 Yeah, thank you very much for the invitation. It's a pleasure to give this talk. So, as Francesco mentioned, I'm going to talk about complex k-theory of dual pigeon systems. So this is a joint work in progress with Xi You Shen and just so you have an idea what this talk will be about or what the structure will be. So at first I'm going to recall the basics of the theory of fixed bundles and the modellized bases and then we're going to talk about a connection with Langland's duality and mirror symmetry. And in the third part we're going to ask the question or post the question where there could be an integral version of the mirror symmetry relation for modellized bases of fixed bundles and we'll see to it's end that the answer is yes with the right notion of integrality and the right notion of the right homology theory and in the last part of the talk I'm going to discuss the proof for given a quick overview of the proof. All right, so let's start with fixed bundles. So at the start we're going to fix a smooth proper curve x and we're going to consider the following data on this curve. So a fixed bundle is a pair consisting of a vector bundle e, a holomorphic algebraic vector bundle e, together with a so-called Higgs field, theta. And so theta is simply an O linear map from e to e tends to omega 1 x. So you can think of this as something like a twisted endomorphism of the vector bundle e. Twisted endomorphism because the sheaf of one forms is a line. So essentially locally you could represent this Higgs field really by a matrix with its entries being one forms. And so the modellized base mx is going to be a variety parameterizing so-called semi-stable Higgs bundles. So there's an additional stability condition that needs to be imposed on the Higgs bundle in order to get sufficiently small modellized problems that can be governed by a variety. But yeah, this is something you're all probably very familiar with so I'm not going to dwell on that point. And so one additional assumption that we're going to make is that the degree of the vector bundle and the rank of the vector bundle are co-prime. And this guarantees that stability equals semi-stability and in fact also that the variety is smooth, the modellized base is smooth. So here are a few facts, interesting facts about this modellized base. So m of x is holomorphic symplectic and there is a map or a morphism called the Hitchin map or Hitchin system which is actually an algebraic integrable system. So this is simply a morphism from m of x to a complex vector space or to an affine space, a complex affine space, such that it's generic fibers in the Boolean variety and furthermore you could show that all the fibers are in fact like rangins so that's why you would call it an integrable system. And yeah, this map is also proper. And so one consequence you can get from that immediately is that this modellized space cannot be compact or proper, it's really just a general quasi-protective variety. And so there's some variants of this modellized space that we're going to be interested in. So the so-called SLN modellized space and the inverted commas will be defined on this slide and so before defining the SLN modellized space I'm going to quickly recall the definition of the actual SLN modellized space. So this one is simply given by the modellized space of semi-stable Higgs bundles with trivialized determinant and where the Higgs field is assumed to be trace-free. So this is called the actual SLN modellized space because those pairs here satisfying this condition can be modeled by SLN torsors with a Higgs field. So that's why the notation is entirely justified. But yeah, the downside of this definition is that modellized spaces are always singular because the determinant condition forces the degree to be zero and hence the co-primality will be violated and so if n is greater than one you end up with singular modellized spaces. But this can be fixed by making the following variation of this definition. So you start by fixing a line bundle L of degree D again these are assumed to be co-prime to N and you define M superscript L SLN of X to be the modellized space of semi-stable Higgs bundles where the determinant is assumed to be isomorphic to L and the trace of the Higgs field is again assumed to be zero. And so this is what we're going to call the modellized space of SLN Higgs bundles under inverted commas. And as a useful shortened notation we're going to introduce M hat of X. So there are also PGLN Higgs bundles and the corresponding PGLN modellized space. This one can be constructed as a quotient stack of M hat. So all you have to do is you have to introduce this group gamma given by the n-torsion point of the Jacobian. So keep in mind that n is equal to the rank of the Higgs bundles we're working with. And this group gamma acts on the SLN modellized space just by taking the tensor product with the line bundle corresponding to such an n-torsion point. If you take the corresponding quotient stack you end up with what we would call the PGLN modellized space or the modellized space of PGLN Higgs bundles of degree D. And another way to define this so this is actually a stack because we're taking the quotient stack so I probably shouldn't say modellized space but yeah by the force of habit this might happen once in a while. So yeah this is really an all before. And another way to construct it would have been to take the PGLN modellized space and quotient by the Jacobian itself. All right so now let me say a few words about this decision map that we've already alluded to on an earlier slide. So I think I already mentioned that locally you could think of the Higgs field as something like a matrix but with the entries being one forms. And so the decision map is simply given by computing the characteristic polynomial of this matrix. And so the actual presentation of the matrix doesn't make any difference