Thank you very much. On behalf of myself,

I would like to thank the organizing committee, which includes myself but did not include myself at the time when I was invited to give this talk. Anyway, it's a wonderful occasion to be here. And happy non-birthday, Maxim, since I understand it's not right now. OK, so the title of my talk

is homological mirror symmetry for affine varieties. And that could include lots of very complicated things and very subtle issues, none of which I will address because I don't understand them. And, instead, what I have in mind is something simpler. And so, among affine varieties,

I mostly want to focus on something very simple, namely hypersurfaces in C star to the n. And, when you need a concrete example, I will, in fact, think of a very simple kind of hypersurface,

namely the pair of pants, which is, after all, a hypersurface in C star squared, defined by the equation one plus x1 plus x2 equals zero,

for example. And then, when I need slightly less simple examples, I will think of, for example, other Riemann surfaces in C star squared or higher dimensional pairs of pants.

So, this is the n minus one dimensional pair of pants. And, it looks like this, but bigger. OK, so what do I mean by mirror symmetry for these things? We have to decide first what the mirror space should be like.

And, conveniently for that, there are candidate mirrors, which I will explain now from the point of view of joint work with Mohamed Abouzeid and Ludmil Katzarkov from 2012.

But, you can also see other competing proposals by Gross, Katzarkov, Rodat, and earlier works by Katzarkov, Kapustin, Orlov, Yotov,

Patrick Clark, and it probably goes back way before. It all goes back. Yes, yes, but they are compatible. The different possible explanations for the same thing. So, they are all consistent. OK, so anyway, what's the mirror of such a thing? So, let's say that we have a hypersurface defined by some

equation, so some Laurent polynomial in n variables in C to the n. So, we'll have a sum with monomials and inside C star to the n.

Yes, ht, for example. And, OK, so quickly speaking, A is a finite subset of integer weights, the exponents that appear in my favorite Laurent polynomial.

Rho is, well, weights associated to each of these integers, a way of basically thinking of a degeneration of these hypersurfaces to some tropical limit. So, there's some convexity property. And, t, depending on your taste,

is either, well, you could think of it as a small or formal parameter. And, in fact, you can think of this as either a degenerating family of hypersurfaces in C star to the n, or as one non-Archimedean

hypersurface defined over the Novikov field. OK, and so, what we've done in there, and was done differently in other places, is construct what I would call a generalized mirror in the sense

of a Stromager-Yauzaslow conjecture to this thing. And so, this thing will be a toric Landau-Ginzburg model. So, a Landau-Ginzburg model for us will be just a non-compact Kähler manifold equipped with a holomorphic

function on it. And, depending on what we want to do, we'll either study the symplectic geometry or the algebraic geometry of its singularities, of its critical values. So, this Landau-Ginzburg model here, concretely, will be a pair YW where Y

is a toric Kähler-Yau n plus one dimensional manifold, which is easiest to describe in terms of its moment polytope. So, how do I describe the moment polytope of this thing? Well, the first thing I will do is tropicalize this equation.

So, let's define, I will call that delta y, but I can't define it yet. So, let's define the tropicalization of f to be a piecewise linear function of n

real variables, xi n to xi n. It will be a piecewise linear function defined by the max of all of these things of the linear function with slope alpha minus the constant rho of alpha. So, if you know tropical geometry, you know how to

come to this from there, and why they are essentially similar. And then, the moment polytope will be the set of all tuples xi one to xi n eta in Rn plus one

where eta is bigger than phi of xi. And, that defines for me a Kähler toric manifold. And, moreover, on that there will be a function, which will be just a toric monomial up to sine, the minus sine, which I don't want to explain. And, it's the monomial with

weight zero, zero, zero, one. You can check that will be a regular function on this thing. OK, so what does that look like concretely? So, if I take the pair of pants in any dimension, so this function doesn't need

powers of t to be tropicalized. It's already there. What the pair of pants looks like is this. The tropicalization phi is

going to be max of xi one, xi n, and zero. Its domains of linearity look like that. I'm drawing the picture for n equals two. And so, here it's zero, here it's xi one, here it's xi two. Now, imagine the graph of this

thing in R3. And, take everything that's above, and that defines the moment polytope for you. What is that polytope? Well, it's nothing but just an octant in space in this case. So, in fact, y will be c to the n plus one. And, what is this function w?

