 This lecture is part of an online algebraic geometry course on schemes and will be about the canonical sheaf, omega x of a scheme x. So we're going to simplify by taking x to be non-singular, that means all its local rings are regular and a projective variety over a field k. You can define canonical sheaves in much greater generality, but I don't want to get involved in all these extra complications. So last lecture we examined the cotangent sheaf, omega x over k, of whose sections are one forms on x. And now if x is non-singular, which we're assuming, this implies that omega of x over k is locally free. In other words it's more or less a vector bundle or rather the sheaf sections of a vector bundle, but there's no great harm in confusing vector bundles with their sheaves of sections. So that means it sort of looks locally like a vector space times x. And from this we can define two other sheaves. First of all we can define the tangent sheaf, which we mentioned a little bit earlier. This is just the dual hom of omega x over k to o of x. So what's going on here? This is the sheaf of differentials. This is just regular functions. So mapping something to regular functions is a bit like taking the dual of a vector space locally. This means the sheaf hom. So if we ask what are the homomorphisms from this sheaf to this sheaf, we can either define a sheaf of homomorphisms or we can take the global sections of that sheaf and define the an abelian group of homomorphisms from this sheaf to that sheaf. And here we're talking about the sheaf homomorphisms. I mean, I think Hart-Shorn writes this in calligraphic letters, but I'm not very good at drawing calligraphics. I'm just using Roman. Anyway, so this is the tangent sheaf of x and the sections are just tangent vector fields. So once you've got tangent vector fields, you can start doing analog of differential geometry. Another thing you can do with this, so this is like taking the dual of a vector space at each point. You'll also take exterior powers. And what we're going to do is we're going to take the highest exterior power of omega x over k where n is equal to the dimension of x. So you can define anything you can do with vector spaces. You can do with sheaves, like taking exterior powers and symmetric powers and duals and so on. And here we're just taking the nth exterior power. This is particularly easy to figure out since we're assuming x is non-singular, so this is locally free. So it just looks locally like a vector space of dimension n. So taking its nth exterior power, it's going to look locally like a one-dimensional vector space. So this will be an invertible sheaf, or in other words, a line bundle. It's essentially a one-dimensional vector space for each point of x, except it's probably twisted a bit. So the sections of this are just n-forms on x. So if you were a differential geometry, an n-form is something you would integrate in order to find the volume of x. Well, at least if x is oriented. So this is the canonical sheaf and is denoted by little omega of x over k. So this is the canonical sheaf. So it's a natural way of getting a line bundle over a scheme x. I mean, we can always get a line bundle by taking the trivial sheaf of regular functions, but that's not a terribly interesting line bundle. I mean, for Vex's project, if it's space of sections, it's just one-dimensional, for example. So what I want to do now is to give a sort of advertisement for the canonical sheaf by quickly giving a quick survey of some applications of it. So these applications are things that come up later in the course, and I will miss out an awful lot of details. So the first application is Riemann Rock for Curbs, which says, suppose you've got a divisor d, and this corresponds to some line bundle l of d, and we want to know what is the dimension of the space of sections of l of d. So the dimension of the space of sections is often written little l of d. So the Riemann Rock theorem states that little l of d can be written, worked out as follows. It's the degree of d plus 1 minus g plus little l of k minus d. So let's explain what all the bits are. Well, I've just done l of d. The degree of d, well, that's easy. If d is sum of n i p i, where these are points, the degree of d is just sum of the n i. So the single point is degree 1. This g is the famous genus of the curve. If you're an analyst, you would define this as the genus of the underlying topological space. In other words, the number of handles of the surface. If you're an algebraic geometry working over a field of characteristic p, you can't do that at least not very easily. So instead, you can define it as the dimension of the space of global sections of the canonical sheaf. In other words, this is just the space of one forms on the curve. And this k here is a divisor corresponding to the line bundle omega x. So k is sometimes called the canonical divisor, which is a bit misleading because there's not really a single canonical divisor. There's a whole family of linearly equivalent divisors called canonical divisors. And what you do to get them is you take a global one form and k is just the divisor of zeros of that one form. And of course, you could choose different one forms and get different divisors, but they're all linearly equivalent. Anyway, l of k minus d, the dimension of the space of sections on this doesn't depend on which divisor you choose as your canonical divisor. So this is well defined. So this bit of the theorem is due to Riemann and this bit of the theorem is due to Roch. And you maybe, I used to wonder why Roch gets his name on one of the most important theorems in algebraic geometry, and then you never hear about them offwards. And it turned out that Roch died very young shortly after proving