 okay okay so you it's recording okay let me find the definition of the Ricci curvature here great okay here is the curvature itself right it's a map from c infinity of e into c infinity of e tensor wedge two of t and dual or c infinity of e tensor wedge two of t m to c infinity of e let's see and then after that we we gave the definition of the curvature explicitly right it's defined like this and then the Ricci should be soon after Ricci where's the Ricci here's the holonomi and here we've got again the holonomi I did the holonomi after the Ricci before the Ricci that's kind of oh here's Levy Chivita oh yes of course yeah all right okay and okay this is again the Levy Chivita so let's see so let's recall the Levy Chivita is a special connection right which is defined which is associated uniquely to the metric and it satisfies you know this nabla g equal to zero and it's torsion free and then the curvature of the Levy Chivita is like this right it's a one three tensor and then we composed it with the right we we used little g itself to define a new tensor it's a zero four tensor right we compose the regular the usual curvature tensor with g tensor the identity right we get something like this and then oh sorry I have to turn off my camera again because it's creating problems sorry about that and then here we have got the I think the Ricci oh it's still not here let's see it looks like I didn't define it yesterday I didn't define the Ricci curvature I thought I did I was positive I did well I can't find it right now maybe I'll just give you the definition again sorry but I'm not going to stop looking through looking through the notes for the curvature I'll just give it to you um okay here it is so the Ricci curvature oops not red all right so recall the Ricci curvature is it's a zero two tensor defined as um so you define we define it point by point at at any point x you get a map you have a bilinear map on the tangent space to m at x into the real numbers and you take two tangent vectors and you send them to the trace of a certain linear map so w maps to the curvature uh okay I'm out of room here let me make this a little bit smaller so this guy maps to the trace of w maps to the curvature of u w acting on v okay so so what do we have here we have this is the bilinear map on the tangent space right so we take two tangent vectors and you send them to the trace of a certain linear map and what's that linear map it's the linear map that takes a vector a tangent vector again and send it to the curvature at that point apply to the two vectors uw and and to the three vectors I should say uw and v right so remember the curvature tensor you can actually think of it as something acting on triples right so this is something the curvature tensor acts on a triple of vectors so and the image is again another vector so this rx of uw v is another tangent vector right so w goes to another tangent vector that's a linear map from tx of m to tx of m and you take its trace all right yeah so that's the uh that's the definition of the Ricci tensor and if you write um if you write um in in local coordinates it's actually kind of easier if you write the curvature tensor r as the sum of r a b c d and then d dx a dx b tensored with dx c which dx d if I write the curvature tensor in terms of coordinates like this then you can write your Ricci tensor um so your Ricci tensor will be a sum of what I will call Ricci ab of dx a tensored with dx b where so then this is as you can see we're going to get exactly the trace this is going to be the sum of r a a c b oh so no no uh I always do this sorry about that r c a c b the sum over c okay so you can it's it's it's kind of easy to write down in terms of coordinates and if you do if you want to do it coordinate free you do it the first way that I just described okay uh sorry I thought I did this yesterday but it looks like I did not actually I think that you didn't yeah yeah I think I didn't yeah I I thought I did something yeah sorry about that okay so so let's see so we were talking about the Ricci flatness so we say something is Ricci flat if this Ricci curvature is just zero right and as I as I mentioned before you can show that the that the Ricci curvature is the actually the well the Ricci form you know I I took the curvature and I associated it to the to to the curvature right I used the metric and I associated the one one form to my Ricci curvature the Ricci curvature itself is actually symmetric it's not anti-symmetric so to get something anti-symmetric you can compose it with the metric um sorry compose it with a complex structure right the complex structure has the virtue that it will turn something symmetric into something anti-symmetric and vice versa so by putting in the complex structure I get something anti-symmetric it turns out it's a one one form and then it that you can show that that is the curvature of the levy-chivita connection on the canonical bundle of your manifold right and then that's how you see that basically if the Ricci curvature is zero then the canonical bundle is a flat bundle from which it sort of follows it's it's in fact trivial and you have a clavial and the holonomic group is contained in SUA all right so that is that is the clavial case and you know a lot of people have worked on the clavial case but that's not the purpose of our lectures here so I'm going to switch the next case which is the hypercalor case okay so the hypercalor case uh is based on the quaternions so let me remind you what the quaternions look like so the quaternions are can be you can write down a basis for the quaternions like this r times one plus r times i r times j and then r times k where i squared is equal to j squared is equal to k squared and is equal to ijk is equal to minus one they are associative right but they're not commutative and uh the you can define the leagroup sbr so sbr is the group r linear uh endomorphisms of h to the power r so h direct sum with itself r times preserving a