 This lecture is part of an online course on algebraic geometry and will be an introduction to the Riemann-Roch theorem. So the Riemann-Roch theorem solves the following Riemann-Roch problem. How many functions are there? Well, that question is a bit vague at the moment, so what do we mean? Well, we mean how many functions on an algebraic curve or a Riemann surface? And that's still a bit of a silly question because there are infinitely many, so we should say with given poles. So suppose we ask how many functions are there whose only poles are, say, a pole of order at most three at this particular point. And the answer is going to be the dimension of some vector space. So we can get the Riemann-Roch formula. The Riemann-Roch formula says this, says L of D is equal to the degree of D plus one minus the genus plus L of K minus D. So what I mean to do this lecture is explain what all the various terms in this formula mean. And then in later lectures, I will show how to apply this formula to various examples and probably prove it in a few simple cases. So the first question is what are we working on? And that depends on whether you are a geometer or an analyst. So if you're doing an algebraic geometry, you work on a projective algebraic curve. And I'm just going to be doing it over complex numbers, although it does work over other algebraically closed fields. And the functions we're interested in are going to be rational functions. On the other hand, you can be an analyst, in which case you work with compact Riemann surfaces. And the functions you work with are going to be meromorphic functions. And it's rather easier doing things with compact Riemann surfaces than projective algebraic curves. And in fact, historically, an awful lot of the theory of algebraic curves was first developed over the complex numbers using analytic techniques. And only later, and sometimes many decades later, did people manage to find algebraic ways of doing things. So let's give some examples of these. So the first example is just we can just take the projective line. So this is the projective line of the complex numbers. So that's what you call it. If you're doing algebra, if you're doing analysis, you call it the Riemann sphere. And this illustrates one of the big differences between algebraic geometers and analysts. They always disagree about what the dimension is. If you're talking about an algebraic curve, a geometry will tell you it's one dimensional, whereas an analyst will tell you it's two dimensional because they're looking at the dimension over the reals. Another example is an elliptic curve such as y squared equals 4x cubed minus g2x minus g3. Well, that's not quite a projective curve. That's just an affine curve, but we can make it projective by looking at z, y squared equals 4x cubed minus g2xz squared minus g3z cubed. But everybody really thinks of it as being that. So this is going to be a curve inside the projective plane. And if you're an analyst, what you do instead is you take the complex numbers and quotient it out by lattice L. So L is the set of all numbers of the form m omega 1 plus n omega 2 for m n integers. Or you could look at the famous Klein cortex. So we have xz cubed plus xy cubed plus yz cubed plus cx cubed equals naught. And this is again a curve in p2. If you're an analyst, you might take the upper half plane, which is just the set of all numbers with imaginary part greater than zero. And quotient it out by some group gamma, where this is a discrete group contained in say PSL2 of R, acting on the upper half plane. So the first question is that we're going to answer is what is the genus of the curve and again, analysts and algebraists have slightly different answers. So for an analyst, G is the number of handles. And so, for instance, for a sphere, it's just a sphere with zero handles. So the genus is equal to zero for the complex plane modulo lattice. If you work out what the quotient is, it looks like a torus. So it's got one handle and the genus is equal to one. For the upper half plane modulo discrete group, the genus is usually greater than one, although sometimes it isn't. Well, if you're doing algebraic geometry, then it's rather hard to look at the underlying topological manifold, especially if you're working over a field of positive characteristic or something like that. I mean, you can sort of do it using et alcohomology, but that's way beyond anything I want to talk about. So instead, we can define the genus to be the dimension of the space of one forms of the first kind. So what on earth does that mean? Well, one form in complex analysis is something that looks like locally like DF, and there's an algebraic analog of it, which is not too hard to define. What does the first kind mean? Well, this is a completely dreadful piece of terminology. This just means it's got no poles. In other words, it's a holomorphic one form. In case you're interested, one form of the second kind means it's allowed to have poles, but all the residues of the poles must be zero. And a one form of the third kind is allowed to have arbitrary poles. So we want to know what is the space of one form. So let's try and figure it out. For P1, we seem to have one dimensional space of one forms because we seem to have one form dZ. Well, however, this turns out to have a pole at infinity. So if we change coordinates, y equals one over Z, then you see this is equal to minus one over y squared dy. So it has a pole of order two at y equals naught or z equals infinity. So here we find g equals zero because every one form has poles somewhere. Here the genus is equal to one because there's a one form dZ. So I'm going to sort of cheat and do the algebraic one form on the complex Riemann surface because the formula for it in algebraic coordinates is a bit of a mess. So we find the genus is one because there's a one dimensional space of forms without any poles anywhere. You can see that dZ is invariant under translations by L. So it gives a one form on the quotient. So here we find the genus is three for this example. One forms and surfaces of genus greater than one tend to