 Yeah, so last time we finished on discussing the relation between the structure of derived categories of various FANA three-folds, and I just reproduced on this blackboard the table that we had yesterday. So let me just remind that, so this is for everything for FANA three-folds of index one of picker number one, and in this case you can specify such varieties first according to the value of their index, of the divisibility of their canonical class, then for the two maximal possible values of the index we have only two FANA varieties, and probably I should also say that for both of them there is a full exceptional collection, and so the interesting part is the index two and the index one case. So in index two we have just five varieties classified by their half-anti-canonical degree, just the degree of the one-half of the anti-canonical divisor, which takes all values from one to five, and we denote the corresponding three-fold by YD, and for index one the corresponding varieties are classified by their genus, which is defined in a way that two G minus two is the anti-canonical degree of the three-fold, and the possible values for this genus, so the genus is bounded from above by 12, from below by two, and genus is not equal to 11, so the possible values are on the blackboard. And then we mentioned this FANA observation that at least for D greater or equal to N3, if you consider the non-trivial component of the derived category of the FANA three-fold of index two and degree D, and the non-trivial component of the FANA three-fold of index one and genus two D plus two, then the non-trivial components are equivalent if you choose the FANA three-fold in their modular spaces appropriately. In fact, there is a correspondence in the product of two modular spaces, which maps surjectively onto both factors, and for which if you take a pair of FANA three-folds corresponding to a point of this correspondence, then these categories are equivalent. Okay, so this is a theorem, and of course it is a, so in some sense this theorem gives like identifications of non-trivial pieces of categories for these pairs of FANA three-folds, and of course one can also ask what happens for D equal two and D equal one, is something similar holds. And in fact, it seems to be, this is not a theorem, but most probably this is what one should expect. So this is an expectation that for D equal two, in some sense the categories of type B and the categories of type A should be just, should correspond to the points of a common family of categories. So there should be a family, in some sense a flat family of triangulated categories, such that E y2 and A x6 are members of this family. So in some sense there, one category is a deformation of the other category, this is just the other way to express this relation. In particular, these categories for three-folds y2 and for three-folds x6, they have the same invariance. So if you consider for instance Hohschild homology, then they are isomorphic, or you can consider the Grottendy group and you can equip it with the Euler bilinear form, then the corresponding lattices will be isomorphic and so on. So, but so far it is not quite clear how such a family of triangulated categories which will include both categories of this type and that type, how this can be constructed and another interesting question whether these two sub-families of this family have some intersection points. Are there some three-folds of this type and can one find a pair of a three-fold of this type and a three-fold of that type such that the corresponding categories are equivalent. This is not quite clear, but this is maybe not too complicated question, so maybe with time this will become clear. And what one should expect for D equal to 1, so here the situation is probably even more complicated. In fact, it looks like the category of type A, so the category associated with the three-fold of index one, AX4, should be degeneration of the category of type X1, of BX1. So, in particular, if you compare some invariance of the corresponding finite three-folds, I mean a reasonable thing to compare is the intermediate Jacobians. So, for instance, if you compute the dimension of the intermediate Jacobian, so here it is Y1, sorry, if you compute the intermediate Jacobian of this three-fold it has dimension 21 and the intermediate Jacobian of this three-fold has dimension 20, so it looks like indeed a simple degeneration. One is a, on a categorical level is a degeneration of the other. So, in some sense there is some expectation on what should be true for these varieties. So, of course, one can also ask what happens for the other finite three-folds of index one. And here the situation is quite well understood for the two biggest values of the genus, for genus nine and genus seven, and it is much less clear for small values of genus. So, let me first say what happens for genus nine and genus seven. So, if you take any finite three-fold of genus nine, then its derived category has a semi-artogonal decomposition in which, again, there are two exceptional objects. One is the structure shape and the other is a certain vector bundle of rank three and the remaining component, in fact, is