 is Giulia Guggiati from ICTP, who will talk about hypergeometric Landau-Gensburg models and anti-economical log-down processes. Hi. Thank you for the introduction and thank you much for the introduction. So what I want to talk about today is a story about mirror symmetry for kind of varieties that are not covered by the standard construction, and especially about this family of malefactor surfaces inside these ways of productive space, where K here is a positive figure. Okay, so let me say first that this is not a random period of surfaces, but in 2001, Johnson and Collard classified all anti-canonical, quasi-smooth, and well-formed, logged effects, F in weighted projected free space. So let me say anti-canonical means that minus K of X is OX1, quasi-smooth and well-formed is technical language to say that they are before and also the catalogue is as I mentioned, so in this case, the point log means that I have to be watching the gallery in this case. And then that shows us the anti-canonical class. Okay, and they find that we have only this period for each case and other 22 for other cases. So in a way, this period over all a bunch of the classification of logged effects. Okay, so why do I want to talk about this? As I said, there is an insert that they are kind of exotic in mirror symmetry. So the plan of the talk, I will start by recalling the context of mirror symmetry we're working on. Then I will explain how hypergeometric give a way to assess mirror symmetry for cases. So I will talk about odd mirror symmetry. And then I will start talking about homological mirror symmetry for people. Okay, so let me start from one. So when I talk about mirror symmetry for fun, in general, I speak about the correspondence that is on one side, a funnel or before, X of dimension M and on the other side, an LG model, YW, which of the same dimension, which for me, here is the non-compact complex manifold with some additional structure together with an homomorphic function. So in a way, I can think of these as one parameter family of n minus one dimension of a right. So the correspondence of the level of odd theory, as we've mentioned, is an identity of period. Here one takes the regularized quantum period of the funnel, which is a generating function for the open invariant. And here we have a classical period of the LG model. So in this period, actually, that is five different questions. So another way to state this is asking for an equality between local system. In here, we have a local system that comes from the quantum model. And what we're asking that is actually geometric and arises as a portion of the local system that gives the variation of the homology on the top. Okay, so this is all geometric, but we also have an homological version, which in this case, one formulation would ask for an equivalent between the derived category of coherency that we have on this side, and some analog of the focaya category for up in black. Okay, this was us who reminded me of them. So this is the picture of mercy maybe about what's actually known. And completely understood is what happens for final, which have an effective anti canonical divisor. For instance, the final search program would start from a funnel and would need a certain topic the generation, which you can actually have only if the linear system of my day is not so what happens is that and also as we saw in some talks yesterday, we would have the orific in homological metrometry that tells us what the anti canonical divisor would correspond to in terms of their move by this here. So what happens if it's minus K of X is actually empty, we do not know what to do in a symmetry and this can happen also in simple cases like complete interaction in way to project which are anti canonical in the sense I said in the beginning. And for instance, if I have all the way AI, they're bigger than one, and I am anti canonical, then minus K of X is O X one, and as a linear system will be just then given because our basis of sections for minus K of X is given by weighted bonomias of degree one in the variables, but if all the variables are greater than one, then okay, and you should look at the figures I brought down in the beginning, we are exactly in this case because all the weights here are bigger than one and I am anti canonical. Okay. Okay. So are there questions so far? Or okay. So in cases like this one, what I want to say is that a way to talk this problem is actually to observe that at least conjecturally, but it has been proven in some cases, the quantum period that appears here is an hyper geometric function. So the alphabetical local system is an hyper geometric local system. So what happens with hyper geometric local system? We saw many, many talks about them for now. And they have a special property of being motivated, meaning that they arise from, from the variations of the homology groups of the fibers of a morphine. And specifically for local system that hyper geometric and defined of a queue, we have a candidate, geometric realization, which is the so called GKZ model. So this is a family of hyper surfaces in a third, the darker structure is just combinatorially out of the electromagnetic data. And okay, this is a story about hyper geometric, but if we specialize these to the hyper geometric