2/5 Stable Pairs and Gopakumar-Vafa Invariants
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Transcript: English(auto-generated)
00:15
Okay, so I want to pick up where I left off yesterday, and so just as a reminder of what
00:23
we were doing yesterday, the setup was, you know, in some generality, we started off with some kind of Calabi-Yau threefold, and then associated it some moduli of objects, sheaves, complexes, whatever, on the Calabi-Yau threefold, and then we, you know, produced essentially a constructible function on this moduli space, and the z-value
00:45
constructible function, which has the property that, you know, if you're in the proper situation and you have these kinds of virtual invariants in the sense of intersection theory, that this constructible function recovers that information. If you kind of, you have some notion of integrating a constructible function where you add up the Euler characters of the strata weighted
01:01
by the value, and this kind of recovers this virtual class invariant that was defined in Richard's lectures. And so what I want to, you know, 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 it'll be something that I'm going to want anyways for later on in this course. And so the kind of moduli space
01:22
I'm going to take is what's called the stable pairs moduli space, or the PT moduli space sometimes, or ponder upon the income. And so the data here is, it's two pieces of data, and this is kind of a moduli 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 sub-sheaf,
01:43
and then you have a section of the sheaf with the stability property, which I forgot to write, that the co-kernel is zero-dimensional. So this has one of these, you know, you know, symmetric obstruction theories or whatever from yesterday, and so we can take its,
02:04
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 n sufficiently negative, these moduli spaces are empty. And the theorem is that this
02:25
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 Coda and Bridgeland. And so what I want to kind of first do is kind of sketch the proof in the kind of simpler case when beta, the curved class,
02:51
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 that appear in this class, beta is not of the
03:00
form beta 1 plus beta 2 with beta i affected. So all the curves that kind of appear which have support beta are in fact integral curves. And the proof in this case, this is actually one of the first, this is kind of originally done by Ponder, Ponde, and Thomas. And what, you know, the reason
03:27
I want to give this proof is really just to kind of indicate kind of the power of this ability 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 moduli space, but we don't really know how to prove it in that
03:41
generality. Thank you. Okay. And so I don't want to 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 enough.
04:12
So for instance, if it were a Laurent polynomial, the symmetry would just say that, you know, the q to the n coefficient is the same as the q to the negative n coefficient.
04:22
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 moduli space, I'll call it p t beta comma n. And so if I compare the q to the
04:44
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
05:08
check kind of strata by strata on the moduli space. More precisely, if I look at this pairs moduli space for n, and the pairs moduli space for negative n, I have a forgetful map where I
05:23
can just forget the section. And this will go to kind of m beta n, which is the space of one-dimensional sheaves with these discrete invariants. And similarly, I have a forgetful map here. And then I have a natural isomorphism between these two spaces, which sends a sheaf e,
05:57
what I'll call e dual, which is this sheaf here. So this is again some kind of pure
06:16
one-dimensional sheaf with the same support. And the way you should think about it is that,
06:21
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 want
06:45
to 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
07:02
difference, you know, fiber by fiber. So let's call this map price of n, I'll call this map price of negative n, using this identification of the two bases.
07:20
And so it turns out, well, what if I pick a point, if I fix a sheaf, the fiber in one projection is the projectivization of global sections of the sheaf.
07:44
You see the support is irreducible, so this business about the co-kernel being non-zero just means that the section is non-zero. And similarly, if I look at the fiber with respect to the, you know, other projection, well it's given by the projectivization of this H0,
08:07
which if you kind of, you know, mess around with duality ends up being naturally identified with
08:20
H1 of the original sheaf. So the idea is that both of these are kind of, you know, you know, the fibers are always projective 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. And then when I want to kind of compare the
08:43
difference, well, 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 Baron function just over
09:03
each of these fibers. Now there's one thing I need, which is that I need to know something about the value of the Baron function here. And the key fact that they prove is that basically that the Baron function on the kind of PT space is just pulled back from the Baron
09:24
function on the base. So it turns out that here I'm going to use some fact that the Baron function is constant on fibers. So it's up to a sine Baron function downstairs. And so as a result, you know, once I kind of sort out the signs, I get exactly the same
09:41
I want. The difference between 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,
10:04
when I kind of add up over all the strata of M, I get exactly this kind of identity.