doesn't affect the characteristic polynomial. So you end up with a well-defined map which is just given by singing out the coefficients of the characteristic polynomial. So those coefficients will now be sections of tensor products of the sheave omega 1x. And so the space of those global sections is by definition the Hitchin base and the map that associates to a Higgs bundle the characteristic polynomial is by definition the Hitchin map. And I already mentioned that the generic fiber was an abelian variety and so this is something you can see quite directly using just these linear algebraic definitions. So in fact the generic fiber is not only an abelian variety it's actually the Jacobian of a smooth curve which is called the spectral curve. And the spectral curve is the dn to one cover of x that you obtain by computing the eigenvalues of the Higgs field. So computing the dn eigenvalues of the Higgs field you end up with an n to one cover and so generically this curve will be will be smooth and the space of line bundles on the spectral curve can be identified with the generic fiber of the Hitchin map. All right so now I want to talk about what happens when looking at two Hitchin systems for Langland stool groups and so we're mostly going to do this for SLN and PGLN which is like the first pair of Langland stool groups that is kind of non-trivial because the group GLN is a self-tool. And if you're not familiar with Langland's duality I assume that you all are but if you aren't then let's just you know say that this is some duality notion that was introduced by Langlands that is based on the classification theory of reductive groups. So from the point of view of a geometry the actual definition isn't very enlightening but it's somewhat fascinating that this notion of duality pops up all over the place and very often it casts a shadow in geometry. So we're going to see an example of this right now. So for instance in the case of SLN and PGLN it was observed by Housel and Tadius that the Abelian vibrations that arise when studying the SLN and the PGLN modular spaces those Abelian vibrations are actually dual to each other. So what does this mean precisely? So if you take a generic point of the HN base then as I told you earlier the generic fiber on both sides will be an Abelian variety and it so happens that if you compute the fiber on the SLN side first that it will be dual to the Abelian variety that arises on the PGLN side. And so this is a result for the SLN and the PGLN modular spaces but something similar is true on the level of GLN and this is an even more classical result because for GLN I mean GLN being a self-dual, Langlands self-dual group you'd expect the generic fiber to be a self-dual Abelian variety and this is indeed known to be the case for Jacobians of smooth curves. And as you know now the generic kitchen fiber for the GLN space happens to be the Jacobian of the spectral curve. And I'm just going to remark in passing that this theorem by Housel and Tadius can be generalized and so this is work by Dunagi and Pantef. So they're observed to the few so there's also a notion of Chi-Hicks bundles for Chi being a reductive group and so they're observed that you can identify the HN bases and so it was a bit liberal with the use of the word canonical I think unless the groups are simple there probably isn't a canonical way and even then you probably only get something which is canonical up to up to scalar. So let's just say the HN bases are isomorphic but then pretty much the same phenomenon happens so if you compute the generic fibers you end up with Abelian varieties that are dual to each other and so now this you can really think of as some kind of geometric incarnation of Langland's duality. All right. How much of this survives for the singular fiber? Yeah very very good point. We're going to talk about this in a moment so we'll see that for for certain singular fibers I mean there's a result by a rink in for instance that works for you know the groups GLN, SLN and PGLN you can extend this duality to some kind of derived equivalence and so this is the case for the so-called elliptic locus where the the spectral curves are integral and then there are also extensions of that result to non-reduced spectral curves. Yeah and so in general you'd expect that this is true everywhere but actually this is some of the yeah I'm going to mention that on a later slide yeah thanks for your question. So given this duality relation between the ancient fibers you definitely expect that this has some influence on the the homology of the the modellized spaces so it's a very reasonable question to ask how the homologies of the SLN and PGLN modellized spaces compare and so this is the question that was asked by Hausle and Thaddeus for Higgs bundles but much more early actually already in 75 how the metasim improved following results which is maybe like the first instance of this kind of mirror symmetry relation but they were working with modellized spaces of just plain vector bundles and so they observed that the singular homology groups with rational coefficients of the SLN space and the PGLN space are isomorphic and so they approved this theorem using arithmetic methods using the the way contractors so essentially they just compared to the point counts and so what's interesting from the point of view of geometry is that since the space on the right hand side of PGLN space is a quotient of the SLN space this implies that this finer group gamma is actually acting trivially in homology so this is definitely an unexpected result and quite a nice one but the same phenomenon isn't true for modellized spaces of Higgs bundles so the action of this group gamma on the modellized space of SLN Higgs bundles is in a way quite complicated and that's also why it's much much harder to prove anything about a comparison between the the homology groups so