 This talk will be an informal survey of complex projective surfaces with kadara dimension equal to zero. So you remember, we had a very rough classification of surfaces according to the kadara dimension, which could be minus infinity, zero, one or two. And in the previous talk, we talked about the surfaces of kadara dimension minus infinity and showed they were all ruled surfaces, or at least stated that. And this time, I'm just going to do a quick survey of the ones with kadara dimension zero. So I'll first of all, give a quick sketch of how you classify them, and then I'll give a few examples. So the classification of these surfaces uses a basic identity due to NERTA. You should say this is due to Max NERTA. The famous NERTA is Emmy NERTA, a father was Max NERTA, who was also actually a pretty good mathematician, although people kind of forget about him because I don't know, don't really know why people forget about him. Anyway, Max NERTA's formula says 12 chi is equal to C2 plus C1 squared. This is actually a special case of the Hitzer-Brook Riemann-Roch formula in higher dimensions. NERTA, of course, didn't state it in this form. Unfortunately, algebraic geometry notation completely changes every few decades, and NERTA's original statement is almost impossible for people to understand if they've grown up with modern notation. Anyway, and just remind what these things are. Here, chi is the holomorphic Euler characteristic. So chi is equal to 1 minus H01 plus H02, where these are the Hodge numbers I discussed in the previous lecture. C2 is the second-chern number, which is equal to the Euler class, or Euler characteristic of the surface. That means it's topological Euler characteristic. And C1 squared is another churn number. And in this case, it's equal to the intersection number of the canonical bundle with itself. Now, in the special case, when the, suppose the Kadara dimension is equal to zero, and you shouldn't confuse the Kadara dimension, which is a Greek kappa with the canonical class, which is a big K. So if we write out what this says, it says 12 minus 12 H01 plus 12 H02, is equal to 12 chi. And if the Kadara dimension is zero, then H02 must be either zero or one. If the canonical class is zero, then H02 is equal to one. And if the canonical class is not zero, then H02 is equal to zero. On the other hand, this number here vanishes if the Kadara dimension is equal to zero. And the number C2, which is the Euler class, is B0 minus B1 plus B2 minus B3 plus B4, where Bi are the Betting numbers, which are the dimensions of the ordinary homology groups. And this is equal to two minus four H01 plus B2 because B0 equals B4 equals one and B1 equals B3 equals two H01, which is equal to two H10 for a complex projective surface. I should warn you that these equalities don't necessarily hold for non-algebraic complex surfaces. Anyway, if we look at this, we can rewrite it as follows. So it just says 10 plus 12 H02 is equal to H01 plus B2. And this number here is zero or one. And these two numbers are both greater than or equal to zero because they're dimensions of vector spaces. So there are only a finite number of possibilities of solutions for this equation. You can quite easily write them out as follows. So here are the possibilities. So H02 must be zero or one. And H01 must be, well, H01 must be at most 10 plus 12 H02. Let me just write this out again so you can see what it is. So H01 is pretty restricted. The only possibilities we get are this. And from this, you can work at the other hodge numbers and the remaining Betty numbers and they turn out to be the following. And these give you the four classes of surfaces. You may think there are five classes of surfaces, but these ones do not exist. And these four classes of surfaces are called Enrique's surfaces, hyper elliptic surfaces. These ones are the famous K3 surfaces and these ones are called Abelian's surfaces. So that's very roughly why you get four different sorts of surfaces of Kadara dimension zero. And what I'm going to do now is just give a few examples of each of these types of surfaces. So first of all, we have the Abelian's surfaces. These are the analogs of elliptic curves. Well, they're the closest analogs of elliptic curves. So in order to classify them, what you do is you map the surface S to its Albanese variety, which is a high-dimensional analog of the Jacobian variety for curves. And it's something called an Abelian variety. And in the case of Abelian surfaces, this map here turns out to be an isomorphin. So Abelian surfaces are Abelian varieties. So Abelian varieties are all of the form C to the N modulo a lattice L, where L is some lattice. And you've got to be a bit careful here because complex manifolds of the form C to the N over a lattice are called complex tori. But the problem is with complex tori is they're not all