 So, this is going to be the last concluded talk by Lucian Zau of the University of Illinois at Urbana-Champaign, USA, who will talk about computing the... Sorry. Thank you for the introduction and thank you for the chance for me to speak here at last. I think it's just because of my last name, it's starting with Z. Okay, so... Yeah. All right, so today I'm going to tell you a story about a curved computing invariant, which is called the Gopakumavafa invariant. So you may have heard of the Grammar-Witzen invariant, but not heard of this invariant, but this invariant is actually a very interesting invariant to study. So first, let me just state the largest conjecture behind this invariant, which is proposed by these two physicists called the Gopakumavafa conjecture. So, it just tells you that. So for x, a caveat three-fold, and you have this kind of equation when summing over all g greater than equal to zero and all beta inside the curved class of the Grammar-Witzen invariant of beta, g and beta of q to the beta and lambda to the 2g minus 2 is equal to some kind of new invariant they have already described. They call the... Of course, they didn't call it the Gopakumavafa invariant, but we call them the Gopakumavafa invariant, which is n of g of beta and 1 over k of sine of k lambda over 2 to the power of 2g minus 2 and q to the k beta. If I'm not writing this wrong, then it should be the correct answer. k is greater than equal to 1. Okay. So why do we need to care about this invariant? As you may know that the Grammar-Witzen invariant may be something like rational number, but now the Gopakumavafa invariant, surprisingly, they are all integers and they also enjoy the more enumerative counting properties like you can extract the 27 lines on the cubic surface by the Gopakumavafa invariant but of course I will not talk about that today because I'm not counting only on the case of local P2. So this is also a physicist's name, which is actually the total space of OP2-neg 3. Okay. So let me just start describing how they define this invariant just in a very intuitive way. So suppose you need to count some kind of curve inside your Carvel 3-fold. Let's say there's only one curve here. So there's only a G and a G curve here and I modulate space of curve. It's only one point. Can everyone see here? Probably not. I should try to write it elsewhere. So let's say this one is our modulite space of curve which is too trivial to be considered in this conference and then now we consider the Jacobian associated to this curve or what the physicist called the D-brains like supported on this curve and actually this will give us a torus of T2G. So how do we count the number of curves? Well, we can count the number of points or we can count the number of Jacobians. But how can we count the number of Jacobians is to consider the Poincare polynomial of this kind of Jacobian and we already know the Poincare polynomial of torus, right? So it's 1 plus 2T plus or 2y plus y-square to power G. So this is the Poincare polynomial of the T2G torus. Okay, so I can now define what I mean by the Gopakumavatha invariant but only the semi-definition because actually this definition lacks many, many conditions that I have omitted because it will evolve something like perversive or perversical homology but now I'll just give you a very, very simple definition but the actual definition you can see on MOLIC and TODA's paper is the title is just Gopakumavatha invariant we are vanishing cycle like 16, I believe. So this is my definition where f is backing from m to b and now you can define what is the Gopakumavatha invariant by the following equation. So it's negative to the power dimension b summation of chi of ri plus so let me just first define big d equals to the dimension of m minus the dimension of b and now ri plus d of chi star of c of m y to the power i equals to so on the right hand side there are our Gopakumavatha invariant where g sum over greater than 0 of y a half plus y negative half to power 2g so as you can see here I have shifted my complex by the dimension just to make sure that the punk rate duality is just y sends to y inverse so this one makes our life easier by just looking at this kind of punk rate duality instead of looking at y to the power dimension minus something, okay? So on the right hand side I have shifted so as well on the left hand side I have shifted some dimensions so that's why we have negative 1 to the dimension b because when you shift a complex the Euler characteristic will be multiplied by some negative number. All right, so this is my definition of Gopakumavatha invariant so it's just a constant sheaf on m so now our m and b are all smooth varieties so I define this like this because then our decomposition theorem will work very smoothly otherwise I'll try to define the decomposition theorem on many of the things like the critical manifold or something I don't quite understand so the fibers, yes so actually the fibers are so what Toda and Molly have said should be something like clavio so it's just telling you that for any point there exists a neighborhood such that f inverse, so k the canonical bundle on the f inverse of the neighborhood will be isomorphic to the constant it's not constant but it's a holomorphic section or holomorphic functions, sorry