 Okay, so I'd like to thank the organizers for the invitation to speak at this wonderful workshop So yes today, I'll speak about Donaldson Thomas invariance of local elliptic surfaces. So I should say this is all joint work with Jim Bryan So I'll start with some general comments on Hilbert schemes on three-folds since not everyone in this audience Might be working with them on a daily basis So we'll start quite general So Let X be a smooth three-fold over complex numbers And we'll denote by beta the two cycle on X and Kai will be some integer Then the modular space will be considering in this talk the Hilbert scheme Curves on X in class beta and with Euler characteristic Chi. So what is this? spell it out so these are Closed sub schemes set max of dimension less than or equal to one Which I mean set is allowed to have zero-dimensional components as well as one-dimensional So I'm not insisting on X to have to be projective. So I'm sure So be fixing some invariance here. So we want the algebraic cycle determined by sets to be beta and The holomorphic Euler characteristic of the ring of functions to be this fixed number Chi So it's the space of all one-dimensional sub schemes in X with this data And the Hilbert scheme So since not everyone may be working with these very often. Let's treat an example. So example suppose The simplest three-fold X is just affine three space Then giving a closed sub scheme of X is the same as specifying the equations which define The sub scheme in other words specifying Z in X is the same thing as specifying the ideal of equations defining X defining Z So I'm using here global coordinates X Y and Z on affine space So here's a picture of a one-dimensional closed sub scheme So here the ideal we're having is the ideal generated by X Y squared Xe Yz and Z squared So what I'm drawing here is the weight spaces of the structure sheaf of Z in the character letters so this guy here has These four generators So there's one infinite leg extending in this direction another one here. So this one has a multiplicity two and Then here we see something special This curve here has an embedded point So what this means is if we look at the ideal of this point Z inside the ring of functions on Z Then the interesting thing is that this ideal is zero-dimensional Rest this object here is one-dimensional and if you take the co-kernel here You get exactly the structure sheaf of the curve which is drawn here. So this is the curve Z this is the curve C We see here that we have removed the embedded point So in mathematical terminology, we say that Z Is not co in Macaulay, which means It's structure sheaf is not pure one-dimensional So which means intuitively that Z may have embedded points or floating points But C on the other hand is co in Macaulay and We will refer to see as the maximal co in Macaulay sub curve of Z So C in Z is the maximal Sub curve with no embedded points or zero-dimensional components How you did how you determine this well in practice that could be quite hard in fact But we'll be mostly at some point in this talk be considering the ones which are fixed by the torus action like the ones I drew here and there it's a bit easier So yeah note incidentally that there is an obvious action of the three torus on this geometry Just a standard action and actually the examples I've given here are all fixed or invariant under this action So these are both Teen variant So I'm only giving you Teen variant examples here because they'll come back later Okay, so this is some general setup. So now let's specialize To the geometry we'll consider today. So today will momentarily go to a surface So S Over some base B is an elliptic surface where We assume two things we assume first of all that the surface has a section so with section B and S section will denote by B as well and We only assume the simplest type of singularities of the codirec classification. So with only rational nodal Singular fibers a little topological computation then shows that the number of these is Exactly the topological Euler characteristic of your surface and we'll assume there's at least one singular fiber so the picture of this guy is as follows We've got a surface here with a section The generic fiber is a smooth elliptic curve and then there's a finite number of these rational nodal fibers And this maps down to some base B. So this is the picture, but then we want to do three-fold theory So the X the three-fold X will be considering Is the simplest Calabi out three-fold you can construct out of this geometry? It's the total space of the canonical bundle over this surface Actually Calabi out won't play a role in any of what I'm going to do in this talk So so what we'll do is we'll study Hilbert schemes On here, so what is the curve class we look at? So the curve class is we always take one time the section and then a multiple of the fiber class So here we see the class of the section Here the class of the fiber so all the fibers have the same class So this is the class will be considering and what we'll do is we'll look at the simplest Invariant you can attach to To this Hilbert scheme will just take the topological Euler characteristic So we'll consider topological Euler characteristic of Hilbert schemes on this X In this class section plus a multiple of the fiber