Well, it's the one that corresponds to the weight vector that goes straight out in this direction, which means it vanishes to order one on all of these facets. So, in this case, w equals minus the product of the coordinates.

So, that's a good example to have in mind. OK, so in what sense is this a mirror to the pair of pants, or, more generally, to hypersurfaces? Well, that's what we're still in the process of finding out. OK, so, up there, I've generalized a mirror. A mirror symmetry is about torus vibrations. And so, you might ask,

how does this work? This guy doesn't even have the right dimension. I started from something with dimension n minus one. I end up with something of dimension n plus one. And, that's where the generalized thing comes about. So, as far as I know, there's not any reasonable torus vibration in the sense

of syz to be put on this hypersurface h. However, there's larger spaces closely related to h that do carry such vibrations. So, in fact, the simplest thing to look at, we'll call x0, is going to be, so the space defined by the

equation uv equals f of x1 xn inside c2 times c star to the n. That's a conic bundle over c star to the n with discriminant locus given exactly by h. So, the typical fibers of this thing look like cylinders.

But then, above my pair of pants or whatever hypersurface I have, these cylinders degenerate to unions of lines. OK, so this is conveniently a nice Calabi or manifold. And, it carries nice Lagrangian torus vibrations. Roughly speaking, all you have to do is pick,

so there's a circle action. So, what you do is you pick level sets of a moment map, which means you choose heights for circles on these conics. And then, you pick Lagrangian space, and you just lift them to obtain Lagrangian tori in the

total space. There's a slight subtlety, but that's pretty close to the truth. What's the slight subtlety? Well, actually, you pick Lagrangian tori on the reduced spaces, which are all carrying slightly different symplectic forms from the usual one of c star to the n. In particular, the reduced symplectic forms are not toric anymore, but they are deformation equivalent to toric ones. So, you still know how to find

Lagrangian torus vibrations on the reduced spaces. The side effect of that is that this vibration is only piecewise smooth. OK, so this thing carries a Lagrangian tn plus one, now, vibration with some

singularities, of course. Singularity is basically when your tori pass through these points, which is singular along h. And, you can use that to build a dual,

in a certain sense, torus vibration on a space which, after you stare at it for long enough, you find that it's most of this mirror space, y, minus some hyper surface, which I will not elaborate on for now. Now, x zero is not the same

area as h. But, what happens is there is a slightly larger space, x, which is the blowup of c times c star to the n at zero times h. And, what this one looks like is almost the same except,

well, except slightly different in what way. So, above a typical point, I just have, so, if I just project again to the c star to the n factor, the typical fiber will just be a copy of c, which, for your convenience, I will draw like this. And, above a point of h, the fiber will be a copy of c

obtained by lifting to the proper transform union a CP1 from the exceptional divisor. And now, you should see that this is exactly the same as that one, just adding one point in So, this is a nice setup for mirror symmetry in the SOAZ sense in that this one is no longer Calabi-Yau,

but we know how to think of its mirror as being essentially the same as the mirror to x zero, but with a super potential added. The super potential, roughly speaking, records, say, if you are a symplectic geometer and doing symplectic geometry here, tells you how adding this partial compactification will

deform Fleur theory for Lagrangians in this space. Sorry, so I should have said x zero will be mirror to something called y zero, which is y minus some hypersurface, not telling you what it is.

The mirror of this one will be again y zero, but now equipped with a super potential, which is exactly this w I told you about. Now, this is not equivalent to h either.

It looks like I'm just playing games and making this space more and more complicated. Next step is I'm going to equip x with its own super potential, wx equals the coordinate from this complex variable. I will call that y. When I do that, I'm deforming again the geometry here. And, the effect of that is to

compactify the mirror to all of, sorry, not compactify because it's still non-compact, but partially compactify to all of y. OK, so now the claim is, sorry, super potential was still negative w. Next thing I need to do is also twist by some class in H two with zero two coefficients

to account for basically discrepancies in the ways that signs are counted for holomorphic curves. That's the bane of symplectic geometers, that holomorphic curves always get counted with the wrong sign. And, the effect of that is to change this to what I wanted. OK, and now the claim is,

so what's the geometry of this function on x? By now, I've probably lost most of you, but I'm going to try to unloose you. So, the function, which is just given by the complex coordinate from C, what is its zero set?