let's first talk about the expectations so as I just mentioned the generic fibers of the SLN and the PGLN modellized spaces are dual abelian varieties so based on yet to I see philosophy you'd expect that the SLN space is a mirror part of the PGLN space and so this is indeed what was conjectured by Housel and Tadius and so based on that expectation they conjectured that the hodge numbers of the SLN space should be equal to appropriately defined hodge numbers of the PGLN space and so since the PGLN space is really an orbifold or a gullumamphid stack the right notion of hodge number is given by what you'd call the stringy hodge numbers or the orbifold hodge numbers and furthermore you have to take into account a chirp that naturally lives on the PGLN side and so this chirpin uses corrections for these stringy hodge numbers and so Housel and Tadius verified in low ranks that these two numbers matched up with the SLN side and so this inspired the conjecture of course a big portion was also this SPIC philosophy which is and the fact that the two spaces have dual abelian variety vibrations. Let me briefly see a few words about this orbifold homology a stringy homology that appears on the right side of the conjecture so if you ignore the chirp this is simply homology of the inertial stack so you just take homology of fractional coefficients of the inertial stack and since the PGLN stack is really a quotient stack you can make this very explicit so this amounts to the following direct sum indexed by elements of the group gamma and you're just taking the homology of the fixed point loci so m hat gamma quotiented by gamma and the rational homology of that so this amounts to the homology of the inertial stack so essentially you're seeing that in orbifold homology you're counting the the inertia strata the fixed point loci with some kind of higher multiplicity and this leads to some notion of homology that for some questions it's just more more appropriate than just the plain homology of the orbifold that would agree with just the homology of the coarse-modelized space if you're using rational coefficients and so something that's interesting to remark here is that those individual summons that appear in the right-hand side can actually themselves be understood as modellized spaces or stacks of fixed bundles on a finite detector x gamma of x or you could also think of them as modellized space of fixed bundles with respect to some group scheme on x that is somehow twisted in a funny way so now what about alpha twisted orbifold homology so orbifold homology that takes the chirp into account well essentially this is some kind of version of the the usual orbifold homology but with respect to a twisted coefficient system such as some local system that lives on the inertial stack and there's a very direct way to produce such a local system so the formalism essentially takes care of it so a chirp you can think of as a morphism from m hat to b2 mu n so yeah those chirps actually always are defined over groups of roots of unity they have a finite order so that's why we can take them up to b2 mu n so this is just the classifying morphism of the chirp and then we apply the inertia stack factor and on the left hand side well we just end up with the inertia stack of the peachy lens stack and on the right hand side we get the inertia stack of b2 mu n which you can identify with okay so unfortunately there's a typo here you can identify this with b mu n times b2 mu n so here there should be a b2 mu n and then you're simply um taking the projection to the first factor so you end up with a map from the inertia stack of the peachy lens space to b mu n and this is the classifying morphism of a mu n torsor and now essentially you just take the associated unitary local system and the homology of this twisted system of coefficients is the orbifold homology that is twisted by the chirp all right and so I'll just explain the full meaning of this conjecture by the Hausland-Tedius and so in 2017 the East Sea Glendai we we proved this conjecture um so yeah this is uh so I'm restating it as a it's a theorem and so we we use um Piede integration that was introduced by Batiref and so previous cases have been known of course so for instance the ranks two and three are very established by Hausland-Tedius in the original paper using a Morse theory strategy and in 2014 Bokolini and Grandi verified the so-called Hilbert scheme cases using a Gretzsch's formula so certain families of modular states and fixed bundles can be understood as this Hilbert scheme to surfaces and so Bokolini and Grandi used this this formula to to verify the Hausland-Tedius conjecture in these cases but unfortunately I think this got never published and then also in 2017 Goldfinn and Olivera proved the cases of Frank II and III for modular spaces of parabolic expundals again using Morse theory so when I say Morse theory it just means you use the natural GM action or circular action that exists on the on the modellized space then essentially you try to understand the fixed-point-low sign computer the comology and so in an upcoming Berg she or Shen proved some Hausland-Tedius conjecture for modular spaces of parabolic expundals in arbitrary ranks again using Piede integration so this is sorry how does the duality work with parabolic structure well it's it's pretty much it looks pretty much the same I mean you just have to define the parabolic expundals and tenders also a HN map and the generic fibers are again dual abelian varieties that's the same parabolic data that you get on the two sides yeah for Esseland and Piedgeland that's pretty much yeah that's it for for general reductive groups I wouldn't be sure yeah but for Esseland and Piedgeland it's the same parabolic data I mean you can like you can do the same trick in the Esseland and Piedgeland cases right that you define the parabolic modellized spaces for