so this here is just ordinary Topological Euler characteristic and we'll group these all together in the generating series Summing both over multiples of the fiber D And over holomorphic Euler characteristic Using a variable P and Q, Q for the fiber And we'll call this generating series DT hat of X So this is the main object of study and we'll also study another DT hat fiber X Which is the same thing except that we take As a curve class not B plus a multiple of the section of the fiber but just only multiples of the fiber So the same generating function but only multiples of the fiber So the reason this is written as DT hat is because these are related to So-called Donaldson-Thomas invariance So if I remove the hat I'll actually be talking about actual Donaldson-Thomas invariance So most of the talk or almost the entire talk I'll talk about these naive DT invariance But how do you get the actual Donaldson-Thomas invariance? So you do the same so again this generating series But instead of the naive Euler characteristic you weigh by constructable function Known as parents constructable function So this is some constructable function which keeps track of the scheme structure of your Hilbert scheme And if you happen to include it you get the actual Donaldson-Thomas invariance But again I'll be talking mostly about these And then of course there's a fiber version of this as well So what is the main theorem I want to talk about? So maybe one more piece of notation For the theorem I'll use two modular forms I'll use datacons eta function and the Jacobi theta function So what is the main theorem? The main theorem is that this generating series DT of X Well let's look at the hat version After dividing out by the fiber part it's just given by Well after eliminating this front factor eta to the power minus the Euler characteristic of the surface And then theta minus the Euler characteristic of the base And the second part of the theorem which I won't talk about Is assuming a conjecture on the bearing function Conjecture star, if there's time I say something about it We can get a formula for the actual Donaldson-Thomas invariance So geometrically dividing by this fiber generating series means You take the connected Donaldson-Thomas invariance Which are only defined formally And it turns out if you include this bearing function You get the same formula up to some signs You get a sign in front times the same formula Where you have to make one replacement You have to replace P by minus P So in this talk I'll be talking about this first item only I should say a few things In some sense the interesting part perhaps of this theorem is not the formula itself But rather the way we derive this So what we do is I'm actually going to present to you a method of reducing this expression here Purely to an expression in terms of the topological vertex So to something purely combinatorial And then that combinatorial expression can be written in this form So this may be a bit surprising because the geometry here is not toric So you'll see how I'll nevertheless force in some toric geometry here So in the case of a K3 surface If you like this gives a new way of getting to the Katz-Klem-Vava formula Which was first derived in primitive classes by this K3 So the first proof in mathematics of the Katz-Klem-Vava formula Was given by Maulik Panderi-Pander Thomas The primitive case Note since I'm not taking a multiple of the section I'll only talk about the primitive case as well And then the general proof of KKV was given not so long ago in a monumental paper by Panderi-Panderi and Thomas So this is all curve classes So what's interesting is that both of these proofs make use of the so called Kavai-Yoshioka formula And we won't be using that So this will be purely a calculation with the topological vertex I should also mention another thing The methods I'll talk about today are also quite effective in other geometries, other geometries, other three-folds with an elliptic vibration. So there have already been applications to axis K3 times an elliptic curve by Jim Bryan, and axis the product of three elliptic curves by Brian, Oberdeek, Pander, Pander, and Yin. OK, so I'll talk now about how to get to this formula. So actually the only moves I'll be making are the most naive moves you can make with a topological Euler characteristic, namely the following. So the topological Euler characteristic E of a space or of a scheme, if you like, has the following properties, the cut-paste property, if you have a closed subscheme of another scheme, then the Euler characteristic of X is just, let's say the Euler characteristic of X removes Z, is just the Euler characteristic of X minus the Euler characteristic of Z. The second property I'll be using is if you have two schemes of varieties, Euler characteristic of the product is the product of the Euler characteristics. I'll also be using that if you happen to have a C star action on X, then the Euler characteristic is just the Euler characteristic of the fixed locus. And the final property is that if you have a bijective morphism, so a morphism which is bijective, so I'm not insisting on an isomorphism here, then their Euler characteristics are also the same. So in particular, if you have a scheme, it's Euler characteristic, it's just the