Well, it's the origin in each copy of C. So, it's these new points I've added to each fiber. Union, well, the whole exceptional divisor because, after all, on the exceptional divisor, the complex coordinate is also zero.

So, you see that it has two smooth pieces intersecting transversely along a copy of H in here. And, in fact, this function is Morsebot with Morsebot singularities along a copy of H. And then, its singularity theory just reduces to the geometry of H. So, there's good evidence for

some of it, well, most of it is not written up in full detail, that there should be a functor from the Fourier category of H to the Fourier category of this

thing. And, that functor should be almost an equivalence. At least, we are hoping it's an equivalence. We don't know how to prove that yet. Sorry, it's an equivalence up to a shift. Sorry, I have a question about this S. Is this a question about being

Morsebot and what the normal bundle is? Exactly. It's Morsebot, but the normal bundle to it is the direction of two line bundles, which have non-trivial seconds to differ with Niklas. And, it's the W2 of those normal bundles. And, similarly,

if you're on algebraic geometry instead, you probably know, similarly, when you have a vibration with Morsebot singularities, I mean, it's a case of basically neural periodicity, as proved by Arlov, that D.B. Singh of this thing, the category that algebraic geometries would associate to

this Landau-Ginzburg model, is the same as the category of coherent sheaves of H. So, for all purposes, this is a good replacement of H. Yes, yes, OK, sorry. I'm not that sophisticated.

OK, oh, I still have a whole blackboard here. I'm not used to having three. OK, so, after a long introduction, now we can talk a bit about homological mirror symmetry. Because, you know, you can construct mirrors by this sort of S-Y-Z argument,

but that doesn't prove, actually, that they are mirrors in the sense of homological mirror symmetry, at least not yet. Someday, probably, the vast program that Kenji Fukuyama and Mohamed Abouzade are pursuing, using family flow homology, will tell us, once you have this, this is good enough, you're done. But, we're not yet there, so I'm still in business. OK, so, one direction we can look at is check whether the

wrapped Fourier category of H is indeed equivalent to the derived singularities, or matrix factorizations of

this toric superpotential w. It should be somewhere up there. Yes, sorry, over there. And, that's been studied in various cases. So, that's known, for example, for the pair of pants and some other Riemann surfaces in

joint work with Abouzade, Efimov, Katzarkov and Orlov. So, what's the game there? I'm not going to get into detail because I want to focus on the other side.

But, so, wrapped Fourier homology is about you take Lagrangian submanifolds, and you perturb them at infinity by a Hamiltonian that grows quadratically. So, that couples Fourier intersection theory with contact dynamics at infinity. So, concretely here, you might be looking at things like properly embedded arcs on

the pair of pants. And, the perturbation that you do when you do Fourier theory is to wrap them around the cylindrical ends of your pants. And then, you look at intersections, and you get something big and infinite dimensional and interesting. And then, the game is to compare that with matrix

vectorizations of, in this case, c cubed minus z0, z1, z2. So, the kind of things you could look at are, for example, the two periodic complexes, sorry, not complexes, different complexes given by,

for example, minus z0 and z1, z2. And, the claim is that this guy and its two friends will actually turn out to be mirror to these three arcs on the pair of pants. And, you can check by brute force calculation that these things match. OK, and then there's work on other Riemann surfaces done by

what? Brooklyn on one side, and also the thesis of my student heavily in progress.