GLM then you impose a restriction on the determinant and then you take the quotient by by some final group to define the Piedgeland spaces but if you really want to see the details you should wait for for she use paper to appear okay and so there are also some some new proofs and methods that I want to to mention that appeared recently so for instance in in 2019 Löser and Wieser revisited the Hauser and Thaddeus conjecture using motivic integration so this means that essentially you know the tool periodic integration gets replaced by motivic integration but actually the the structure of the proof changed as well I mean they it took a slightly different viewpoint on the whole thing and the option is that now you don't have to use arithmetic methods so there's this is a proof that truly purely works in over the complex numbers and in characteristics 0 so there's no detail why a arithmetic geometry needed and then another new proof that appeared recently in 2020 is the one by Davesh Maulik and Jun Liang Shen and so this is really a completely different approach so they're using perverse sheaves and so essentially there's a correspondence the hemoscopic correspondence that was defined by Engel that can be used to relate certain portions of the comology groups and then using the formalism of perverse sheaves and vanishing cycles they could show that this correspondence induces really an isomorphism as it should and so this is really quite amazing because not only does it compare the Hoche numbers or the Betty numbers of both sides but actually constructs an explicit isomorphism that is well defined up to a complex scalar and so yeah that's really wonderful to have actually an explicit map that allows you to go from one side to the other to kind of inspired by the existence of such isomorphism you could ask the question like what's the most general coefficient system where this mirror symmetry relation holds because how's the tannious conjecture holds so for instance could you take singular comology with with integral coefficients right i think it's a natural question to ask but yeah i mean you get shut down pretty soon because on the right hand side you find even in already in degree two you find non-trivial torsion classes but the left hand side can be shown to be torsion free in general so it's clear that because of this difference here there cannot be a mirror symmetry relation for integral coefficients and so the fact that the left hand side is torsion free is actually the first theorem of Xi Jinping and myself that I want to to mention so yeah essentially this is true for both the s l n and the the chi l n spaces but not for the p chi l n space but if you take singular comology with integral coefficients then the these comology groups are torsion free and in low ranks this is solved was explained to us by by tom bird and i'm going to say a few going to make a few remarks about how we are approaching this and how we're proving this statement for general ranks and i'm going to do this on a later slide so at first i want to kind of continue with this philosophical detour about what kind of integral mirror symmetry relation one should expect so i just explained that this hazel tedius conjecture doesn't hold with singular for singular comology with integral coefficients and in a way that probably won't be surprising to the experts because the experts will know that any can for instance construct as an example of derived equivalent free faults that have non-isomorphic torsion parts in degree free singular comology and similarly treiman in 2019 constructed a two a t dual pair of flat free faults such that the degree two and degree three torsion parts of the comology don't match up various directional comology spaces happen to be the same as you would expect from mirror symmetry but there's a fix to both of those observations so if you replace singular comology by complex k-theory then in the case of edinkin's example you get that the complex k-theory groups are the same and this is in fact always the case for forimokai partners you can just use to forimokai kernel and it induces an isomorphism in complex k-theory more about that later and similarly in treiman's case so treiman proved that the degree zero complex k-theory of the first flat free fault set hat agrees with the degree one complex k-theory of the second flat free fault so you have this kind of degree shift and that's something that you'd expect just from the usual mirror symmetry relation for for free faults and so kind of inspired by this she and I started thinking about an analogous result for complex k-theory of the s-land and tepeche land space and this is indeed something that we that we could prove so this is our main result so there is an isomorphism of the complex k-theory of the s-land space with the complex with the twisted complex k-theory of the the p-tree land side and so here this is really a shorthand for gamma-equivariant alpha twisted complex k-theory of the s-land space or you could also think of it as alpha twisted Jacobian equivariant complex k-theory of the g-land space that's just because you really have to take into account that m-chec is a is an obi-fold is a a dirty monthly stack so this is really some equivariant k-theory group on the right hand side but you don't have to add a subscript like obi-fold or stringy because equivariant complex k-theory is automatically the right obi-fold version to work with that's like one of the the marvelous things about complex k-theory and so i want to emphasize that actually we do a little bit more than that so we actually constructing an equivalence of spectra in the sense of omega spectra the sense of stable homotoby category in the sense of stable homotoby theory and so yeah the statement we actually proving is that the complex k-theory spectrum of the s-land side agrees with the twisted complex k-theory spectrum of