10:27
And so really this is kind of the, you know, why this is such a useful idea. You kind of just focus, you know, fiber by fiber, and you can actually then turn that into an argument about global invariance. So let me just say a couple of remarks about this. It was maybe not clear from
10:58
what I said, but I was using the fact that the curve class was irreducible, you know,
11:02
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 Paul algebra technology of the kind that, you know, is hopefully the subject of
11:26
Veronica's lectures. Second, yes? There's a question about the proof. So is it obvious that the main function pulls back? No, it's not. This is something that, it's not hard,
11:44
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 prove using, you know,
12:06
the fact that the Beren function behaves well with respect to smooth maps and so on. And the fact that the Beren function in some sense is really, you know, can be detected at the level of the schemes without keeping track of all the abstraction theory and stuff like that. But this really requires an argument. Okay, so maybe the second remark is that,
12:29
you know, this rationality is much more general. This is what I said before. This should always hold. Once you kind of put some, you know, cohomology classes to cut the virtual dimension
12:46
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
13:03
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 Beren function. So
13:22
here I was really, you know, the Beren function was really along for the ride. What was important was that I was kind of could kind of study this question constructively. And really then I needed to check something about the Beren function, but basically I needed to check that the Beren function was kind of, you know, constant on fighters.
13:41
And so in particular, this entire argument would work if you replace the Beren function with something that was constant. So it would work if you somehow, you know, replaced this function that you're weighting everything by, by something constant or maybe, you know, a sign or something like that. And so in particular, this rationality, I mean,
14:03
this was actually originally what you can over prove. This rationality of this series is actually something about the actual topological Euler characters, because it's also not just the virtual invariance. Tomorrow I will kind of state a stronger rationality conjecture,
14:32
meaning a more kind of constrained version about what these rational functions are,
14:43
which only holds. So unlike this kind of weaker statement, which this only holds in the virtual setting and it's still open, even in the, even in the Calabi-Yau case. Okay. So I'll,
15:06
I'll talk about this tomorrow. Actually, Andrei, could you do me a favor? Could you,
15:31
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, to that, the backup. Could you repost that?
15:42
So I tried using it, but I don't see, I don't see, is it not working? Oh, hold on. Is it not working for you? I just see the initial state. The link you shared, can you access it? The host one or the participant one?
16:02
Well, the one that you sent to everyone. The Google drive? The Google drive. So I just see the, oh, I see it now. Yeah, it's working now. Yes. Okay. All right. Good. Could you just repost the link actually? I'm not sure if it will. Yeah. Okay. So what I'd like to talk about for the rest of today is, so, okay, so in general now,
16:33
let's go back to this kind of generality where you have X, we have some moduli space, and then we have this kind of constructible function on it.
