Stefan Schreieder, Leibniz Universität Hannover, Germany.
From: The Crafoord Prize Symposium in Mathematics – Algebraic geometry and Kähler geometry, 2024-05-15.
Read more: https://www.kva.se/en/event/the-crafoord-prize-symposium-in-mathematics-2024-2/
[Music] so this is joying to work with Olivier de gay fortman who did his PhD in Paris and a post talk with me in Hanover so I’ll start um very very basic a let this in a g dimensional complex Vector space is a free subgroup generated by some R bases so as an example for gals 1 C is just a plane and we pick two linearly independent vectors which generate a lce like this now Associated to such a lce as a complex Tauros which is the quotient of this Vector space um by translation with the lce so in this example you take a fundamental domain and then you have to glue opposite sides if we do this we get a Taurus that would be the topological picture and so for g equals 1 and that’s um the higher dimensional case okay and so two t are isomorphic if and only if there is a linear map from CG to itself that carries one letters to the other so in in general um for G at least two if you take a general letters then this will be a nice complex manifold but um it the the geometry will not be very rich so for instance for General choices there will be no curves on this thing there will be no meromorphic function so one has to add some positivity property to get rich geometry in particular to make these things are bre and one way of doing this is to ask that they carry a principal polarization so a principally polarized ailan variety I will just preate this with ppav this is a pair a comma Theta where a is a complex Tauros so essentially determined by some letters inside C to the G and Theta is a divisor or rather divisor class up to numerical equivalence it’s a divisor on a um that has only one section so this is the additional piece of data that um makes these these guys hyic and gives them special properties instead of writing down the deviser you can just write down linear data that that implies the existence of such a deviser so this is equivalent to um to a positive definite her Mission form h on this complex Vector space of course this alone is not a condition such that the imaginary part of this form restricted to the l l is integral so it takes on integral values and it’s uni modular so the corresponding Matrix is determinant one okay so this is just to say that this additional data is really linear algebra data now the motivation for this definition comes from the examples if C is a smooth projective curve over the complex numbers then we can look at the integral chology and this will be a lce inside the real chology so this is a vector space of Dimension 2G and this is a um generated by an R basis of this Vector space essentially by construction but then so this is purely topological but then since a smooth projective complex curve H Theory gives ionically complex structure on here this is isomorphic to the space of 01 forms there’s a way of writing down such an isomorphism and now this becomes a complex Vector space of Dimension G and so we can take the quotient so this gives the Taos and then the polarization comes from an additional inine algebra data so some pairing and this is just um cup product and punity essentially so the polarization is induced by um cup product well yeah and conjugation um plus pun rity so essentially think about it the the UN modular pairing that you get on on the first chology just by cup product and and Punker okay so that’s the first example and then maybe the relevance of this example the theorem of torelli from 1913 who proved that two curves are isomorphic as smooth projective varieties if and only if these jacobians are isomorphic and here an isomorphism between the jacobians means that you have an isomorphism of Tor in this sense so linear map that Maps one lce to the other and it should respect this additional linear algebra data so the question of isomorphism classes of Curves is translated into linear algebra via the theorem it’s natural to wonder whether this works for other classes of varieties and here is another famous example we take a smooth cubic three fold then we can look at the third chology and of course this is again a lce inside the real chology but now because it’s a cubic hod Theory so the outer hod numbers vanish and this will there will be natural isomorphism to the one two part of the H composition and this is has now a complex structure and it has complex Dimension five so this has real Dimension 10 and this is set to the 10 so we can Define as before the quotient and this has this is a complex Taos of Dimension five moreover there is a natural polarization and it’s just defined as before and it’s a principal polarization so there is a principal polarization induced by cup product plus Punker Duality right so the cubic has Dimension three so real Dimension six so cup product on H3 um gives pairing now you can ask the same question and in fact the same holds true so two cubics are isomorphic if and only if two smooth cubics if these intermediate jacobians with