algebraic. The ones that are algebraic are called Abelian varieties. And there's actually a subtle condition. The lattice has to satisfy. In order for it to be algebraic, there has to be a Hermitian form so that the imaginary part of this Hermitian form is integral on the lattice L. Now, if N is one, if you're working in one dimension, it turns out that every lattice it's fairly easy to find a Hermitian form with this property. In fact, there's more or less a canonical one. But if N is greater than or equal to two, then this is no longer true. And there are quite a lot of complex tori that aren't algebraic surfaces. They're analytic, but not algebraic. So there are two slightly different classification problems. You can classify the analytic tori or you can classify the algebraic tori which are the Abelian surfaces. Anyway, we can give some examples of Abelian surfaces. First of all, we can just take the product of two elliptic curves. There are also a few examples where you take a product of two elliptic curves and divide out by a suitable finite group and that's still an Abelian surface. We can also take the Jacobian of a genus two curve. So you remember that any curve has a Jacobian whose dimension is the genus. So if the curve happens to genus two, we get a two-dimensional surface which is an Abelian variety. People don't study Abelian surfaces all that much because pretty much everything you can say about Abelian surfaces is actually a special case with some more general theorem about Abelian varieties. So the theory of Abelian surfaces is more or less a special case of Abelian varieties. I think I forgot to mention, you can also write down the underlying topological space quite easily because it's just a product of, so as a topological space, this is just a product of two N copies of a circle. So an Abelian surface just looks like a product of four copies of a circle in the same way that an elliptic curve looks like a product of two copies of a circle. Okay, well, there's a sort of variation of Abelian surfaces which are the hyper-elliptic surfaces. So we now got hyper-elliptic, just this. These are also sometimes called bi-elliptic surfaces and there seem to be two sorts of algebraic geometers. Those who call them hyper-elliptic and those who call them bi-elliptic and there seem to be not on the speaking terms with each other. Anyway, these are all constructors follows. What you do is you take an Abelian surface and you quotient out by finite group acting fixed point freely. And sometimes when you do this, you get another Abelian surface, but in general, you don't get a, but sometimes the surface you get isn't an Abelian surface, but is something called a hyper-elliptic surface. So let's just see some examples of this. So first of a typical example might be you take a product of two elliptic curves and quotient it out by a cyclic group of order four. Where the cyclic group of order four acts as follows. So I'm going to take E1 to be the complex numbers modulo one I. And I'm going to take the automorphism of order two where you map Z to IZ. And I'm going to take E2 to be C modulo one omega where omega is any old non real complex number. And I'm going to let Z map Z, that the automorphism be Z maps to Z plus a quarter. And then this gives you the quotient, this quotient gives you a hyper elliptic surface. It turned out to be seven families of hyper elliptic surfaces because there are seven different sorts of groups you can have in here. And the groups you have in here are all products of one or two Abelian groups and they can be written down in obvious notation as these seven groups. So people don't really study hyper elliptic surfaces very much because they're more or less just special cases of Abelian surfaces with certain rather special automorphisms. Next, we come on to the famous or notorious K3 surfaces. So the funny name was chosen by Andrei Wey who named them after Kummer, Keila and Kadyra who all worked on them. So let's have a few examples of these. The first examples are Kummer surfaces which are constructed as follows. What we do is we take an Abelian surface and we quotient out by a group of order two. So the Abelian surface is C squared modulo ellatus and we quotient out by the group of order two, we quotient out by the group of order two which just takes any point x to minus x where x is something in C squared. And this surface has 16 conical singularities and we can just resolve these singularities by blowing them up. And if we do that, you get a Kummer surface or K3 surface. Well, there are quite a lot of other ways of constructing K3 surfaces. So I'll just quickly mention four of them. First, we can take a branched double cover of the projected plane branched over a degree six curve or sex stick. Secondly, we can take a degree four hypersurface in P3. For example, w to the