the right hand side and we're not just doing this out of vanity i mean spectra really play an important role in our argument it's necessary to pass through stable homotoby theory to get this statement that the actual k-theory group size are more fake so omega spectrum are really an integral part of the argument and are indispensable and in case you're unfamiliar with spectra you could think of them i mean this is this dangerous of course but you could think of them as something like a a chain complex but in principle it's like a it's a very complicated topological space so for which in particular you can define a notion of homotoby groups and those homotoby groups will always be abelian and they are defined even for negative values and in a way this is kind of similar to the homology groups that you can compute for a chain complex but a spectrum is really a more refined object and this in this case it's it can capture a lot of information that a normal chain complex wouldn't be capable of capturing so that's in particular related to complex k-theory so we know a singular homology for space can be computed by taking the homology groups of a chain complex the singular chain complex but this statement isn't true for for complex k-theory there you really have to replace chain complexes by by spectra okay um so yeah let me continue with some remarks about this this theorem so the the left hand side can be shown to be torsion-free actually this is going to be um so i thought that there's a a question just going to quickly finish my sentence so the left hand side is torsion-free because a singular homology is torsion-free and torsion-free-ness of singular homology always implies torsion-free-ness of complex k-theory i'm going to explain the the argument in the latest slide and since the left hand side is torsion-free you'd expect that the right hand side is also torsion-free and so that's that's the case and it's actually something that we need to establish first in order to deduce that the map we're constructing is actually an isomorphism and so now there's a question uh is there an explicit description of the the loop spaces of the spectra um so i would say yes i mean even of the the spectrum themselves i mean so i guess you're referring to uh those spectra right the k-theory spectra of some space n and so you can really think of this as some kind of mapping object or mapping spectrum from n hat to k-theory to to the complex k-theory spectrum and this complex k-theory spectrum you can describe quite explicitly for instance by taking um the space of thretom operators on a separable hibbled space and a bunch of other kind of explicit descriptions like that that you could use okay so the the point that i was trying to make earlier is that so since the left hand side is torsion-free you have to show that the right hand side is torsion-free too and so this happens to be the case uh which is you know quite uh quite remarkable it kind of fits with some general phenomenon that is observed once in a while that passing from singular homology to complex k-theory sometimes annihilates just the right amount of torsion and so yeah this is really quite a quite a remarkable thing and i want to make a few few remarks about that so i planned this quick detour which is actually unrelated to the the topic of this of this talk but it somehow fits with the overall philosophy um so um it's about current search activity so you might recall that uh in the 80s francis carven proved us this nice theorem about Hamiltonian spaces so if you take a a compactly group g and a Hamiltonian g space m which has a proper moment met mu and zero is a regular value of mu then uh carven proved that um the equivariant k phi sorry the equivariant singular homology of m with rational coefficients so it checks on to the homology of the symplectic quotient again with rational coefficients and probably as was already known to to francis carven at the time so this search activity statement fails with with integral coefficients you can salvage it a little bit by imposing some additional conditions so this is something you can find in the paper by toman and bytesman from 98 but there is no statement as general as the theorem for rational coefficients unless you're brave enough to take another homology theory and so this is what uh Harada and Landweber did in in 2005 so they proved that carven search activity holds for complex k theory and this time it's not necessary to to rationalize or to excise the torsion part so in a way this is really in line with what i just said earlier that because you pass the complex k theory you're really annihilating the right amount of torsion from from singular homology the right amount meaning that a statement like current search activity is true in this case so now i i want to give you a quick fact sheet about complex k theory because um i just want to make sure that we're all on the same page and for the the second half of this talk it will really be essential to know a few things about how this homology theory is actually constructed and just know a few facts about it so um k u star is a two periodic homology theory so this is also referred to as a spot periodicity so you're free to either think of it as being graded by the integers model 2 or you can think of it as a as a homology theory that is graded by all the integers but which is too periodic and so explicitly an element of k u of 0 of x can be constructed in terms of topological vector bundles complex vector bundles on x it's now here i'm using x to denote a finite c w complex and so such a vector bundle e gives rise to an element that you usually write as brackets e and its construction is compatible with direct sums so the bracket of the witness sum agrees to just the sum of the two two brackets if you construct a universal group that is holds the group group then you've got your definition of k