 This talk will be an introduction to the complex projective surfaces. So some examples are as follows. So first of all, we could have the projective plane or we could have product of two curves. So you might, for example, take a product of a projective line times a projective line. Another obvious collection of examples are just hypersurfaces in P3. For example, we could take w to the four plus x to the four plus y to the four plus c to the four equals zero, which is a simplest example of a K3 surface. And we have the following basic problem, which is classify all surfaces. Well, of course, classifying all surfaces is hopelessly difficult, but we can make some sort of progress on it and get at least a rough idea of what all surfaces look like. So in order to see what the classification of surfaces look like, we might start by looking at the classification of curves. So let's just recall what curves look like. So first of all, any curve is birational to a unique, non-singular projective curve. And we can see this by embedding the curve in projective space. We can then, we can see the curve in the projective space we can then take its completion, which makes it projective, and we can then get rid of the singularities by any of the two dozen or so known methods for resolving singularities. So all we need to do is to classify the non-singular projective curves as long as we're only worried about classifying up to birational isomorphism. And there's a basic invariant of non-singular projective curves called the genus, which you remember is a non-negative integer and can be any of these. And for each genus, there's a modular space, which is some sort of variety whose points correspond to isomorphism classes of these curves. So I guess it's a course modular space. And its dimension looks like zero, one, three, six, nine, 12, and so on as was sort of found by Riemann, although it's not clear how rigorous his argument was. However, so this gives a sort of picture of the classification of curves. It's given by a countable collection of varieties and we at least know the dimensions of the various bits of it. However, this isn't really the best way to look at the classification because there are three completely different sorts of behavior. Genus zero, genus one, and genus greater than one should be sort of a three different classes. So from the point of differential geometry, you would say that these ones have curvature that's positive, these ones have curvature that's negative, and these zero, and these ones have curvature that's negative. Well, curvature works fine in differential geometry, but isn't a terribly easy property to look at algebraically. So instead, we can divide these up by looking at the number of sections of the canonical class K, which you remember is essentially the cotangent bundle of the surface. More generally, we can look at the number of sections of powers of the canonical bundle for n greater than or equal to zero. Now, for genus zero curves, there are no sections of the canonical class or any of its powers because there's a negative degree. For elliptic curves, the space of sections is one dimensional and the same for powers. For genus greater than one, the number of sections of the canonical class is just the genus. More generally, we can look at the number of sections of K to the n for n equals zero, one, two, three, four, and so on, and it grows linearly. So it's one dimensional for naught, G dimensional for K equals one, and after this, it becomes three G minus one, three for five G minus one, seven G minus one, and so on. So what we have here is linear growth. It's not exactly linear because it sort of goes wrong for n equals naught and one, but it's sort of bounded by a linear function. And here we should think of this as being constant growth. It's bounded by a constant, and here it's just identically zero. So there are three different sorts of behavior that you can get by looking at the number of sections of the canonical class. And this can be used to give a morphism from the curve to a certain variety. So these numbers actually are called plurigenera. So we have the plurigenera, are usually denoted by Pn of the dimension of the space of sections of the nth power of the canonical bundle. They're called plurigenera because P1 is just the usual genus. So they're a sort of generalization of the genus. And we can look at the following graded ring. We can just take the sum over all n of the global sections of K to the n. And this is a graded ring, and so it will usually give rise to a projective variety, at least unless the genus is zero. And the dimension of this projective variety is related to the growth of the plurigenera. So the dimension is for G equals zero, one or greater than one. For G equals one, we just get a point. So the dimension of this variety is zero. For G greater than one, we have lots of sections. So the variety is dimension one. And here the variety is empty, which conventionally has dimension minus infinity. I guess varieties can't be empty, but never mind. And this number here is the dimension of a canonical model. Well, at least it's a canonical model if the genus is greater than one. But in any case, we have a map from our curve to this variety X, which has dimension minus infinity zero or one, and will quite often be an isomorphism, but there are some special cases when it isn't. So this number here is called the kadera dimension of the variety. It was actually introduced by Tarka rather than kadera, but it was based on kadera's work. So now we're going to do the same thing for surfaces. We want to classify surfaces into several different classes according to kadera dimension. So let's look at what happens. So now we're going to study the same thing for surfaces. Well, first of all, we can look at the cotangent sheaf, which can be denoted by omega to one. And the problem is this is not a line