the polarization are isomorphic okay so there is a very nice analogy between these two sides now you may ask well is this really a new example or maybe I’ve just written down the old example in a in a fancy way so maybe I get the same principally polarized varieties but that’s not the case and that’s the second part that’s another theorem in the same paper so this intermediate Jacobian is not isomorphic to any choban of a curve so these examples are really new and this theorem the relevance of this theorem is that it implies that the cubic is not rational so if it was rational then you could start at P3 and you blow up curves and points and blow them down and arrive at Y at some point but that just means that this intermediate Jacobian has to be isomorphic to the Jacobian of a curve one of them that you blow up and since they prove this is not the case it’s irrational so one motivating question of this talk is how far are these two examples from each other you can ask okay they are not isomorphic but you can ask other questions and ask yourself and how far they are apart and we will see later that this is um closely related to more refined rationality questions on the cubic to stable irrationality in particular but I’ll come back to this okay so two complex Tor are isogenous so I tried to write down a weak a weakening of the um of of isomorphism and one weak version of isomorphism as an isogeny so complex story are isogen genus if um there is a finite ital map between these quotients so that in topological language that just means that one is a topological cover of the other and okay so isogenous will be denoted by a TI or equivalently more in linear algebra language there is a linear map from cg2 itself that carries the Letts Tor Q to the other LS t or Q okay so if the lses are carried to each other and theor are the same and if this is true rationally they are isogenous essentially because q and R is dens um for any principally polarized ailan variety a z the locus okay and I dropped the polarizations because they’re not so important anymore from now on the Locos of all principally polarized varieties that are isogenous to the given one in the modly space of a billion varieties of Dimension G is countable and tense so in other words if you have two such guys then up to changing the lce a tiny little bit they they are going to become is um isogenous maybe not isomorphic but isogenous that’s that’s what this is saying okay so maybe the most na question would be is is any principally polarized abidan variety isogenous or even isomorphic um to a Jacobian and the answer is yes if the dimension is at least three and no otherwise and the reason is a dimension con plus a pair category the category argument so more precisely the modly space of genus G curves say Dimension 3 Gus 3 if G is at least to and this is strictly smaller than the dimension of the modly space of principally polarized bilan varieties which as Dimension G * G + 1 / 2 so these two numbers agree for g equals 2 and 3 and that’s essentially the proof why the answer is yes for G less or equal 3 but starting at the value gals 4 this number is bigger and now the suos of all principally polarized aan varieties that are isogenous to jacobians is going to be a countable Union of strata and each strata has this Dimension because it comes from from curves well and that is not going to eat up all the space by Bare category theorem so this this is easy it’s easy because we work over the complex numbers you can ask the same question over other fields then the answer is not so easy but still true so there is some famous work on this problem chai or proved this under the rean hypothesis and then simmerman was the first to to prove it unconditionally oh it’s just to it okay okay so instead of looking at this question over other fields I want to keep the complex numbers but change the question so that the dimension count doesn’t work anymore so this is question one so start with a principally polarized bilan variety a is there is some integer K such that a to the K is isogenous toobian okay so instead of asking that the principally polariz bilan Vari is isogenous to toobian maybe some power is and now of course a dimension count does not work because let’s see so we want to understand all principally polarized bilan varieties that are isogenous to a case power of a principally polarized variety of Dimension G so this guy is going to have Dimension K * G and the dimension of this Locus is just a dimension of the locus when you ver a so it’s just a dimension of H so it’s a countable Union of L side of Dimension um G * G + 1 / 2 but there is also the image of mg via the torelli map so the the locus of principally polarized varieties that are jacobians and this has Dimension 3 Gus 3 it’s an injection by the Torell theorem oh sorry it’s not 3G because now the dimension is K * G so there’s a k here and now of course for K very large this is much bigger than this number and so a priority for Dimension reasons there’s no obstruction that a component lies in this in this Sky okay there there has been some work on this problem