 I'll be talking about, well, obviously there'll be some amount of introductory material, but the new work is joint with several co-authors, so there's Mark Gross at Cambridge, Sean Keele at UT Austin, Konsevich at IHES, and Bernd Siebert in Hamburg. So those are my collaborators for the joint work. So just to give a little bit of background, there's a notion in commutative algebra and combinatorics called a cluster algebra. So these were originally defined in a very algebraic fashion, and by Fomin and Zelovinsky. And later, Fok and Goncharov gave a more geometric description of them. But right from the outset of the subject, there was a conjecture that a cluster algebra had a canonical basis. So it's some algebra over a field, say the complex numbers, and what they were conjecturing is some way to give this algebra a canonical basis as a complex vector space, some infinite set, which is a basis of this algebra. And they were motivated in part by some examples which they discovered, which they called finite type cluster algebras, where there was a distinguished set of elements called the cluster monomials, which were already a basis. However, there are some more complicated cluster algebras where these cluster monomials are linearly independent, but they don't span the whole algebra. So they were sort of looking for some additional elements to add to this set to form a basis. And sort of one of the reasons they expected this would be true was some work of Lustig. So in representation theory, Lustig had described something called canonical basis for representations of algebraic groups. And so they were partly motivated by that work to introduce the notion of a cluster algebra in the first place. So the idea that there should be a canonical basis around was also inspired by this work of Lustig. So the new insight which led us to work on this problem is that this should be regarded as an instance of mirror symmetry. So cluster algebra is the ring of global functions on some variety, a cluster variety. And that variety is, in some sense, a Kalabiyaw. And it's not a compact Kalabiyaw, it's a non-compact Kalabiyaw. And if you apply mirror symmetry to this variety, then this canonical basis arises naturally from a construction, a symplectic construction on the mirror. So that's a very rough overview of what I want to talk about. But this is at least to motivate what I'll do today is I'll talk about what is the correct notion of a non-compact Kalabiyaw variety. That's what's called a so-called log Kalabiyaw variety in our context. And how do cluster algebra fit into that framework? So that's what I'm going to talk about today, log Kalabiyaw varieties. So let's have a definition. So what will it be? Well, it'll be a quasi-projective variety U, and it will have the property that's given by, so we take a smooth projective variety X. Let's work over the complex numbers. And I remove a divisor D. So D will be a normal crossing divisor. So locally it just looks like a union of coordinate hyperplanes in Cn. Let's also fix a bit of notation, let's say the dimension of X is n. And it should have the following property. So KX plus D should be 0. That's the log Kalabiyaw property. And so equivalently, it seems that we have a holomorphic n form that was called a holomorphic volume form on U. So what do I mean by that? So that's just an n form, top form, which is nowhere 0. And it has simple poles along the boundary. So with simple poles, D is a sum of irreducible divisors DR. That's the notion of a log Kalabiyaw variety. And I'm not going to, that's right, effective reduced no multiplicities. That's right, that's right, yeah. So for various reasons, this is regarded to be the correct notion for correct generalization of Kalabiyaw's this open non-compact setting. So one reason is related to the Stromanjiyaw's as low conjecture that if you want to have this so-called special Lagrangian torus vibration on U, it forces the form to only have simple poles along this boundary divisor. So for various reasons, this seems to be the correct generalization to non-compact setting of Kalabiyaw varieties. Okay, so a couple more definitions. We'll say it has maximal boundary if this divisor D has a zero strata. In other words, a point cut out by n branches, analytic branches of D. So in other words, symbols a point in D, so locally isomorphic to just the origin in the union of coordinate hyperplanes in CN. And one more definition we'll say it's positive, this variety U is affine, or more or less equivalently D supports an ample divisor. So for instance, the case that Alessio and Al have been talking about, if you have a far-known variety and you have such a divisor in the anti-canonical linear system, that's an ample divisor, so this will be a positive example. Yeah, so I say a log-Kalabiyaw has a maximal boundary. If when I look at this boundary divisor, there's a point, so just sort of a boundary has a zero-dimensional strata. Is it clear? I'm sorry. Ah, good question, yes. It's independent of the choice of compactification, actually. So that's not, that's a theorem, but yeah, so that's right, that's right, yeah. Let's just give an example. So, you know, so here's an example. I can take P2 with a smooth elliptic curve, or I could take P2 with a rational nodal curve. This guy's maximal. Was there another question? It is a property of U, yeah, I won't prove it. It's not too hard. Basically what you have to do is show that this property is invariant under blowing up. Yeah, I'll probably abuse notation a little bit and talk about U or the pair X of D. But I will allow, yeah, so it's