equivalent to the derived category of a curve of genus three. So, C3, the curve of genus three. And a similar decomposition exists for the three-fold of genus seven. So, in this case, again, there is a pair of exceptional vector bundles. One is the structure shape and another one is a vector bundle of rank five and the remaining piece is the derived category of a curve of genus seven. So, this is how it looks in this case. So, these two cases are also understood. But the remaining three cases are much less understood. For instance, I don't know whether there is, I mean, of course, the structure shape is an exceptional object in all these cases, but it is not clear whether it can be extended to an exceptional pair. So, it would be nice to construct a second exceptional object in the derived category of these three-folds and then to understand the orthogonal subcategory. In fact, one can say something about genus five case. One can give another interpretation for the orthogonal to the structure shape in terms of a certain shape of algebra on a projective plane, but so far it doesn't help much. So, this is what we know about the structure of these derived categories. Now, of course, the next question to discuss is what kind of geometrical consequences these structural results have. So, what are geometric implications? Yes, yes, yes. This is one of the ways to recover it, yeah, indeed. Another way is that you can consider, you can choose a particular modular space of vector bundles on the corresponding three-fold and this modular space will be this curve and then the universal object for this modular space will give you a functor from one derived category to the other, which gives you this embedding. I mean, from the point of view of intermediate Jacobians, it is quite hard to construct a functor, a reasonable functor, but interpreting this in that way, you can do it. In fact, this is typically what you should expect in general. If you expect that derived category of one variety contains the derived category of another variety, SSM orthogonal component, then you should have the embedding functor. And there is a very nice Arloff's result that says that such an embedding functor is always a 4MOKI kernel. So, this 4MOKI kernel should give you, is an object on the product of these two varieties and you can think about it as about a family of objects on one variety parameterized by the other variety. So, if you want to construct such a decomposition, you should look for a nice mod-lay problem. But also, I mean, in some sense my point that you can also work in the opposite direction. As soon as you know that there is some decomposition of this sort, then you can somehow consider any mod-lay space on this variety, on the big variety, and then if you take any object parameterized by this mod-lay problem, you can also project it to this subcategory. And then you will get, you can consider also the mod-lay space for these projections of the object. And the projection functor will give you a morphism between these mod-lay spaces. Of course, in general, it will be rational because, I mean, for a priori, for different objects, you can lend into objects with different classes in the growth and de-group. No, no, this is, of course, not true, but, I mean, some of them may be stable for your stability conditions, some may be not. I mean, the map in general is rational, but let us pretend that it is regular. Then typically, in this way, you will have a description of this mod-lay space in terms of that mod-lay space plus some additional data. And this is, in many cases, this can be observed. So, for instance, in these three cases, in the cases of these equivalents, it is natural to expect some relation between mod-lay spaces on one three-fold and on the other three-fold. And maybe let me mention one of such relations. So, I assume that d is greater or equal to n3 and x sub d and, I'm sorry, y sub d and x sub 2d plus 2 is a pair of three-folds such that the corresponding categories are equivalent. Just a pair of three-folds corresponding to a point of this c that lives here. Then one can consider two mod-lay spaces. One mod-lay space is the Hilbert scheme of lines on yd. So, this is a very natural variety associated with the three-fold, just the Hilbert scheme of lines. And for the three-fold of the other type, you can consider the Hilbert scheme of conics. And the statement is that they are just isomorphic for any such pair. So, the variety that parameterizes lines on one variety also parameterizes conics on the other variety. And in fact, the way how it can be proved is just by observing that if you consider, so this is not quite proved, but this is an explanation for this coincidence is that if you take a line on yd, then its ideal is an object of this subcategory, in fact. In fact, it is very easy to see that the ideal of a line is semi-artogonal both to the structure shift and to the line bundle of one. So, it lives in this subcategory. And if you take a conic on x to d plus 2 and again you take its ideal, then this ideal is in this subcategory, x to d plus 2. And also you can check that the equivalence of this theorem just takes ideals of lines to ideals of