um and one I want to mention is by Lu and zo and 19 so what they prove is that if e is a very general elliptic curve then the case power is not isogenous to a choban for all K bigger or equal than 12 so C is here a smooth projective curve not isogenous yes sorry thanks okay there there is a bound here but if K is sufficiently large um this K’s power is not isogenous toob over curve and of course the question is subtle because it is even isomorphic to the Jacobian of a singular curve you can just take the CIP curve and glue it to each other like this so this is a singular curve in the Jacobian if you do that K times it’s exactly the sky but nonetheless there’s not going to be a smooth curve that is isogenous to this okay and the theorem as well as the question are related to a famous conjecture they call manord conjecture which predicts something like that and in particular predicts that um this shouldn’t happen if a to K has at least mention 8 so maybe a remark so conjecture predicts no if Dimension is at least eight okay but what does it mean it just means well no let’s let’s that’s what kulman predicts okay so our result is as follows let a be one of the following either very general principally polariz the bilan variety of Dimension at least four or to the intermediate choban of a very of a very [Music] general cubic threefold then no power of a is isogenous to a choban of a curve okay now now let’s let’s see if the dimension is at least four that’s our assumption then in the case where the power is one the question was easy it was just a dimension count so the first non-trivial Cas is when K is 2 and indeed 2 * 4 is bigger or equal than 8 so we’re exactly in the colort range no no so in in the coor conjecture or here ah it doesn’t matter so if you have a product of jacobians since a is very general this is simple so if this is isogenous to a product of jacobians then each Jacobian has to be isogenous to some power so this two statements are equivalent yes yes absolutely yeah sorry the or contion says yes it says something about Str and the Mod Space so it it okay maybe I should have just written the conjecture it says that uh special sub iy for in the Mod Space AG for G at least eight special sub varieties are not generically cont Jacobian locost and special means a subvariety cut out by the condition that’s some chology tens are at H and the the L that you get by asking to be isogenous to a power gives a special Locus but only if you have it for very general because then you can move a and you get a whole Locus that’s the point yeah it could be true but so their their techniques are not going to show this no no otherwise we didn’t have to do anything here no no it’s possible that the statement still holds but the the complex statement is not hard and that took us 50 pages or so um plus lots of previous work okay yes yes we we we don’t know so these guys are certainly not defined over qar our our examples are defined over very transcendentally extensions of qar so you yeah yeah I understood the question I think this is open yeah there meth well I’m not an expert on the method but I’ve talked to some people and I don’t think it’s obvious that all right so this this is about one which is um related to colan ort and and yeah okay but then there is part two and part two is not related to col manord nor to this work of tan because this is really not it’s not a special the locus of intermediate jacobians of cubic three volt is not a special sub variety so this tors do not apply it’s it’s a different kind of a statement okay the the motivation to look at this comes from a result of clasa who proved that if Y and P4 is a smooth cubic then the following implications hold so you can first ask whether this is a DEC composition of the diagonal and this is an interesting condition because um if Y is St rational so say y multiplied with P1 or with another protective space becomes eventually rational which is a famous open problem we don’t know but if it was true then there had to be such a decomposition so a natural approach is to prove there is no such decomposition but we just don’t know and this is equivalent to um the question whether the this this class so there’s the Theta divisor and you take the force Power and then it’s a general fact that this is divisible by four factorial and integral chology and you want that this is the cycle class of a set combination of curves so it should be an algebraic class this is this is equivalent so to prove that there is a DEC composition of the diagonal or there is to prove that there is one you have to prove that this is algebraic now how to do that and in the same paper um CLA found many cubics that have that property and what she did is she constructs an isogeny of odd degree from a Jacobian of a curve and if you have that then algebraicity follows because two times that class is algebraic by the prim construction so you only need a odd multiple you need to show that an odd multiple of this class is algebraic and if there’s an odd degree isogeny from Jacobian of a curve then you can use that the classes here are algebraic push