bundle. So taking sections of it doesn't obviously give a projected variety. Well, that's not a problem because we can take the second exterior power of omega one, which is a line bundle called omega two called the canonical bundle. And we could call it K because writing omega two is a bit annoying. And now we can copy what we did for surfaces. So we define plurigenera to be just the dimension of the space of sections of K to the N. And we can form a graded ring like this and ask what happens? So here we get a graded ring and this gives us a projected variety. And we can now ask what is the growth of the plurigenera? So these are called PN. And there are four, now four cases. We could have P to the N are all zero. So the kadera dimension, which is usually denoted by little Greek kappa is now going to be conventionally denoted by minus infinity. All the PN can be bounded. They're not necessarily constant, but if they're bounded, then we say the kadera dimension is zero because in this case, the projected variety here is dimension zero. All the PN can be bounded by some linear function in which case this variety is dimension one or PN sort of grow quadratically in which case the dimension of this variety is two. So this gives us the four basic classes of surfaces. And now we have to discuss the problem of singularities. So we know any curve is isomorphic to a unique non-singular projective curve. And we can ask, is the same true for surfaces? Well, no, it isn't because any surface can be isomorphic to many different non-singular projective surfaces. And the reason for this is that we can blow up a surface. So given S and a point P in S, we can blow up S at the point P. This means P is roughly speaking replaced by a copy of the projective line. And if we call the blow up S hat, then there's a regular map which is also birational from the blow up to S. And so, and of course you can blow up more than one point and you can blow up a point on the blow up and so on. So there are enormous numbers of ways of getting surfaces that are birational but not by regular to other surfaces. So S is called minimal if S is not a blow up of some non-singular surface. And we can ask, any surface is isomorphic to a minimal surface. This is any non, I should have said non-singular surface is isomorphic to a non-singular minimal surface. And we can ask, is this unique? Well, this is almost unique. It turns out it's unique if the Kedara dimension is greater than or equal to zero but it's not usually unique if the Kedara dimension is equal to minus infinity. However, it turns out it's not terribly difficult to classify the different minimal surfaces that are isomorphic if the Kedara dimension is minus infinity. And so, we now have two slightly different approaches for classification. The original approach was to classify surfaces up to birational isomorphism. It seems to be more convenient to, instead of classifying surfaces up to birational isomorphism to classify minimal surfaces. So for Kedara dimension at least zero, these two problems are essentially equivalent. For Kedara dimension minus infinity, it's slightly nicer to classify the minimal surfaces. And from that, you can easily read off the classification up to birational isomorphism. So now let's give an outline of the classification. So this is an outline of the Enrique's Kedara classification. So these are going to be classification of minimal, non-singular, projective, complex surfaces. So first of all, we have K, the Kedara dimension can be minus infinity. And for these, there are essentially two things. The surface can either be rational, which means it's birational to P2. So it could obviously be P2, but there are also some other ones called Hirtser-Brooke surfaces. And these are surfaces that map to the projective line and the fibres are also projective lines. These surfaces were probably well known before long before Hirtser-Brooke studied them, but they're named after him anyway. So next we have the ruled surfaces. So these are examples of these are just P1 times a curve. And more generally, we could have a surface mapping to a curve with fibres P1. So Hirtser-Brooke surfaces are special sorts of ruled surfaces in fact. So the Hirtser-Brooke surfaces and P2 are all the minimal surfaces birational to P2. And the ruled surfaces again are all for a given curve are all birational to each other. And they form a class of minimal surfaces that are birationally isomorphic. So next we look at Kedara dimension of zero. And here there are four classes. First we have the so-called Abelian surfaces. And these are the surfaces that are Abelian varieties. An Abelian variety is a projective variety that's also a group. So in one dimension, the Abelian varieties are just elliptic curves. In two dimensions, the analogs of elliptic curves are called Abelian surfaces. And we could take a product of two elliptic curves. That would give one example with E1, E2 elliptic. Or we could also take a Jacobian of a genus two curve. Next, we have the very famous example of K3 surfaces. A typical example of these is W to the four plus X to the four plus Y to the four plus Z to the four equals zero. More generally, any non-singular quartet curve is a K3 surface. These are the Kalabi Yao surfaces. So Kalabi Yao manifolds in high dimensions are of very great interest in physics at the moment. So there's been a lot of study of K3 surfaces because they're essentially the simplest non-trivial examples of Kalabi Yao manifolds. Then we have the hyper elliptic surfaces. And people don't really pay much attention to these because these are just Abelian surfaces quotient out by finite group. And if you know the Abelian surfaces, it's not too difficult to figure out what the hyper elliptic ones are. Then finally, we have the Enrique surfaces. And these have the same relation to K3 surfaces that hyper elliptic surfaces have to Abelian