 Thank you, Maxime, for the introduction. And I would like to thank the organizers for inviting me. It's a great honor for me to speak at the Professor Kashwara's birthday conference. So the title of my talk is The Frobenius Structure Conjecture in Dimension 2. It is joint work with Sean Keogh. So let me first give the plan of my talk. First, I will state the Frobenius Structure Conjecture of Gross, Hacking, and Keogh. And then I will state the main results of the talk. So the part, I will explain structure constants involved in the conjecture via no Archimedean geometry. And first part, some finiteness theorems. And the last part, I will speak about compactification and extension. So let me start with just a statement of the conjecture. Statement of the Frobenius Structure Conjecture. So here is the setup of the conjecture. We start with y. It is a connected, smooth, projective variety over complex numbers. And then inside y, we have a divisor d in the anti-canonical class of y is an effective SNC divisor. So we assume that d has zero straight term, containing zero straight term. So let me draw an example. We have y, like two-dimensional example. We have y like this and d, anti-canonical divisor. Inside, containing a zero stratum, it's just like a chain of rational curves. Like this is an example of d. What do you mean zero straight term? Sorry? Zero stratum is just here, like a zero-dimensional stratum point. So because d can be singular, so we have stratification. We say there is a point inside. SNC, at least. Yes? And we take a complement of d, u to be complement of d. This complement is called log-calabi-yaw variety with maximal boundary. So this is just a name. Maximal. You assume that the divisor of a convention is reduced when you say symbol of normal costing? Yes. Yes. Reduced, yes. So this is the geometric data we start. And then from this, we consider b is due intersection cone complex of d. So it's just the cone over the due intersection complex. So in this case, b is simply this, like a fan. This is b. So in this case, b is just the fan where each ray corresponds to irreducible components of d. And each two-dimensional cone corresponds to the zero stratum. It's very simple. Yes. And inside b, we have integer points, bz, just integer points. And I just want to remark that integer points in b, they can be thought of as divisorial valuations on the function field of y. You can think. But yes, so it's just a remark. So yes. And then let me introduce a ring R. R is the ring generated by the monoid Ney, where Ney is the monoid of curves. Ney is just a monoid of curves in y, modular numerical equivalence. So the usual, if you know Ne, it's the usual Ne. And I define from now, I introduce one last notation, a to be a free R module generated by bz. So in other words, a is just a direct sum of R over bz. And let's denote the basis to be theta p. So it's free R module generated by bz, I write it as. It's just a direct sum of that many copies of R. So this is just the basis, theta p. Yes, so I hope you are not yet confused. So I introduced seven notations. The first three is just the geometric data, let's say. This is geometric data we start with. And then we have two b, this simple combinatorial scene. And from this geometric data and the combinatorial data, we define two algebraic data. One is a ring R and a free R module A. So from this growth and hacking keel, they observe a natural R linear map, R multilinear map by this from A to the power N to R. To R for every n greater or equal to 2. So they observe there exists a natural R multilinear map given by counting rational curves in Y. So I will tell you what is this R multilinear map in five minutes. Is it really good, the completion of R? No, just R. Is it really finite linear combination? Oh, let me assume that is a good question. Let me assume that D supports an ampoule divisor. So maybe I don't write. Yes. Maybe I write. So let me assume, yes, thanks for Maxim's question. So we assume U is a fine. Now there is no trouble here. So they observe there is a natural R multilinear map given by counting some rational curve in Y. So I will tell you what's this map. But let me first state their conjecture. Then they conjecture the following. So first, they say that this R multilinear map is non-degenerate. Is non-degenerate. And then they say that there exists a unique commutative R algebra structure on A such that first the unit for the R algebra structure equals just such that unit equals this base theta 0. And it's compatible with the multilinear map, which is if I take product A1 to An and I take trace, which means the coefficient before theta 0, this is given by the multilinear map. And the third part of the conjecture is that spec A to spec R. This restricted to the torus associated to the pica group of Y. The torus associated to the pica group of Y lives in spec R by definition. And if we restrict this map to this torus, then this is a family of a fine log-calabi of varieties with maximum boundary also, maximum boundary. And this family is called a mirror family of U. So let me just recapitulate. We start with some log-calabi of variety, which is a complement of some divisor in Y. And from this, we build a ring and a module. And then by counting some rational curves, growth, hacking, and keel, they defined a multilinear map on this module. And then they conjecture that this multilinear map actually has some underlying Frobenius algebra structure inside. And that algebra structure makes this algebra into, again, a family of log-calabi of varieties. So. What is trace that you wrote in the formula? Trace is just the coefficient. You're right. Sorry? Leave a trace on the blackboard. Not naming. Oh, you mean I write the explanation. Formal trace is equal to coefficients. It's just a coefficient before theta 0. Yes. So the conjecture is really about, although it's phrased in terms of this multilinear map, it's really about rational curves, the geometry of rational curves in Y. So let me explain now how