 Okay, let me get started. I just wanted to mention a correction from the last lecture and I also wanted to apologise because several people in the audience tried to correct me at the time. There was a question about where did some minus one curves come from in this surface and after the lecture I realised they had mangled the construction actually to produce this surface in the first place, one has to blow up. So let me just explain. We started with the singular cubic surface in P3 and we resolved some singularities and I claimed at that point that these curves were minus two curves and there were some interior minus one curves. That was false. Actually what one needs to do is blow up those three points so the strict transforms of the lines are actually minus one curves and then blows those points up to arrive at the surface XD. So several people asked where these minus one curves come from and so they had identified there was an error in the construction. So I apologise. Also maybe you want to mention one source of the confusion maybe is that this elliptic vibration should be regarded as the mirror of P2 of a smooth elliptic curve. Very often in literature this surface is forced off as the mirror of P2 but if they would be more precise what they would mean is the mirror of P2 together with its toric boundary, a triangle of lines. So somehow under mirror symmetry passing from the triangle of lines to a smooth elliptic curve in a word smoothing the nodes of that triangle corresponds under mirror symmetry to these three blow-ups. So that's something that can be made precise. So I apologise for the omission last time. Okay so what I want to talk about today is mirror symmetry. So we had cluster algebra yesterday and I want to connect that with mirror symmetry and so this will be a sort of overview of the philosophy of mirror symmetry and in particular I want to discuss quite a bit of symplectic geometry but sort of necessary to motivate our construction. And so for that purpose I wanted to mention what I myself found a very useful reference. So Denier Rew has several expository papers on this topic. One of them which I recommend is this one. So I'm just going to give you the archive preprint number. So this is a good reference for this topic. Okay but let me begin with just a sort of general comments about mirror symmetry. So what does mirror symmetry say? Pairs. So let me talk specifically about the log context but of course historically the case of compact calabiows was considered first. So the repairs I'll use the same notation as yesterday so u will be a log calabiow except minus d and v another log calabiow with the following properties such that the symplectic geometry of u is in some sense equivalent to the complex geometry of v and vice versa meaning the complex geometry of u is equivalent to the symplectic geometry of v. So the first thing I need to say is well what do you mean symplectic geometry? Yesterday when we were talking about log calabiows we were just thinking them as complex varieties. So here what I've done is fixed a cala a symplectic form on each side so let's just talk about x so omega on x. So what's a symplectic form? So it's a smooth real two form closed the omega is zero and everywhere non-degenerate. On every tangent space it's a non-degenerate skew form and it's assumed compatible complex structure let's call let's write j for the complex structure on the tangent bundle. So what we require so it's invariant under j if I look at a form given by g is omega dot j dot this is a metric so this is positive. So if we're in the case we were discussing yesterday where we really have an honest complex manifold this is just the notion of a cala metric a cala form. But one thing that's important from the symplectic point of view is you might want to consider j being what's called an almost almost complex structure. So it's an endomorphism of each tangent space that squares to minus one but it's not necessarily integral in the sense that it doesn't actually necessarily come from the structure of a complex manifold. So I gave the definition in this form partly to explain that we can work in that more general context where we don't necessarily assume we have a complex structure just this j the almost complex structure. Okay and so you know so what's the sort of typical way an algebraic geometry would get such a thing? Well it'll take my variety x and embed it in projective space and I take the restriction of the natural form on projective space that's the so-called Fubini Studi form. So then the class of omega will just be the first sharing class of the ample line bundle defining the embedding in H2. It's roughly the same as choosing an ample line bundle given by ample line bundle A. Okay and so maybe I'll just say since this is a school another way to think what's this omega Fubini Studi. So this is just the natural choice of a scalar form on projective space. One way to think about it is you know projective space of course is Cn plus one minus zero mod C star. But another way to think about that is well look at the sphere. So I guess it will be the sphere in this R2n plus two so S2n plus one mod S1. So if we take this