this crucial bit of the Riemann-Roch theorem. So the next application of the canonical bundle is a sort of generalization of this, which is ser duality. And the point is you should think of the canonical sheet as a sort of dualizing object. It's called dualizing object D. So generally, if you've got something like vector spaces or a billion groups, you might have these objects A and you might define a sort of dual of A to be the homomorphisms from A to some dualizing object. For example, for vector spaces, the dualizing object would be the field you're working over with, and that defines the dual of a vector space. And for finite abelian groups, the dualizing object might be, say, the circle group, and that defines the Pontreorgan dual of a finite abelian group. And for sheaves, there's a sort of more complicated duality where the dualizing object turns out to be the invertible thief. So we get, for example, we get ser duality, which says that a cohomology group of X with coefficients in the sheaf X is isomorphic to a sort of dual cohomology of X with coefficients in the dual of F, except that's not quite right because we have to tensor this with the canonical sheaf and you have to take the dual of that as well. So the the canonical sheaf kind of appears here as a sort of playing a similar role to these dualizing objects, except in a slightly more complicated way. You can think of this as being a sort of analog and algebraic geometry of Poincaré duality. You remember if you've got a manifold X, then if you take the cohomology of X with coefficients in R, then this is isomorphic to the dual of the cohomology of X with coefficients in R, except you have to take the dual of this and you may think what has happened to the canonical sheaf. Well, it turns out this this only happens if X is orientable, and if X is non orientable, then you have to sort of twiddle this by some sort of orientation sheaf. So so ser duality is sort of formally quite similar to Poincaré duality, except you need to put the the the canonical sheaf in. The reason why this is related to the Riemann-Roch theorem is the Riemann-Roch theorem is really two different theorems bundled together. So the first theorem tells you the Euler characteristic of L of D, which is a zeroth cohomology of L of D. Guess what the dimension of this minus the dimension of the first cohomology of L of D, and it tells you that this is this is degree of D plus 1 minus G. So so this is the form of the Riemann-Roch theorem that hits a book generalized to higher dimensions. So so that in order to get the Riemann-Roch theorem, what you do is you apply ser duality to this, and this just says that this is isomorphic to H naught of omega time tensor with L of D minus 1. So ser duality says that these two spaces are dual, so they have the same dimension. So if you combine the hits of Riemann-Roch theorem for curves with ser duality, then you get the classical Riemann-Roch theorem for curves. And the hits of Riemann-Roch theorem and ser duality both generalize to higher dimensional varieties. Third thing you can do is do the define the canonical embedding of a variety. So we have the following problem. Classify varieties X. So this is the sort of fundamental problem in algebraic geometry in some sense. We want to classify all varieties, and obviously this is hopelessly difficult in general, but we can try and do it with two steps. Step one, embed X into projective space, and step two, classify subvarieties of projective space. And both of these are impossibly difficult, but never mind. Still you can make some progress on classifying subvarieties by using things called Hilbert schemes, which in some sense parametrize all possible subvarieties of PN. So there's some sort of machinery for dealing with the second problem. For dealing with step one, we've got to find a way of giving a variety. How do we embed it into projective space in a comical way? And what we can do is we can pick a line bundle L on X and pick some sections, say all sections, and this gives a map from some open subset in X to projective space, which if you're lucky will be an embedding of X in projective space if you've got enough sections of the line bundle. So what we need to do is to find a line bundle. So given a variety, how do we find a line bundle on it? Well, there's one obvious way, which is we can just take L to be the chronicle line bundle to the power of some integer. And so if you take M to be a positive integer sufficiently large, this gives you things called the canonical embeddings of a variety into projective space. Okay, this leads to the question of when does a line bundle, when does the canonical line bundle have enough sections that you actually get embedding? And this gives the fourth application of the canonical line bundle, which is the Kadyra dimension. So we have the following question, does the canonical line bundle, or its powers, let me go to the i for i greater than zero, have lots of sections? So if they have enough sections, then we can get an embedding into projective space. And the Kadyra dimension, usually denoted by a little Greek kappa, is the dimension of the image of X under the canonical embeddings of omega to the i for i sufficiently large and having the right congruence properties. Actually, it's a little bit complicated because the dimension of the canonical embedding can, the dimension of the embedding can jump up and down a bit as i increases, but roughly speaking, there's a sort of maximum dimension and that's the Kadyra dimension. It's also possible that none of these of any non-zero sections, in which case we say K is minus infinity if none of the omega to the i for i greater than or to have sections. So we can see some examples of this. If we take p1, we