 All right so what I'd like to do today is just kind of pick up where I left off yesterday because to talk a little bit more about this conjecture and in particular what the evidence we have is and like I said there's kind of a bunch of examples and there's kind of one kind of really general piece of evidence in favor of this conjecture. So let's just remind ourselves of where I kind of left things yesterday. So we started off from Kolabi'a threefold and then I took some specific moduli space. I took a moduli of one-dimensional sheaves with some stability condition and the idea is that the support of the sheath is this class beta that I singled out. So beta is an element of H2 and I want to kind of try to do some kind of like curve counting theory in this class beta and then I fix height one and then there's some stability condition which I won't go into again and the point is that this has a map of this chow variety which just parameterizes one cycle. It's not schemes just curves with multiplicities and then I kind of you know made this definition which was I basically I have this kind of dt sheaf upstairs that we kind of constructed on Tuesday. I push it forward and then I take its perverse comology sheaves and the claim is that the Euler characteristic of these perverse comology sheaves if I kind of repackage as a linear combination of the comology of tori this is the definition of my of these kind of integer invariance and then the conjecture with Tota is that you know this is in some sense kind of recovering what you get from these other approaches to curve counting for instance the one that I've been using in these in these lecture courses is approach via stable pairs invariance and then I mean actually I made a mistake so when I spoke about this yesterday I kind of assumed Rahul had already started talking about the relationship with Gromov Witten theory but I think maybe he did it today and not yesterday but then the idea is that this also these these theories also equivalent to the Gromov Witten theory I should be equivalent to the Gromov Witten theory of the three cold and so these numbers if you believe everything I'm saying would kind of control the Gromov Witten theory too and so I mean so when I say it determines that you know there was some kind of explicit formula where you have to kind of basically take the log of this kind of generating function and I didn't say this explicitly last time but you know the reason this log is there is that the stable pairs theory the support of the sheath doesn't have to be connected you can have kind of disconnected support and that that's no problem stability on the other hand in the sheath theory will basically will force the support of this sheath to be connected so going from connected to disconnected you expect some kind of logarithm to show up anyways one question that I kind of haven't discussed yet and I really I will say for tomorrow is you know what's special about taking chi equals one there in my definition on a modular space why not some other value of chi and in fact it shouldn't matter actually what we expect actually anything I'll discuss this tomorrow but basically it shouldn't matter but the fact that it doesn't matter is kind of a non-obvious constraint and then you know that's kind of something that you know you can kind of explore separately so what I'd like to talk about today is to kind of just talk about the evidence for this conjecture so in general I said this last time you know I really think of this conjecture as kind of you know some kind of incomplete onsots for trying to understand these numbers in the sense that you know there are cases where we can kind of study this very well and you know then we have kind of good evidence but for a kind of more complicated geometries it's still hard to get our hands around so it's not clear to me that this is going to be the kind of final state of affairs in particular this question about whether these numbers as I've defined them are definitely our deformation invariant is not at all clear to me and that's obviously a prerequisite because these stable pairs numbers or the drum width numbers are deformation invariant but let me give one example so this is an example that you can open I like a lot which kind of illustrates um you know some of the some of the subtleties in this definition kind of show up already and so so before I kind of talk about more general evidence I want to do this one kind of example is just state what the numbers are in this case so this is what's called the Enriquez Calabi out this is a this is a nice geometry that you know back in the day Rahul and I spent some time um playing around with the invariance of this geometry from the drum of wooden perspective and so um the starting point here is going to be uh an Enriquez surface s which is a surface that you get by taking a k3 and acting quotienting out by a uh fixed point free involution you can find k3 surfaces that have such involutions and then the quotient you get is an Enriquez surface and then the corresponding Calabi out three fold that I want to take is the following I'm going to take e is going to be an elliptic curve I'm going to take my k3 the pre-quotient s tilde cross e and I'm going to mod out by an involution that acts by a fixed point free involution on s tilde and acts by you know plus or you know acts by sending you know x to negative x on the elliptic curve and this is a Calabi out three fold this is a nice example and the way you can think about it is that if I take x and I just kind of project onto the e factor I get them at e modulo plus or minus one which is just p1 and so this generically has fibers that are this k3 surface s tilde but then it also has four you know double s fibers this is a k3 vibration but the singular fibers are kind of non reduced now s itself the Enriquez surface just here's just some some facts about Enriquez surface Enriquez surfaces themselves are fibered over p1 and these are elliptically fiber and s itself when I think of it as an elliptic vibration also has some doubled fibers so I have you know where c is you know again a genus