And, there's work on higher dimensional pairs of pants. For example, Nick Sheridan's first thesis result was the compact analog of that, looking at compact

Lagrangians in the higher dimensional pair of pants, and matching that with singularities, I mean, basically the skyscraper sheath of the origin in here. OK, but the other direction, which is more what I would

like to think about today, is slightly more mysterious. This one is far from being completely done, but it feels like it's almost under control in some ways. And, the one where I think more work remains to be done is the opposite direction. Now, how do I,

OK, so the other direction is about comparing the derived category of coherent sheaths of H versus some sort of

Foucaultia category of this Landau-Ginzburg model. And, what the issue is is that this should be some sort of fiberwise of wrapped Foucaultia category.

And, well, such things are not defined in general, at least not yet. But, in this case, fortunately, we can do it. OK, so what I'm going to say from now on is actually a joint work with Mohamed Abouzade very much in progress.

It's more in progress than the last time that I gave a similar talk in that now we've actually started writing. But, we are still, well, still in the geometric preliminaries about whether the Lagrangians we are going to look at in a moment are geometrically bounded, and in whose sense exactly?

OK, and that's probably others too. OK, all right, so what's the main steps that

I want to discuss? So, one is can we actually define and construct and calculate in some way the

fiberwise wrapped version of theory in this case? And, the answer is yes, we can in a very limited context, and for very specific Lagrangians. So, I'm going to do this only for a very restrictive kind of

category. There's a more general approach. Well, so I'm not familiar with Gabe's work, actually. So, I don't know how much he constructs. OK, so, OK, and my students actually then is also working on a generalization of this,

which should work but might be too complicated to use. So, who knows? Anyway, in this particular case, we can do it. And then, we can construct an object. Yes, yes, sorry. So, the problem is, OK, so I'll explain in a moment what the issue is. But, we are going to deal with non-compact Lagrangians of a certain kind.

And, we'll want to do something that's halfway in between. So, when you have a function with, basically, a set of critical values is proper, you can look only at Lagrangians that are fiberwise compact, and escape to infinity in an orderly manner. And, basically, this is the setting of Paul Seidel's framework.

Well, after, I guess, the ideas of, probably I should have, well, I should probably have started with saying that Maxim had the first idea. But, anyway, we should have, OK, categories for such pairs. Sorry, the generic concept originates with Maxim.

OK, so in Koncevich, then Seidel, and so on, one looks specifically at things like thimbles for electric vibration, Lagrangians that are non-compact in the total space, but fiberwise compact. The Vrap2K category looks at non-compact Lagrangians, and applies a Hamiltonian flow that perturbs things in all directions at infinity. And, what we want to do is a

consider the fibers of this function, and only slight perturbations in the base directions. That's what this is about. OK, so having defined fiberwise wrap theory with very restrictive assumptions,

the next step is to see if there's any Lagrangians of interest that actually satisfy those assumptions. And, the claim is there is one. So, I'm going to call, maybe, whatever. We'll call it w of y and w. And, I could conjecture that this object generates the category since right now it's

the only object I know how to put in there. It's obvious that it generates the category. But, that would not be a very fair claim to make. OK, anyway, I will still write should generate because, I mean, so the goal, I mean, the purpose of this guy in life is to be the

mirror to the structure sheaf of h. And, on an affine variety, the structure sheaf generates the derived category. So, this should be the only Lagrangian I should care about.

Well, so I don't have a good one because I don't know how to do these things. There should be one. What you're going to see is that this L0 turns out to basically be skeletal in nature ultimately in that it looks like it's defined very smoothly. But, if I try to draw a picture of what it looks like at infinity, you will not believe that this

thing was meant to be smooth. So, I'm sure that a skeleton description might be possible. I don't quite know how to do it at this point. So, I'm not going to say anything more. And, the first step, of course, is to calculate endomorphisms of L0 in this fiber-wise drop sense and

find that it does match with endomorphisms of the structure sheaf. And so, this tells you that the derived category of h embeds into this category.

And, actually, well, there's nothing else right now because we don't have other objects in there. But, again, that's not exactly, well, we don't know other examples of objects in there. That doesn't tell you exactly what you want to know. OK, so let me say a little bit of content about what goes into there. OK, so what's the basic idea

of a fiber-wise dropped category of these things?