16:41
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 cohomology theory associated by moduli space with the property that the, you know, the virtual invariance will just be the, you know, alternating some of the
17:11
betting numbers. And so what we'll actually do is we'll actually kind of, you know, construct
17:25
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 this will end up being a perverse
17:40
chief on M, but then, you know, to first approximation, you could first think of this as a, like a constructible sheaf or 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
18:00
stockwise Euler 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 cohomology and I take the Euler 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,
18:22
is that, you know, on the one hand when, you know, when we do these moduli problems, we're doing moduli of coherent 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? It has to do
18:55
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
19:02
smooth variety with a function on it, and then the kind of moduli space I was looking at was the critical locus of f, which is just the zero locus of the differential of f. And then in this setting,
19:21
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 if I want to kind of promote this down to
19:45
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,
20:26
using this notion of what are called 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,
20:42
vanishing cycles measures the, you know, difference in homology between the singular fiber and the smooth general fiber. The way that kind of set that up, if I have v over A1, which is my function, unless I'm interested in the fiber over zero,
21:04
which I'm going to assume is singular, but first of all the complement of zero
21:20
and its universal cover, the corresponding Cartesian diagram, and let's call, you know,
21:41
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. Okay, if I can do the following, I'm going
22:08
to take the constant sheath on v, and then I can, you know, pull it back and then
22:21
push it forward back to v, and then restrict this to the central fiber. So this is what's called, I'll call this psi q, 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 cohomology of this millner fiber at that point,
22:48
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 cohomology, 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
23:10
category of constructible sheets, and then I can just take the cone, and so again up to
23:22
shifts and so on, this is really the analog of taking, you know, this reduced millner fiber to get the baren function, and again what this is, you're supposed to think about this, measuring the kind of the difference in topology between the nearby fiber and the kind of closed,
23:42
the singular closed fiber over zero, and so once I throw in a shift, these two operations,
24:03
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 v zero, and in fact, kind of the surprising thing is that they preserve kind of this
24:26
abelian category of perverse sheaves, so I have phi sub f which goes from perverse sheaves on v, keeps on v zero, and also psi sub f, but this is the one that I'm going to be interested in.
24:51
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
25:01
Euler characters and replace it instead with reduced chromology, and then imagine doing that in families to get an actual complex of sheaves, 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
25:36
perverse sheaf on the zero fiber that's supported on the critical of this, and in particular it has
25:47
the property, I'll just call this, you know, phi sub m, it has the property that it's stockwise Euler characteristic is exactly this Beren function, and so the goal in which I want
26:06
I'm going to 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, because I
26:43
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 moduli 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
27:08
to work analytically or you know kind of formal locally, so for instance in the kind of gauge
27:20
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
27:42
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 an integrable d bar connection on sections of e, and so if you you know write any such kind of you know zero one
28:04
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
28:36
holomorphic Chern-Simons functional which is some explicit integral over x, x is collabiale
28:47
you just kind of cook up some some zero three form to kind of compensate it, so this is this
29:02
was 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 a construction that you can do that picks out exactly the condition that you want and this is modeled off of an analogous condition in for real three manifolds, but this is not the only way of thinking about this actually for
29:25
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 moduli space so some object on x
29:43
then if I look at the formal completion of my moduli space in this 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
30:07
some x1 and then you can actually write down explicitly a power series on this tangent space
30:23
that whose critical locus cuts this out so the idea is that if I look at the x algebra of my object it carries a product the yonada product but actually carries operations basically massive products induced by the fact that the way I get this is I take
30:49
the cohomology of some differential braided algebra so you have these higher operations go from
31:06
once you write them down they you can go from they go from you know symmetric power of x1 to x2 and then you can just write down a formal function on x1 which is just the sum
31:21
what I might screw up the denominators but I think this is right and here what's going on is here this pairing that I'm using is exactly the Serre duality pairing and so what you then show
31:45
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 you know the Calabi-Yau condition showed up in some pivotal way here it showed up because I was taking a