sphere inside Cn plus one you can take the obvious scalar form on Cn plus one. So omega is just some dxI dyI, Cn plus one. Restrict to the sphere that's S1 invariant. That descends the omega Fubini Studi on Pn. This is a special case of what's called symplectic reduction in symplectic geometry. Let me also mention there's something called the B field. You can also introduce a B field. So what's that? A B will be again a smooth real two form. Closed, the db is zero. And it's considered up to translation by an integral class. So when we talk about the class of B, we really think of it as a class in H2 Duran, XR mod H2 XZ. So this is something that you know was news to algebraic geometries when it was first introduced. Somehow under this mirror symmetry correspondence, the complex moduli of X is supposed to correspond to the symplectic moduli of Y. But in order for the number of parameters to match up under that correspondence, you had to have more parameters than were available by just varying the symplectic form. So you had to sort of so-called complexify the symplectic form. So omega in the story becomes replaced by B plus I omega. So called complexified caliform. So mostly I won't talk about the B field, but when it's no bother, I'll include it for completeness. So let me talk now about various formulations of mirror symmetry. So the one that's really motivating almost all of our work is what's called the Stroman-Gieau-Zaslo conjecture. So let me review that now. An extremely beautiful and influential conjecture. This was a way to understand geometrically what is a mirror pair. So they assert UV form a mirror pair precisely if, well maybe let me just say it one way. If UV are a mirror pair, then there should be dual special Lagrangian torus vibrations. So let me unpack that. What does that mean? So I've got two maps, U and V, mapping to the same base B. So these just, at the beginning, are just continuous maps. There's a common locus, let's call it B0 inside B. It's a common locus of smooth fibers, a smooth meaning in a C infinity sub-manifolds. And each smooth fiber, so let's call these maps F and G. So F inverse of B is a real N torus. So you know just S1 to the N. Lagrangian, what that means, you've got my symplectic form. When they restrict my symplectic form to the manifold, it's 0. So again, remember L is half the dimension. Oh, I should have said so. Here N is the complex dimension of these guys. 2N is the real dimension. So this is a half dimensional sub-manifold. And my symplectic form restricts to 0 on it. So just think about the linear algebra of skew form. So I've got skew form, non-degenerate skew form on an even dimensional vector space. What it's saying is that the tangent space of the Lagrangian at a point is a maximal isotropic subspace of the tangent space of the ambient manifold. Special, so special Lagrangian, I won't say much about this. It's a further condition involving the holomorphic geometry. So the imaginary part of the holomorphic volume form restricts to 0. So as I said, I won't say too much more about that. And what's this thing about the fibrations being dual? So if I take the corresponding fibre of G, it should be the dual torus. So the inverse of B, the corresponding fibre of the other map, is the dual torus. Now what do I mean by that? So somehow there's only one real N torus. I mean it's sort of canonically the dual torus. So if I think, without choosing any basis, I can identify L with H1XR mod H1XZ. Oh, I'm sorry, L. I guess strictly speaking I have to choose a base point on L to do that, but that won't be important. So this is just if you like this lattice, H1LZ, tensored with S1. An L dual corresponds to the dual lattice. Define it in an invariant way that does not depend on the choice of bases. Another way to say it is that if you look at the family as a whole, let's take the push forward. Let me use this O to denote the smooth locus. So look at the smooth locus, push forward Z from the family, the constant sheet Z. So that's just the local system whose fibres are H upper 1 of the fibres of the map. So these two local systems on the base are dual. So this looks like a lot to swallow, but if you unpack it, it's really a very precise description of how to build a mirror. Of course, I should say this conjecture came, really came from string theory, and with all sorts of additional things that I haven't mentioned which give further clues about how to do this in practice. Sorry, was there a question? So I'm just saying that I take a fibre of U and the corresponding fibre of L. The two fibres are these dual torus. Oh, the same conditions on V. So this L dual should also be a special Lagrangian torus in V for the corresponding data on V, so the volume form and the sublective form. No, actually they should be the same. So I should say one of the things about mirror symmetry is I said right at the beginning that the original statement of mirror symmetry, you have two calaviand manifolds which are both complex and symplectics. You've chosen a kale structure on both manifolds, and it should be true both ways, that the symplectic geometry of A is equal to the complex geometry of B and vice versa. So