conics. So, it gives you this identification. So, this isomorphism is induced by this equivalence. And of course, if you consider some moccasin complicated modular spaces associated with these two varieties, you probably, I mean, both, then you will have a picture of the sort that I just described. There will be a morphism, maybe a rational morphism from a modular space corresponding to this variety to some modular space of objects in the corresponding subcategory b, which will be identified with some modular space of objects in the subcategory a to which some modular space of objects on x will be mapped. And so you will have a certain nice correspondence between these modular spaces. And if you are lucky, I mean, in the best situation you will have just an isomorphism of this type. But maybe you will have a morphism and a description of fibers for this morphism or something of this sort. But you see that this geometric isomorphism is a very nice manifestation of this equivalence of categories. So, you have some result on a categorical level and then of course it gives you many results on a geometrical level. Okay. And maybe a good question in this direction, which gives a link to the lectures of Arendt is, I mean, imagine that you have such a category. Then if you want to speak rigorously about modular spaces of objects in this category, you need some stability condition on this category. So the question is whether these categories admit stability conditions. And as far as I know, it is a very, very, very recent result that the derived categories of these finite three-folds themselves have some stability conditions. This is a recent result by Emanuele Macri. So I think also Marcella Bernardara and I don't remember the other people. So maybe Paul Stollari also who will come for the third week. So maybe we can ask him about the progress. And of course it would be very natural also to try to construct some stability conditions for these categories because this will make this relation between modular spaces more rigorous. Okay. Okay. So this is what we have for these pairs. And of course we expect something similar in these two cases. For instance, it is natural to expect that the Hilbert scheme of lines for the three-fold of type Y2 is probably a deformation of the Hilbert scheme of ponics for the variety X6. And the similar degeneration result for D equal 1. But this I cannot say whether this is true or not so far. Okay. Now what about these two varieties? About genus 9 and genus 7 three-folds. Here we have this description of the derived categories in which this curve appear. And of course it is natural to have some relation between modular spaces of objects here and on the corresponding curves. And in fact there are there is a series of papers by Daniele Fiance and Maria Chiara Brambila who discussed this question. So they construct some relation, very explicit relation between modular spaces of vector bundles on the three-folds XG and modular spaces of vector bundles on the corresponding curves. And I mean one of the particular cases of this result which I remember is that if you consider genus 7 case and if you consider the Hilbert scheme of ponics on the corresponding Fana 3-fold then it will be just isomorphic to the second symmetric power of this curve of genus 7. So in this case this is just the surface which parameterizes pairs of points on the corresponding curve. So this is another way how the curve can be reconstructed. Okay, yes, sorry, Brambila Fiance. Okay, so yeah and as I said for smaller values of the genus it is hard to say anything because we don't have a good description of the category. Okay, so this is what do we have for three-folds. Let me say some words about higher dimension. Of course I mean in general what one can say in this direction is limited by our knowledge of the structure of semi orthogonal decompositions. So in cases when we have some knowledge of this type we can investigate these problems, we can find relation between model spaces and so on. So this is why I don't consider just arbitrary variety of some dimension but restrict to some examples in which cases there is a good description. So a very interesting variety of dimension for which some version of the story exists is a cubic four-fold. So let me say something about this case. So by definition this is just a hypersurface of degree 3 in P5. So degree of axis and let us restrict to the smooth case. So a smooth hypersurface of degree 3 in P5. So how one can construct a semi orthogonal decomposition for such a hypersurface. Of course I mean we can use the same idea. We can find some exceptional collection and then consider this collection and its orthogonal subcategory. And since the canonical class in this case is just O of minus 3. So the Kudaira vanishing says that there is an exceptional collection of length 3. So O of 1 and O of 2 they form an exceptional collection. And as usual you can extend it to a semi orthogonal decomposition and there will be a decomposition of this sort, d of x with the four components, with these four components. Three components are very simple generated just by line bundles, each of them equivalent to direct category of vector