them forward to get that an OD multiple of this class is algebraic so this holds for um co-dimension 3 Losi in the mod of cubics now one one could one could write a slightly weaker statement that still implies this so instead of looking at the intermediate choban you look you look could look at a power and the implication would still be true but now it’s less obvious it follows from Forman and pman so one natural way of producing of of proving that this is algebraic would be to find such isogen is which was the motivation to for f the theorem so a shows that three um does not hold in general because in fact there is no isogen at all so in particular not no no OD degree isogeny okay the case k equals 1 goes back to to nanua that will be mentioned in a second so now how do we prove this theorem do follows from another result okay so we take some subari in the Mod Space of of genus G curve and we take a curve and set that is very general and we ask that Z contains all hypoelliptic curves so it should contain the hyperelliptic Jacobian the hyperelliptic locus so what that just says is that you have a very general curve well you have a curve of genus G and it admits a specialization to a very general hyperelliptic curve of that genus that’s what this assumption is say and then maybe there should be an assumption on the genus the genus should be at least four now let’s assume that the case power of the Jacobian is isogenous to a product of jacobians of Curves where C1 to CN our arbitrary curves of arbitrary genus then the statement says that K is n so the factors on both sides the number of factors are grease and X is isomorphic to CI for all I so what what is this saying this is some kind of strengthening of the of the Torell theorem so if the curve X is sufficiently General say a very general curve of Gus G or very general hyperelliptic then you can recover the curve just by the isogen type of any power of its Jacobian for okay so theorem A and B for k equals 1 is due to Naro um exactly was 16 and an earlier paper by belli perola and 89 so this is the base case and some of we be built on that so I have some time to explain something about the proof the first thing I want to explain is why B implies a so what’s b b is the statement that if you take a power of the Jacobi and then it’s isogeny type more or less determines X so if if this isogenous to a couple of to a product of other jacobians then the curves have to be the same and you want to conclude this statement that a under this assumption is not isogenous no power of it is isogenous toward a coin of a curve okay okay so assume there is an isogeny from the jacoban of a curve to some power of of of a where a is as in theorem a okay so the first step is to prove that as complex Tor they must be isomorphic okay so how how do we do this first observation is that it’s enough to show that the image of the push forward map is m m * the space for some Matrix some K * K Matrix okay so let me explain what I’ve written here we have this cover this this map this isogen from the Jacobin of the curve to the case power this gives a push forward on on cohomology or homology as you wish it goes from the first cohomology of the Jacobian to the first um chology of a to the K but the chology the first chology of a to the K is just the case power of the first chology so this this will be some suets and I want that it’s not any sus but I want to say that the suus is exactly the image that you get if you take if you apply some K by K Matrix to to the space in the obvious way okay so why why is this enough this is really the trick that goes back to belli PIRA in the case k equals 1 and it’s it’s a version of this so if this holds then f is just this isogeny and why can I tell well you dualize both isogenes and then you notice that two aogist that start at the seilan Variety coincide if and only if the kernels are the same and to compute the kernel you just have to understand this image but both have the same image and so both isogenes and the other way have the same um kernel so they coincide in particular the stoian has to be a to the k so what’s good about this Criterion is that this is now invariant under the deformations you just want to prove that the image of the push forward map has a certain shape and for this you can deform the theban VAR a and prove it on some deformation okay so we can deform a to JX where X is a very general hyperoptic curve okay so why can we do this well our assumptions imply this so by one it’s clear if a is just very general of Dimension four you can specialize it to hyperoptic Jacobian and in the second case it’s less obvious but it’s work of Kino that we can do this and then we get an isogeny now what happens to the curve we deform the curve the curve may split up but the Jacobian remains compact type so the left hand side splits up into a product of Jacobian and we get an isogen electus okay but now we are exactly in the situation of theorem B and B says now that all the curves are isomorphic to X and N is K well but that says that F0 is a map from the case power of this Jacobian to itself so it’s an