one curve and so what I'd like to do is I'd like to take the example where I take my curve class is just one of these you know half fibers I take this curve class in my Enriquez surface and then I push it forward into x you can think of this as the class c0 it's going to come from taking um c and the Enriquez and then kind of projects to zero on this projection down to p1 and so there are some cases that you can look at oh whoops yeah yeah sorry um all right so far everything I said is correct so the first case we can look at so then we want to figure out what are you know what are these invariants in this setting these variants as defined in this crazy way um so the the first case is when I take s to be a generic Enriquez Enriquez surface is common module common families and the behavior of this structure they're talking about changes depending on which one I take so the first case is when s is generic this double fiber if I look at the underlying reduced fiber is smooth in that case uh the corresponding moduli space um if I look at first the chow variety there are only four um points in the chow variety so in this case the chow variety in this class beta there's just four points corresponding to these four double s fibers so we just get chow variety looks like this each of these points corresponds to one of these you know smooth genus one purse in the corresponding Enriquez fiber and if I look at the moduli of one-dimensional she's I just get a copy of that genus one curve in the moduli space so we're kind of in this kind of ideal situation and so this case is really easy to do the calculation for uh if you kind of spit it out you get that ng beta is you know four if genus is one and then zero otherwise and particularly this matches with the calculation that you get if you do the kind of Grim of Witten or PT calculation but if s is not generic uh then the behavior becomes more complicated so for instance the simplest case is if this curve right so this curve c again is this genus one curve which is you know with multiple c2 is you know one of these doubled fibers of my elliptic vibration and so in general what this you know in the non-generic case this c is still a genus one curve but it can be nodal or in general it can be you know some kind of cycle of rational curves that looks like this there are different types that you get depending on which Enriquez you pick so let me pick this one this is what's called the i2 Enriquez and so in that case the chow variety is no longer uh just four points anymore the way to think about it is that this colabi out three fold well before i kind of admit the projection onto p1 but it also admits a projection onto my Enriquez and well okay this the way to think about it is that you know this whatever this this vibration is when i restrict it to one of these two components it becomes trivial and so each of the components of this um copy of c inside of x can kind of move in families and they may or may not kind of you know link up again so the chow variety it turns out ends up being a copy of e cross e and then what is the moduli space so the the way to think about this e cross e is that each of these p1s can kind of move in a family parameterized by e and they can move independently and so inside of e cross e well first of all you could ask um yeah so first of all which of these uh cycles inside of the chow variety uh are actually connected because the support of my sheaf will have to be connected and it turns out what it looks like is it looks like um so i'm going to just do this case where c is i'm just doing this case where c is uh two theoretical components it looks like the diagonal you know the locus of points you know x comma x and the kind of anti diagonal where kind of you know inside of e cross e and so these intersect at four points and then m beta on top of it so what the what the one cycle looks like at these intersection points the one cycle is a copy of c and so these are correspond to those four double enriquez fibers if i'm at some other point here that corresponds to um a partial normalization of c where i kind of separate out one or the other note and so if i look at the modular space of one-dimensional sheaves with those possible supports um i get something that looks like this let me try to i'll draw the picture this is what the modular space looks like so there are two copies of e and then over each of these four points i have this configuration here and the singularity in each of these kind of singular points here looks like three axes in you know three space meetings so this looks like the critical locus of x y z everywhere else it's smooth and then if i did one of these more complicated uh configuration so you know another kind of non-generic enriquez i would get kind of a similar picture except it'd be like e to the n you can kind of draw it would be kind of more complicated but you can still do the calculation i'll explain kind of today how to do kind of do some of these kinds of calculations you can do this calculation and what's interesting is that so the chow variety has kind of changed it's jumped in dimension from zero-dimensional to two-dimensional in this case and the behavior of the image and the behavior of the modular space has also kind of changed a lot but when you do the calculation you in fact do get the same answer as in the generic case you still get n g beta four g is equal to one zero g is not equal to one and so we do have some evidence that even kind of really crazy you know relatively kind of you know uncontrolled uh deformations that we still have some kind of invariance of the the calculation but it's it's it's not something that we have any kind of systematic understanding of okay um let me just say one more thing which will be relevant for a one-on-one talk about today so the way i formulated the conjecture was kind of global on the modular space i took the invariant i took the modular space and i just produced some numbers but because everything is defined using constructible sheaves or constructible functions you can kind of do a local version instead of working with the entire