spaces and one more component in which all the non-trivial information about the cubic is concentrated. So what one can say about this component? So in fact it has very interesting properties. So first of all one can consider its serfunctor. So serfunctor I did not give you the definition but this is the functor that encodes information about serduality. So maybe let me write the definition here. So if you have a nice triangulated category T then one can define its serfunctor by requiring that there should be bifunctorial isomorphism between homes from F1 to F2, dual and homes from F2 to S sub T of F1. So this is precisely what happens for serduality when T is d of x. So in fact such existence of such a functorial isomorphism determines this functor in a unique way. There is, if the serfunctor exists then it is unique. And for T equal to d of x, S sub T is isomorphic to the tensor product with a canonical class and the shift by the dimension. So in particular if we think about a triangulated category as about a non-commutative variety then the information about its serfunctor encodes in some sense its canonical class and its dimension both the first and the second. So what one can say about the serfunctor of this subcategory? In fact what one can check is that this serfunctor is just isomorphic to the shift by two. So in particular if we think about the serfunctor as about tensoring with canonical class and shift by the dimension then the canonical class should be trivial in this case and the dimension should be two. So if we think about this category as about a non-commutative variety this variety should be a surface with a trivial canonical class. And what kind of surfaces with this property do we know? Of course in the usual commutative world we know only two classes of surfaces of that type basically k3 surfaces and abelian surfaces. And so the next question is how to distinguish between these two classes. So should we think about this category as about a non-commutative k3 or should we think about it as about a non- commutative abelian surface? And to distinguish between these two one can consider the Hohschild homology of this category. And again for a general triangulated category one can define this Hohschild homology by considering the identity functor of this category, by considering serfunctor of this category and by considering all x groups between these two functors. I mean of course this definition should be modified to make it rigorous. One should explain why and how one can consider x groups between functors and the functors of a triangulated category. It requires some work but I probably don't have time for this. Just believe me that this Hohschild homology can be defined, that it is a nice invariant. In fact maybe I should say that just two properties of this invariant. So the first property is that for categories corresponding to varieties, for derived categories of algebraic varieties, this gives you some well-known geometric invariants. So if t is d of x then the Hohschild homology of t, so of d of x, in degree k this is just the direct sum over all q minus p equal to k of h p q of x. This is just the Hohsch homologies of your variety and this is called Hohschild constant Rosenberg isomorphism. There is a natural isomorphism of this type. This is one important property of Hohschild homology and the second important property is that Hohschild homology is additive with respect to semi orthogonal decompositions. So if t has a semi orthogonal decomposition with several components then Hohschild homology of t is just the direct sum of the Hohschild homology of the components. So in particular if you want to compute Hohschild homology of this particular category this is very easy. So first you should compute the Hohschild homology of d of x and for this you have to know the Hodge numbers and the Hodge diamond in this case looks just like that and all the other Hodge numbers are 0. So the Hohschild homology of d of x in this case, so by this formula to get the dimensions we should just sum along the columns of the Hodge diamond. So we will have 1 in degree minus 2, we will have 25 in degree 0 and 1 in degree 2. So this is just c in degree minus 2, c 25 in degree 0 and c in degree minus 2, in degree 2. So the formula will be like that and then if by using this additivity property we should just subtract from this the Hohschild homology of these three categories and since each of these categories is equivalent to the derived category of vector spaces by again for instance by Hohschild constant Rosenberg its Hohschild homology is just c sitting in degree 0. So we should just portioned out by a three dimensional vector space in degree 0. So the Hohschild homology of this category what we get as a result will be just c in degree minus 2, c 22 in degree 0 and c in degree 2. And clearly you have the same answer for a k3 surface. The Hohschild, the Hodge diamond of a k3 surface looks like that and it gives you precisely the same numbers for the dimensions of the Hohschild homology and if you consider the Hodge diamond of an abelian surface it will be completely different. You will have non-trivial H1 and HH1 and HH-1 for instance and HH-0 