So, first of all, I could try to define that in more generality, but really, I will be in this direct setting. So, maybe the example you should have in mind is the one I explained, the mirror to the pair of pants, Cn plus one mapping to C by the

product of a coordinate. So, this is something where the only critical value is zero. Although zero will have something which is just the union of all the coordinate hyperplanes, the singular locus is the union of the coordinate, sorry, the union of all the strata of codimension two or bigger. And, the other fibers will be

smooth and look like C star to the n. So, the kinds of Lagrangians we want to look at will want

properly embedded Lagrangians of many forms satisfying various conditions. So, the first one is about in which directions they can escape in the base.

And, that condition is similar to what happens in the more familiar case of things that are fiber-wise compact, so things like thimbles and so on. You want, roughly speaking, your Lagrangians to fiber over arcs, at least outside of a compact subset. So, one way I could state it is that my Lagrangian L

projects under something which, well, I don't care what happens in a compact part, but outside of a compact subset, this is going to look like a union of radial lines. And, there's a forbidden direction which for me will be the negative real axis. OK, so the image of L under

this projection map outside of a compact subset is a union of radial straight lines and not real negative.

And then, there's going to be, of course, some other asymptotic conditions. Sorry, there's going to be some other reasonable conditions that we want to impose. So, we want these things to

be properly embedded. And, we want, when they escape at infinity fiber-wise, to have some bounded geometry property in a fairly weak sense, that they don't look too bad in small balls,

just enough to be able to do some Fleur theory. And, we'll want some extra conditions, some of which are going to be, well, I think I need to wait a bit to explain some of them. I don't want my Lagrangians to bound any holomorphic disks. That helps with Fleur theory. Well, the next condition is

just too technical, so I will not explain it yet. First, I need to explain what kind of, I've lost a blackboard. So, how do we want to perturb

some of these Lagrangians when we are going to take their Fleur homology? However, what kind of Hamiltonian perturbations do we want to introduce in Fleur theory? No, we should undo that.

What kind of perturbation should we do to the Lagrangians? That's how we actually do it in this case. It should be equivalent, but we don't know how to do it the other way. OK, so, two things I need to do are perturb in the base and perturb in the fiber. And, perturb in the base, well, let's pick a flow on C, which is identity on a compact

subset, and maps, well, let me just draw a picture of what it does. So, it's identity inside. And then, if you take a collection of radial straight lines, what it will map them to is it will bend them

towards the real negative direction, and then go straight again. So, this sort of thing. OK, so all the radial directions remain radial at infinity, but they've all been bent.

Bent, except, sorry, the real negative direction is fixed. And, all the others bend. Eventually, they accumulate towards the real negative direction. OK, so now this flow, well, I could try to work harder to make it area preserving, but I don't actually care because what I'm going to do to it is take a horizontal left of rho t to

the symplectic orthogonal of the fibers. And, the claim is I can do that. I will obtain a flow upstairs, which is autonomous. And, it's not a symplectomorphism,

but it maps fiber Lagrangians to fiber Lagrangians. So, in particular, it maps these kind of Lagrangians of the same kind. OK, it's still an admissible Lagrangian. And then, there's something

else I can do, which is I can define phi t to be the flow of a Hamiltonian on the total space, which is invariant under parallel transport,

and fiber-wise proper. So, what this will do is it will preserve all the fibers, but within each fiber, it's going to rotate things at infinity.

And, what I want to say is I want this guy to have linear growth in a certain sense. So, for those of you who already know wrapped-flow theory, this is the linear growth kind of Hamiltonians in wrapped-flow theory. And then, these two flows actually are both autonomous, and they commute. This one is not symplectic,

but actually it shouldn't bother us because it maps Lagrangians to Lagrangians. And so, now we'll define lt to be the image of l under both of these in either order.

We are going to make it isomorphic by definition. But, see, the point is the area on C is not relevant because area in the total space is very different from

area in C. So, you could change the symplectic form of total space, but I don't know if that would be really legal. No, because then that will not preserve fiberness. If you just take the pullback, you will have to rescale the flow by an amount which varies

in the fiber. So, I don't know a better solution, but I claim this should not bother us. Well, sorry, lt of l is Hamiltonian isotopic to l just by a Hamiltonian which I do not want to make explicit because it's a Lagrangian isotopy. And, you can check easily that it will be exact.