endomorphism of JX to K and this is just given by multiplication of K by K matrices because um JX is simple so there are no non-trivial endomorphisms of JX so this gives the Matrix and that’s how you prove that this push forward is really given by multiplication the image is given by multiplication of a matrix so that’s very Matrix comes from so that’s that’s theorem B okay now the second step and positive definite and irreducible means that this space cannot is not isomorphic to a product of lower dimensional such inner product spaces so you just have a letter this together with such an inner product and there there is an one toone correspondence between irreducible principal polarizations here and such spaces and of course that uses that a is very general so a itself is only one polarization so here’s a sketch of the proof a polarization is a numerical equivalence class of a ample line bundle so a class and Nar a very group and for any such class we get a homomorphism from the Aban variety to its dual by mapping a point to the degre Z line bundle that you get by pulling back the devisor by translation of X and subtracting D now the Dual of a to K is just isomorphic to a to K and this uses that a is principally polarized so it’s isomorphic to its dual so what is written here is really the endomorphisms of a to De K but then as before endomorphisms of a to the K are just given by matrices because a was efficiently General so that it’s simple it has no endomorphisms no non-trivial endomorphisms so a potential polarization gives us a matrix and then you just prove that this Matrix is symmetric and the determinant one and it’s positive definite to give you such an inner product space that’s a check okay so surprisingly this it’s an open problem to classify the right hand side so I think this is done up to Dimension 20 something but in general it it grows exponentially and it’s not classified it’s a it’s a hard problem the problem is that this is integral that’s the problem so it is classified in low Dimensions but not in general okay so what do we get by step one this case power was just isomorphic to the Jacobian and by step two as a complex Tauros by step two the principal polarization here on the right hand inside it gives some principal irreducible polarization on the right so it has to correspond to some in a product space okay okay and now we we specialize again so we specialize a to e * B where e is an elliptic curve and B has Dimension One Less with the property that there is no homomorphism from the itic curve e to B again we can do this um in case one and two there are such degenerations now what happens if we do this what happens to this isomorphism now of principally polarized to bil varieties on the left hand we have e * B still polarized but it’s in a product space in a product space is not going to move if we just specialize the variety since this splits into a product that is really sorry Case power that is really e to the K Lambda time B to the K Lambda and the right hand side will now be a product of Jacobian of curves so the curve C May split up into a compact type curve so the right hand side may be may become a product of jacobians but Landa was an irreducible in a product space so this is an irreducible principle polariz staban variety so it has to match an irreducible Factor on the right so EK comma has to be isomorphic to one of those jacobians for somei okay so now we remember I need one more board okay so some case power of the cpic Curve is isomorphic to a Jacobian I’ve said earlier that there were were some results that something like that cannot happen indeed by L so this cannot happen if K is at least 12 then e to K is not even yeah so this elliptic curves should should be very general so that J invariant should be transcendental and then then if K is at least 12 then the 12 the K’s power of E is not even isogenous to choban so certainly not isomorphic to it so K cannot be too large and then I said that these inner product spaces well they disappeared but this inner this inner product space that gives the polarization they are classified by KES up to Dimension 14 so we are lucky and we look at the list and then we get that K has to be eight and this inner product space has to be the E8 letters okay well okay so there’s one case so it could be that case8 and the Eight’s power with the E8 let is polarize the theob of a curve okay so let’s let’s see um to rule that out we look at the automorphisms of the sky so the automorphism of this a power of the elliptic curve if the E8 let is polarized is just the vile group of E8 on the other hand this is if there is an isomorphism like this then this is isomorphic toout morphism group of the curve now let’s look how large these groups are and this group is a big group so it has that many elements but the automorphism group of a curve can never have more elements than 84 * the genus minus 1 now we know that this is an eight part of the elliptic curve so the genus of the curve is eight so this is eight and then this number is much smaller than that number and this is a contradiction so yeah I stop here