also will be different from that. So clearly from this point of view AX should be thought of as a non-commutative k3 surface. And in fact there are let me say a couple more things about this category as a kind of explanation why it is very reasonable to think about this category in this way. So first of all in fact for some cubic fourfolds or some x in fact there is a k3 commutative k3 surface, y such that AX is equivalent to d of y. So for instance one of the examples is the following. Let us take Grasminian 2,6 so maybe let me take a vector space of dimension 6 and let us consider Grasminian of two-dimensional subspaces in it and so by definition it lives in the its pluker embedding embeds it into the projectivization of the second wedge power of this space W and let us take some subspace of co-dimension 6 in it. Co-dimension of VW also be equal to 6. Let us consider this general linear subspace of co-dimension 6. Then we can define y to be just the intersection of this Grasminian with this linear subspace in this pluker space. So in other words this y is just linear section of co-dimension 6 of this Grasminian. Then in fact y is a k3 surface and this is very easy to see if you use a junction formula to compute the canonical class and if you use left-shut's theorem to control the co-homology then you get this result for free. So this is a k3 surface and now let us also see that it is naturally associated to a certain cubic four-fold. In fact to do this let us consider so our space V was a subspace in lambda 2W so we can consider the dual spaces lambda 2W dual. There will be this embedding will give you a subjective map to V-dual and the kernel of this map will be six-dimensional. So let me denote it by V-perp. So this is just the kernel of this map and this is because of this co-dimension condition this is six-dimensional vector space. And now let us consider the projectivization of this space lambda 2W dual and it's the projectivization of this linear subspace in it. And now that in this big projective space there is a natural cubic hyper surface. So basically this space parameterizes skew forms on a six-dimensional space and we can consider just all degenerates skew forms. For a skew form the condition to be degenerate is controlled by the determinant of this skew form and since the dimension of the space W is even the determinant of a skew form on this space is equal to the square of the pfafion of this form. And since the determinant of a six-by-six matrix is a polynomial of degree 6 in its elements the pfafion which is the square root of this determinant is a polynomial of degree 3. So its vanishing gives you a hyper surface of degree 3 which is usually called the pfafion hyper surface. So this is the pfafion cubic hyper surface in this projective space. And then you can define x to be again the intersection. So it will be the intersection of this cubic hyper surface with this p5 space. And of course it will be a cubic four-fold. And then one can check that one has this equivalence. So this is an example when something like that holds. And in fact there are also some other examples when such an equivalence can be constructed. But in general the general picture which is not yet proved but again which we expect is the following. So let us consider the modular space of all cubic four-folds. Let me call it just in for simplicity. This is a certain modular space of dimension 20. So it is easy to see that the dimension of this modular space is 20. Just if you consider what is the data you have to specify a cubic four-fold you need to specify a polynomial of degree 3 of six variables. And the space of these polynomials has dimension 56. But also we have gl6 group acting on the space. So the quotient of the space of all polynomials by this section of the group it gives you this modular space. And its dimension is 56 minus 36 which is 20. So we have this 20 dimensional modular space. And what we expect is that there should be countably many divisors. Divisors say d sub d are parameterized by some integers in this modular space. Such that for any x for any cubic four-fold that corresponds to a point of this divisor there is a k3 surface which is a polarized k3 surface of degree d. So it corresponds to a point of a modular space of polarized k3 surfaces of this degree such that the corresponding categories, sorry, ax is equivalent to d of y. And the case that was described here corresponds to polarization of degree 14. And in fact this is the smallest value of d for which such situation is expected. The next case should be d equal to 26 that was described recently by Nick Adington and his co-authors. So next case. So the case d equal to 14 is well understood and it is on this black board. So the next case is d equal to 26 and was done by Adington and his co-authors. And the next case is d equal to 38 in which case we don't have a description yet. So you see that in fact for many cubic four-folds we expect this category to be in fact commutative but in fact it is very easy to see that for a general cubic four-fold this category is not commutative. So for general x in fact ax is not equivalent to d of y for any k3. In fact for any algebraic