local system right into a mafano, we would get an LG model, which is the one given by the GKZ model. And the expectation is an identification between these local system and the variation that's called the dimensionality of the field B, the middle field B in the in this sheet. But so if you look at our definition that we have there of mirror symmetry, it looks like a mirror. Correct. But the problem is that it has the wrong dimension. So what's not right here is to ask, well, can I extract from these algae models, another one, which now has the right dimension. So these would give us indeed, like these algae model must have the process that this variation is that they have a portion of the other variation. And if we could find something like that, then this would be a mirror of the right dimension. In general, let me say that for a complete interaction like that, if they dimension it and then we could dimension it, these are dimension and plus. So we would need to reduce the mention. Okay. And this is the general idea that one can use to construct mirrors for panel without effective and canonical. And what I want to do in part two is to say what we get for this certificate. Please stop me if you have questions or things are not clear. It's a sort of a story. I'm not getting too much. Okay. So yeah, so these local systems have monodromes that is generated by matrices. And you can write them out combinatorially. So those are, they have a secret portion similarities. Because by asking that they are quite smooth, I'm asking that the only similarities that they have are the ones that are inherited by the action of the start that defines the product is created. And specifically, this one have similarities that are strictly locked terminal. So non canonical, they are disturbances bigger than minus one, but non canonical. Okay, so if you apply this reasoning to this case, I'm thinking, okay, what we get is a local system of rank, if K plus two, and wait one. So we do expect a pencil of current as a mirror in the sense of the definition. But these GKD model would be four fold overseas. So in other words, a pencil of three fold. And and the theorem I have with Alexia Corti is that indeed there exists a surface which has this shape, where my superpotential here is the projection to the variable P. And we do prove that the idea and table delta is complicated. But now I'm going to say something about it. And we find a period of this family, which is indeed the so called I function of our fun. So the point to construct these is exactly to extract it from these GKD models. And maybe I'm going to say some words about it. But let me first notice that if I fix it, what this is, is an hyperalytic curve because of this pressure. So this is a pencil of hyperalytic curve. And it's a pencil of genus Pk plus one. And because indeed the variation of homology must match the local system that comes from the bottom. And what will be strange about the the math W that we find is not all around. So the dependency of the variable on the variable T is higher. So it's really out of the usual kind of picture. Okay. Okay, so the way to construct this is indeed to find an LC model, which for each key realizes the following equality of structure. And it's really done in the case in the way we did it in this paper by analyzing the geometry of these people. So just to give an idea, these threefold here, as I said before, are either certain in glory. And what we do is to run a minimal model program that constructs a partial compactification of those. When you have this partial compactification, actually, you are, you're happy because the 13th in the history of the compactification is the same as the one I started from. So I can consider this in the same way for my goal. But this one has the on it, the structure of other types of vibration. So actually, instead of studying this group here, I can study two other factor sequence, only the h one over the base, which is a start with with coefficient in this sheet here, because I am panel. So the other contribution would be the player. And now when you have these, what you really need to study is the local system for the vibration. But what is the local system? This is the local system. We'll stop at a point. That is the h two of the fiber, which I will denote here, that's like this, because it's over a point factor, with coefficient in q. But again, in time, you know, this is just the picture group of the fiber. So one time, the picture group of the fiber, I would be happy because I would understand the global system. And the picture group that we have here is nice, because if you look at this, please, the question of the surface, you find that the picture group as rank two. And I have two distinguished generators, we plus and minus and minus that depends on a polynomial delta in the variable p and x. So now if I construct a two to one cover of these p with variable x, which is given by that polynomial there, sorry, I changed the variable. These two to one cover would be really a parametrization of the picard group that I have here. In other words, I would have an isomorphism between the local system that I guess on this side, given by this map, and the article five star q that I had there. But then here, I'm happy because instead