u 0 of x and there's a spectral sequence the so-called attia hirzburg spectral sequence that relates complex k theory with singular homology the second page of the spectral sequence is given by singular homology with coefficients in k u q of a point which is just a funny way of writing the integers of q is even and zero otherwise and so yeah the spectral sequence converges to the the complex k theory groups and there's a funny thing happening namely after tensoring with q the spectral sequence degenerates on the the second page already so all of the differentials vanish after tensoring with q and so this implies that the the differentials can only be non-zero on torsion classes so using those properties you get the following so first of all you get this very nice identification of rationalized complex k theory with a two periodic version of rational singular homology so k u star of x tensored with q is isomorphic to h star of x with coefficients in well you could think of it as a as a chain complex essentially so this is something like a hypercomology group so q you just take the q algebra generated by some degree minus two element beta and so this essentially imposes a two periodicity on on singular homology and thanks to the two periodicity you can relate it with rationalized k theory so on the numerical level this implies that the ranks of the complex k groups can be identified with sums of petty numbers are you either restrict to petty numbers of even degree or odd degree and another thing that you obtained from the athea-hützeburg spectral sequence is that torsion freeness of singular homology implies that complex k theory is also torsion free so this is a property that i'm already i've already alluded to in an earlier slide but i want to emphasize that it's possible for k u of x to be torsion free even if integral homology isn't so this is really you know confirming that passing to complex k theory can annihilate some some nasty part of torsion that you wouldn't know how to deal with on the integral homology side so the great thing about complex k theory is that there's a very nice equivariant version of it so if you have some group acting on your space x then you can simply replace plane vector bundles by by g equivariant vector bundles and this yields equivariant k theory and so informally you should think of this as being the right k theory of the the corresponding quotient stack or obi fault if g is a finite group and furthermore if you have a a chirp which is g equivariant on x then you can define alpha twisted g equivariant complex k theory which is defined by using alpha twisted equivariant vector bundles and so it's pretty much the the same idea and so here maybe i let's just if you're unfamiliar with what these words mean concerning chirps and twisted vector bundles then i will propose that you just ignore them ignore the chirp and yeah because unfortunately i didn't prepare any slides explaining this in more detail but if you just brush the chirp under the rug you you won't lose a lot about it it's just in order to get the right in order to get this isomorphism it's really necessary to have the chirp there and so there's a a very nice result by Fried Hopkins and Telemann from 2002 which relates this alpha twisted g equivariant k theory after complexification with um complexified two periodic obi fault chronology of the the quotient stack so here i'm assuming that she is a she's a finer group because otherwise i mean there's still a statement i mean they prove quite a general result also for for compact key groups but i i shouldn't call it obi fault chronology in this case so that the formulation would be slightly different but for g being a finer group and so that's the case that interests us this is this formulation is fine and so now if you if you contemplate this result by Fried Hopkins Telemann and you compared us with the isomorphism that she would like to produce and you see that my result with she was really something like an integral analog of the household tedius conjecture because after rationalization what you obtain is just an isomorphism between the two periodified complexified obi fault chronologies of the pitchland side with the two periodified complexified singular homology of the s land side okay so that's essentially concluding the motivational section where where i explained why um this is really a an analog of the household tedius conjecture for for complex k-fury now i want to say a few words about the strategy that we use to to establish this result um so in brief the idea is to to use a furimokai kernel to construct a map between the k-fury groups and then you just have to come up with a way of showing that it's actually an isomorphism and so in this case if you have two smooth schemes which are proper of a common base x and y then you can take a furimokai kernel and rather than directly writing down a map between the drive categories you can also use it to define a map between the k-fury groups and here essentially you're replacing all you're doing is you're replacing you push forward on a proper map by some kind of wrong way map that exists in complex k-fury but otherwise it's the same construction as the one used by by mukai so you take the pullback of some class e on x you tend to with the kernel and then you push the whole thing forward to y and this defines a map from the k-fury effects to the k-fury of y and the beauty of it is that this also works equivalently and in a twisted setting so something that is a little bit less obvious is the fact that this map is also using a map of sheaves of spectra relative to the base s so here i mean sheaves with respect to the standard topology on the associated analytic space of s so you take complex points and you use the standard topology in that but yeah if you just it's a bit of a leap of faith but one can show that this furimokai construction also