variety. Yeah, yeah, and this is one of the conjectures that a cubic four-fold is rational if and only if this holds. If there is a k3 surface equivalent derived equivalent to this category. So let me mention since this was asked. So this is a conjecture that x is rational that this category controls rationality. And only if ax is equivalent to d of y for a general for some commutative k3 surface y. Let me maybe mention that this result in fact is very simple. You can just see how the numerical groten d group of this category looks. And it is easy to check that for a general cubic four-fold the its numerical groten d group has ranked two. While for any k3 surface it has ranked at least three because there is h I mean this is clear. So for a general cubic four-fold we cannot have anything of this sort. So this observation shows that this conjecture would imply non-rationality of a general cubic four-fold which is very old conjecture which we can't prove yet. Okay and maybe yeah also probably I should have said before that of course you could ask in this direction maybe we can somehow also decompose this category. Maybe this decomposition is not good maybe we can find something orthogonal decomposition for this piece of this category and get a more precise information. But in fact from this property we see that ax is a Kalabiyao category and so in fact the bridgeland argument shows that this category is decomposable. Let me mention it here. Composable. And since it is decomposable if this category would be equivalent to the derived category of any algebraic variety then it follows that the variety should be a surface with zero and canonical class and from this it follows that it should be a k3 surface and then you see that for a general ax this does not hold. Okay so this is a short summary about this category. Now let me say a couple of words about geometric implications of these for modular spaces. So what do we have about geometry? Of course since the essential piece of this category is a k3 category we should expect that the modular spaces behave like modular spaces of shifts on a k3 surface and one of the nicest observations about modular spaces of k3 surface is that all of them carry a symplectic form. So we should expect to have maybe not symplectic but maybe a two form on any modular space on a k3 surface and indeed this is what happens. So if for any modular space shifts on x on a cubic fourfold there is there is a natural halomorphic two form on it and again the picture you should have in mind here is the following. So we have this semi orthogonal decomposition so let me maybe denote by alpha the embedding functor from this component to d of x then we can consider the corresponding projection functor say alpha star from d of x to ax and so as we discussed it induces maybe a rational morphism from some modular space of shifts on x yes so we can specify some rank c1 c2 and so on we can consider some modular space of shifts on x and there will be a morphism to some modular space of objects in this category and this modular space in fact should carry a symplectic two form and in fact one can check that the kernel spaces of this halomorphic two form constructed on this modular space they just correspond to the tangent spaces to the fibers of this morphism so tangent spaces to fibers are the kernel spaces for the of this theorem and so in fact this symplectic two form on this modular space can be thought of as a form induced by this two form and projection and again in so maybe let me give some examples we see a bit better what happens here maybe the most well-known example of this sort is if you consider again the Hilbert scheme of lines for this cubic fourfold so if you consider the Hilbert scheme of lines of x this is just an example of a modular space then in fact if you consider this projection functor it induces an isomorphism to some modular space of objects in this category AX and in particular the two form that can be constructed on this Hilbert scheme of lines is itself symplectic so this is a this carries a symplectic form and in particular since this is a smooth projective variety with a halomorphic symplectic form it follows that it is hyper-hyper-calar in fact this was first observed by by Will and Danagi so this is a very interesting hyper-calar variety of dimension four so now one can ask what happens for curves of bigger degree so the next case one that one can consider is the Hilbert scheme of conics in fact in this case the map to the modular space is not an isomorphism but in fact it can be thought of it factorizes through a natural map to the Hilbert scheme of lines which as we have seen can be considered as the modular space of objects in this category and in fact this map is very easy to describe geometrically so if you have a conic on a so if you have this cubic fourfold and if you have a conic on it then every conic generates a projective plane so we can consider the projective plane in the ambient P5 which is just the linear hull of this conic and then we can intersect this plane with this cubic hyper surface of course the intersection contains a conic so but the intersection should be a curve of degree three it means that it