modular space i can fix a point i could fix a one cycle on x take the corresponding point on the chow variety and then just take the contributions of this one cycle gamma to both this kind of you know goku kumar of off end variants as i've defined them and to this kind of and to the stable pair invariance so for instance the way you do that on the on the kind of goku kumar of off a side is instead of taking the Euler characteristic of this perverse homology sheet i could just take this perverse homology sheaf and then restrict it to the point that i'm interested in on the chow variety and then do everything that i was doing before this kind of local invariant similarly on the kind of stable pair side if i wanted to define you know i could just take my pair's modular space which again also maps to the chow variety and i could just take the um you know the preimage of this point the fiber of this point that i'm interested in the chow variety and then just integrate the baron function over that and so then again you can kind of you know ask for some version a local version now of this relationship between these two kinds of sets of invariance before i had this exponential what that would correspond to in this case is that you know if i'm interested in the pair's invariance for a given gamma i should look at not just gamma but also all the kind of effective sub cycles of gamma but the cleanest version happens if gamma happens to be irreducible so some irreducible curve with multiple c1 and then i just get something that looks like um something that looks a lot like my original mcdonald formula in the smooth case and so really it's really this local one that's kind of you know this local one implies the global one i just have to kind of integrate over the chow variety and this local one then you can kind of try to understand cycle by cycle so what is the evidence for this conjecture so i what i'll do is today is i'll kind of focus just on the case when gamma is an irreducible cycle and then tomorrow i'll talk about the evidence which is not as strong for kind of more complicated cycles you know reducible non-reduced and so on and so the main one i want to talk about which is the kind of you know the main general theorem we have so far is in the case of a local surface so let's say s is some you know smooth projective surface for reason kind of annoying technical reasons we kind of impose this basically a simple connectivist condition and then the colabi out threefold that i'm going to take is just the total space of the canonical bundle so some non-compact colabi out threefold which has a projection to s and so then the theorem is i suppose we have a one cycle on x such that the projection to s is uh irreducible then the relationship between this kind of gopokumarov often variants as we've defined it and the stable pairs invariant that holds for this cycle so what is the picture here the picture here is okay so here is my surface s and then i put it in you know think of it as the base of this non-compact colabi out threefold by taking the total space of a line bundle on it and then the cycle i have kind of lives up here so you know maybe and the assumption is that the projection to down to s you know i projected down to s and i need to get an integral curve when i project it down so so this is a stronger hypothesis than just saying that gamma is irreducible it all i also need its projection down to s not to have any kind of you know not to be you know have positive degree onto its image on the other hand um the singularities so this is gamma and downstairs is pi lower star of gamma the singularities of gamma can be quite nasty so for instance this projection is a curve inside of a surface so pi lower star of gamma is a locally planar singularities in particular it's gorenstein lci blah blah blah but gamma itself can be quite nasty and in fact i mean what i think is true is that pretty much you know any you know any kind of space curve singularity so embedding dimension three say some kind of formal singularity type like that can be approximated arbitrarily well by another formal curve singularity that appears in this theorem and so in particular you're going to be able to get you know non-gorenstein examples non-lci examples and so on and so really this i mean this gives some kind of evidence for a kind of extremely nasty singularity some version of this theorem being true and so again you know because the cycle is irreducible when i say the groom of wooden sorry the gobu kumar vatha stable pairs conjecture it's really in the irreducible case so so because we're in the irreducible case this formulation of this conjecture is kind of simpler to write down so let me just write it again if we're summing i'm integrating my baron function over the locus of stable pairs that kind of have support gamma and i'm saying it's basically equal to you know the Euler characteristic of this kind of stock of these perverse homology sheets up to some normalization so this this i interpret is again this kind of version for very singular curves of of this mcdonald's formula and i mean the thing i'll point out is that the the fiber that uh of the the fiber over this point in this modular space and beta is um it's some analog of like the compactified jacobian of this jacobian of this singular curve so this is really you know taking rank one torsion free sheaves like sephirically the same as taking rank one torsion free sheaves on this reduced curve gamma but because the singularity is just so bad so bad because we're in the non-born skin setting this we don't really have any control over this kind of space so for instance there are examples very simple examples where you can get the space to be a reducible i think non-equid dimensional and so it's really important that we're taking this kind of the homology that we're taking is the homology of this um of this dt sheet as opposed to like the constant sheet so i want to say something really for say something about the proof of this theorem so again the statement of the theorem is that um for these kinds of curves this correspondence is going to be