surfaces. They're just K3 surfaces divided by group. And this group always is order two. So you can sort of study K3 Enrique surfaces by studying K3 surfaces. For Kadara dimension one, you get the so-called elliptic vibrations. Here the surface maps to some curve and the fibres are mostly elliptic curves. However, you can have a finite number of fibres that aren't elliptic curves, but as something degenerate like a projective line with a node or something like that. So typical examples of these will just be an elliptic curve times any other curve, which is particularly simple example, because here all the fibres will be isomorphic. So you can have a finite number of fibres will be isomorphic and there are no singular fibres. Incident elliptic vibrations are very similar to elliptic curves over the integers. So if you take the spectrum of an elliptic curve over the integers, then it maps onto the spectrum of Z. And the fibres over points of Z are elliptic curves over finite fields or the rational. So the theory of elliptic vibrations is very similar to the theory of elliptic curves over the integers. In fact, what happened historically is these seem to have been studied independently for a few years before people noticed they were more or less the same thing. So Kadara classified the possible exceptional fibres in this case and Neuron in this case. And then people noticed the classifications were the same. And finally, we get onto a K equals two. So these are surfaces of general type. And there are enormous numbers of examples of these. So a few examples, we might take C1 times C2, where C1 and C2 have genus greater than one. We might take a hypersurface of degree greater than four in P3, provide it's non-singular, that will be of general type. And there are a huge number of other constructions that people have come up with. However, so what we have is for these ones here, the classification is sort of known. So we have a pretty good idea of what the different modular space and of these things are and what the dimensions of the various components of the modular space and so on are. For these ones, the classification is just wide open. We don't, I mean, we have lots of things, enough examples to show the classification of general type surfaces is incredibly difficult, but we still don't know even what a typical surface of general type is. We don't even know if there is a typical surface of general type. In fact, calling this a classification of surfaces is a little bit misleading. It's a bit as if, you know, we had an Enrica-Cadara classification of say birds. So what is the Enrica-Cadara classification of birds? Say it says that there are three sorts of birds. There are penguins, there are ostriches and there are birds of general type. Well, obviously you wouldn't count that as a classification of birds because you're just having this general type of birds which has an enormous number of birds that you don't know how to classify. And you should think of surfaces of general type as being like that. They're just almost all surfaces are of general type and we don't know how to classify them. So you should take the claim that there's a classification of surfaces with a big grain of salt. However, in practice, I must say the classification is actually pretty useful because most surfaces you've come across in practice tend to be of Cadara dimension less than two. I mean, people spend most of their time studying things like Abelian surfaces and K3 surfaces. So I'm going to finish by giving an example of a non-algebraic surface. So I said I was just classifying algebraic projective surfaces over the complex numbers. You can also classify surfaces over fields of characteristic greater than zero which whether classification looks similar but with a one or two minor extra complications. You can, if you classify non-algebraic surfaces you get some lots of completely new phenomena. So there are some types of non-algebraic surface that have no resemblance to any algebraic surface. And one of the most famous examples of these is the hop surface. So the hop surface is constructed as follows. You take C squared and you remove the origin. And this is acted on by the integers because we can just map Z1 Z2 to two to the N Z1, two to the N Z2. So we're just sort of, there's an obvious automorphism where we just double all components. And there's nothing special about the number two here. You could take any other number. And we can work out what this looks like topologically because we can write this as it's homomorphic to S3 times the positive reals. And Z is acting on these as multiplication by powers of two. So if we quotient this out by the action of the reals, this is isomorphic S3 times S1 because this is the positive reals modulo, the multiplicative action of powers of two. And this can't be any algebraic surface. And we can see this as follows by looking at topologically here, the betting numbers which are the dimensions of the homology groups over the reals are 11011. So this is for H0, H1, H2, H3, and H4. On the other hand, the betting numbers, B0, B1, B2, B3, and B4, for a compact non-singular algebraic surface must have, well, there are several constraints on these. First of all, B1 must be even. Secondly, B2 must be greater than 0. And you can see the hop surface fails in both ways that H2 is zero, which is not possible for an algebraic surface. And H1 is odd, which is also not possible for an algebraic surface. So this fails rather drastically to be anything like an algebraic surface. Kadera sort of made great progress in classifying the non-algebraic surfaces. And we mostly have a pretty good idea of what the non-algebraic surfaces look like, but there are some classes of them for which we still haven't quite figured out the classification. So the next few lectures I'll probably be discussing the various types of algebraic surface in a bit more detail.