so this is some kind of hidden assumption I have got but now I just forget all these kind of assumptions and so our definition of the invariant, so this one n is called the GV type invariant so not our real invariant because we cannot just get some random spaces here and there and then go get so beautiful formula right here so our GV invariant is actually defined by saying that m is equal to moduli of sheaves supported let's say n beta sheaves supported on this beta with Euler characteristic equals to 1 so this is our higher space and the lower space is just the chow variety of this beta so when you define the so then you can define this kind of pushing forward and then you can define these numbers and these numbers and G are just called the GV invariant so this definition sounds a little bit weird but I can actually tell you a very very easy example which is just P2 with degree 1 curve so for P2 we can consider the modular space of degree 1 curve which is just another P2 and we can also consider the modular space of sheaves supported on this degree 1 curve which is again a P2 so for P2 with only degree 1 curve it's just an identity map so we can apply our formula which just says that the only thing that is non-zero is i equals to 0 but now also d equals to 0 and we also have this sheaves right here so the only thing that is non-zero is chi of P2 equals to 3 which means that our KUMA-VAPA invariant for genus 1 and genus 0 and degree 1 is equal to 3 and you can also consider degree 2 curve and then it's a P5 goes to P5 and you can calculate that NZ2 will be equal to negative 6 when you see this formula and it perfectly match up with our groma-wetton prediction as inside the formula I have already erased but now this comes a very strange problem because when a like M degree goes up like higher than equal to 3 we cannot have so nice description of our modular sheaves so we need to use my computation method which is something I will tell you like the second biggest topic is called the McDonald formula so what is the McDonald formula well probably most of you or many of you have already known that it's just telling you the cohomology of some symmetric products on curve when the curve is so C is our smooth curve of genus G and now we consider the symmetric product and you can just calculate by counting these classes that is symmetric and its summation from I sorry so here is I here is K summing from 0 over I over 2 and then which product of I minus 2K of H1 of C if I remember it correctly so for McDonald formula there are many ways to prove that but there is a very simple way to see it looking at the Jacobia so M beta where beta is the curve class you are considering so beta is inside our Cavio 3-fold so I'm not sure what you mean by the fiber class because our starting point is a Cavio 3-fold and we choose a curve class inside the Cavio 3-fold but X is our starting point it's a Cavio 3-fold a little bit Russian here so we start by X Cavio 3-fold and then we choose a beta inside our curve class the second homology is H2 of C and then we define the modular of sheeps supported on this class and that is our M beta and of course we have to assume that the Euler characteristic is one so B is the of this beta sorry it's writing a little bit low so it's hard for you to see alright so now I need to say what is the McDonald formula which is in my main calculation so the natural question to ask the first what I promise is a proof of this kind of formula so suppose D is large enough and B is 2G-1 then the Abel-Jakobi map mapping from the symmetric product of D points going to the Picard group of D is a P D-G bundle so then you can just push forward the Abel-Jakobi map of the C and then sorry is equal to summation of C of negative I 2K goes from 0 to D-G so you just calculate the cohomology of P D-G and then push it forward and then you can see that so for Picard D when D is large enough it goes to our Jacobian and you already know the cohomology of the Jacobian which is just much product of H1 and then you just combine these two facts together and you will see the McDonald formula okay so I shouldn't have erased my definition right here so let's keep it here is the McDonald formula so then we need to see what's going on when we have a family of curves so it was by Mignolini and Shende that Mignolini and Shende that they have generalized this formula to a family and also Molly can be to the family of integral curve that we as well some kind of this property here so let's say C to B is a family of integral integral curve and now we need to generalize to something like the related Hilbert skin so let's define C of D to be the related Hilbert skin that each fiber is this guy H of I H sorry C C B to the the Hilbert skin of D points of density on this curve skin and then we can also have this kind of formula right here just by looking at the I's push forward pi star D so this means that map from C of D to B the push forward will be exactly quite similar to what we have right here like the bundle structure so it's R of I minus 2K pi star J of C J is the relative compacted by Jacobian like for each point on B we associate the Jacobian of this curve and then we can also form C here which they call the relative compacted by Jacobian yes K is going from 0 to D minus G I believe