somehow it's usually, however, in practice what people usually do is break that into two pieces, just consider one side as a purely symplectic object and the other side as a purely complex object. So, for instance, in the original sort of in a green placer, et cetera paper, they would take a variety and compute Grom of Wooden Invariants, and now Grom of Wooden Invariants can be phrased entirely in the symplectic category, so that was their symplectic side, and then on the other side they're computing periods of a complex manifold. That's the complex side. But if you write down those manifolds correctly, so writing down a symplectic and complex structure on both sides, you can have it both ways. That's what I'm asserting here. Okay, but let's see some examples of this sort of structure. So what's a special Lagrangian torus vibration? So I should say immediately, warning, nobody knows how to construct these vibrations in dimension bigger than two. So these vibrations are generally very, very hard to construct in the compact case. However, in the non-compact case, there are lots of easy examples. So let's give some. So here's a simple example. Let's just take the torus. What's the SYZ vibration? So I haven't told you what the symplectic form is. Let's write one down. Let's take the obvious one. So let's say here of coordinate z1 up to zn. I can take as usual some dxj dyj, where these are just the real imaginary parts of the coordinate zj. Then SYZ vibration is just quotient by the compact torus. And so, you know, if you want to actually sort of identify this, for instance, you can identify it with Rn greater than zero, just by sending z1 to zn. Maybe I'll take the norm squares of these functions. So that's an example of a special Lagrangian torus vibration. Well, I didn't tell you what a holomorphic volume form is. It should be usual thing times some scalar, which I didn't figure out before the lecture. It'll be some power of i to ensure that this property is true. Okay. So that's an example where you can... Of course, it's not a very interesting example. But that's an example where you can write the thing down. Let me also mention maybe the case of K-3 surfaces, because that's another example which you can understand. So that's a compact collaviow of dimension 2, and simply connected. So what happens here is this is actually a hypercaler manifold. So you've got three complex structures, i, j and k, satisfying the quaternion relations, and it just so happens that if you have a special Lagrangian torus vibration in one of these complex structures, if you pass to one of the other complex structures, it becomes a holomorphic vibration. So, of course, it depends on our normalisations, but passing from, say, i to j under appropriate normalisations, this is what will happen. And so this will just be equivalent. And so, of course, it's easy to construct an ellipse vibration on a K-3 surface, so the Italians probably knew that 100 years ago. So this gives you examples of special Lagrangian vibrations on K-3s. And as we know, so what does that look like? What does a holomorphic elliptic vibration of a K-3 look like? Well, it's an elliptic vibration over CP1, and generically, there are 24 ordinary singular fibres, i1 fibres, they're just a pinched torus. And you have the usual sort of monodromy, so Picard-Lefchertz monodromy. You have a nearby fibre. There's a vanishing cycle, delta or something. And the monodromy on H2 is, oh sorry, H1 rather, is alpha goes to alpha plus or minus the pairing with delta times delta. Again, the sign depending on, your conventions, I guess. So this is, of course, not news to people that know complex geometry, but what I'm saying is this is telling you exactly what a special Lagrangian torus vibration looks like on a K-3. So it's a real two-torus vibration over S2. Generically, you'll have 24 singular fibres that are pinched real two-tori, and you'll have this monodromy as you go around the singular fibre on the, on the co-amology H1 of LC. So that's a case where, you know, so by some good luck in low dimensions, we can construct these vibrations, but in general, you know, it's a very, very hard analytical problem, and so one tries to proceed in directly using this conjecture as a guide. I'm sorry. Oh, so the mirror of a K-3 is another K-3. So topologically, it's the same, but it's got a different complex structure. Yes, actually, there is a very nice paper of Mark Gross. I mean, I'm very fortunate to work with Mark Gross because he's been working in the mirror symmetry for 20 years, and so he seems to know absolutely everything about the subject. But, you know, so for instance, there's this paper called Special Lagrangian Fibrations Geometry, and if you look in the last section of the paper, which is essentially self-contained, he just explains completely this construction of mirrors of K-3s from this perspective, you know, in a purely algebraic and differential geometric context. Yes, so I'm afraid, just to answer Don's question, that the first example also has the same defect, but the mirror is again C star to the N, so somehow it's not very exciting from that point of view, but at least it is an example. Okay, so