of considering these comology group, now to this isomorphism, I can consider this one, which is really just the comology of the curve. Okay, but this is a really explicit and adopt analysis in this case. So one has to find ways to generalize it. If we want to study this problem for every convenient section. Okay, so now we do have mirrors for those. And they are strange, because they, they are not plural polynomial. And also, they are not the left side vibrations, because they are hyper geometric with a very weird fiber over zero. So one can ask, okay, we did construct odd mirrors for this, but can we actually also prove homological new proximity for those? Can we study the categories that we have in this context? Okay, so I'm going to do that now, if I manage to raise the board. But if you have questions about this part, please ask me. Another base of that is one. Yeah, because you have an academic family, which comes on academic. So you, you do have a complex variable, you can see that the differential equation over p one, we play it in the light. You can normalize to zero one and you know, well, yes, if you want a local system, yes, because on p one, you would, you're not a local music, we would have not in this case, no. Yeah. But you should, you should do study this problem for computer tracks and endotic varieties though, not in a way to exactly say, then you would have a DKZ family over multi variable dollars. So in that case, more complicated stuff. Okay, so you, what do you mean by higher dimension? No, I think it would work that either you find the tenor strategy or you do it case by case, which is a bit annoying, because of the case, you have to study the geometry of the model. But actually, I do have now some techniques for which from the polytope of the, from the combinatorial data and from the polytope of the different varieties, you can actually read off already what it will be if this polytope is lower width. So for width one, it would be a blow up along some complex extraction of something or for week two, it would be something like your quadric bundle. So you already have a domestic structure that comes in the other form. But in general, yeah, yeah, in this case, you have an hyper geometric function and you just write down the hyper geometric od that that satisfies and it's the same that you see the same. Yeah, yeah, so we okay, you had the x a k plus four inside the way to do that with space and you just take the, the edit function to factor it with the entry that the factorial ratio with a k plus four at the numerator and one and at the denominator, you have two to take that one. So it's really like factorial ratio coming from the degree and the weights. Okay, just to be sure. How much time do I have? Okay, I will take time. Okay, so now we want to start the homological mirror symmetry for this family, also because my final goal would be to understand what happens in this case, about all the urethanes that I have about young canonical divisor. And also, as I mentioned, but I will say later, my progression that that I have in this case is not flagship. So I don't know which category to put on the inside. Okay, so, so what we want is some equivalent like this, we're in general one here considered that there are a category of the ground advantage. Okay, so let's say that we have problems on both sides in this case, because our surface is here, that we had to start with, where so they're not dark and they're not. They are hyper surface in a way that I could say so I could consider the exceptional collection that comes from here, but this would not be full here, I would have a feedback category. So it's not clear how to like explicitly describe this. And, and here is even less clear, because my pencil of hyper elliptic curve has three single fibers as every metric. But first of all, this case is singular. I have in Ardon, yes, now he, okay, plus one, plus one singularity at the point, zero, zero, zero. And also the fiber over zero is, is a cat. The other fiber over C is no, that's fine. So I don't know which category should I consider here because this is not defined in final work. Okay, so what we did until now is to study the B side of this. And with the Rota, we find a description of this category. And this is a work in progress with Rota and Atherman. And hopefully I will have a So I start from here and I will give the description of the category that we think. So, so our main theorem. So in here, I will denote fairly after the stock associated to the surface and by after the surface and singularity. So in this formulation of metacymeter, it is important to consider these as a stock and not as a single variety. So the, the object I need to study here is really the direct category of X as a stock. Okay. And what we get is that for each K, the direct category that I have here, admit full exceptional collection, which are given by 10 plus 12k objects for each K. And we manage also to describe them explicitly. So the main, the starting point that we use to prove the theorem is the derived GL4C Mekai correspondence in the thickly case as given by Ishi and Ueda in 2015. So this correspondence has that if I have a surface with thickly portions in the lattice, I can consider it's minimal resolution. And also the canonical