four plus x to the four plus y to the four plus z to the four equals zero. Thirdly, we can take the intersection of a cubic and a quadric in P4. Or fourthly, we could take the intersection of three quadrics in P5. And people found these four families of surfaces and they were a bit puzzled by them because they all seem to be very similar. They've got the same invariance. And moreover, it turns out there's a 19-dimensional family of each of these. You can see this quite easily. For example, for sex sticks in P2, the number of coefficients of a sex stick is six, sorry, seven times eight over two equals 28. So we seem to have a 28-dimensional family, but we should subtract one because if we multiply a degree six polynomial by constant, this doesn't affect the curve. And then we should also subtract the dimension of the automorphism group of P2, which is eight-dimensional. So we get 28 minus one minus eight, which is 19 dimensions for this one. And if you calculate the dimension of all these, you again get 19-dimensional families. In fact, Enriquez found a 19-dimensional family in projective space of dimension G of degree two G minus two for any G greater than or equal to one. So you can see these four examples of the first four cases that Enriquez found where these ones have degree four, six, eight, and so on. Well, this was a bit of a puzzle and it was sort of explained by Cadiara as follows. What's going on is that we have a, there is a 20-dimensional family of possibly non-algebraic K3 surfaces. And this contains an infinite number of families of algebraic K3 surfaces. So the picture you have is something as follows. You've got this sort of 20-dimensional space and inside it, there are all these hyper surfaces of degree 19 corresponding to algebraic K3 surfaces. So you can't really understand what's going on with algebraic K3 surfaces unless you include non-algebraic K3 surfaces. And then you find that K3 surfaces are really all part of a single irreducible family of surfaces. I should mention that K3 surfaces are Calabi-Yau manifolds. This means they've got a calymetric with vanishing Ritchie curvature. Calabi conjectured this and Yau was able to prove it a very famous result. And Calabi-Yau manifolds seem to be used quite a lot in physics because a manifold with vanishing Ritchie curvatures are rather nice thing to compactify on. So K3 surfaces are the simplest, non-trivial examples of Calabi-Yau manifolds. It's possible to be in varieties of Calabi-Yau manifolds. It kind of depends on how you define Calabi-Yau manifolds and some people insist they should be simply connected. So K3 are the easiest examples of simply connected Calabi-Yau manifolds. So the final class of algebraic surfaces are the Enrique surfaces. So the original example is most evaded by the following question. So Castell-Nuovo proved that a surface with irregularity and second plurigenus equal to zero implies the surface is rational. And this was a little bit of a ugly result because instead of using the second plurigenus, it'd be much nicer if you use the first plurigenus. So he asked if the irrationality and the first plurigenus, which you remember is the same as the geometric genus, are both zero. Does this imply the surface is rational? And Enrique's showed that the answer was no. And his original example is as follows. What you do is you take a tetrahedron, which is going to look something like this. I forgot how to draw a tetrahedron there. So we're algebraic geometers. So if we were geometers, you would draw the tetrahedron like this, but algebraic geometers, think of these lines as going on indefinitely. So the tetrahedron looks like this. And what we do is we take a degree six surface with double lines along the six edges of the tetrahedron. And this surface is singular, but normalization is an Enrique's surface. I've no idea how Enrique's came up with such a bizarre example. I suspect what had happened was that Enrique's had studied thousands and thousands of different sorts of surfaces and just happened to notice this one answered Castle Novo's question. There's another way of forming Enrique's surfaces, which you should take a K3 surface and to quotient it out by a group of order two acting fixed point freely. So Enrique's surfaces have the same relation to K3 surfaces that hyper elliptic surfaces have to abelian varieties. You take the K3 surface or the abelian surface and just quotient out by a finite group. So the theory of Enrique's surfaces turns out to be very similar to the theory of K3 surfaces. Their classification is reasonably well understood. There's a 10 dimensional irreducible family of Enrique's surfaces. And it can be described in a reasonably explicit form. Okay, so that's enough about surfaces of cadara dimension zero and the next talk will probably be about surfaces of cadara dimension one and two.