a, yeah, so I'll allow myself to pass between different choices of X, D of the same U fairly frequently. So I think really one should think about U as the object. Okay, or it's roughly equivalent to say that D sports, certainly if D sports an ample divisor then U is a fine, yeah. And the reason I mention this is often that, you know, you'll, how do you construct such a thing? You know, you start with your X and you choose your divisor D. So if you happen to know D is an ample divisor then, okay. So let's see, what about some more examples, more sort of general examples. So the first example really to be example zero is that U is just a torus, algebraic torus. So here, my holomorphic volume form is just coordinates in this way, DZ1 over Z1, wedge, et cetera, et cetera, wedge DZN over ZN, where Z1 up to ZN are the characters of the coordinates on this torus. And so here, you know, XD, what can I take? Well, any torus X is any smooth, projective torus variety containing, you know, with torus C star to the N. And D is just the so-called toric boundary to complement the torus. You know, if you want to be completely specific, you could take X to be projective space and D to be just the union of the coordinate hyperplanes. So that's somehow the sort of base case which is rubber easy. Maybe I'll just say immediately, so I said that, you know, these varieties in some generality tend to have a canonical basis of global functions. So in this case, one can see it immediately. Look at the global functions on this variety. Well, those are just given by the canonical basis, just given by the characters torus, so all the mononials coordinate ZI. So this is a canonical basis determined up to scale. So why is it canonical? It's just exactly the units in the ring. So that's the characterization of the spaces. Of course, this isn't a very interesting example, but we'll sort of see in more complicated examples. This basis can be very interesting. So example one, let's suppose I have a log club. Yeah, I want to construct another. Here's one way to do it. So let's pick a subset of co-dimension two and I want this to be contained in a unique component of D. And so I am assuming I want everything to be smooth and normal crossing. So I should say Z is smooth and intersects all the boundary, intersects the remaining components transversely. Then let's blow up Z. So define X tilde, D tilde, B the blow up Z and D tilde will be the strict transform. I'll write D prime, strict transform of D. So then check that this is again log club. Yeah, so KX plus D tilde is zero. So let's just give the picture at least dimension two at least looks something like this. Here's my X D I choose. So in dimension two, I'm just choosing a point contained in a unique boundary divisor. I blow up that point. There'll be an exceptional divisor E over the point P and D tilde is the strict transform of the boundary. So E is not contained in D tilde. The higher dimensional case is very similar. That's right. Yeah, so if you do this enough times, you'll lose the positivity property. Yeah, so maybe I should just say, why did I write these two conditions down right at the beginning? So our results will be the strongest under both of these assumptions. So in case you have maximal boundary and positivity. Questions so far? That's right. Yeah, that's actually that's my next comment. So yeah, so I'll refer to this as a non-toric blow-up. Yeah, so let me make the following remark. So here to happen to the interior U, so U tilde. So this contains U. So pi, this blow-up, it's called pi the blow-up, is an isomorphism over U. So just the inverse of pi includes U and U tilde. But if we look at the complements, what's that? That's this exceptional divisor E intersected with U tilde. So this is a line bundle over Z. So we've added something to the interior. And so there's some sort of, in symplectic geometry, this is some sort of surgery. So sometimes called a handle attachment in symplectic geometry. So Weinstein. Well, I had a P1 bundle and I removed the zero section. Oh, maybe you're right. Okay, maybe I should say an affine. What do you call that? Okay, that was probably worse than it was, right? Thank you. Okay. Yeah, so I just wanted to go back to Sasha's remark, is of course we could but also blow up a stratum of D of any dimension and define, so this will be X tilde goes to X, and then D tilde should be the full inverse image. So we include the exceptional divisor in D tilde. So then, of course, U tilde hasn't changed. So we regard this as sort of a, from our point of view, the most important thing is the interior. So we regard this as some sort of trivial operation that we should be allowed to do whenever it's convenient. So this is what we call a toric blow up. We'll do this whenever it's convenient. Because, as I said, U is the main thing and U is unchanged. Okay, so now if you know a little bit about rational geometry of surfaces, here comes the first exercise. So first of all, we say a toric model of a local R-V-L-X-D is a diagram. Start with R-X-D. We do a sequence of toric blow ups, let's call that X tilde D tilde. And then we do some non-toric blow downs. In other words, we have some X bar D bar and some non-toric blow ups here, sequence of non-toric blow ups. Such that this eventual variety X bar D bar is a toric pair. Toric variety with its toric boundary. Oh, you can imagine there's some kind of roofs. To be honest, yeah, well, yeah, no, it's not a silly question. I mean, so this is a big theorem if you just ignore the boundary by who is it, and that if you have any birational transformation between