saw the canonical line bundle is just the cotangent bundle, which is o minus two, and has no sections other than zero. So the Kadyra dimension is minus infinity. If we take an elliptic curve, then again the canonical line bundle is just the cotangent bundle, which we saw was trivial, so it's o zero. And the space of sections has dimension one and it has dimension one even if you take some power of this. So this means we do have a non-zero section, but we're only getting a map from the elliptic curve to a project that the zero-dimensional projective space, in other words a point. So the image of the elliptic curve under the canonical map is just zero. We don't get an embedding of the elliptic curve. On the other hand, if you take a curve of degree four, such as x to the four plus y to the four plus c to the four equals zero, then in this case we find that we will calculate the canonical line bundle a little bit later, and we find omega has lots of sections, and in fact it has enough to give an embedding of this into the plane, so we find the Kadyra dimension is now one. So roughly speaking, if a variety is dimension n, then the Kadyra dimension can be minus infinity zero, one, two, up to n. And these behave a bit like varieties of various curvature in differential geometry. So you remember in differential geometry, sometimes you can give varieties a Romanian metric and sometimes this is positive zero or negative curvature. So these are sort of analogous to positive curvature manifolds. These ones are sort of analogous to zero curvature manifolds, and these ones are sort of analogous to negative curvature manifolds. The relation between curvature and the Kadyra dimension shouldn't be taken too seriously, it's just a sort of rather informal analogy, and these ones here are sort of mixture between zero and negative curvature in some sense, so negative curvature sort of tend to be things like hyperbolic manifolds. So there are some formal similarities between varieties of Kadyra dimension n and hyperbolic manifolds. And this is sometimes used to be a classification of manifolds. So for example, you'll hear about there being a classification of surfaces due to Enrique and Kadyra, where you first will look at the Kadyra dimension and then you classify, you have a pretty good classification of the surfaces of Kadyra dimension minus infinity zero and one, and the surfaces of dimension two are called general type. So that's the Kadyra, so the Kadyra classification of surfaces sort of classifies the surfaces not of general type, but calling it a classification of surfaces is really a bit misleading because almost all surfaces are of general type in some sense and we still know very little about the classification of surfaces of general type. By analogy, we can say that calling this a classification of surfaces would be like having a classification of birds that classifies birds into penguins, ostriches, and everything else. And obviously that's not a very convincing classification of birds because the category everything else is nearly everything and the classification of surfaces is a bit like that. General type is sort of everything else includes almost all surfaces and we don't know how to classify it. But anyway, so that's the sort of advertisement for the canonical sheaf turns up in lots of places. Well, now we should give some examples of how to calculate it. Well, first of all, we've done P1, we calculated the canonical sheaf for P1 and found it was just O of minus two. Now let's do n-dimensional projective space and figure out what the canonical sheaf is. Well, for this, we use the exact sequences of sheaves we worked out earlier. We had the cotangent sheaf of P to the n and we saw this map to O minus one to the n plus one and this map to O of zero, the sheaf of regular functions of this map to zero. And let's just write in the ranks here to make it slightly easier to see what's going on. So this is rank n, this is rank n plus one, and this is rank one. Now if you've got an exact sequence of vector spaces, A goes to B goes to C goes to zero, suppose these have dimensions A, B and C, then we know from linear algebra that the topic stereo power of B is canonically isomorphic to the topic stereo power of A times the topic stereo power of C. Now we can do the same thing for sheaves and what we find is that the highest exterior power of omega P to the n tensile with the highest exterior power of O of zero is isomorphic to the highest exterior power of O minus one to the n plus one. So this means the direct sum of n plus one copies of O minus one, not the tensile product of n plus one copies, it's rather bad notation unfortunately. Well, this thing here is the canonical sheaf of P to the n and this thing here is just trivial because if you take the exterior power of the the trivial line bundle, you just get the trivial line bundle and here we're taking the n plus one exterior power of a sum of n plus one copies of a certain line bundle and that's the tensile product of n plus one copies of these which is O minus n minus one. So this gives us the canonical sheaf of any projective space. In particular, we see that the Kadara dimension of projective space is minus infinity because none of these sheaves O minus n minus one times m of any global sections and that's if n is greater than zero, I guess. Well, next we can calculate the Kadara dimension of a hypersurface y in P to the n. Let's assume this is non-singular and for this we're going to show that the canonical bundle of y is equal to the canonical bundle of P to the n, tensile with L, tensile with O of y where this is regular functions on y and this is the canonical sheaf of P to the n and this is the line bundle corresponding to the divisor y. So since y