let's talk a little bit more or sort of loosely now. What about if we just want to know the topology of this vibration? So let's say I take X-dialog clabiaw with maximal boundary. So remember, that means that my boundary divisor is a normal crossing divisor, and it has a zero stratum, a point cut out by N branches of the divisor. So just write it down again mathematically. So we've got a point in the divisor locally. It just looks like zero, the origin inside the union of coordinate hyperplanes inside Cn. So what's true is that the class of the zero stratum of the class of the fiber of the SYZ vibration on U, so this is some class, let's call it gamma in HN of UZ, is given by this real N-torus. So what do you do? Remember I removed the coordinate hyperplanes, so this is just C star to the N again. Locally, I just take the fiber of this moment map close to the zero stratum. So the idea is that near the zero stratum, you're sort of back in the toric picture and that tells you at least topologically what the SYZ vibration looks like. Of course this won't be true on the nose that this thing is automatically special Lagrangian and any properties at all, other than it represents the correct class in homology. And just as a remark that this isn't completely nuts, in fact this is uniquely determined. So of course the way I described it, it appears to depend on a choice, namely the choice of this zero stratum. But what you can show is that any two zero stratum are connected by a chain of Cp1s. So you only have to show that a one stratum, this is a result of a collar actually. Whenever you have a log collar VR with maximal boundary, have two different zero stratum, you can get from one to the other by a chain of Cp1s. Now it's easy to see, I write down this torus, let me just draw the two dimensional case. I write down this torus here corresponding to P and a torus here corresponding to Q. But if I now look at these meridians of this sphere and take the real two torus in the normal bundle, you can see that these two things are homologous. In fact this is a well-defined class and that's another reason why we like maximal boundary, that we can identify the fibre of the SYZfibration in this simple topological way. Okay, so now a slightly more interesting example, the blow-up of a toric variety. So for instance, remember this includes all the cluster varieties, a start of a toric variety and I just start blowing up. So what we're going to do is just modify the moment map, mu. And again so I'm going to give a complete treatment here, but there is a very complete treatment in this paper, Abuzade, Aru, and Katsarkov from 2012. But let me at least give you the idea. What do you do? So let's draw dimension two. So you start with the image of the moment map. So that's, sorry, so the moment map mu, I've got my toric variety, it's mapping my toric variety X or maybe it should be called X bar maybe to a polytope. And so corresponding to the boundary divisor that I'm going to blow up, there's an edge of this polytope. This is sort of a mu of the boundary divisor. And I'm picking a point to blow up. So the image of the point is one of those. So over this, so maybe I'll just say it for a second. Over the general point here, there's a real two torus. And over this point or any point on the boundary, this collapses onto an S1. This is the image of the, this is the polytope P. So what do I do? So I'm going to blow up and I want to construct a special Lagrangian vibration on the blow up. So here's a nice construction of it. What I do is I just take a little triangle in this moment polytope. So I'm imagining this as, so I guess maybe I'm just thinking about this as C cross B star mapping to R cross R greater than zero or something. And so I can just use the coordinates here on the image. And let's just take this to be distance epsilon and this to be distance epsilon in the X direction. Okay. And so I remove this triangle. And what I'm going to do is I'm going to do a surgery on the total space, which corresponds to, again, just removing this part of the vibration. But now I'm going to glue this by a simple shear. So glue this map. So if I treat this as the origin. So for the purposes of this discussion, I'm just going to take this to be the origin and then just writing down the linear map. So what is it? So I guess one zero is constant and it's the opposite of the usual direction. Okay. And so that produces some singular base. So this is no longer embedded in R2. And so I've produced a manifold which has a torus vibration at least away from this point here. And it's got monodromi around this point given by this matrix. So topological T2 vibration over this base, maybe B minus the point, with this monodromi. And then the claim is that you can actually make this into a special Lagrangian torus vibration on the blow-up of the toric variety. So for a suitable choice of some plectic form, this is Lagrangian vibration on the blow-up of x at the point. Of course, x tilde. And the way to think about it is what you did was you took the omega on x and at least in terms of classes, you just pull that back and you subtract a little bit of the Poincaré jewel of the exceptional