smooth projective varieties, it can be factored into a sequence of blow ups and blow downs. That is a theorem, I think. Yes, yes, right, right. Yeah, but you know, all I'm saying is that's a non-trivial theorem and I haven't even thought about whether that's sufficient to, yeah. But let's, I mean, so this is some kind of definition which may or may not be correct. Yes, yes, right, right. So I'm, as I said, maybe this needs slightly modifying, but this is the essence of it. And then the lemma I want to mention, just to give you some idea, is that if we're in dimension two with maximal boundary, then there is a toric model. And so, of course, in dimension two, birational geometry is very explicit. Any birational map can be factored into a sequence of blow ups and blow downs of points. That's a very classical result using that it's not hard to prove this lemma. How to say? So one can always blur up the bound, you know, do toric blurts on the boundary. Oh, I'm sorry. Yeah, so no, there's no uniqueness. Right, that's a good point. Yeah, this is highly non-unique in general. Yeah, so sorry, thanks for the comment. Let me just give an example. So for instance, yeah, so maybe the best example would be something like this. If I just take, let's say I take P2 of an elliptic curve, actually, no, I don't want to do that, rational nodal curve, and I blur up a bunch of points here, then it's sort of a classical fact, I'll get some minus one curves lying over these points. And so that is an instance of a toric model. But there'll be lots and lots of minus one curves on XD that have absolutely nothing to do with the original exceptional curves. So there'll be all sorts of different ways of blurring this down to P2. So, you know, so for instance, as everybody's aware, you know, if you have a cubic surface, that's obtained from P2 by blurring up six points. But there are 27 minus one curves on the cubic surface, there are many sets of six curves inside that 27 that you can choose as the exceptional curves that will blow down to P2. And so, you know, that's an example, a very classical example, you know, which sort of, if you sort of study that, it's related to the value of V6 and so on and so forth. So somehow the point is that there can be many, many different toric models and it's a choice to choose one. Yes, sorry, I wanted to take a rational nodal curve here. Yes, sorry. The point is, oh, sorry, sorry, so that's not a toric model, sorry. I'm sorry, I'm sorry. Yes, so let me call it D bar or something. And I'm sorry, that was my fault. Yes, so blur for a bunch of points here. I'm sorry, this is supposed to be an example of a toric model. What I'm saying is it's not unique because there are lots of minus one curves here which can be blown down instead. Yeah, thanks for the correction. Sorry, okay. Yeah, so let's just do a quick example. So here's something where, you know, the toric model is not completely obvious, but it's not hard if you think about it. So for this picture, actually maybe I'll slightly license here. Yeah, I mean, actually I'm always going to assume that D is non-empty. Yeah, but another example would be something like P2 of an elliptic curve. So somehow that's never going to have a toric model because you can't get rid of the elliptic curve by blowing down. What I'm saying is that, you know, if you have this toric model picture, there's going to be a relation between the two boundary devices. They're going to differ from each other by a bunch of P1s. So, yeah. Okay, so this is an example of a toric model. So I'll label the self-intersection numbers of the curve. So I've got a conic and a line in P2. So where's the toric model for this surface? Well, let me describe it in the following way. So first I'm going to blow up this point here. One of the intersection points doesn't matter which one. And now I want to blow up this point here, Q. Now this looks suspiciously like the surface F2. So in fact, there is a minus 1 curve like this and that can be contracted to the toric surface F2 with its toric boundary. So this is the F2, it's just a rational ruled surface with a negative section of self-intersection minus 2. And so where did this curve E come from? So if you sort of trace it back, what it was, if you take the tangent line, it's the conic at the point Q, that's what this curve is. It's the strict transform of that line. So after one blow up, that line is here. After one more, it's meeting the boundary at a smooth point. Oh, so I'm just drawing the self-intersection of the curve. So it's the, yeah. It's just a convenient way to keep track of, yes, sorry. And so here's my exercise. Exercise one, a find a toric model P2 with a rational nodal curve. Rational nodal cubic. And let's just one more exercise. It's no vibrational java cube smooth surfaces. It's fairly easy, but this is the case. So show that if B is smooth, Xd log Cy2, then that does not exist at all. Incidentally, perhaps I should have said, it's easy, but let me put it out. Note, by the adjunction formula, Xd log Cy2, Kd0, in other words, the arithmetic genus of the curve D1. And so D is either smooth elliptic, rational nodal, or a chain of a wheel of P1s, a cycle of P1s. So we're assuming, remember, it's nodal. So we don't allow, for instance, a cusp. So let's also give a sort of negative example in this direction. So let's have a log Cy3, which is irrational. So certainly if it's irrational, so it's not birational to P3, then it certainly doesn't have a toric model. So if you know a little bit about Farno-Freefolds, this isn't hard. Just by definition, I assume that D is a normal