this multilinear map is defined. So pick one of the same thing as Neurons-Evry. No, it's dual. Pick one Neurons-Evry. Here, yes, in this case, they are the same. Modular. Yes, they are the same. Yes. Yes. So now let me define this multilinear map. So given n points, p1 to pn in the integer points in B, and beta curve class in the monoid of curves, Ne of Y, we make some blow up called Toric blow up pi from y tilde, d tilde to y d, which is just blow up some strata in Y. This is called Toric blow up. Like in this two-dimensional case, the only thing we do is just blow up this zero-dimensional, zero strata, blow up some points. So we make a Toric blow up such that each pi has divisorial center. Ah, yes. Divisorial center, let's denoted d pi in d tilde. So it's really easy. We have, like, after blow up, we have many, many components. And now we assume that pi has divisorial center on some components. For example, this is dp1, this is dp2, this is dp3. Because I said that integer points correspond to some valuation. Divisorial valuation. Yes. So now we define let Cp1 to Pn beta be the number of rational curves in this space that looks like this. So we just let this to be the number of rational curves that touches dp1, dp2, dp3. And also of this curve class beta. So to make it precise, I say that that counts a map from P1 with n plus 1 marked points to y, such that satisfying the following condition. First, we want the point pi touches dpi with some multiplicity, which means that we want pre-image of dpi is mi times pi, where mi is the multiplicity of bigger pi. So where bigger pi equals mi times the primitive vector. I just wanted to touch with a multiplicity. And then I want the curve class to be beta. So essentially, that's the condition we want to put on the curve we want to count. But we don't get a finite count at this stage. So we must add some extra condition just in order to get a finite number. So we also assume that the domain curve has fixed general modulus and fr equals a fixed general point y inside y tilde. So I just add that condition to be more rigid. So then one can show in the proposition that this set is a finite set. It's finite. So the number that we want to count is well-defined. And we define our multilinear map just by this number. So the multilinear map from a to the power n to r, it's just given by it sends the basis theta p1 to theta pn to the sum over, since r is the ring generated by curves. So just the sum over curves of this number. I denote z to the power beta to be basis of this r, basis of r. So this is the definition of the multilinear map. So this would be a non-negative integer. Yes. This is just a finite set. And this number is just the cardinality of the set. So it's a non-negative integer. So I hope now this conjecture is precise. So the multiplicity is supposed to be 1 for something. It's continuous. So you mean, as a modular space, this set? Yeah, it's just a reduced scheme, zero-dimensional. Yes. So yeah. Why is it finite? So the maximum question. Yes. Why is this some finite? Because I added the assumption that u is a fine. In this case, the boundary d will support an ample divisor. And using that ample divisor, we see that if, because we see that here, the intersection of the curve with the boundary is fixed. So when the boundary supports an ample divisor, that fixed intersection number means there are only finitely curve class. So do you want to sum up the given or? They are marked points on the rational curve. So you give it? Not on target. Not on target. So you choose some points already. In the domain, yes. And I ask that point to go to the divisor, yes. So this sum is also finite. So I just summarize that we count some rational curves in our y, or some blow up of y. And that count is a really naive count. And using the naive count, we define a multilinear map. Then the conjecture says that the count satisfies mysterious properties, especially it gives back again some kind of family of logarithm varieties. OK, so you have a question. Yes, so the main result of the talk is that the following theorem that the conjecture holds in dimension 2. So let me explain the idea of the proof. So we will construct this Frobenius algebra structure by counting some other things, which means we construct the structure constants for the algebra of A by counting so-called non-archimedean holomorphic disks. Using some techniques I developed in my thesis, which was under direction of Maxim. So let me explain how we get the structure constants again by some enumerative geometry. So for P1 to Pn in BZ, we want to figure out what is the product. So we want to know how to multiply things in A, because A was defined as an R module, and we want to define multiplication. So we want to know what's the product. And since the product is an element in A, and A is a free R module generated by BZ, so I can write this element as sum over the basis. And so the coefficient before the basis is an element in R. Since R is generated by the monoid of curves, I can write an element in R as sum over its basis, which is the monoid. And then the basis we denoted by z to the power gamma, z is just a symbol. And then the coefficient before we write it as P1 to Pn q gamma. So this formula is nothing. Just write it out. And our goal is to define this number mu P1 to Pn q gamma using non-archimedean geometry. So now I explain how to get this number. So since we want to use some non-archimedean geometry, we use a non-archimedean field. The field of formal Laurent series. It is a non-archimedean, because we have the absolute value given by valuation of t. And then I take uk to be base change from complex number to our field k. And just as in complex geometry, we can analytify this k-variety and get analytic space over k. So we get a k-analytic space. We use the approach of Belkovich here for the analytification. So roughly, the analytic space is just the points. They are just some kind of valuations on ring of functions. So we have a natural embedding from B to