if I'm going to look at the following, you know, moduli space of sheaves, I'm going to look at, I'm going to look at sheaves,
52:40
which are one dimensional sheaves where the support is C one. So that kind of, I take the non-reduced component and I just take the underlying reduced curve, which is just a P one. And I basically, you know, I'm just going to set my discrete invariance to be whatever the train character
53:02
of the structure sheaf of this curve is. So the support of the sheaf is C one and the Euler characteristic is one. And so I can look at the corresponding moduli space of sheaves on X and set theoretically, it's just a point. This is the only object in it,
53:21
but it turns out, and this is the calculation. This is from Richard, you know, 30 years ago or something, is that this moduli space scheme theoretically is non-reduced.
53:42
As you can see what this is, is first of all, I mean, it's cut out the equations, U squared, V squared. So this is the same as looking at the critical locus of the function U cubed plus V cubed, because you know, these up to three, these are the partial derivatives. And so, okay, so then, you know,
54:02
so this is a pretty explicit function. You can work out what this miller number calculation gives you. So if you, well, it's just a point. So the value of the Baron function at this unique point ends up being negative one squared,
54:22
one minus negative three. So this negative three is exactly the Euler characteristic of the milliner fiber in this case, which is value four. On the other hand, you know, this thing is zero dimensional. So the virtual dimension equals the actual dimension.
54:44
So the virtual class in this case, if I just calculated it, this is just the length of this zero dimensional scheme, which is four.
55:01
So, all right, so this is the main thing. So what I'd like to kind of do in the,
55:22
I guess I'll start this now and then I'll continue this tomorrow is I wanna kind of sketch, you know, how this, you know, give some indication about how this theorem gets used. This ends up being a really useful theorem for understanding, for calculating these numbers.
55:57
And so, okay, so what I'll maybe do is
56:00
I'll just do one example now, and then I'll just, I'll start it now and I'll kind of finish it tomorrow. So let me state this theorem. And so this is gonna be a theorem about these, you know, these stable pair invariants.
56:25
This space of stable pairs, and then we can define a generating function where I fix beta and then define what I'll call the PT series,
56:48
the virtual number of these spaces summed over two,
57:00
which again is kind of, should be reminiscent of the kind of generating function that I started this talk with, where I took the summation of the Euler characteristics of all the Hilbert schemes for a fixed pair. So this is gonna be the kind of general version for an arbitrary Claudio threefold. And so the theorem is that, so again, the X here was my Claudio threefold,
57:24
is that this generating function is the Laurent expansion Q of a rational function,
57:43
a symmetric with respect to Q goes to Q inverse. In other words, it's basically built out of things that look like, you know, Q to the one minus R,
58:02
one plus Q to the two R minus two. You can express this generating function in terms of a specific rational function. And so, okay, so what I wanna do is I wanted to kind of sketch the proof of this in the special case when beta is inducible and the proof is due to Ponder, Pandey and Thomas.
58:21
And what's kind of nice about this, so I won't do it now, I'll do that tomorrow, but let me just say why this is a nice result, which is that, you know, this rationality is something that we expect for any threefold. But right now we can only prove it, this is expected.
58:48
And we can prove it, you know, for things like, you know, complete intersections and so on. But in terms of a really general, this is a really general statement. If we, the only case where we can prove it really in some kind of generality
59:01
without knowing something really specific about the geometry of X is in this Calabi-Yau center. This is not only known, but they kind of, you know, for general, in general, we don't,
59:28
what's special about the Calabi-Yau case is precisely we get to use this kind of constructional approach to prove this kind of theorem. So I'll maybe say like, you know, a few lines about this argument tomorrow.
59:41
Let me stop you there. In the examples you wrote down in the variant function, is it worked out by going through this?
01:00:00
sketch you outlined, or the functions could be found differently? You mean the examples of what the Beren function is? Yeah. Yeah, so right, so in the examples I wrote down, you can calculate, this Milner number is something you can calculate. We have techniques for calculating it.
01:00:20
So that's basically how you compute it. So like for instance, you know, it turns out that finding the Milner fiber, the Milner number for like ukiplas, that is something you can more or less see by hand. And then you just work it out and you get this, you know, negative three popping out. And in general, there is kind of a procedure for calculating it.
01:00:40
You know, if you have like a function and you have like a lot of time, then there is like an algorithm for calculating what the Milner number is. But for, you know, usually the geometry we're interested in get, you know, get larger and larger dimensional, and so getting your hands on it isn't really feasible in practice. If you want to do like, so an example of, here's an example
01:01:01
of something that I, you know, that we like is, you know, if you do like the Hilbert scheme of points on C3, the function that you would want to find the Milner numbers for is in some sense pretty explicit, but involves, you know, three n by n matrices. And so if you wanted to actually kind of, you know, do that for any given point in the Hilbert scheme,
01:01:21
this is actually quite difficult. More questions? Yes, we have one. So if I take an Arkinian scheme or set point, is it true that the bearing function is bounded above by the length
01:01:40
of these set points? Sorry, say that again. If I take a set point, so a spec of an Arkinian ring, so I have only one close point like in the example by Richard. So is it true that the length of this set point is an upper bound
01:02:01
for the bearing function? Yeah, I mean, again, up to a, I mean, so I think, so I mean, if the, you know, if it's coming from one of these, maybe it's just equal to it even in general. It's definitely equal to it if you know it's, it's coming from a, coming from a critical locus. That's just what the theorem says.
01:02:21
But if it's not coming from a critical locus, there's still a definition of the bearing function, and then you know, maybe it's still equal to it in that case. But in the cases that show up naturally, it'll always just equal that length. Right, certainly it can be smaller. I was wondering if it can be bigger as well. Probably not. No, no, yeah. I think that's right.
01:02:47
Thank you.