be a little bit complicated right down on here on quotients that they are playing by discrete groups. The one forms are more or less the same as things called modular forms, which are rather complicated and extremely interesting objects, which I'll probably give some lectures on later, but I'm not going to discuss them right now. So next we can ask what is a divisor d? Well, this is a divisor d is just a formal linear combination of points. So you generally write it as a sum like this sum of n i p i with this has to be a finite sum only a finite number of the n i and none zero. Each p i is in your surface or curve or whatever you want to call it. And each z i is in. So each n i is in Z. And the n i can be negative. So informally you think of this as being a finite collection of points, but you've got to be a bit careful because you're allowed to have sort of negative numbers of points at various points if you want to. And if we look back at this, we see we have the divisor d appearing there and there, but also the degree of the divisor will the degree of D is just the number of points. Counted with multiplicities, which is just the sum of the n i. And now if f is a meromorphic or rational function. So if f is a rational function, then it has a divisor f, which is just the zeros of f. Let me put that in inverted commas because it's not quite correct with their multiplicities. In other words, it's sum of n i times p i, where n i equals the order of zero of f at p i. And the reason I was putting that in inverted commas is that you have to remember that poles are the same as zeros of order less than zero. So it's not quite the zeros of f, it's the zeros and poles of f. For example, you know that on a compact Riemann surface, the number of zeros is equal to the number of poles counter with multiplicities. And we can express this by saying the degree of the divisor of f is equal to zero. That just says the number of poles is the number of zeros. So next we have to talk about the canonical divisor. Everybody calls it the canonical divisor, but this is really misleading because there's no such thing as the canonical divisor. It is really a canonical divisor. There are lots of canonical divisors. So this is equal to the zeros of any meromorphic one form. Actually, if the Riemann surface is not the Riemann sphere, you can choose the one form to be holomorphic. You only need meromorphic for the Riemann sphere. And the problem is it's only defined up to linear equivalents. So we will explain what this means. So you see, there may be many different meromorphic one forms and they were all different zeros. So calling it the canonical divisor is really a bit of an abuse of language. So divisors D1 and D2 are called linearly equivalent if the difference is the set of zeros and poles of some meromorphic function f. Now you notice that if you've got two different one forms, suppose you've got one form divided by another one form, then locally this is going to look like dy by dz, which is just a function, possibly meromorphic. So the quotient of any two one forms is a meromorphic function. And this means the difference of their divisors is going to be the zeros of that meromorphic function. So there's not really a canonical divisor. It should really be called a canonical divisor class. So if we look back at the Riemann-Roch formula, we've now explained the canonical divisor and we've done everything except explain what this symbol L of D is. So L of D is just the dimension of the space of functions. That means rational meromorphic functions, of course, f with informally poles only on D. Again, that's not quite correct, as we will see. More precisely, this means that f plus D is greater than or equal to zero. So what does greater than or equal to zero mean? Well, a divisor sum of n i d i is greater than or equal to zero means that all the n i are greater than or equal to zero. So these divisors are called positive or sometimes effective divisors. And let's think about what this means. What this means is that f is allowed to have poles along D. For instance, suppose D is equal to 3P plus, let's take 3D minus Q for P and Q points of the Riemann surface. Then this condition says that f has a pole of order less than or equal to three at P because provide the pole is order at most three. This thing, 3P minus at most 3P will still be positive. And it must have a zero of order at least one at Q in order to make the coefficient at Q positive. So in general, L of D will tell you the dimension of a space of functions where you allow it to have poles at certain places and also insist that it has zeros at other places. For example, we see that L of D is equal to zero if the degree of D is less than zero because this says the number of zeros is greater than the number of poles, which is impossible for a function on a compact Riemann surface. You can also see that L of D is equal to one if D is zero, that sum of zero times PI of course is the zero divisor. And here this is because the only function with no poles on a compact Riemann surfaces are just the constant functions. So there's a one dimensional space of them. So those two cases are easy to work out and the problem is of course what happens if the degree is positive. So let's first look at some of the easy consequences of the Riemann-Roch theorem. So first of all, in the Riemann-Roch formula, we're going to take D to be equal to zero. Let's take D equals zero and see what we get. It says L of naught is equal to the degree of zero plus one minus G plus L of K minus zero. And we can work out what all these terms are. L of naught is just one as we mentioned, the degree of naught is obviously just zero. One is one minus G is minus G and this is just equal to L of K. So we find L of K is G. In fact, we defined G to be earlier to be the dimension of the space of holomorphic one forms. And this is more or less the same because suppose we take omega to be a fixed possibly meromorphic one form, then it's canonical divisor is K. So L of K is the dimension of the space of functions f so that f plus K is positive. In other words, f omega is