let me check sorry I over 2 yes okay so now we have these two formula but why do we need to care about this formula right here well it's just because when our curve is smooth the moduli of sheeps supported on beta when you look at the fiber of these kind of moduli of sheeps it's exactly the Jacobian so that we can actually use this formula to calculate many of our Gopakuma Lava invariant right here so you can see that to describe this moduli of sheeps it's pretty easy so let me just show you an example in my very limited time to say why this kind of formula is true so of course in the first glance you have assumed that it's a family of integral curve but now for our case the curve may not be integral but the miracle is that this formula also holds in many cases so now let me just show you the example where you can do by hand even in home so now we consider p2 with degree 4 curve I have just skipped degree 3 because for degree 3 the moduli of sheeps is exactly c1 so this will not show the formula because we already know what is c1 so let's say degree 4 curve which we have a very hard to describe moduli space so now our child ring of degree 4 curve in p2 is just given by so we just consider all this kind of curve that is of degree 4 of ax to the power 4 plus vx to the power 3 y plus something to the power 4 and then you collect all this kind of possible coefficient in the front and then you can form this kind of moduli space by just collecting this coefficient and the number of coefficient is 4 plus 2 choosing 2 is 15 so then since scaling by a constant is just the same equation so then you can you need to subtract by 1 so it's a p14 case so our base base is p14 now you can calculate what should be the answer of the first gopacuma vapa invariant so in this case our genus is equal to by the degree genus formula over 2 is equal to 3 so then the highest gopacuma vapa invariant should be n of 4 of 3 so here I didn't remember the so I believe I write down genus first so genus is 3 and then 4 is a degree so how do we calculate n3,4? well if you look at the definition of the formula which I'll just tell you that it's actually r0 of pi star j or pi star m in our case of c and then by this kind of McDonald formula you can just plug in d equal to 0 in this formula and see that r0 of pi star 0 is exactly r0 of pi star j so this is our first equation and then you can see that since r0 of pi star 0 pi star 0 is exactly our p14 so r0 of pi star m is also exactly just a constant sheet on p14 which is just chi of p14 equals to 15 so this is our first calculation we can just do this kind of r0 by considering the first equation of the McDonald formula so let's switch to the second equation so what is the second equation which is chi of r1 pi star of m of c and now we need to calculate by this formula and then we see that r1 of pi star 1 is exactly r1 of pi star j so then we have to like check just check the r1 pi star 1 by using the Ray's spectral sequence so remind that chi of the base space which is our b of the r pi star is equal to chi of our higher space which I call it m so then we can check by this equation that it's actually chi of r0 pi star m c2 times because we have r0 and r2 and by punk radiality they are the same minus chi of the universal curve c1 I believe it's just giving you 2 times 15 which is 2 times the type 14 minus 3 times the universal curve because it's very easy to compute it's negative 12 okay so this is how it goes by calculating all these kind of numbers and you can collect all these kind of numbers and place in the polynomial I have written so it's 15 y to the negative 3 minus 12 y to the negative 2 plus dot dot dot equals to summation of ng of y a half plus y negative half to power g 2g so then you can calculate what are all these ng's and I'll just write down my answer right here so so ng so n3 as our we have already seen it's 15 and n2 when you calculate this kind of coefficient it's negative 102 and 1 equals to 231 and n0 equals to negative 222 so these are something just coming from the direct calculation by applying this formula but now here comes the problem because physicist says that n0 should not be this number but rather n0 is negative 192 so what's going on in this formula as I have already reminded you that our formula is based on the assumption that all the curves should be integral so now here comes the non-integral part which is that our degree 4 curve splits into a degree 3 curve and a degree 1 curve so when you consider these kind of two modular space like degree 3 curve is actually a p9 and degree 1 curve are p2 and you calculate the other characteristic of this modular space it's exactly 30 so this discrepancy 30 is exactly coming from this degeneration but when you do the perverse sheaf calculation what we can prove is that it's actually the correction is coming from the degeneration into the degree 3 and degree 1 curve probably the main theory of this stage but now we are going to go to even higher degree and try to prove more things so this method now only works for degree up to 5 but for degree 6 we have more degeneration like going into a degree 4 curve and 2 degree 1 curve and now I'm not quite sure what should be the thing that we should do correction in this case so I think that's all of my presentation thank you