32:03
holomorphic three form here was showing up because I was using the fact that I have a pairing a natural pairing between x1 and x2 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
32:21
bit confused as to what f is here oh right so okay yeah maybe I maybe I shouldn't have gone into this I what I was trying to do is I was trying to kind of give some idea of why the modulized spaces that are showing up are actually critical loci at least locally right and so in the in the if you're willing if you're kind of comfortable in the gauge theory world you can see it for vector bundles just by explicitly writing down this
32:44
functional if I took the derivative of this functional with respect to a this is some infinite dimensional thing so you have to kind of you know deal with that then you get exactly the integrability condition which is something like you know d bar a plus you know a commutator a is equal to zero so that kind of falls out exactly here what's less clear is why this kind
33:04
of formal thing also works but I just wanted to indicate that there is a way of making sense of this purely algebraically without thinking about gauge theory but so maybe maybe maybe it's better best not to get into this but I just want to indicate that there is a picture
33:23
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 of information theory yeah that's exactly what I mean this higher operations is I'm thinking of this as an a infinity algebra and then the the Sarah duality is giving me the structure of what's
33:44
called a cyclic a infinity algebra and so then there's some kind of general theory about you know understanding your moduli problem as in terms of the intern in terms of exactly the critical in terms of the critical of this thing so there's some let's see a reference for
34:03
this there's you know basically there's some nice notes of Khonsavitch and Sogelman from like the 2000s that kind of discussed this but I'm sure 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 collabiau if I want to understand what are the equations cutting out this formal completion inside of the tangent space
34:23
you can always you're always just looking for the zero locus of these you know of these formal functions which go from x1 to x2 and then what's special in the collabiau case is that you can take all of those functions and group them together as coming from derivatives this one power series okay well okay maybe maybe maybe that was a bad thing to try to
34:49
emphasize because these kinds of 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 Panta, Talin, Vachier, and Vassozi
35:07
and then Chris Bravo, Vittorio Busci, 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 the kind of the
35:25
scheme setting so M of x is a scheme there exists a Zariski open chart for my point which can be
35:40
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 you know small maybe and 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
36:03
result is that you really get kind of a Zariski open set where it's true and this this proof is really much harder you see the idea is that these guys the the first paper introduced this notion of what are called minus one shifted symplectic structures which is really a notion
36:27
that's kind of properly in the world of derived geometry and then the second group kind of proved that there was a essentially a local structure theorem a Zariski local structure theorem for whatever these symplectic structures are what I like about this result by the way
36:49
other than the fact that it's just like a cool result is that it really kind of in a strong way uses derived geometry in the sense that you know a lot of the other constructions in the
37:00
subject like you know all the stuff about like obstruction theories and virtual classes are kind of you know dodges to get around 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
37:24
in an essential way so let me just state we've seen already some example of so for the most part
37:42
this theorem tells you that you have these Zariski local kind of critical charts where you have this critical locus 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 it explicitly we've seen some examples already so for instance there was an example
38:02
from yesterday where I had this kind of you know 3-3 hypersurface inside of p2 cross p2 and then I wrote I just get some explicit moduli space and I described it as a critical locus of a function another example that I like a lot where again you can actually
38:26
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
38:47
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
39:01
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
39:27
gln so this cyclic vector just means that if I take x y and z and I apply it 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
39:44
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
40:05
the critical locus condition is I just take the derivatives and set them to zero it exactly picks out the condition that the matrices commute and this is a really good this is a really nice example because you know it's explicit enough that you can imagine
40:22
you can actually try to like 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 any too much detail about here
40:48
but you know you know what I've said already is that m of x is basically covered by you know what we call these critical charts which again just some description open sets which
41:04
are described as critical loci and then you know not surprisingly there's also some kind of compatibility on overlaps this you know one way of thinking about this compatibility
41:27
is what's known as the notion of what's called a d critical scheme which is basically some notion that was developed by uh 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
41:49
so on each of these critical charts I have a this perverse sheath that I've constructed on on u if the chart is given by this data of u v and f where u is the actual open set of m of