induces a map of sheaves of spectra but you take the push forward of the complex k-fury sheaf on x to the base and you take the push forward of the complex k-fury sheaf on y and then this furimokai transform induces a map between the two and so it's it's not at all obvious that this is true because yeah i mean at least to me it's not obvious because i mean like in the definition of these furimokai transforms you have these wrong way maps and it's not at all clear that they are sufficiently coherent to actually write down a morphism between sheaves of spectra and so because of that we're using a different viewpoints to actually show the existence of this this morphism of of sheaves so we use a theory that was was developed by by blah and by by moulinos so i'm going to start with blah's results so by a dg category you can either mean the the classical thing complex linear dg category which is pre-triangulated or you could think of a c-linear stable infinity category but whatever you choose them so blah proved that there's a functor from dg categories in this sense to spectra so this is a a functor between infinity categories and this functor is called a k-top top logical k-fury and it has the property that it sends perfect complexes on x used as a dg category to the k-fury spectrum of x and so this really means the k-fury spectrum of the analytic space so this means just the complex points of of x and there is a generalization of this result to a relative setting and so this is due to tasos moulinos so if you prove that if you take some complex space scheme s then there is a functor from the infinity category of s-linear dg categories two sheaves of spectra on complex points of s so this functor is denoted by k-top subscript s and it has the nice property that it sends alpha twisted perfect complexes on x to the twisted k-fury sheave on on x so here i should have probably written alpha twisted perfect complexes on s the sense to the twisted k-fury sheave on s that would have made more sense okay yeah sorry for the for the typo here the moulinos also proved that this construction has a certain compatibility with proper push forwards so this compatibility is not entirely general unfortunately but so what is true is that if you take a proper morphism s to s prime and your view perfect complexes on s as an s-prime linear category then topological k-fury relative to s-prime can be shown to be isomorphic to the push forward of the sheave of spectra given by topological k-fury relative to s of perfect complexes on s and that's by definition of by by this statement here about that's the same thing as just the push forward of the complex k-fury sheave so there's another question whether sheaves of spectra are related to parametrized spectra on the spot i i can't give you an answer unfortunately so the way the way we're thinking about sheaves of spectra and that's the same for glom and moulinos we're just using the notion of you know of sheaves on an infinity top so this sounds really really fancy but essentially that's the notion that you can find in either high octopus theory by lyrie or in spectral algebraic geometry but you know herstically speaking you're just associating to you can think of it as a functor from the the category of open subsets ordered by reverse inclusion to the infinity category of spectra so this gives you the definition of a pre-sheave and then you also have to have some kind of hyper sheave condition that needs to hold yeah all right so now essentially this theory by by blah and moulinos is really saving us here because now we can just apply this functor k-top s to the actual fury mokai transform so it's through the actual map between dici categories and then some for a little bit of work this gives us exactly the smallest sheaves on complex points of s that we wanted so this way we really get this this fury mokai this this relative fury mokai transform between the push forwards of the k theory sheaves and and so that's also an advantage of working with that formalism this also works in a twisted equivariant context so indeed equivariant context one has to be a little bit careful um so in our case this is fine because we're working with um equivariant k theory with respect to a finer debilion group and so in this context it's easy it's sufficiently easy to kind of generalize the results by blah and moulinos to to this setting here but in the general equivariant context with twists and this will be will be challenging so now um so where are we now so we've got this map between the the sheaves of spectra now we can just simplify the whole thing by by centering with the rationals so this means we take the the smash product with the like the eilberg-mclane spectrum associated to the the rationals and so now by essentially the identification that i stated earlier of the k theory of complex k theory rationalized with two-periodified singular comology you can think of this simply as a map between sheaves of chain complexes so that's not just an a map or morphism in the derived category of chain complexes of the rationals so now we just get this morphism from rf star qx uh beta-beta inverse to rg star qy beta-beta inverse so this is now definitely a much much simpler object to to work with and so at this point uh because we are now working with rational coefficients it makes sense to evoke the decomposition theorem by valence and bernstein-deine and caber so according to which um the derived push forwards are sums of shifts of semisimple peri-sheafs and a semisimple peri-sheaf being the middle extension of a semisimple local system with some locally closed sub-righty that is called the support it makes sense to actually you know concentrate the whole analysis of this furimokai equivalents on the support of these these locally closed sub-righties so in particular if you assume that you can show that this furimokai transform is