stock associated to it. And what what they prove is that there is a fully faithful factor by that embed the direct category of the minimal resolution into the one of the stock. And moreover, there is a semi-automated composition that gives the derived category of the stock as an exceptional collection. And then the image through this factor of the derived category of the minimal resolution. Okay. So in general, when you have the SL2 Mekai correspondence, you would have that these two derived categories that are equivalent. But in this case, the single IEA as I mentioned before are different. So they are log-terminal. So the group is not SL2 but GL2. And so I do not have an equivalent to two categories, but just an embedding. And I have a feed that comes an exceptional collection that comes from the singularities of my variety. So these results already tells us that since log delpates are rational, I will have overall a full exceptional collection, because here I will have a full exceptional collection and these are exceptional also. But if we need an explicit description, especially for homological neurochemistry, we do need to understand what the minimal resolution of F is. Okay. So that's what I'm going to do. I'm going to try to understand this F-field, which is a quasi-delta, because it's rational, but it reads higher. And also this raise is also an interesting question, because if you can construct a quasi-algae model for this, do we see a relation between the mirror of the resolution and the mirror of F to the stock that I talked about before? Well, I do know the monodroming already because of this hyper-geometric kind of technique. So that's nice in this case. I don't see that for the moment. Yeah. Okay. So now I'm going to describe how F-field looks like. So for this, I need you to remember the weight and the degree in that, because I need to understand what my singularities are. And the question of X, if you write the possible monomials, will look like something like that. So if I call the variable over there, I call them X, Y1, Y2, and Z for the weighted productive space, I can have monomials of these types. So the first variable to this power, then I can have a quartic in the second variable. And then I can have this product. And then I can have, so this is all I can get. And by looking at the polynomial, you realize that the point which is the origin of the last chart, so the point 0001 belongs to X and in X is a singularity of this type. So by combinatorics, this will give a curve F, which is a minus 14 plus one curve. And also the line X equals that equal zero, which is singularity, will interact X in four points, PI, which in here are singularities of these types. So if you resolve that, you get the one minus three curve that I will denote by K0. And then K minus one curve that are minus two. And I denote with PIK, where PIJ, where J goes from one to K minus one. So I said that these are four points of this type. This is probably a bit confusing. So I'm going to draw a picture. I said that we have a point P, which is singular and four other points. And actually, we have lines connecting them, because if you put X equals zero, you see that these are given by four lines in X and they connect one. So now when I blow up the point P, I get a curve F, which is the one I'm mentioning here. And when you blow up each of these points, you get a minus three curve, which I want to follow. And then you get curve, which are minus two. But then you have the three transform of those, which are minus one in here. And you have four objects like that. So this is the exceptional curve in there, but it doesn't tell us who X is as a circuit. But the theorem that we prove here is the following. So there exists a more cell that goes from the minimum resolution to X prime, which is a model path of degree two, with a generalized vector point. Moreover, if this X tilde here was the surface cut out by this polynomial that I brought here, this one is the one cut out by the same polynomial where I now put k equals zero. So if you put k equals zero in there, you find a 40 in P 2111 and that's the effect of the degree two. Okay, to draw this to make understand what I'm doing. What I'm saying is theorem that if you start from here, and you contract these four minus one curves, you do get the same picture where the curves above becomes minus one. And that is the curve that was minus 4k plus one becomes minus 4k plus one plus four. Right. And you can do that k time until you get the picture where you only have F and the zero that now becomes minus two curve. And F is now a minus one curve. But then you can contract this curve again, and you get to a picture like that, where these curves are minus one, and they meet in a single point. And I'm saying that this is a del factor of degree two, and is actually the del factor of degree two cut out by the same polynomial where I took h equals zero. Okay. So this is good from our goal, because not only now we know that that is rational, but we know that is a block a bunch of times of a del factor of degree two. So for a del factor of degree two, we have exceptional collection made by 10 of them. And in this way, we can construct explicitly through these blocks and all of formula and exceptional collections of