true independent of the singularity how bad the singularities are and the reason why i want to say something about the proof is that actually the philosophy of the proof is actually something that shows up a lot in this kind of flavor of donelson-thomas theory and it's actually it shows up in other kind of parts of algebraic geometry too um so let me just say something about the philosophy of the proof maybe philosophy is too kind of pretentious of where let me just say strategy which is that you know you first kind of prove the theorem in some kind of very nice setting some kind of ideal setting which in our case is going to be which in our case is going to be a situation where you know all this kind of virtual structure this bearing function this sheath etc are all trivial in our case what we're going to do is we're going to prove the theorem i'll tell you what that very nice setting is in a second but the way we'll prove it in this setting is to really use some kind of properties of perverse sheath so this has been used this kind of notion of what's called perverse continuation which is a piece of lingo introduced by mgo and then we're going to kind of bootstrap from the very nice setting to the general case and the idea here is we're here we're really just going to use some kind of nice functorial properties and vanishing cycles and so here you know that what the kind of key idea is that you know even if we have a very singular very badly behaved modular space since we're writing it as sitting inside of we can write it inside of a very nice modular space as a critical lobes and that kind of will allow us to kind of transfer the nice results to the ugly situation using this kind of the fact that vanishing cycles is in fact a functor so okay so i'll explain exactly now what i mean by this kind of two-step strategy but let me just mention that you know the strategy this idea kind of shows up in other context so so other examples this idea kind of shows up for instance this is where you know i learned it from is in this kind of work of davison and mineheart where they were really working about quivers with potential so this is in this very much in the setting of ranike's lectures and they were able to prove theorems about the kind of donalds and thomas theory of quivers with potential by reducing to the case of quivers without potential and there there was already some work of mineheart and ranike to go from step one to step two then they just you know hit the theorems and of of mineheart and ranike with this kind of function proves the theorems in their setting another example we heard of i mean if you were at brooch nigg's talk earlier this week this kind of strategy shows up it can be very useful when you study kind of higgs bundles and i'll maybe talk about that tomorrow a little bit if i have some time so brooch and he used it and this also showed up in some work i did with juliang shen where we were kind of interested in some conjectures by higgs bundles and the idea was that again you find some nice setting where the theorem is kind of easy and then you hit it with this kind of functor to kind of prove it in the case where the theorem is less easy and then actually kind of in a in the non-comological setting this kind of approach has been used for instance also in the derived category so for instance um this work of halper and leisner he has some results on like derived categories of moduli of k3s and you know the kind of key structure result that lets him um you know prove his result he he applies this kind of approach where instead of vanishing cycles he's using this kind of notion of a singularity category or category of matrix after addition but the same kind of principle where you have some kind of dream situation where everything's kind of easy and then you kind of hit it with this construction to kind of prove things for more complicated derived category so let's see how to use this strategy in the case at hand so again remember the theorem that i want to kind of explain the proof of is this one here kind of irreducible one cycles on these local surfaces and so what are the two steps so again the first step is to find a really nice situation where there's no kind of virtual structure going on where you can prove the theorem and for us what that will be is you're going to consider a a versal family of integral with locally planar singularities so locally planar means that the all of these the fibers of this map they're all can locally be embedded inside of a to so the embedding dimension is always two for all the singular points of the curves in question versal means that um if i pick a singular curve and i look at a take a singular point on it the you know universal deformation of that singularity there's at least locally on b there's going to be a smooth map from a neighborhood of this point on b to this universal deformation of the singularity so another way of saying that is that you know if i give you a singular fiber and i look at one of the singularities all the kind of you know partial all the kind of partial smoothings of it will also show up in the in the nearby fibers of this family in particular so this is nice in the sense that it's kind of very far from our setting the total space of this family is going to be smooth and in fact all the modular spaces that we would want to associate to this family are also going to be smooth so for instance on the modular space of one-dimensional sheaves i have this kind of object here which is just kind of is the family of if you like compacts pyjagovian so this for if i fix a point on b and i look at the corresponding curve i'm taking rank one torsion free sheaves on the fibers the fibers of this might be singular but the total space is smooth on the stable pair side what i can do is i can take the relative Hilbert scheme of points and again the fibers of this might be singular because the fibers of my original family of curves are singular but the total space is smooth and again that's something that's implied by this condition on the deformations of the singularities and so the theorem in this case which is