holomorphic. In other words, this is just the dimension of the space of holomorphic one forms. The other easy case we can work out explicitly is if we put D to be the canonical divisor K or a canonical divisor, whatever. Then we find L of K is equal to the degree of K plus one minus G plus L of K minus K. And again, we can work out all the various bits of this. We've just worked out L of K and found it was G. Degree of K, well, we don't know yet, plus one minus G plus, well, L of K minus K is L of naught, which is one. And if you look at this, we've seen who made some sign errors because these ones ought to cancel out and these Gs surely ought to cancel out. In fact, they don't cancel out. And what we find is that the degree of K is equal to 2G minus 2. So we know the degree of the canonical divisor. And the behavior of Riemann surfaces depends very much on whether this number is negative, zero or positive. So let's just quickly look at a few examples. For G equals naught, we get the degree of the canonical divisor is equal to minus 2. So it's negative. And you remember, we tried to find a meromorphic one form, dz. But at infinity, you remember it had a pole of order 2 at infinity. So in other words, the canonical divisor is equal to minus 2 copies of the point infinity. And we see the degree of the canonical divisor is indeed minus 2. If you look at an elliptic curve of genus one, here it says the degree of K should be zero. In fact, not only is the degree of K zero, but K can actually be taken to be zero because we have the one form dz has no zeros or poles on it. So K is actually zero and the degree of K is indeed zero. And as I said, one forms on surfaces of genus greater than one are a little bit tricky to write down. So I'm not going to give examples of them. I'll just finish by having a brief mention of some generalizations of Riemann-Roch to higher dimensions. So first of all, there's the Hertzabruck-Riemann-Roch theorem. This works for complex varieties, or more generally complex varieties, non-singular ones of dimension greater than or equal to one. So it's a high dimensional version. For example, a divisor sum of n, i, p, i, the p, i are no longer points, but they are now co-dimension one irreducible subsets. And what Hertzabruck found was a formula for things I'm going to rather sloppily specify as h naught d minus h1d plus h2d minus h3d and so on. So this is a sort of Euler characteristic, or at least it would be if I took the dimensions of these spaces. And here, instead of writing d, I should really write a line bundle associated to d, but I'm being a bit lazy. So how is this related to the classical Riemann-Roch theorem? Well, h naught of d is essentially the same as the number I called l of d, the space of functions with poles only at a certain place. hn of d turns out to be more or less closely related to l of k minus d. And this relation is a special case of something called serduality. Serduality is actually a generalization of Rock's contribution to the Riemann-Roch theorem. So if you go to the Riemann-Roch theorem, who did what? Well, Riemann did this bit of it. What he had was an inequality saying l of d is at least this term here, and Rock found the missing piece. If you wondered why Rock has his name to one of the greatest theorems of mathematics, but otherwise you've never heard of him, the answer is he died very young shortly after proving the Riemann-Roch theorem. So he's one of these people like Abel or Galois or Ramsey who died when they were very young. Anyway, so serduality identifies the highest cohomology group. In fact, it halves the number of cohomology groups you have. So in dimension one, the hits of Riemann-Roch theorem gives you a form of h0 of d minus h1 of d, which corresponds to l of d minus l of k minus d. In dimension two, people got very confused at first because it gives you a formula for h0 of d minus h1 of d plus h2 of d. And h2 of d can be identified as h0 of k minus d. So what people had in the original days was a formula for h0 of d, which was easy to understand, and h0 of k minus d, which was also easy to understand. And there was this very mysterious term that people didn't know what to make of. It was even more confusing because the first surfaces that people looked at, this term was actually zero. So it looked as if you really did get a precise formula for this. And then when people looked at more complicated surfaces, they discovered the formula broke down and doubtless spent several hours double checking their calculations to make sure they hadn't made a mistake. But anyway, what they did was they called this term the irregularity. And it was only when chief theory was introduced by Hirtse-Brook and Samson that people actually managed to figure out that the irregularity was really a special piece of a higher... It was the dimension of a higher cohomology group. And of course, in higher dimensions, you really need these higher cohomology groups because trying to interpret them directly becomes almost impossible. Finally, I should just mention that there's a further generalization of the Hirtse-Brook Riemann-Rock theorem due to growth and dick. So growth and dick's version of the Riemann-Rock theorem. Instead of working with one manifold or variety, growth and dick realized that the theorem is best formulation in terms of a map from one variety to another. And what it does is it relates certain derived functors of f, the images of certain derived functors of f to so-called churn classes of sheaves on x, which I'm not going to explain because this takes an entire one semester course. The special case of the Hirtse-Brook Riemann-Rock theorem is where you take y to be a point. So y being a point gives the Hirtse-Brook Riemann-Rock theorem because these derived functors of f then turn out to be just these cohomology groups that I mentioned when discussing the Hirtse-Brook Riemann-Rock formula. Okay, so next lecture I will probably be discussing the Riemann-Rock theorem in the special case of genus Nort Riemann surfaces when you can work out everything completely explicitly and see what's going on.