crossing divisor. And you might say, why don't you consider the cusp? So in fact, if I had a cuspiddle curve, and I tried to resolve the boundary to normal crossings, then the boundary would, if it stays anti-canonical, I would have to have a curve of higher multiplicity. So there would be a curve in the boundary where the volume form has a pole of higher order. And so that's kind of out of this realm where we understand how to do mirror symmetry. Yeah, sorry, I meant to add one thing, so this is with maximal boundary. Necessary condition for a toric model. Sorry, I'm saying that this is necessary to have a toric model. That's right, yeah. So what I'm saying is, oh, sorry, let me just continue. Yeah, if I did say it, I didn't mean to say it. I'm saying, no. No, no, because... Yeah, I'm not claiming any kind of if and only if here. I'm just saying that certainly a necessary condition, both of these things are conditioned, so both of these things are necessary to have a toric model. First of all, your variety should be rational, and secondly, the boundary should be maximal. So I'm just saying, of course, in dimension three, we know a lot about pharno-manifolds. There are some examples of irrational pharnos. So, for instance, I can take X, be a smooth quartic, in P4, that's irrational according to Iskosky and Manin, but I still need to cook up a maximal boundary divisor. So what's my D going to be? Well, it's a hyperplane section, so it's in the anti-canonical linear system, which is just O1. It's a hyperplane section. Okay, so this presents a problem. How are we going to make it have a zero stratum? Well, in fact, what we'll do is make a singular example here that can be resolved. So what I'll do is take D to be singular with what's called a cusp singularity. So let's be specific about this. So for instance, this is my quartic as an example, and D will be U equals zero and singularity, so the cusp singularity, of P and D. So I'm setting U equals zero and let's look in the affine piece where T is non-zero. So then this becomes zero to the four plus Y to the four, X, Y, Z zero inside C free. So that has minimal resolution, a cycle of P ones. That's an example of a so-called cusp singularity. Okay, and so now what you can do is take this pair, X, D, and do some blow-ups, so resolve to get this guy with maximal boundary. So the picture is in X itself, so X is smooth, but this guy has some kind of nasty singularity. When you blow-up, you do just one blow-up, you get this picture where you have two divisors that meet like this. You've got two divisors meeting along this triangle of P ones, and if you sort of do a couple more blow-ups, you get to this guy X to the D. You can kind of see, quite normal crossing, but here's, these guys are going to become zero strata here. Yeah, so it's not too hard to produce examples which you absolutely cannot have a toric model. Something similar will work, yeah. Yeah, yeah, I think basically the same thing will work. So let me just make this remark that, roughly speaking, a toric model or the existence of a toric model is the same as having a a torus, a full-dimensional inside U. So certainly if I have a toric model, I get this torus, so if I have X, I've got the torus in here, and that lists to here. So this map is an isomorphism over the torus, and so I get my torus in X. So certainly this direction is obvious. The other direction, well, this is sort of related to the discussion we had earlier. What do we know about birational geometry in higher dimensions? So you probably need some sort of minimal model program, and you need to think about singularities. But something like this is morally true. Okay, so now I can tell you what a cluster variety is. So two conditions. So the second one we've already discussed at length, cluster variety is a log collabial U, such that so first of all, there's a two-form, so sigma is going to be a non-degenerate holomorphic two-form on U with log poles along the boundary divisor. Again, so I'm writing U is X set minus D and again this condition will be independent of D, basically due to the lean. So what does it mean to have log poles? So I've got this sheaf on X of holomorphic one-forms with log poles along D. So this is just locally generated by so yes, at a point P so here I'm saying that P in D in X, where D has our branches so I'm allowed to have D log of Zi for i corresponding to a component of D and have the regular a DZJ for the other J so then my sigma should be in a global section of this wedge two of the sheaf. It can locally be expressed in terms of these differential forms at a point P. There should be a Turing model. So this condition one you should think of as some non-compact analog of a holomorphic symplectic or hypercaler variety so I like to call this log holomorphic symplectic and so of course an immediate example is that in dimension two this is nothing new so in dimension two cluster well the first condition is vacuous because you can just take sigma to be equal to omega n is equal to two and so cluster is the same as the existitoric model which as we just saw is the same as maximal bound. But this is a non-trivial condition in higher dimensions the same as in the compact case yes it wouldn't be because the boundary is certainly not an empty you see yes so no it does not okay so there's a little lemma really clarifies this first condition so so what does it mean to have a Turing model so the blow-ups are very constrained a Turing model you know for cluster variety so remember we've got some picture like this X bar do some blow-ups and this is a Turing variety and