quaternionic uh her mission form uh which here I will denote that by q so what do I mean by quaternionic her mission form I mean that it's the same thing as for being complex her mission but it's it would respect to the quaternions so it satisfies the equation q of a v b w is equal to a bar b q of v w for all a b in the quaternions and v and w in h r okay and this is and the bar is um so if a is lambda plus mu i plus mu j plus rho k then a bar is what you would imagine lambda minus a i minus mu j minus rho k okay so this is um quaternionic her mission and if you have a an r linear endomorphism that preserves this form you call that quaternionic her mission and then um such a q can be represented by a matrix by an r by r matrix which I will call a with entries entries in h such that a times a bar transpose is the identity of h to the r okay and um so now what you can do is each time you choose um each time we choose sorry sorry yes go ahead in the previous page there is a little typo uh in the previous page there is a yeah uh huh where uh a bar is equal to lambda minus mu oh yes of course yeah thank you sorry you're welcome minus mu thank you okay right now each time we choose some uh i in h with square equal to minus one then we can embed we get an embedding of s pr inside su2 r okay and um here is how we do this so we complete i to a triple to a to what we call a quaternionic triple i j k as before right which means that i square j squared and k squared and i j k are all equal to minus one right so that's the quaternionic triple we do that and then uh we represent uh multiplication by these guys um on h r by matrices that i will denote just by the capital letters i j k right and then you can show that you can write that you can decompose the matrix a as a sum of some matrix h plus another matrix omega times j where uh where this matrix h is her mission now this is this is just complex her mission with respect to i and this omega here is anti-symmetric and it with entries so the entries of omega are not just arbitrary quaternions they're actually in some sense complex numbers um where you where you use the complex numbers that are that are expressed as in terms of little i only so you have a copy a copy of the complex numbers inside h uh you know by using little i here right r plus r times i that's a copy of the complex numbers on this matrix omega has entries only in that subfield of of the quaternion so anti-symmetric with entries only involving i so any um so you do this and then um you can see that then sp of r can be represented by um so a matrix so a matrix an r by r matrix right with the entries in the quaternions right you can show that an r by r matrix belongs to sp r well first of all by definition basically it means that it commutes with a and then you can show that this this is equivalent to it commute commuting with with h and omega okay and this is when uh and you so then this will give you the embedding that you want uh if you think of um so maybe that's getting a little bit too complicated but but anyway you can use this sort of let me not get into any more details because i'm not sure i'm i think i'm going a little bit too fast for people to be able to really follow this argument it's not very complicated but you know if you sit by yourself with a piece of paper and a pen you can you can work it out so you can from this you can get this embedding that i was talking about of spr into su2r by thinking of sp sp r as um so think of uh i can get your embedding of sp r into su2r by thinking of um su2 i mean um no there's no s there sorry let me correct that so this is u of two r by thinking of u of two r as the group of transformations of hr commuting with h the matrix h all right okay and as i say if you don't mind i i'm not going to get into the details because i'm going a little bit too fast to really uh make that absorbable in such a quick time all right so um okay but why so why are we doing this so so given so now let's go back to our Riemannian manifolds right so we have we have our quaternions uh you know we we went over a little bit of stuff about the quaternions so now if we have a Riemannian manifold right given a Riemannian manifold m with holonomic group uh contained in sp r we can identify the tangent space to p at m with h r um to obtain a sphere of complex structures and how what are these complex structures so um so we said that the holonomic group is contained in sp r and yet right and we can naturally think of sp r as a group of automorphisms of h r right so that's how we do the identification of the tangent space to m with h r right so we have then these uh the holonomic group acts on tpm and it it's sort of it commutes with its action but with the action of sp r on h r right and then um so you have uh you have this identification and then you can define this sphere of complex structure in the following way so what are your complex structures so you write lambda equals a times i plus b times j plus c times k with a squared plus b squared plus c squared equals one so now you see that your abc the pair abc naturally belongs to the sphere s2 right inside r3 there's just a there's just a unit sphere right s2 inside r3 so that's why we could we talk about a sphere of complex structures right and so each time you have a point of the sphere you get the complex structure like this which is given as a linear combination of three complex structures i j and k and what are these complex structures um i j and k um so you have your um what you started out with a with a manifold which was a complex manifold right so you you have a given complex structure i right and then this this quaternionic structure allows you to um to define these other complex structures so these basically so i and j and k are obtained from the identification of tpm with h to the r right so you have an action of the quaternions on h to the r and you kind of transfer that to tpm and you you define as i said i'm not doing all the all