the analytic space. B was our combinatorial thing, the dual intersection comb complex. And there are some valuations. So we can embed. And moreover, we have a retraction from tau from this analytic space to be a continuous map. So now that's the setup we use for non-archimedean geometry. And now I just say that we define this structure constant, mu p1 to pn q gamma. They just count holomorphic disks in I do the same base change and the analytification to y or y tilde. They are almost the same, which looks like this. Which looks like, so we have dp1, dp2, dp3. Let's just do 2. And we count holomorphic disks, which looks like this. So it touches dp1 and dp2. And then let me give the preset. And of course, it has a curve class gamma. And let me give a precise meaning of what I mean by counting such disks. It just means we count webs from disk. Delta is just closed unit disk. Disk with n plus 1 marked points, p1 to pn to r on the disk and a map from this pointed disk to our analytic space. Such that same conditions, almost the same. First, I want to touch these divisors at marked points, p1, p2. So I want a pre-image of dpi equals touches. Just touch the divisor with some multiplicity. First condition. And the second condition. So curve class is gamma. And then for? As a pre-image, can you surface with a boundary of all these things? Delta times plus and second homologation. Yes, but do you want to know what it means? Maybe not yet. So maybe I'll tell you later. Yes, I will explain it later because it will be a digression to explain. Let me just say it makes sense curve class equals gamma. Yes, and for other conditions, we will use this retraction map to put more conditions. So we have a retraction map from this to b. And so in b, we have this origin. b, I'll draw there. It's just this fan. So we have this array in the direction of p1 and another array in the direction of p2 and array in the direction of q. So we ask that first, maybe I first say, first I want the boundary under the retraction map goes to somewhere. So I want the boundary go to some fixed point b near the ray oq. So it just means that I fix some point b near the ray oq. And I want the boundary goes there. And then I also ask that if I take a neighborhood of the boundary and under the retraction map, then it is just some segment starting from b in the direction of q. So I ask a neighborhood of the boundary goes to something in the direction of q. Yes, so that's almost the only thing we fix. But there are still too many. So in order to try to get some finite count, we still need to just fix some extra condition, which is domain has fixed general modulus. And f of r is a fixed general point equals a fixed general point. So we say that we define this structure constant just be counting this kind of thing, satisfying this condition. But unfortunately, it's not as easy as in the previous case. We have this proposition that we get a finite count. So here we have trouble one is that if we try to count these things, the space of all such disks is infinite dimensional. Because we can vary still vary many parameters. So in order to get a finite count, we just use a solution I did in my thesis is to impose an extra condition on the boundary, which I call a regularity condition on the boundary. So by this, I mean so we have our disk. And since it's analytic geometry, we can do analytic continuation. We can try to do analytic continuation at the boundary. And by regularity condition on the boundary, I just mean that when we do analytic continuation at the boundary, it should extend straight. So what I mean is that by analytic continuation at the boundary, our disk extends all straight. So more precisely, it means that its image in B by our retraction map tau is straight with respect to the so-called integral fine structure on B. So now, good news is that theorem 2 is that the space of holomorphic disks satisfying all these conditions, the space of holomorphic disks satisfying all the conditions star plus regular boundary, this is a finite set now. Yes, it's really a finite set. So since it's a finite set, we can just count its cardinality and we get a number. And we define this number to be our structure constant. However, there is another trouble, trouble 2 is that when we extend our disk at the boundary, we have some ambiguity. It's that extending straight on the left of this OQ may be different, may differ from extending straight on the right of OQ. It's just the meaning of straight on the left and the meaning of straight on the right. They are a priori different. So let me just draw a picture. So for example, if this extension is straight seen as on the left of OQ, it might not be straight seen as on the right of OQ. If we extend straight on the right of OQ, it may extend like this, for example. It's just because integral affine structure. It may, like on the left, it may go like this. On the right, it may go some other way. So left and the right, they are different. So this says a priori that our count depends on the choice of whether we pick B on the left or pick B on the right. So fortunately, they are the same. The way we can prove that they are the same is that we define yet another regularity condition for the boundary using some, not using this, but using some Toric model. So I don't have time to explain. I just say that counts using left regularity condition equals counts using another regularity condition called Toric regularity condition equals counts using right regularity condition. So corollary is that our structure constants, mu, is well defined. So yes. So now we defined the numbers mu. They are the structure constants. They are supposed to be the structure constants of the algebra. Then, which means that now we know what is this product of elements in A. And then our natural question is whether this product, this multiplication, is commutative and associative. So commutativity is easy. Just follows from the construction. But associativity is more difficult, which is a theorem just like