actually refinement of what we need was something that was proven by uh myself and joey yun and then also um miglerini and shende and it basically says that on the left hand side i can take the um constant sheet on the smoothing and push it forward let's call this you know pyh kind of the Hilbert scheme and here i'll call this pyj because it's like the compact pyjagovian so just think of this as some kind of direct sum you know index by n of all these you know push forwards that this is the same as taking on the right hand side i'm going to just take the the single space i'm going to take its perverse homology sheets by q okay so what does this mean on the right what i really mean is that you know this thing's in the denominator i can kind of expand out as a generating series i can do both of them and then you know for any given you know power of q there are only going to be finally many terms like this that show up and so concretely it's saying that it's saying that the homology on the left hand side of one of these singular curves and i take its Hilbert scheme of points and i sum over n is basically determined on the right hand side by taking the kind of perverse filtration of the co-homology of the compact pyjagovian and again this thing has this this theorem again has this kind of strange feature that on the left hand side the number of points is indexed by q on the right hand side the kind of perverse degree is indexed by q because this is the kind of thing that's kind of predicted from this kind of gopokarvafa stable pairs correspondence and this is kind of the first case where one can kind of prove it and it's really important that you're taking the perverse filtration here that's kind of that'll be kind of important in the proof and i ask the question about this formula yeah so is there a natural way to hold a filtration on the vector spaces in the right hand side so that the associated gradient is the left hand side and some some symmetric algebra let's see so i guess i'm wondering if one can categorify this statement oh yes yes yes okay right so uh categorify it let's see so i the statement that maybe uh is relevant here and maybe maybe this is what you're asking but uh after you know these papers uh yorgan uh renamo kind of wrote a paper where what he basically did was um he constructed an algebra action of you know he basically constructed raising and lowering operators on on these spaces uh and then if you look at kind of the the primitive elements with respect to that you know vial algebra action that basically picks out the comology of the compactified Jacobian and so a statement like this is basically saying that you know these are the generators and then this denominator is coming by kind of freely acting on it with the kind of raising operators in your algebra is that kind of what you were asking about yeah that's perfect thank you so you know this is kind of a concrete theorem in the sense that you know these compactified Jacobians are for these you know nice planar curves with planar singularities are pretty explicit and these Hilbert schemes are also pretty explicit um and what's interesting about it from my point of view is that you know it forces you to really see these perverse filtrations on this space of compactified Jacobians so they kind of show up in a pretty from a pretty natural starting point you know the left hand side it's something you could really try to calculate and then immediately let you calculate the perverse filtration on the right hand side even though the perverse filtration itself to try to define it from first principles is a little bit mysterious so the proof what is the idea of the proof um so again the kind of you know buzz phrase that I like to use is this idea of what's called perverse continuation if I give you let's say I have just any proper map from x to y where x is smooth and I take the push forward of let's say the constant sheet I can study it I can study its perverse homology and one of you know the first things interesting things that happens in this situation is that we actually have something stronger than just looking at the perverse homology sheaves uh the perverse homology sheaves determine this complex in the sense that um we have a decomposition theorem that tells me that this push forward is actually a direct sum of the perverse homology sheaves shifted so that they live in the right degree now each of these perverse sheets I could look at its kind of you know simple constituents and then we say that f has full support if each of these pieces are all supported on y and when I say support I mean really kind of strictly supported they're not pushed forward from something lower dimensional this is kind of a strong condition on this um on this map so for instance if I like take y and I just blew it up at a point that would not satisfy this property but if you are in this situation then the perverse homology sheaves pk are in you know a precise sense determined by the restriction to some kind of you know to large open subsets of you if you shrink you enough you can assume as a local system or maybe a shifted local system and determine in really some kind of algorithmic sense this is what's called this notion of the icy extension and so the way you can use that is that imagine you're in a situation where you have two perverse sheaves which are you know fully supported which have full support in the sense in the sense that they're determined algorithmically by their restriction to this open set then if you have uh let's say an isomorphism of this generic restriction then you can promote it to an isomorphism over the entire base even though you don't know I mean when I said it's algorithmically determined that's kind of a lie it's not like you could like well I don't know I think it would be hard to do it by hand and actually see what the fibers are over the complement of you so even though you know the fiber of p and p prime over these kind of you know the boundary could be quite complicated you still know kind of just on on general grounds that they're the same and so the way to prove this theorem in this