Euler characteristic of the underlying reduced scheme, the underlying reduced variety. So this is all we need. So now let's go to the geometry we're considering. We're having X, the total space of the canonical bundle over this elliptic surface. So this has an obvious C star action, namely scaling of the fibers of the vector bundle, of this line bundle. So this action lifts to the modular space to the Hilbert scheme. So here I'll write the bullets by which I mean I actually group them all together. So the curve class is fixed, so I'm grouping them all together just for a briefity of notation. So we have this torus action. So what we do now is we take an element in here, some sub-scheme, and we assume its torus fixed. So remember we only have to calculate the Euler characteristic of the fixed locus by this property here. So let's see what these sub-schemes look like. So suppose we have a one-dimensional curve here, a one-dimensional sub-scheme which is C star fixed. So what can we say about it? So the first obvious thing is that the underlying reduced curve has to lie in the zero section of your vector bundle. Now comes the important one. So what can we say? So if we look at the ideal, which is defining Z, this ideal has a C star action, which decomposes into weight spaces. So we have a decomposition into weight spaces. And the summons of this decomposition have a specific form. They are themselves ideal sheaves of schemes on the surface, up to some global twists which you need for gluing. So there is a decomposition like this where these Z-i's here appearing at the different weight levels actually form a nesting. So we have a nesting here of one-dimensional schemes. And finally what we know is the curve class is the section plus a multiple of the fiber. So we know that if we take the sum of all the classes of these Z-i's, we have to get our curve class back. So in particular, this sum can only be finite. So this decomposition here only goes up to some level. So you should imagine this as a stack of pancakes which become smaller and smaller. So one, the bottom one, has to contain the next one, et cetera. So this is still quite general, so still quite hard, but it becomes easier if we look at Z without embedded points. So now we're going to look at C star fixed curves, which are cone Macaulay. So let's see the maximal cone Macaulay sub-curve of Z. So I'm removing all zero-dimensional components and all embedded points. So this C actually can be drawn in a very geometric picture. So I'm now going to draw a picture which will be the main picture. So imagine the zero-section S surface inside our three-fold. So here we have the section B. And what does such a C look like? Well, the underlying reduced curve has to be just one time the section and then a bunch of fibers, let's say fibers above points, so a bunch of smooth fibers, fx1 up to fxm, and then a bunch of singular fibers, fy1 to fyn. OK, this is the underlying reduced curve. So what does the actual cone Macaulay curve look like? What you do is because of this nesting property, in the cone Macaulay case, each of these here are just divisors on the surface. So the way you should imagine this is let's start with this fiber here. You thicken this fiber at weight level zero, some direction in the surface. Then at weight level one you do the same, et cetera. So what I'm doing here is I'm thickening this fiber here infinitesimally in both the surface direction and the fiber direction according to some partition, some partition lambda one. And then I thicken the whole curve according to this partition. And the same for all other fibers with possibly some other partitions here. These partitions are just the cross sections of what your cone Macaulay curve looks like. So this is the picture of a fixed cone Macaulay curve. The section only appears with multiplicity one. So here you should think of like a little box. So this is just not thickened at all. So here I'm putting the partition of size one. So the thickening is really, in terms of the ideal, if say locally this is defined by some x, you just take x to some power L and do that at each level in the surface. So these are all just infinitesimal thickenings. And these are all possible pictures of all maximal cone Macaulay subcurves. So the idea now is to just stratify the entire Hilbert scheme according to these pictures. So note, by the way, that the sizes of these partitions all have to sum up to the fixed curve class, which I just wiped out. So now the main claim is that if you look at, OK, so let's give some name to the stratum of all sub-schemes with underlying fixed cone Macaulay curve. So whatever I fixed there, I fixed a bunch of points above which we have smooth fibers, a bunch of points above which we have singular fibers. So this is at most the oil characteristic of the surface. And then I fixed a number of partitions. So I'll denote by sigma the stratum of all elements in the fixed locus of the Hilbert scheme where the underlying cone Macaulay curve is that picture, by maximal cone Macaulay curve determined by the points x, y and the partitions lambda and mu. So this is the stratum of all one-dimensional schemes where the underlying cone Macaulay curve is this picture. So now the main claim is that when we fix this data, the oil