this, that this multiplication is associative. Yes. So that's the theorems I want to explain just for the structure constants. And now I go to the next section, which I called finiteness theorems. So as Maxim and Professor Kashiwala already asked about finiteness, and now we have another question about finiteness, which is whether these two theorems, they are finite. Because if they are not finite, they are not really structure constants. It's not an algebra. Then it's some formal algebra. So a natural question is, are the two theorems in this equation finite? So this finiteness follows. It's a bit more complicated than that finiteness. But it is still true. It's still true. So we can prove, which I call finiteness part one, is that the two theorems, both theorems are finite theorems. So for this, we use some convexity property of the image of this disk. So in this case, the image is something that looks like this. So we use some convexity property of these green trees. And so we get finite sum and we get algebra. Then the second finiteness theorem, which is important, is that the first finiteness just says that we get commutative associative algebra. And the second finite says that a is a finitely generated. So for the second finiteness, one has to use some more structure on a that maybe I don't have time to explain. But we really have two finiteness results. And it shows that the algebra structure we get from counting these kind of disks, they are reasonable. OK. And last section, compactification and extension. So we have constructed, so until now, we get this reasonable a. So since a is reasonable, we can take a spec of a. And it goes to a spec of r. And now we want to say more about this algebra a. So in order to say more about more information, because in the conjecture, it says that this family defined by these counting curves, it's really nice. Like the fibers, they are logokalabiol varieties. But until now, we have no idea what are the fibers of like, what is this family? After all the effort, all the things we know is just a is a finitely generated r algebra. So in order to obtain more information about spec a, we need to compactify the fibers of this map, and as well as to extend this family to a bigger base. So we need to do two things. First is compactify the fibers, which means we compactify spec a into some x such that this is proper. So the way we compactify the fibers is again to define this compactified thing by something like this. But we no longer take a spec, we take a proge. So we compactify the fibers. And after compactifying the fibers, we can extend. Now we can extend the base. Maybe I cannot explain how to extend the base, but let me just say that the base is a spec r. And this just since n e, the cone of curves is due to the net cone. So this can be seen as the toric variety associated spec of r is just the toric variety associated to the net cone of y. And that thing is naturally embedded into the toric variety associated to something called the moriphane of y. So the net cone is just a cone in a bigger fan called moriphane, and our family x can also be extend. This extension exists. So maybe I put it as a theorem that this extension exists. So using this compactification of the fiber and the existence of the extension, we can deduce the final theorem about the property of this spec a. So we show that. So this family, let me denote it again by curly x. So one can prove we have curly x and so this is also. We have a pair since it's. Is this just the part of the infinity? So final theorem, let me state is first this family it is. So first it is a flat projective family of surfaces with veil divisor. And second with veil divisor z, maybe with veil divisor. And then this z, the family of z over the base, is just a trivial family, a cycle of rational curves just as what we started in the beginning. And the third, we can show that the fibers over the fibers of the family, they are semi log canonical. And if we take an anti-canonical class plus z, it's a trivial. It's again the same as we started with. D is in the anti-canonical class. So last, the fibers over the picatorous, this is better. It is log canonical. And the complement of z and a fine canonical log calabria surface. So thank you very much for your attention. Thank you in two questions. Why is more or less p2, isn't it? Why is more or less some blow up of p2? I mean, suppose y is p2, then what is x? It's again the same in this case. Is there anything simple, I mean explicit? Yes, all the computation here can be done explicitly. I mean, x is also p2 if y is p2. Actually, if y is p2, x is always over some bigger base, x is a family. So if y is p2, then x is a family over the just, yeah, I think it's still p2. One has to check. But the TORRIC case is not an interesting case for this question. Because what is an interesting case? Interesting case, they are blow up of TORRIC varieties. While I say TORRIC case is not an interesting case, because in the TORRIC case, all these numbers that we can count, they are just, so this multiplication we get, it's again in the TORRIC just a monomial, like multiplication of monomials. I mean, there are no interesting sums, always like just one term. So the more interesting case are non-TORRIC, and everything can be computed explicitly using the wall crossing formula of Maximian, yeah. I have a question. How do you understand the surfaces as smooth, just because it's... It's not as smooth, it's canonical. Ah, OK, so it's... It's less smooth, but they are really non-smooth things. And yeah, it's not as smooth. I see, I see. This is the best one can get. And you get it in real examples, you have got some... Singularity. Singularity, I see. Yes, but from what the counts I described, we can see that actually we don't need to assume why it would be smooth. So it's symmetric as a picture. Ah, I see. Because the counts we get finite set and we can avoid the singularities of why. OK, other questions? Then thanks to the speaker and to the questioner.