is a non-singular hypersurface, we can think of it as being a line bundle and therefore it's got a divisor attached to it. So we're sort of restricting the canonical bundle of P to the n only we have to kind of twiddle it by this line bundle. And in order to see this, what we do is we start with the exact sequence, nought goes to i over i squared, goes to omega P to the n, tensile with O of y, goes to omega of y, goes to nought which I haven't proved, but you can find in Hart-Shorn, I think it's Hart-Shorn 8.17 and I'm not going to give the proof of this because I'm getting a bit bored of proving things about exact sequences of sheaves. And again, if we look at the dimensions of these, sorry if you look at the ranks of these vector bundles, this is rank one, this is rank n and this is rank n minus one. So we're going to do the same trick of taking exterior powers and what we find is that omega P to the n tensile with O of y is the canonical bundle of y tensile with i over i squared. So here the i over i squared comes from there and the omega y comes from there and this bit here of course comes from this. And well, we know the canonical bundle of P to the n, so this is just O of minus n minus one because we just worked it out. So we've got to figure out what this thing here is. So i is the ideal sheaf of y contained in P to the n, so it's just the sheaf of functions vanishing on y, roughly speaking. And i over i squared is just equal to i times i to the nought over i to the one. And this is just i times, well, i to the nought over i to the one is just the is just the regular functions on y because we're quotient out regular functions on a projected space by functions vanishing on y. And i is equal to l with d to the minus one where d is the divisor corresponding to y. I guess I could write that as l of y to the minus one. Yeah, except well whatever. And that's because sections, so i is equal to l of minus d which is equal to l of d for minus one, so the minus one should be outside that. That's because the sections of this, the sections of both this thing here and this thing here are just functions vanishing on y. So if we put everything together, we find that the canonical bundle of y is the canonical bundle of P to the n tensored with regular functions on y, tensored with l of d minus one, which is equal to o of y, tensored with o of minus n minus one, tensored with o of d, where we notice that the sheaf where the bundle l of minus d corresponding to y is just o d if y is a hypersurface of degree d. So we find omega y is just equal to o of y twisted by d minus n minus one. So this is the final result for the canonical sheaf of a hypersurface of degree d in n dimensional projective space. And let's see a few examples of this, let's just take n equals two, so we're looking at hypersurfaces in P2. And for d equals one, y is just a line P1 in P2. And we see from the formula, so the formula was omega y equals o y d minus n minus one. So omega y is o of, this is going to be one minus two minus one, which is o of minus two, which is the result we formerly got for the canonical sheaf of P1. If we take d equals two, then y is a conic in P2. And we find that omega y is now o of two minus two minus one, which is equal to o of minus one. And a conic on P2 is just isomorphic to P1. So it looks a bit funny here because we now seem to be saying that the canonical line bundle on P1 is now o of minus one rather than o of minus two. So what's going on here? Well, the point is this o of minus one is actually o of minus one on P2. And if we restrict it to P1, it actually becomes o of minus two on P1 because we're mapping P1 to P2 by a degree two embedding. So we take x, y to say x squared, x, y, y squared. You might call this rst. And then sections of o of one are going to be given by these functions r, s, and t. So if you restrict them, they'll be given by things like x squared, x, y, and y squared. So if we take o one on P2, this restricts o two on P1. And similarly, the duals o of minus one restricts o of minus two. If we take d equals three, here y is now a cubic curve, say y squared equals x cubed minus x or something like that. And now we find that omega is o y of d minus n minus one, which is now o y of three minus two minus one, which is just o of y. So omega is isomorphic to the ring of regular functions on y. So for example, the dimension of the space of one forms is one because the space of sections of regular functions on y is just one because y is a is a projective space. So we said the genus is the dimension of the space of sections. So the genus is equal to one. We can also work out the cadara dimension. So omega to the n is again going to be o of y, and the space of sections is always going to be one. So in the map from elliptic curve to a projective space given by sections of this, the image is just a point. So the cadara dimension is zero. It's the last example. Let's just very quickly look at what happens of d is equal to four. Here y is aquatic. For example, it might be extra four plus y to the four plus z to the four equals naught, for example. And we can work out the canonical bundle is now going to be o of y twisted with four minus two minus one, which is equal to o y twisted by one. And every if you twist by positive integers, you increase the number of sections. So we see from this that the number space of sections of one forms is at least three because o one on p two has the three dimensional space of sections. And you can restrict them to this degree four curve and get a three dimensional space. In fact, the dimension of the space of one forms having to be exactly three, but that takes a little bit more effort. And as I seem to be this lecture seems to have gone on too long, so I'll stop there.