characteristic does not depend on the precise location of these points xi and yj. So this is perhaps not so surprising, but that's somehow the main claim. So the oil characteristic of this stratum, so I'm using here abbreviation. This bullet, remember, is I'm summing always over all holomorphic oil characteristics. So these oil characteristics only depend on the partitions, so how many there are of each type and what partitions they are. So this only depends on lambda and mu. So I'll call this quantity here since it doesn't depend on x and y. I'll just call it e of lambda and mu. And the form of variable we're having here is p. Yes? Yes, so this stratum is this stratum and with the dot I actually mean that I'm taking the union over all possible holomorphic oil characteristics, chi. So chi, I'm grouping them all together. So the curve glass is always fixed, but those I all group together. So what does this imply? Well, since our modular space is just the union over the stratum, the disjoint union, you already get some sort of solution. So you choose any number of smooth fibers, m, and then any number of singular fibers, n. Remember, you can only choose at most oil characteristics of the surface singular fibers. And then what do you get? So then you get a big sum over partitions lambda one up to lambda m, mu one up to mu n. So I'm summing here over all possible strata sigma that can occur. And then what do you have? So you just have q to the total size of these partitions. The total size of these partitions determines the multiple of the fiber times, so this is the number of ways in which we can choose m smooth fibers, so oil characteristic of the base minus the number of smooth of singular fibers. So this may be negative, so then I'm using the usual conventions of binomial coefficients. And the number of ways in which we can choose a singular fiber is just oil characteristic of the surface choose n. And then the final factor is this e here. So I've done nothing really here. Just stratify the modular space into these strata and sum their oil characteristics up. So the key point now is to prove the claim and find a formula for the oil characteristic of such a stratum. Any questions at this stage? Yes, which is in here? OK, so let's calculate this. OK, so I should keep this picture. OK, so we're now reduced to a stratum of the modular space where the underlying structure is located. And then we're looking at all possible ways of putting in embedded points and floating points, and we want the oil characteristic of that. So we are reduced, so fix such a stratum, sigma xy lambda mu. So in particular this fixes some cm curve c, determined by this data, so the picture you see here. So we now want to calculate the oil characteristic of this stratum. So the point now is that we're going to use a cover, an open cover of our three-fold. So take an open cover, which I'm going to describe, u alpha of our three-fold x. So what are the elements of this open cover? So first of all, so what we're going to do is we're going to take really interesting points around all the interesting points in this geometry, which means around all the singularities of the underlying reduced curve. So we take a small ball everywhere where the fiber hits the section, and we take a small ball around the nodes, so small open balls, the singularities of the underlying reduced curve. So actually we're doing algebraic geometry here, so taking small balls is perhaps not the right thing. So really what I'm doing here is for the algebraic geometers I'm taking a so-called Fpqc cover, Fidelman plat, a quasi-compact. And so what it means here, instead of open balls, really what I'm taking is I'm just looking at the local ring at those points, and I'm taking their completion. I'm just taking speck of the completion of the local ring. So really what this just is, if you choose coordinates, this is speck of the formal power series ring in three variables. So instead of open balls, I'm actually really taking formal neighborhoods around all the interesting singular points of the underlying reduced curve. Because now we have a three-dimensional torus which is acting on here, just a standard torus action. So that is where the topological vertex will come from. So what else am I going to take? I'm going to take tubular neighborhoods of what's left. So I take this curve and I remove all the singularities. So then you get like these open tubular neighborhoods, et cetera, also here for the singular fibers. So really, again, I'm just taking formal tubular neighborhoods of this underlying reduced curve after removing the singularities. But I just want you to think of it as small analytic open neighborhoods. And then what's left? There's one big, risky open left, just the three-fold to remove the entire underlying curve. So this is the cover. So now let's see how we can calculate on this cover. So the point now is that we have a restriction map. So we have our stratum here. We fix this underlying curve. And we've got a restriction map to a bunch of Hilbert schemes on these local pieces of the cover. So I'm just taking some one-dimensional sub-scheme with this underlying cone Macaulay curve. And I'm looking at what it induces on all the pieces of this cover. So now comes the key observation. Since we already