kind of nice case is by one matching the kind of Hilbert scheme side and the kind of compactified Jacobian side over some large open set and if the open set where the kind of curves are smooth the family by assumption was versatile so on an open uh over an open subset all the fibers will in fact be smooth and then two showing that both sides have full support because once I do that once I show once I've matched them on this open set then kind of automatically by this principle I match them over the entire base so um so for two how do you show something like this has full support well this is hard this requires a theorem so this is actually essentially this is something of the techniques for doing this were developed by Ngo and his work on the fundamental lemma that's on the compactified Jacobian side the Hilbert scheme side you basically just can reduce to that case and this is you know this is I think is hard proofed by a passage to care for 6p and it's really some kind of property of the fact that generically these are kind of you know abelian vibrations some has something to do with the fact that these fibers generically look like abelian varieties what about the first part once you do that though then you reduce to a question about smooth curves and this is essentially this is basically just the comological version of mcdonald's theorem again now these Hilbert schemes are just symmetric powers of a smooth curve you know part of what he did he didn't really just prove it there about oil characters he proved this theorem much more generally which is that you can kind of calculate the comology of the symmetric power of a smooth curve in terms of the comology of the compactified Jacobian so this is like I said so this is the kind of first step this is the nice case which is where your family of curves happens to be just extremely well behaved all the modernized spaces are smooth there's no sheaf of vanishing cycles anywhere here and then what lets you do it is that even the total space is smooth but the fibers are singular but your situation is so nice that you also have this kind of continuation property where you can even just reduce it to studying a question about smooth curves and what about step two so step two is where you kind of go to the general case now I have a curve sitting inside the total space as is just some surface it doesn't have to be phono or anything like that and the only kind of really assumption is that it's the projection of this is still a kind of integral curve inside of s and so then the claim is that we have the following diagram and this diagram will allow us to reduce step two to step one let's see how do I want to write this so first I'm going to write down t mapping to a one let's see so there exists so for further exists a versatile family of deformations of c c again is this locally planar curve which is the image of gamma such that okay so and a function on the base so t amounts to a one some function g and then I can look at the picture from step one look at the families compacted by Jacobians and I can look at the family of relative Hilbert schemes that I called pi h i j before and I could take these compositions so I'll call you know f to the n f j such that if I look at the pair space on x p t space this is the critical locus of this composition and m beta x is the critical locus maybe early maybe the precise statement is that the pair space kind of in a neighborhood in a neighborhood of the fiber of this in the chauvin in other words I can kind of construct critical charts that are built out of the kind of nice situation in step one and so the the picture you end up with is that here's kind of m beta of x the chauvriety of x at least after I maybe shrink it sits inside here and then similarly for the kind of pairs space now constructing this diagram takes a certain amount of effort and this is kind of you know what we did in our paper but once you have it then you're in good shape you see we know two things about this vanishing cycles construction which is that when I take vanishing cycles of a function it commutes with proper push forward and second it preserves because it's a sense per verse she's to per verse she's it preserves taking perverse homology and so what I can do is I'm interested in for instance taking the constant sheaf here and taking its vanishing cycles and then pushing it forward let's say to the chauvriety or maybe all the way to t but instead of being doing it that way I can first push forward to t and then take vanishing cycles with respect to this function g so for instance on the stable pair side and on the on the one dimensional sheaf side I'm interested in maybe this object and maybe pushing this forward but this is the same as taking phi sub g of and then similarly for this relative Hilbert scheme of points and so then to prove the main theorem you just take the main result of the first step which is some equality of sheaves on t and then hit it with this functor phi sub g and so here it's important in particular that you know phi sub g isn't just a sheaf it's a it's an operation that you can stick in any kind of complex of sheaves and produce another one that's kind of what we're using here I have some isomorphism on t I apply it with this functor and I'll get an isomorphism on whatever that supports um all right um I'm almost out of time I was going to do try to do an example so you know in this generally you know you kind of it's kind of an abstract argument but you can actually use this to do some calculation so um this principle uh so let me write down the example because it's kind of fun to just to see how this all works out explicitly um which is that you could take a curve in a surface where c is like some rational genus one curve so maybe it has a node or maybe it's a cusp and then you can kind of rig the surface so that the normal bundle is you know degree zero but nontrivial and what that means is that this curve is rigid on the surface so chow data s it's just a point but if I now take the total space of the canonical bundle uh the chow variety the the curve can deform off the surface um so