divisory. So this is exceptional divisor. That has the effect. So this is the usual form. You don't want to have an ample divisor on x. You want to construct an ample divisor on the blow-up. Take the pull-back of the ample divisor. You just subtract a small multiple of the exceptional divisor. And of course, you can assume, of course the Poincaré jewel, of course it's a class, but I can represent it by a form that is supported in a neighbourhood of E, a sort of tubular neighbourhood of E. And in fact, you can see E in this picture. So what happens over p is we're going to fill in a pinched torus fibre over this point. So yes, so maybe the picture is like this. Here's this pinched torus fibre. Over this boundary divisor, I've just got a cylinder. So one of the directions of the torus has collapsed. And the exceptional divisor, that's a CP1, looks like this. So the class that collapses on this torus and the class that collapses at the boundary is the same. And so I get this CP1, this is E, copy of S2, fibring over the interval in the base. OK, so that's probably a lot of information, but if you want more details, check out this paper. It's a very nice description. But also since algebraic geometries often have seen this picture, let's just compare it to this well-known picture in algebraic geometry. So if I have the moment polytope for a toric variety, and I want to blow up a torus fixed point, so this is like, now this is the case, and this is the fan, and this is the corresponding polytope. And I want to blow up this point. So everybody knows how to do that in algebraic geometry. It's easy. What you do in terms of the fan is you just add this ray, which is the sum of the... generated by the sum of the two primitive vectors along those two rays. In terms of the polytope, so remember this is some normal polytope to the fan, so what you're doing is just chopping off the corner. And again, so when you draw a polytope for a toric variety, implicitly there's an ample divisor in the picture. So here the ample divisor is replaced by the pullback of A with a multiple of E subtracted. And again, what will happen is that what you've done here is removed a triangle of side length epsilon. So you can see, if you knew about this picture, it's not too much of a stretch to see that there's a similar picture when you blow up a non-torus fixed point. So of course it's harder because you can't use the equivalence under the whole torus, but you can use equivalence under an S1 in that case. And so somehow one can do it. Oh, one more comment about this example is that if you remember, so remember Al's discussion of mutation of polygons. So the first type of mutation Al discussed. Oh, sorry, Al Casbrick. I don't know, I just fell his name. I'm sorry, is it Casbrick? No, no, sorry. Al Casbrick. So mutation of polytopes. So what I mean is he had this picture, so he would take some polytope and what he would do is roughly speaking there'd be some origin. He'd remove a triangle here and sort of insert it on the other side. Maybe I'll draw a square so my analogy will be easier to visualize. It's not an analogy, it's really explicit correspondence, but I want to show what would happen for the square. You would remove, so here's the origin, you'd remove this triangle and then you'd insert, if I understood incorrectly, you insert the triangle above. So what happens is, yes, so do this. That's our picture. And then you glue and you insert it upstairs. So somehow you do this. So this will be the mutation. So what I want to say is that what we did was sort of stopped halfway. So the halfway point is our example. In other words, if you just cut out this triangle, glued it together and didn't do anything on the other side, then you'd have exactly our picture. But if you go the whole way, you'd have two of our operations. First blow up, then blow down on the other side. This is exactly what we saw last time, just an elementary transformation. So somehow this is another connection with the other series of lectures. Questions about this example? That's right, so you have to work. You have to produce the symplectic form. I've really just described it at a topological level, but if you look in this paper, you've used the Lagrangian... Oh, in Dimension 2, yes. And they say in higher dimension they don't quite have it, but they expect it to work. So I defer to the symplectic geometries on that point. OK. Next I want to talk about homological mirror symmetry. So that's Konsevich's categorical formulation. Konsevich. Extremely beautiful conjecture. So what do we do? So we've got these two mirror pairs, two mirror collabiales. So again, this was originally formulated in the compact case, but there's an analogue in our context as well. So on the one hand it's something that we understand. So it's the derived category of coherent sheaves on the variety V. Just the bounded derived category. On the open manifold V. It would be isomorphic to some form, or equivalent as categories, to some form of the Foucaire category. Of course I'm sweeping a huge amount of stuff under the rug here, but let me just add the words. So first you have to actually take the derived version and you have