fixed the underlying cone Macaulay curve, there is no gluing here anymore. So when you do the restriction, you lose no information. The gluing is only determined by how the underlying cone Macaulay curve glues, but that's when we fixed already. So by restricting to these open pieces, we already have here a bijective map. Again, the underlying cone Macaulay curve is all that comes in when you glue because when you look on the overlap of patches, the embedded points and the floating points are simply not there. So this restriction map is bijective. So to be precise, it's a bijective constructable morphism. It's not actually a morphism, but for Euler characteristic reasons. This is fine. This means it's a morphism on a certain stratification of particles. So we reduce now to just calculating the Euler characteristics on these local pieces. So let's do one example. Let's look at, for example, this open neighborhood here. So I'm going to call this point P here. And let's call this partition simply mu. So I'm dropping the superscript. So here we have some open neighborhood u alpha. Let's see what happens on there. So what happens on this open piece? So we're now on this open piece around such a node with the partition mu coming in. So what we want to calculate is the Euler characteristic of this local piece. And as I said, now suddenly we have a full three-dimensional torus. So this is just the Euler characteristic of the full C star cubed fixed locus. But this modular space has isolated fixed points which are just parameterized by 3D partitions. I'll draw a picture in a minute. So this is just the sum over all 3D partitions pi with asymptotics and then pi to the size of P. So let me draw the picture. So the C star cubed fixed elements are just all 3D partitions, which at some point limit to our fixed partition mu on both sides. And here there's no lag. So the underlying cone McCauley curve locally just looks like making this come together. And then the way you put in embedded points is you just stack boxes here further on top. So these are the elements we sum over. So this object is very well known in physics. So I should maybe also say what is pi here? So pi is the size of the partition, which is a little tricky for infinite lags because this extends infinitely. So this is what's known as the renormalized volume. So it's the sum over all boxes. If a box appears in more than one leg, you subtract the number of legs it appears in. So this is the well-known topological vertex, which you can find in, for example, you can find in Okunkov, Pechetikin, Vafa. So the reason I'm putting here this symbol in front is that the two coincide up to some overall factor of P, which has to do with the way you normalize. So there's some overall factor of P here in front. So now what have we achieved here? Say again, what's asymptotic? Sorry, I couldn't hear. Does the partition need to have, oh, what does the asymptotic mean? Oh, so I just wiped out the picture. So it means, so we fixed some partition mu, 2D partition. And in this case there are two infinite lags. And if you take a cross section of these two infinite lags, it has to be this fixed partition, this fixed 2D partition mu. So that's what it means. So only in a finite box and outside that box, you just have mu extended and empty extended. I mean, in the paper a priori you could put here any lambda mu nu. But for this geometry, this is the most complicated which comes up. And for those working with topological vertex, if you follow the conventions in here, you have to take strictly speaking the transpose partition here. So what do we get now? We get now for the Euler characteristic of this stratum. So this also proves that it will be independent of the locations of our points. So what is this Euler characteristic? Well, so we just take a product over all the opens in our stratification. So then we, sadly I wiped out this picture, but because of lack of blackboard space I had to. So I'm now just writing the local contribution for all the small balls where the fiber hits the section. Then there's a local contribution. So where a smooth fiber hits the section, then there's a local contribution where each singular fiber hits the section. Remember, we also chose a small ball around every singularity of a singular fiber. So this is the total contribution of all the small open balls. But then, of course, we didn't talk at all about the tubular neighborhoods. So there is a contribution of the tubular neighborhoods and the contribution of the big Savisky open subset, which is the complement of the curve. So this is the other contributions of the other opens. But again, it turns out that you can, by looking at such a tubular neighborhood and recording the locations of the embedded points, you can also use this stratification to also express those in terms of the topological vertex. So if you put it all together, you get the following final formula for the DT invariant. So I'm taking the formula which I derived earlier, the one where E was still appearing. And then I plug in this formula. And then what do you get? You get some front factor, this front factor. Don't worry about those halves. Those halves disappear. And then there are two contributions, one from the smooth fibers. So the sum over all possible 2D partitions