the picture you get in the nodal case is like this and then it can deform off the surface in a one-dimensional family and the modular space of one-dimensional sheaves in this case looks like the kind of thing that we got in that Enrique's example you have when the curve is on the surface itself then you have a genus one curve and so the compactified Jacobian in that case is just a copy of that curve itself when the curve moves off the surface and drops in genus from one to zero then you just get a single point forever so you get something that looks like this and again the singularity type here looks like the critical locus of xyz and the map to the chow variety is i just kind of contract this nodal curve to a point and i get a map to just align and so now you can actually use the procedure to actually compute um the go the kind of the goko comarov often variants in this case but in fact again this procedure works much more generally so so the case i was kind of highlighting was really and computationally you know i've just been doing nodal and customizable examples but the strategy really works much more generally so you you know the situation that i like for instance is you could do something where on the surface i have like you know just a bunch of branches coming together but then in the total space of the three fold they can still come together but now maybe you get a embedding dimension three so you get something that's really far from um gorenstein then it's a little harder to do this calculation explicitly i think but i think you're gonna still do it if you all right since i am out of time uh let me stop here so what i'll do tomorrow then is i'll talk about the evidence we have this everything here was about an irreducible one cycle and so you can ask well what do we know when you allow multiplicities and reducibility and so on like that so i'll talk about that and then i'll kind of talk about some kind of other directions that you can try to investigate all right let me stop here thank you this picture where you can you realize both the Hilbert scheme and the modular spaces is pretty close in one generality on gamma and c can you state this oh yeah right so actually we right now you know in our paper we just did it for um in our paper we just did it for the integral case i think it's probably fine but we didn't actually do this i mean in fact in general i think this whole strategy let's say the curve was reduced reducible but still reduced then their analog of step one is already in the books and i think there would be some analog of step two as well but we never worked it out very carefully so that's something that i think is doable once the curve downstairs has multiplicities then i think it becomes harder and then i'm not actually sure either both steps one and step two are still kind of mysterious so for instance forget about the surface for a second i mean forget about the three fold for a second just in the case in the locally planar case if my curve is non-reduced uh then we would expect this theorem to be modified in some way uh even in the reducible case this theorem has to get modified and then then that would kind of filter down and kind of this diagram in step two would i think also become more complicated but i i don't think there is any i i think it's it's not crazy to imagine that this for in the local surface case some version of this diagram might still work but i you're going to require more argument even in the reducible case we never wrote it down carefully and if you ask the final question for a curve in a three fold instead of a curve in a in a local surface yeah then i don't know that's for you in the general case really for a general three fold then i then it's harder to see what's going to happen um then you run into these questions about like so you know what basically what's going on here is that we you know we have this notion of a critical chart can you find a a description of the critical locus that is you know you have a you have a cover of your moduli space by these things that are you know critical loci function and what's going on in this step is that we found a critical chart that's big enough that it contains the entire fiber of the map to the chow variety and so the question is whether you can do that in general like if i give you just like a random curve and i you know possibly with some multiplicities and i give you all the sheaves that have that support is there a big enough critical chart that will contain all of those sheaves at the same time if there is then i think then i think some version of this argument would kind of you could then try to start using it but these theorems that show the existence of the critical charts don't really give you a sense of how big those charts are you know there's a risky open but do they contain this entire fiber and that's kind of really what's kind of going on in this step two that's very special and i honestly don't know the answer to that in general any other questions yeah yeah um so uh you show that this moduli space of sheaves on on x is a critical locus is it is it how do you compare that to the other critical locus description as like uh this that it's just a moduli space of sheaves on a three fold like is there some comparison is it the same oh yeah yeah yeah that's right yeah so this critical structure agrees with the one that you get like on just general abstract grams yeah that's right and that's important you actually have to you have to make some argument for that uh because you know we're using a very specific one here and you need to match it up and ultimately comes down to you know it'll come down to for instance you need to use the fact that uh you know there's this very specific calabi out three form on the total space of chaos and that has to be the one that you use in that abstract general nonsense approach too so at some point you need the fact that you pick the correct calabi out three form and that lets you show that you can match the the the critical structures as well any other questions let's take the chat well see how this is it this time i'll finish again