to take it split closure. You have to add extensions. I'm sorry, you have to add direct summands. What else? Yes, and because we're in this non-compact setting we have to take what's called a wrapped invariant of the Foucaire category. This is in the non-compact case. But anyway, so that's just like the small print. What is the Foucaire category? Well, roughly speaking, it's a category whose objects are Lagrangian submanifolds. So the objects, well, yeah, it's a Lagrangian submanifold. So what do I mean? Again, it's half-dimensional. It's omega, the symplectic form restricts to zero. Together with some additional data. So together with various ways of saying it, one way to say it is a U1 local system. In other words, that's the same thing as a flat complex line bundle with an invariant metric, or let's just be completely basic about this. How do I define such a thing? I have my L, I just have a representation of the fundamental group of L into U1, up to conjugation. Now U1 is commutative, so that's just the same as an element of H1. So this is just, you know, I've got this complex line bundle. If I go around a loop, this tells me the monodromy of the connection, the flat connection around that loop. And the morphisms, you know, I won't say anything about it really, but they're given by what's called the Lagrangian blurcomology. And so it's some sort of quantum version of intersections of Lagrangians. Okay, and so again, so there's a very nice preprint by Aru. So if you want to know more about the Foucaille category, I can't recommend a better reference. It's called the Beginner's Guide. We'll get the archive number right. But that's a nice reference for all this. Okay, but why am I telling you this? So the point is this sort of clarifies to some extent the SYZ conjecture. So why on earth should the SYZ conjecture be true? Well, if you think about this conjecture of consavage, it just somehow sort of explains. So the relation to SYZ. So here's the idea. So if I take a point in V, then I can consider the skyscraper sheaf at the point, I'll just write OP, but that's an element of the derived category, of course. And the idea is that under mirosimetry that should correspond to an object, the Foucaille category, which is given by a Lagrangian L. Together with this local system, I'll just use a symbol rho for the associated representation. Where, so remember our picture. So what was the SYZ picture? I've got these two vibrations over the same base. So if I have a point in V, I get a Lagrangian, so L will be the corresponding fibre of F. Okay, and I'm supposed to give myself a local system on it. So rho, remember that's in H1LZ into U1, but that's just the dual torus. That's the dual torus of L. And so the idea is, so that's where my P lives, so that should be what rho is. Okay, and so maybe to say this a little bit better, so more generally, we expect, it's not proved, but this is kind of the heuristic that we should have in the back of one's mind, is that this equivalence is given by some kind of real Fourier transform. Fourier mucai transform. Is there any more chalk? I'm running out of chalk. Oh, maybe this is chalk here. So let me just try to explain what I mean. So everybody's probably, well, I hope some people are familiar with this construction of mucai, the original construction. You have an abelian variety and the dual abelian variety. There's no way to go from sheaves on the abelian variety to sheaves on the dual using what's called the Poincaré line bundle, using the fact that the dual is a modulai space of line bundles on the original abelian variety. And so that's sort of similar here that we have a torus and the dual torus given by a modulai space of local systems on the original torus. And so one can try to use some kind of Fourier transform to take a bundle on L and produce some bundle or constructible sheave on L dual. And so somehow, you know, this picture here that when we took a point on one of the tori, on the other torus it was supposed to correspond to a local system, a rank one local system, is the analog of in the mucai story if you start with a point on one of the abelian varieties it corresponds to a degree zero line bundle on the mirror abelian variety. So, you know, if you want to see more about that then there's a book, Alexander Polishuk, a name I can never spell. Let me try. On abelian varieties. Thanks a lot. And the Fourier mucai transform. OK, but that's an aside maybe. But what does this tell us about, back to our problem about the SYZ conjecture. So why does the SYZ conjecture look natural now? So somehow what we're saying is, of course what is V, or just completely tautologically, is the modulai space of points. In other words, it's the modulai space of the objects OP in the derived category of V. So that's, I'm not saying anything interesting there, but on the other side, so it should be a modulai space of the L row in the fucaya category of U. So in other words, where L is a fibre, an SYZ fibre of U, and rho is some arbitrary U1 local system. So that's sort of somehow how to think about the SYZ conjecture at a conceptual level. I think I should stop here. I'm starting at 12, right? OK, thanks a lot.