should be composed as a union of this conic and the line and so in this way we can associate with every conic on X a certain line of course there is a subtle point here that this plane can be contained in the cubic fourfold and in this case this space this map will not be defined at this particular conics so in general this map is not regular but only rational and this is just an example that such a projection is not regular in general but I mean for instance we can restrict to considering only cubic fourfolds that do not contain planes a general cubic fourfold has this property so then this map will be regular and in fact this map I mean let us pretend that it is regular maybe by considering sufficiently general cubic fourfolds then in fact this map is a P3 vibration and these fibers this P3 they just correspond to different planes passing through a given line and the tangent spaces to the fibers are precisely the kernel spaces of the two form on this Hilbert scheme of conics so this is the second example maybe let me mention two more examples because they are also quite interesting so first of all one can consider next case which is the Hilbert scheme of twisted cubic curves maybe I should say rational cubic curves in X and then it turns out that this is a nice ten-dimensional variety this is a nice smooth ten-dimensional variety again for a sufficiently general X but the two form on this variety is not symplectic it is degenerate in fact it's core angle in a general point equals two so from this picture we expect a vibration with two-dimensional fibers and in fact one can see one can check that there is a picture like that so there is maybe let me write m prime of AX to distinguish from that modular space and here m prime AX tilde and this map is a P2 vibration here the blow up here this is a recent result by Christoph Lien Manfred Lien Christoph Zürger and Duker von Stratten and this modular space that they construct as a contraction of this Hilbert scheme of rational cubic curves is in fact a nice eight-dimensional hyperkeller variety and there is one more interesting hyperkeller variety recently constructed for a cubic four-fold in a similar fashion this was done in a recent paper by Rado Laza Julius Saka and Claire Wozien in fact they proved that there is the following nice hyper hyperkeller variety so if you consider a cubic four-fold you can also consider it's a hyperplane section so let me denote it by XH which is a cubic three-fold of course and then one can consider a certain special modular space of ships on it that is isomorphic to its intermediate Jacobian in fact for a cubic three-fold you can realize its intermediate Jacobian as a certain modular space of ships I guess that this is not true in general but this is true in this case and so one can consider the modular space I mean in this way we have a five-dimensional space of cubic three-fold sitting inside this given cubic four-fold just all hyperplane sections and for each of them we have its five-dimensional intermediate Jacobian so we can consider so to say universal intermediate Jacobian of all hyperplane sections this will be a variety which will map to the space of all hyperplanes in our projective space with fiber being the intermediate Jacobian of the corresponding hyperplane section and they show that this ten-dimensional variety is also hyperkeller and in fact this is a very interesting hyperkeller variety I mean in the previous two examples the corresponding hyperkeller varieties for deformation equivalent to Hilbert squares of K3 surfaces but this one I mean here to the Hilbert square and here to the Hilb 4 of K3 but this one is not a deformation equivalent to Hilb 5 of K3 in fact it is a deformation equivalent to one of those paradigms or gradient hyperkeller for varieties so you see that you can find very interesting geometric information which which follows in some sense from this K3 property of this category and I wouldn't be surprised very much if some other modular spaces of shapes on on a cubic four-fold will give in the end some new examples of hyperkeller varieties I think this is a very interesting direction of research okay so yeah I guess that I don't have much more to say and my time is over so maybe I should also say that there are some other very interesting high dimensional varieties that behave in a very similar way so there are other examples of varieties whose derived category is generated by some exceptional collection plus a K3 category and also one can investigate modular spaces of these varieties and try to find some hyperkeller structures on them so I mean maybe one should somehow summarize the point of my lectures as saying that if you know or if you can guess how the derived category of your variety looks like it gives you a lot of evidence about how modular spaces of objects on this variety should what kind of properties they should have just from from the structure of the derived category and this this is very useful if you want really to describe this modular spaces or if you are searching for some interesting example this is like I mean derived categories should give you some light in the dark forest of modular spaces let me stop here