So, that's probably the best answer. It is a Hamiltonian isotopy of l, just the Hamiltonian is not the same for all l. The key one will be exact.

It will be actually contractable topologically. There are others I want to look at which would not be exact, but they don't bound any disks. There are no disks in the base. That's a very good point. For the pair of pants, everything is fine. If we are not with a pair of

pants and a few other examples like that, the total space is not exact. So, that doesn't happen. And, yes, sorry, so that means I didn't say exactly in what sense we had things. So, in case I don't get there, the last step is in the case of a pair of pants, it should be completely clean. In cases that are other than the pair of pants,

this is modulo some correction terms and some kind of written theory corrections. The calculation of the endomorphisms of L0 being what you think. There's a calculation of the floor differential. Let me just try to get there.

It's true that it's well-defined. And, to calculate it, you need to use things that are well-accepted but not as clean and elementary as the others. OK, so if you set this up properly, the various things that will happen are that,

OK, so I said the fibers, they look like C star to the N's. And, what my flow will do, really, is some rotations inside the argument directions of C star to the N's, the phi t, the fiberwise flow. And, I can arrange that this flow happens always at unit

speed in one of the directions. So, that means points can come back to themselves in times that are other than multiples of two pi. So, the claim is I can arrange that the distance between LT and LT prime at infinity is bounded below

uniformly whenever t minus t prime is not a multiple of two pi. That's a specific feature of the Hamiltonian I know how to build in this toric case. And, I will not care about non-toric examples. So, that's a big restriction in generality.

For a fixed t and t prime, it's bounded below infinity? Yes, yes, sorry, yes, uniformly by a quantity which depends on the distance of t minus t prime to two pi z. OK, and there's other reasonable

features, which I'm going to just skip. OK, oh, sorry, but important. So, the other thing I should discuss is what happens since I'm doing this linear perturbations.

So, what happens fiberwise is that in a small amount of time, I have a Lagrangian, which might look like this in my C star to the N. And, I will start perturbing it. And, of course, as I perturb more and more, it's going to intersect more and more because the ends are rotating at a constant pace.

OK, but for small times, there's only one intersection, a distinguished one. In the base, if I take something that fibers of an arc, and I push that arc by a whole t, I will just push the arc around. And, for very small times, as long as this angle doesn't hit that angle, I'm going to still have a distinguished intersection. So, the claim is for small t

minus t prime, there's a canonical element, intersection point of lt with lt prime. And, we are going to call that the identity of this thing, which is kind of a strange thing to say because

they are not the same Lagrangian anymore. The correct thing to say is that we will define Fleur theory by basically letting this flow operate, and considering that we can replace, freely, l by lt for large values of t. So, technically, you set this up as a limit.

I don't know whether it's direct or inverse. Actually, oh, direct. Yes, OK. OK, so what we're going to do is really look at some direct

limit of the Fleur complexes of l. So, I'm going to take, well, since I said two pi, I could say probably two pi n plus pi with l prime is what I will define to be hub l l prime in my category,

where the connecting map is given by multiplication by this thing I called identity. OK, so, OK, how do we find Lagrangians of this kind in here? And, what's interesting? So, if you've seen Lefschet's

vibrations, you know what's very tempting to do. When you have an isolated singularity, what you do is you take the vanishing cycle in the nearby fibers all over a path that starts at the critical value and continues forever, and you get a Lefschet's symbol, and that's a very nice thing to play with.

So, in this case, we have singularities along the union of various strata. In the C cubed case, that's just the union of the coordinate axes in C cubed. And, the first obvious thing you might want to do is say, hey, this is a Morse-Baud singularity everywhere except at the origin. So, I can take a circle in

one of my coordinate axes, and push this by parallel transport along the straight line in the base to get something that will look like, so in this case, the vanishing cycles are S1's for each of the points. So, you'll get a torus in the nearby fibers. And, this will produce for you a Lagrangian R squared times

S1, a solid torus. And, these are interesting things to look at, but they're not what we want. They're actually going to be the mirrors to skyscraper sheaves of points on the pair of paths.