the details here but um um you sort of you spread the the complex structures on hr along m you know on the tangent space is to m right and then gives you complex structures on m and then um you can you can verify that all of these complex structures are what we call constant with respect to the levy levy tvita connection so we can check that that nabla applied to lambda is zero for every lambda complex structure as before so then what you get is that g is caler by our definition of caler g is caler with respect to all of these complex structures okay so we have a sphere of caler structures meaning a sphere of complex structures that are all caler right okay so um so you see so in the calabi al case the holonomic group was potentially bigger right so um so let me just make a comparison here so comparison uh calabi al case let me assume that not that whole g is not just uh contained in sum but it's actually equal so it's uh actually let me do sorry oops what happened oh nothing okay all right because the calabi al case was sum so if you actually have equality then it means that your holonomic group is not contained in sbr right it's it's it's actually bigger strictly bigger so it's equal to sum so here you have only one one a unique complex structure which is caler so unique complex structure complex caler structure right in the in the hyper caler case so we have that whole g is contained in sbr right so we have at least a sphere of complex structures that's what we have and um in fact if uh so um here is our our next definition so definition is is that we say that m is irreducible hyper caler if the whole the holonomic group is actually equal to sbr in such a case there is exactly one sphere of complex structures uh sorry caler ones of course one sphere of caler complex structures so uh all right now this is the case where at the other extreme you have you also have um the case where so the extreme case if the holonomy is zero then you can have lots and lots of caler complex structures right so more caler complex structures an example of that is the complex torus so if m is a complex torus then the holonomy is zero and m has a lot of you know a lot of um a lot of these caler complex structures okay so um let's see how much time have i got uh we started at five minutes to 11 so i've got another maybe 15 17 minutes okay so um so let me now talk about the decomposition theorem okay that's one of the main things that we're interested in so the decomposition theorem this is for uh retrieve flat manifolds okay so suppose that m is the is a complex compact complex manifold which is also caler so what i'm giving myself here i'm giving myself m i'm giving myself a complex structure i and i'm giving on it on i'm also a romanian metric g and i'm assuming that my metric is caler with respect to this particular complex structure okay so compact caler we're going to also assume that it's complete and we're going to assume that it's richly flat so if we go back to our um list of uh holonomy groups right we can go back and look at our list of holonomy groups where is it here it is so i said it's um compact complex complete caler and richly flat right so if you look at the list here i'm not in the s1 case uh but i could be in case number two where the holonomy group is u of m or i could be in the case three where the holonomy group is s u of m but actually no i can't be in case two either because case two is not richly flat okay so i cannot be in one or two because one is not caler and two is not richly flat so i'm assuming i'm richly flat so i can be in three because i'm three i am caler and i'm richly flat in four i'm also caler and richly flat in five i'm not caler right and then in six and seven i'm not caler i'm not i'm not caler either right so then you see that the the cases the only possible cases are the holonomy is the holonomy is s u m it's it could be sp r but of course it could um it could also just be zero right i mean if it's a zero it it will uh satisfy what we want so here's so here's what the what the decomposition theorem tells you it tells you basically what you would suspect you know from the from the description of the holonomy groups if you have a compact killer and which complete and richly flat manifold then these are the this is what happens to it so the universal cover so here we're using also the the deram decomposition theorem right so the deram decomposition theorem was for simply connected things right so we take the universal cover of m then we say that we we have the the statement that this is isomorphic too now the deram decomposition gave you a euclidean space but this is a complex manifold so the euclidean space had better be a comp over the complex numbers so this is a ck times a product of some manifolds vi and then times a product of some manifolds xj right where so ck has the standard calorimetric of you know just the complex numbers um and for all oh sorry no just i and j um let me just say for all i first for all i uh vi is um compact simply connected with holonomy and of course irreducible because the irreducibility for follows from the fact that the holonomy is irreducible right so with holonomy su mi right so these are calabi house the holonomy is su mi and for all j xj is compact simply connected again irreducible in parentheses because that follows from the from the description of the holonomy with holonomy sp of rj for some rj sorry and that should have been did i put mi here yeah i did okay so the holonomy groups are of the type su here and for these guys are the derivative type sp which means these are hypercalors right because the holonomy is sp of something so that's the description of the universal cover and then the second part says something about m itself so there exists a finite et al cover let's say m prime of m isomorphic to a product t times the product again of the same vi's and then the product of the same xj's where now t is a complex torus okay so um so this this gives you a