these categories. And so combining these pieces that are described communatorially, we do get an exceptional collection for the whole stock. And if you sum up the numbers, you realize that gravity is made by this number of books. Is this okay? Your question before I read? Also, let me observe that these four curves that I drew here, I said that they are the divisor x equals zero in the surface. And so at subs in the variable as way to, they are not an element of minus k, but they are an element of minus k over. And also, when we blow them up, we obtain all three of them. So a question that is not really is, can, can this replace the anti canonical effective divisor that I usually have in the, in the mirror to make a picture? So I have five minutes, but a bit more. Okay. So in the last minute that I have, I want to say something about how to approach the A side, given the fact that it's really not. So for this case, I said that what is studied in the literature is this category, which is constructed for electric vibration. So one needs to have only notes in the fiber for curves. And then I need to choose a reference fiber and then some parts in the base and move to the key value. So I will have vanishing cycles, that will be the opposite of the category. And then I need a more logical to come to the more busy. So the problem in this case is that the mirror that I constructed is indeed something where this point here is the singular in the surface. This fiber is a cap. And then I have a note over my debate. So the, what the philosophy that was constructed to us was to start from the algae model and actually deform it to a lapse of vibration through a process that I think is called in general multiplication. And then once I have it, I can actually consider the associated category of the grand vanishing cycle, which now, since I do have a depiction of the other side, I can compare with the direct category of coherency. Okay, so in practice, since my why was given by something of this form with just the projection to key, we know that the point t is critical when in this case, when these delta is multiple groups. And in particular, the fiber which will be bad, the more than nodals when these are the rules of multiplicity bigger than one. So a way to, a naive way to classify this is to construct white field by a new polynomial delta, where now these delta defined, let's say, n to one cover, I did not think of the plane T with the ramification point of multiplicity at most too. Because now if I do use this new polynomial delta tilde, then this one will be a lapse of vibration. But if one does that, you know, naive way by adding the small epsilon terms to the original polynomial, what we get here, in here, we did in the written progression, the discriminant was something like p to the 18k plus nine plus times the nodal value that I had here. And here, I do obtain a separable polynomial of degree 18k plus 10. While from my previous exception collection, I expected 12k plus 10 critical values, because each of them will create a vanishing site. What we do observe in this picture is though, that these 18k plus 10 values, please, in the state that are very close to the origin, last 12k plus 10, which are these things. So one could wonder if the right category to consider here is this category, but we don't have an equivalent but that's an embedded one. And what we're doing at the moment is actually to go back to the case, k equals zero, which would give the surface. And for this surface here, these, as I was saying, these are the effects of degree two. So we have many mirrors. We have the homological mirror given by Urus Azarov and Orlo. We have the rationality surface that one gets from the Laurent polynomial of the Fano program. And then we do have the mirror that I construct with for the case k equals zero. And I'm trying to compare the three of them because here we have a categorical statement and here we have a period statement and we want to show that they are the same. And yeah, and potentially give some intuition on that. I guess I can stop questions for speaker. Do you have a guess as to the meaning of the six k? No, I have no idea why it's k. It's an especially thick is an under that was not coming from any pieces of the decomposition. So it's really weird. Yeah, because what I was hinting at earlier was just that given that you have the exceptional collection, by considering automorphisms, the van der Rijf category, coming from usual functorial operations, tensoring whatever the usual stuff, or in Mackay, you can get predictions on monitor on the other side. But it shouldn't just be the three, right, you should be getting more matrices coming out of this. And so then you could sort of tease out using structure those matrices. Yeah, yeah, connection to this. Yeah. And also, like in this case, like you have a very, very worked out with the matrix factorization and continuity. So that's like, yeah, but I would be I would be bolder. I would just I would just take take what you've got the theorem you've already proved, work out the matrices, and then try to struggle with your directly with your 18k plus 10. Yeah, I did work out the matrix. Yeah. No, I don't know. Yeah. Yeah. Okay, no questions online. Any questions? Let's thank you. Thank you.