So, this corresponds to skyscraper sheaves of points on, actually, on H in general. This will be the same. I will have just taking tori inside the smooth strata of the critical locus, and pushing them gives me

the mirrors to points. OK, so, in fact, part of the conceptual problem was that for the longest time, we're kind of stuck with this issue that we thought what we should be doing for the structure sheaf is instead take the real positive part of this critical locus.

So, in this case, say, R plus inside each axis, parallel transport that, and then you get a piecewise smooth Lagrangian. And, we don't know how to study it for fluoromology, and we don't know how to smooth it. That was the problem of smoothing this thing in NC cubed was assigned to several of

my graduate students who went on to instead write very long, complicated things about infinity structures, and decided that algebra was easier. So, I still don't know how to do that. Instead, the new idea is that we don't need to obsess

about what you would call maybe generalized symbols. And, if you really can't deal with the singularity, then maybe it's best to just bypass it. OK, so what do I mean by that? So, we have a singular fiber, which is the union of coordinate axes. But, we're not going to look at it. We're going to look at a smooth fiber. So, remember, my potential is

minus the product of the coordinates. The other fibers are C star to the N. And, specifically, the fiber at a real negative value, say, minus one is VC star to the N, where the product of the coordinates is one. So, that one has a well-defined positive real part. So, let's take that for that

kind of arbitrary, but it's more elegant. OK, so in this one, I will look at R plus to the N. And then, I will take a path in the base that does something like this, goes around the origin, and escapes in two directions, kind of to the right.

And then, I will parallel transport this along that arc. And, what that looks like, well, down here in the middle, it looks like a nice, smooth thing. And then, at infinity, it looks like a crazy thing, which is really, well, the best I can do is, OK, so the claim is I will get something that will be an arc,

sorry, isomorphic to R to the N plus one. And, if I slice it by setting one coordinate to be constant, in the case of C cubed, then I will get something that's like a Lagrangian R2. And, that Lagrangian R2, the way it looks, you'd think, OK,

take a very nice straight Lagrangian, parallel transport it along a smooth curve. This is a very simple, explicit thing. What could go wrong? Well, what it looks like when you slice it by setting one of the coordinates constant equal to some larger value is where

moreover these two things come together as you go out to infinity. Anyway, it's messy. It looks very piecewise linear. That leads credence to the idea that there is a skeleton hiding behind there, and that, in fact, this is not so far from the one I had complained was bad

over there. But, anyway, here it is. Technically, it's smooth. Even for infinity, it looks less and less smooth. The curvature is unbounded. Injectivity radius is going down to zero. It still has barely bounded geometry fiberwise in the sense of C curve. I don't think he invented that

condition, but in the sense of the technical condition, he uses to study holomorphic curves with boundaries and Lagrangians. And so, the claim is this thing actually exists and is suitable for doing floor theory. This is what I will call L0. Maybe if I were slightly less,

well, it's OK. Why is this not completely unreasonable? Maybe I should point out.

This construction of taking a Lagrangian far out at infinity and moving it around all the singular fibers and bringing it back to infinity has been known for a little bit to be a mirror to the inclusion functor from the mirror of the fiber to the mirror of the total space.

And, in this case, because of a strange way that we arranged for this generalized SYZ construction, the inclusion of, well, the fiber is mirrored to C star to the N. And, the inclusion of C star to the N into H is, of course, secretly into this bigger space

is exactly what we want to do. What we do actually is we include C star to the N into this big blowup space I was considering, and then we restrict to the exceptional divisor which gives us H as the image. Anyway, so there is some justification for why this should be the right thing. But, of course, we need to calculate. And, I will not do the full

calculation in front of you, but I want to convince you at least that the answer might look like what you want. So, the question is what do the endomorphisms of this object

look like? So, to answer that question, I need to take it and its image under this flow that I was talking about, which means in the base, I will see one copy, L0, and the other copy, L0, T for T very positive will look something like that. But, both branches will have been pushed past by this flow,

rho T in the base. And, fiberwise, what do I get? Well, so all the intersections will happen over these two points. So, I might as well focus on these. So, this one, what I will see is I mean C

star to the N. And, inside that, L0 was just the real part, just the real part. And, its image under wrapping, so that was something that was just going to twist around a lot. I think I drew it the wrong way, but let's not worry. And, here is essentially a similar picture.