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x I've produced this object which is this vanishing cycles for essentially the constant sheath and you would like to glue these together but there's a problem with doing this and this
42:40
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 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 okay so the critical locus of
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s in this case is just v itself and then this kind of you know vanishing cycles object I'm going to produce is just uh it's just a constant sheath with again with this shift but there's another way of getting v as a critical locus which is that what I can do is
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I can take let's say uh I'll call l to some z mod 2 local system on v so assume v
43:41
has non-trivial local systems and then I can take the corresponding you know two torsion line bundle and then I can write down a function on the total space f tilde which is basically just fiberwise as you know t going to t squared so using the fact that
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the l squared is trivial and so it's not hard to see that if I take the critical locus of this
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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
44:43
associated to l so depending on how I described um you know v is a critical locus I either got a constant sheath or I got this kind of z mod 2 local system and so this is going to cause
45:04
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 there's you need a little bit of extra structure
45:21
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
45:46
and what we call the 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
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is what's called an orientation it's just a choice of square root of this line bundle
46:23
and it turns out once I pick this choice globally then uh then I can kind of solve this problem about doing for the theorem uh which is due to let's see braves uh bussi duPont joist and centroid because if I take my moduli space and choose an orientation choose the square root
46:56
then you can kind of solve this glint problem which we'll call this dt sheath which is now
47:15
some perverse sheath that lives globally on your moduli space and this choice this choice really
47:27
makes a difference so first of all there's a question about like why does why does the square root even exist and then if you know choices will differ by like two torsion line bundles on m how does that affect the answer and then the first thing the point is it really does affect
47:40
the answer so different choices 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 cohomology they'll totally just change to what the cohomology is what is this orientation in the kind
48:07
of the critical case so for instance if m is just globally a critical locus then uh well so
48:21
the obstruction theory in this case you might remember it looks like um this two-term complex coming from the hessian of f if I take the determinant of this the canonical bundle of v
48:44
restricted to m squared and so then the kind of the natural orientation if you have a global critical locus is just to take k k v restricted to m is kind of the natural choice but of course
49:09
depends on my description as a as a global critical locus but for instance in the examples like the Hilbert scheme or something where I gave you some kind of nice function that cuts it out then that in particular gives me also a nice orientation that cuts it out another example
49:30
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 Michael Avia threefold is the total space of the canonical bundle on the surface so again this is the kind of geometry
49:45
that shows up in showed up in Richard's talks last week on the Waffenwitten theory in that case if I consider for instance you know sheaves on x that are kind of proper over s
50:01
I have a map where I kind of take this sheave on x and I just take its push forward to s and I get a point in the moduli stack of sheaves on s so e maps the high lower star e
50:23
and let's just assume I'm in a situation where this push forward is you know you know coherent interest then you know there exists a natural orientation on m of x that you can think of just by looking at what the the how to think about the
50:58
obstruction theory on m on m of x so you this arham is what you know basically the determinant
51:09
of this is going to calculate my obstructive my uh virtual canonical bundle and I can trap this
51:20
in terms of two complexes that are you know built out of that so on one hand I can this is built out of arham f f and then the kind of other piece as the triangle is arham of f f tensor ks with a shift here and so then well if I combine ser duality with just taking you know
51:48
determinant of this is going to be the product of the terms of these two things I combined that with ser duality I get that the canonical bundle on m of x is naturally the pullback of the virtual the virtual canonical bundle on m of s
52:09
squared 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 this is
52:21
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 if I had more time I would kind of explain the proof of this um due to nekrasov and okunkov which is that orientations always exist and their argument
52:42
is in kind of very great generality anytime you have some kind of you know moduli of objects on a kalabi out in a kalabi 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 years ago of dominic joyce and marcus
53:05
upmeyer is that okay so I'm not actually sure about with a precise hypothesis so maybe maybe it's important for them that it's a projected kalabi out threefold so maybe maybe this might not be exactly correct they kind of provide that kind of a canonical
53:21
choice and in particular what's great about what they've done is that it's 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 their proof is heavily gauge theoretic I have to confess I