complete description right of compact uh caler richie richie flat manifold right so um you can see now also you can see also what happens with what the holonomy group of such a manifold what are the possibilities for the holonomy groups right because you know the holonomy groups of these factors um you know that t has holonomy group zero and then the vi's have holonomy group su of mi and xj's have got sp of rj okay so this is our um so this is our description right and uh so from now on then so so if you have uh so this is the description of these manifolds so the the next the for the rest of these lectures i'm going to concentrate on these uh hypercalor manifolds right which are the ones with holonomy sp of r for some r right um and uh you know uh describe some some things about them are there any questions before i uh switch gears and get into uh hypercalor manifolds more okay all right and i have got another maybe 10 minutes or something okay all right all right so the first thing about um hypercalor manifolds right so hypercalor manifolds are are what we call holomorphic symplectic okay so what does that mean it means the following it means that suppose that you have a manifold mij um compact scalar and simply connected and richie flat of dimension two r with uh now the dimension here two r over the complex numbers okay before before we were talking that about the dimension over the real numbers now from now on our dimensions will be over the complex numbers because everything is going to be a complex manifold so over c and with holonomy group um sp of r okay so suppose that we have these things then we have the following conclusions number one there exists a holomorphic two form phi on m which is non degenerate everywhere what do we mean by that we mean that it's non non degenerate as a map um uh from um uh let's see as a map from uh it's a two form so sorry it's as a map from tm to tm dual okay so um right so basically this this holomorphic two form what it means is that basically phi gives you an isomorphism between tm and i will write it now as omega one because that's the holomorphic with tangent bundle okay so phi case basically gives you gives you an isomorphism between the the complex the holomorphic tangent bundle and then it's dual so that's what we mean by phi being non degenerate right so there is this the holomorphic two form like this and it's it's unique up to multiplication by a scalar by a complex number uh scalar right and then number two is for all p between zero and the dimension h zero of m omega two p plus m two p plus one of m is zero and h zero of m omega two p of m is equal to c times phi to the p i guess it was actually the uniqueness is contained in the second statement right um so uh right so this is kind of nice so this is what we mean by being holomorphic simplistic which means that we have a holomorphic two form which is everywhere non degenerate and um and people you know even have let me give you actually the definition so a smooth complex analytic manifold uh let me just say then a complex manifold so a complex manifold is called holomorphic symplectic if there exists a phi from tm to omega one m holomorphic two form so and everywhere non degenerate holomorphic two form okay um if uh so let me call this m again as usual m is called irreducible if this phi is unique up to multiplication by a scalar all right uh do i have more time uh how much more time have i got if you're a couple of minutes but you can okay all right um well um well actually now i want to do examples so uh let me maybe i can just say something about the case of surfaces and then we can do the other examples tomorrow so okay so examples um so for instance surfaces for surfaces what do we have well you have that sp1 is actually equal to su2 and um so so this was this was the hypercalor case and this one was the calabiol case so what does this mean so this means that the hypercalor in this case is equal to calabiol and what are these guys so these are k3 surfaces so these are we have k3 surfaces well these are the irreducible holomorphic symplectic ones right but then you have so these are the ones where whose holonomy is actually equal to sp1 or su2 but then you we also have complex tori of course and that's it so that's all we've got so in dimension two we just have k3 surfaces and complex tori and what's a k3 surface right let me just give the definition of the k3 surface and then I'll stop a k3 surface is a compact complex manifold of dimension two such that omega 2x the holomorphic the the vector bundle of holomorphic two forms is isomorphic to oh it's trivial and h1 of x o x is zero okay so one can so if you if you just make the definition like this you can you can prove a lot of things you can prove that these are these these are simply connected you can prove that their integral homology is torsion free these are not too difficult and then you can prove that they are caler this one is hard okay uh in a unique way with a with a unique actually what what do I mean by unique way um remember these are all hypercalor so they have lots of scalar structures but they have a unique calor metric so the metric is unique the complex structures are not right so for each calor metric you have a sphere of complex structures right that are calor for that metric but um uh but once you if you fix the complex structure you have a unique calor metric so there's only one yeah one kilometer and uh all right if you don't mind I'm just going to give examples of k3 surfaces and then I'll stop so the easiest examples of k3 surfaces you can do uh double covers of p2 branched along smooth sex sticks uh you can do a smooth cortex in p3 uh you can do two three complete intersections in p4 and then you can do two two two complete intersections in p5 and these are these are uh these are the easy ones enough to this uh it's going to get a little bit more complicated okay so I will stop here sorry for having gone over a little bit thank you thank you