OK, so what are the intersection points that we see come up? Well, we get two copies of a wrapped-floor complex of the real part in C star to the N. And, that is already understood. So, in the case of a cylinder, for example, this wrapped-floor complex has z-worth of generators.

And, you can calculate explicitly the multiplicative structure and find that what you will get for your floor homology, just for this picture of r plus to the N and C star to the N, will be Laurent polynomials in N variables. For each cylinder factor, you get things that I would call

maybe one here in the middle, x, x squared, and so on, x inverse, x minus two, and so on, where one is actually the one that's in the interior. You have a minimum of your Hamiltonian. And, the same for all the others. OK, so that means here I will get a copy of Laurent polynomials as my floor complex.

And, here I will get another copy. So, generally, I will call H. OK, now, except the multiplicative structure,

of course, I need to now work in the total space. But, fortunately, I'm in a situation where the projection is holomorphic. So, holomorphic curves project to holomorphic things with boundaries on these things. That means the only holomorphic curves I need to look at are either entirely within the fibers, or they project to this bygone in this case.

And, when I multiply, I will need three copies, and then will project to triangles with corners at the intersection points between these arcs in the base. So, it's all going to be explicit and very computable. Well, almost computable. Anyway, so the claim is the multiplicative structure is, in fact, we need to write

Laurent polynomials in n variables, which all have degree zero. And, with an extra generator, we will call H, which is degree minus one odd and H squared equals zero, as you would expect for an

odd generator. OK, so the multiplicative structure on this floor complex is, on the nose, something reasonably pleasant. And, there's no higher operations, mu three and beyond. However, there is a differential. I didn't tell you about the differential yet.

So, what's interesting, and where really the meat of the argument ends up being, is in the calculation of a floor differential. So, the main part of the calculation is actually going to be knowing what is the

floor differential applied to H. So, what this means in practice, whoops, yeah, it doesn't look the same. It's the same, but it doesn't look the same.

Zero was here, and I'm going to draw it similar. So, I want to count over this region, which encloses the origin. I need to count pseudo, but actually in this case really I need to count the holomorphic sections over this

thing with boundaries on, well, here, L zero, here, L zero T, and with corners at, well, this corner needs to be at the guy, say, it was in the middle that I called H. And, this one can be wherever it wants. I will get some Laurent series in principle,

but polynomial once you know that you have compactness. In the variables xi. OK, and now how do we count sections of these things? Well, this fits in the general framework of these ideas that Seidel and then James Pascal have in his thesis and so on have developed for

computing. Roughly speaking, what you do is you push the origin out of here and try to understand what happens when you do that. And, the answer is some disks bubble off. Sorry, not in this case, but I'm skipping a step. I think I should anyway. Anyway, so the claim is this can be done completely for the case of Cn plus one,

and modulo some Gromov-Witten theory in the general case, Gromov-Witten theory of a sort that can be found in the

papers of Chan, Lau, and Long, or other papers of that sort. And so, what you find is that d of h will end up being exactly, there will be one term in this for each toric divisor. And, the toric divisors of your toric variety do correspond exactly to the

monomials in your defining equation. And so, labeling them in the same way, you will find things that are of the form, so one plus dot, dot, dot. This dot, dot, dot is some Gromov-Witten invariants which are going to be correction terms, t to the rho alpha, x to the alpha, all that times e.

And so, now that tells you that if you take the Fluhr cohomology, so the cohomology of this thing, h, of course, doesn't survive. In fact, this piece maps injectively into here. So, this all dies. And, what dies in here is

exactly everything that's divisible by h. So, the cohomology is exactly Laurent polynomials mod this defining equation that I called f, I think, initially.

And, that completes the proof, except I didn't give you any, of course, of the actual arguments and details. OK. Oh, this is all in degree

zero. So, it's automatically formal. Yes, yes, it's just a wink.