53:52
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 it's like they what they do is
54:00
they take the kalabi at threefold and they cross it with a circle and then they kind of do some analysis about the corresponding real seven manifold and then some kind of special holonomy for those things so at least from my point of view I don't really under I don't have a good feel for what's going on in this theorem in this second theorem
54:24
all right 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 need is that you know we started off with x we took m of x we maybe have to choose the square root you know
54:46
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 at some perverse that lives on my moduli space and then if I want to for instance kind of get a cohomology associated to my
55:08
moduli space I can just take the uh you know the hyper cohomology of of this perverse and so what properties does this satisfy well first I mean the first one is the thing that
55:31
I said at the beginning which is that if I take the stock wise Euler characteristic I get exactly the barren function that's really cooked into how we kind of pick this choice of
55:41
feed of glue if I instead take you know the global cohomology and I take the other characteristic there well this is the integral of the barren function which in particular is my
56:05
virtual invariant what else well we get some properties that just come from the fact that
56:20
you know vanishing cycles has some nice properties so for instance this dt sheath is closed undertaking duels verdier duels which concretely means that if I take the kind of cohomology it's naturally dual to the kind of compactly supported cohomology
56:43
so for instance if m is proper you basically get quancary duality what else I mean this won't be relevant for me but this is extremely
57:05
interesting and useful in general is that everything here can be kind of you know decorated with hodge structures so the kind of formal way of doing it is that you know this phi lifts to this category of what are called the mixed hodge modules and but concretely what
57:29
that means in practice is that then when I take global sections this carries a natural mixed hodge structure and this is extremely useful for calculations and so finally let me just
57:51
state a non-property which is that you know one of the things I mentioned in class
58:00
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 out of the
58:23
baron function that's something kind of very non-obvious because the actual family of 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 m is proper then by virtue of this kind of you know index formula
58:44
from yesterday this weighted Euler characteristic is a deformation invariant as x varies but that's
59:03
not going to be true for this kind of combological thing you can you know a simple example to see things can go wrong is where you consider the following family these aren't really moduli
59:27
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 I'm just going to take a Riemann surface m
59:41
a v c and then I'm going to just take omega to be a one form on it one more one form and then I'll define a family of spaces which is just the zero locus of
01:00:02
T times omega. So when T is non-zero, I just get a bunch of points corresponding to the zeros of my, of omega.
01:00:23
And in that case, you can just see it. Well, the, you know, everything is isolated. So the, the, the cohomology is all supported in degree zero. And you just get, you know, here to the two G minus two degree zero,
01:00:48
but T equals zero. I get the entire curve. The vanishing cohomology in that case, the C in that case is just the constant chief up to a shift. And then now when I take the cohomology,
01:01:01
it lives in different degrees. You know, it gives a 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 threefold, it's a, not at all, you know,
01:01:22
it doesn't really make sense to talk about in general, these kinds of, you know, these kinds of cohomological invariants for a modulase-based sheaves on a quintic threefold. You have to tell me which quintic threefold. Like in general, there's no reason to expect the answer to be insensitive to it. All right, let me stop here.
01:01:50
We have questions. We have one from the chat. I can try asking it myself, but I'm not sure I agree with it.
01:02:01
So the question is, if the, I actually think that the question is, if the orientation data on the, on the stack of compactly supported complexes is, if it comes from the global one of choice up minor.
01:02:22
Oh yeah, that's a good question. So this is like, I mean, so I, I actually don't know if, can I see this Q&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?
01:02:40
I mean, he would know better than me, but, let's see, make sure I try to understand this question. I was wondering, oh, I see, through the canonical Hilbert scheme orientation data come from the global one of Joyce up Meyer. I have no idea. I mean, again, I think he's in a better position
01:03:01
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, like 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
01:03:22
or something like that. But then you're going to 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, but it's not clear to me that their construction will agree with this kind of, what I was calling the canonical one,
01:03:40
which is maybe not the right choice of word. But so, you know, so I'll pre-orient a 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 modularized spaces. And then it's not clear to me that those two orientations agree.
01:04:01
But again, I'm not really the person to ask there. One more question in the Q&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 the choice of compactification.
01:04:23
Okay. So the question is an answer, I guess. Yeah, okay. All right, any other questions? If not, then let's thank Ramesh again, thank you.
01:04:40
Thank you.