so what I eventually want to get a form on X or equivalently on X tilde sigma but of course that will define a form sigma bar on X bar because outside co-dimension 2 I can just take it to be sigma and then this will extend over co-dimension 2 the usual sort of Hartog's property so if I have a form here I'll certainly have a form sigma bar on X bar but now we know this is a Turing variety so what it follows is that sigma bar is actually constant coefficients so this is just given by a skew matrix so this is a non-degenerate skew matrix so I want this to be have log forms along the boundary that's right so what I was trying to explain just then maybe it wasn't clear let me say it again if I have the form on X or equivalently X tilde I can get the form on X bar automatically because after all if I just remove the co-dimension 2 subset of X bar then that will be an open subset of X tilde I can just restrict the form no no no it's okay so this is a bunch of blow-ups so there's an open subset in here let's call it maybe V which maps isomorphically and this is complement is co-dimension 2 so I have the form on V it will automatically extend to just by this Hartog's property so what I'm saying is if I have sigma up here I certainly have this guy down here okay but the why is that a good thing is because on the torus we know that this let me just write it down if I take this log forms on the torus that's just a free module it's a trivial sheaf so it means that when you take global sections you're just going to have so this form just becomes a matrix so now what's the condition so now I say okay now start with this form and start blowing up what's the condition that this form lifts that's a non-trivial condition so let me tell you what it is oops where are my notes so what's the condition so I must blow up the following type of center so it's a co-dimension 2 center so C will be this center will be the intersection of some component the boundary with the hypersurf is given by a character of the torus so let me write this down and explain so here where so what's C so C inside D is a component of the boundary chi is a character of the torus so you know this is just a monomial lambda is just some scalar non-zero scalar and chi is determined by following condition so I have this form sigma bar I've got a divisor I can take its residue along that divisor so now I'll get a log 1 form on the boundary divisor that should be equal to the log chi up to a scalar so roughly speaking what I'm saying is if you take the two form on the toric variety the boundary divisor take its residue you'll get a 1 form so that defines some kind of foliation of that boundary divisor you must blow up a hypersurface contained in that foliation sorry this is just a scalar so yeah that's the condition there's no condition on lambda that's right yeah I mean the whole thing remember you have to remember that we're in this sort of situation we're on a torus this form is constant coefficients so I know that when I restrict to this boundary divisor I'm going to get something with constant coefficients that's right yeah there is some rationality involved yeah exactly so somehow a priori when you restrict this you just get a 1 form with arbitrary complex coefficients but I'm saying up to a scalar those have to be multiples of a rational vector yeah do you see what I'm saying there is a non-trivial condition here if this divisor when you restrict if you get something which is a tuple which is not a scalar multiple of a rational vector then you cannot blow up there's no character somehow if you think about in terms of foliation that doesn't close up and you don't get algebraic manifolds there's a non-trivial condition hiding here if this guy is not rational then you won't be able to do any blurb at all and so what I want to I'm out of time but let me just say so what's the picture so the picture of course is changing coordinates without loss of generality let's say that this component is given by z1 equals 0 and this chi is given by say z2 so then what are we doing so up to what we get it's just the dimension 2 picture across a trivial direction but remember dimension 2 is just this simple thing you've got a a toric boundary divisor you select a point and you just blow up and so what we're saying is that in the cluster picture just look at a single blow up it's just exactly this dimension 2 picture across a trivial direction of course that's locally so for a single blow up but globally of course I see I sort of like to draw this picture here's a picture of your toric variety of its toric boundary what you'll have is that these guys that you blow up they'll be now hypersurfaces in the boundary divisors but they won't line up so it won't be globally a product but it will be locally along this stratum a product so that's kind of the the condition that comes out of this log holomorphic symplectic condition under the assumption that you have a toric model so what I've presented here is a sort of algebraic geometry version of what the guys in cluster algebras were doing to begin with so they had you know they in their combinatorial description they had this skew matrix and then there was I'll describe more next time there was this process of mutation and the notion of seeds and so on but somehow this is the interpretation in algebraic geometry of the combinatorial data that goes into the cluster algebras the skew matrix corresponds to defining a holomorphic two form on your log collabi alve variety okay so I should stop here but I'll continue in this vein tomorrow um actually this afternoon yeah so thanks