of some factor. So let me comment on this. So which part have we seen? So we've seen this part. And all the rest you're seeing here appears from the fiber contribution. So this comes from the fiber, from the tubular neighborhoods and this one here too. And this one comes from the tubular neighborhood of the punctured section. And then what's the contribution of the, and this also comes from the tubular neighborhood of the section. And then what's left for the singular fibers is this expression. So again, the part we've seen is this part. This part here you don't have to worry. That was, I was telling you that the topological vertex and the Euler characteristics of the strata differ up to some overall power of q of p. That's this power here. Again, this comes from the section. This comes from the punctured fiber. So this is the final formula. And these are purely combinatorial objects. So in principle you can put this in the computer. And then you'll find numerically that this is the KKV formula for K3, for example. But to actually prove that takes some work. So it turns out that if you express this one in terms of sure functions that this one is actually known in the literature. This one appears in a paper by Bloch Okunckoff. And it gives rise to a theta function. Now this one turned out to be new, or at least we couldn't find it in the literature. So it took us quite a while to get this one. So we actually needed a specialist here, namely in joint work with Ben Young. This one was expressed in terms of an operator on Foxpace, the infinite wedge space. So this part was really largely due to Ben Young. He got the main ideas there. Then if you calculate this one you again see some explicit expression with thetas, and it all combines to the theorem in the beginning. Now, since I see that there are a few minutes left still, oh, in actually calculating this one? No, so it turned out what you could do here is, so this entire expression turns out to be, you have this infinite wedge space. You write down some operators on there. Because of this mu here, this q to the mu, you can somehow see there's some natural operator, which has this as its expression. And then the way to calculate it is you play, so once you've written it in this way, so there's a bunch of operators here, let's say a, b, c. And all you have to use is the commutation relations between these operators, which are already known in the literature, because most of them are just the ordinary vertex operator. And then there's one which has to do with this blog of operator. So you use two ingredients, the standard commutation relations, and the cyclicity of trace. And it turns out that only these two ingredients, by playing them out against each other in the clever way, which was Ben Young, you actually find that expression. So there's actually a separate combinatorial paper on this. So maybe in the remaining four minutes, let me only do one thing. Let me just write down for completeness for the geometers. I was telling to get the actual Donaldson-Thomas invariance, we need a conjecture. So all I'll do is write down this conjecture. And the Toric case of this conjecture is known. So let X be any three-folds. This time we do want Calabi-Au. And let C be any Cone-McCauley curve in it. And to be on the safe side, we assume that the only singularities which the underlying reduced curve has locally look like something Toric. So either locally there's some smooth curve or a node or three coordinate axes coming together. Conceivably there may be more general situations where this holds. Then let's look at a stratum of the Hilbert space of the entire Hilbert scheme. So we look at, again, as we did here, so we look at all sub-schemes where the underlying Cone-McCauley curve is our fixed curve. So this lies inside the entire Hilbert scheme. I'm suppressing here that we also have some curve class. Maybe I should put that in. And then what is the conjecture? So then the Bearent function, we want to be on the safe side again, so we integrate the Bearent function over this stratum. So this is the Bearent function of the entire Hilbert scheme on the three-fold. And the claim is that this is just the Hilbert scheme the Bearent function at the underlying Cone-McCauley curve times minus one to the number of embedded points times the actual Euler characteristic of the stratum. So you could even conjecture something more refined. You could even say the Bearent function at Z, where Z lies in here, is nothing but the Bearent function of the underlying Cone-McCauley curve times minus one, that would be an even stronger thing to conjecture. And the final thing which we can do actually is this part we can compute. So this gives rise to this front factor. So this actually calculating the Bearent function at these curves turns out to be quite delicate. Unexpectedly these curves actually turn out to be smooth points of your Hilbert scheme. And it leads to this particular sign. But we can't, this is all we can do. And of course, I mean, you can list some evidence for this irreducible curve classes. This appears in work of Poundary-Pounder-Thomas. The TORI case, this is known. Perhaps in a stable Paris case, more is known. But yeah, for us it's just we conjecture this and then we get the formula. Okay, that was it.