an equivalence over every peri-support then this implies right away that the rationalized furimokai transform on on the level of k-fury rationalized k-fury will be an equivalence as well over the full base this time this is this kind of amazing uh comological spreading out phenomenon that happens when you use peri-sheafs that you can kind of prove general statements about singular fibers by restricting attention to fibers with tame singularities and kind of singularities that are easier to manage so now under those assumptions so what we've got is that this furimokai transform is an isomorphism after rationalization and with a little bit of extra work so this doesn't directly follow from that but with a little bit of extra work which can show that actually this original k-fury I think furimokai transform induces an isomorphism of the non-torsion parts so like the integral lattices of k-u star of x and k-u star of y so in particular if you know for another reason that these k-groups are torsion-free then you actually succeeded in showing that uh this map here is an isomorphism and by by the white-head lemma it implies that the method of spectra is is also an equivalence so this is really a kind of a straightforward construction that is based on rationalization on the decomposition theorem and of course there's a kind of more profound parts that luckily for us we're already taking care of by other people so one big problem here is you have to choose a the furimokai kernel which doesn't have to you know satisfy any conditions on the entire base but on a sufficiently big open subset it needs to be a direct equivalence and then on the same open subset as hard you want to show that all of the purse supports are contained in here in order to actually apply the program I just sketched and then the third step and so this something that we have to do ourselves in this for this project you have to show that torsion vanishes on both sides so yeah let's see whether this program can be can be realized in the case of the h-infibration so the answer is yes and no so it doesn't work directly for the case of the h-infibration you're interested in but it it works to to a certain extent than in the other cases so first of all the furimokai kernel so here you can use the kernel that was constructed by by a rinkin they just used the derived equivalence over the locus known as the elliptic locus so that corresponds to the locus for the spectral cursor integral and so rinkin constructs the derived equivalence of the h-infibers there which can be used and then you just take an arbitrary extension of a rinkin's integral kernel to the full h-infase so that's actually a funny feature of our work that essentially you just need this kernel to be an equivalent in a sufficiently big open subset but otherwise you can just extend it arbitrarily and it will always give you an equivalence in in k theory and so now what you have to do next is you have to slightly alter the modulite problem so you have to work with fixed fields that allow to acquire certain poles so the easiest way of just describing that is by working with fixed fields that are twisted not by the canonical divisor but by some arbitrary line bundle d which is supposed to be of even degree and the degree has to be at least equal to twice the genus because in this case the pair supports have been completely determined by decaldo decataldo and they're known to lie within the elliptic locus and it's it's still an open problem to actually determine those pair supports completely in the case where d is equal to the canonical divisor so that's the actual modulite space of fixed bundles people usually work with but yeah if you kind of alter the modulite problem slightly it's possible to understand the supports and now in order to work around this issue that the pair supports aren't known for actual fixed bundles we use a method that was introduced by by davis maulik in julien shen in the 2020 paper so this is the vanishing cycle method and it's it's a very nice very nice observation here that you can essentially embed the modulite space of fixed bundles into a bigger modulite space for which the pair supports are known and you can actually embed it as the degeneracy locus of some regular function and now this opens up the possibility of using vanishing cycles to understand the comology of the modulite space of fixed bundles and so this is what maulik and julien shen did so to show that on if you work with fixed bundles with respect to the divisor d plus a point then there's some natural function f on the hitching base such that the sheath of vanishing cycles apply to the push forward along the hitching map agrees up to shift with the push forward along the hitching map of the constant sheath which is now supported on the modulite space with respect to defined with respect to divisor d and not d plus p so in a way applying vanishing cycles allows you to remove a point from the divisor this way you can increase the degree or decrease the degree and essentially still prove interesting statements using the decomposition theorem but this time for the higher degree versions of the modulite spaces and so essentially what she and I had to do is we had to prove an analogous result of this for for sheath of spectra so here you also take the push forward of the constant k-fee or sheath of spectrum on the ESLN modulite space defined with respect to the divisor d plus p and if you apply the vanishing cycle for the same function f that was defined by pandekimchen-limchen then you get up to shift just to push forward of the k-fee or sheath on the modulite space of externality defined with respect to the divisor d so notice that the point p has disappeared when you apply the the vanishing cycles and the same thing is also true with a sheath theoretic version of albatwistered equivariant k-fee so that's the part in red and there is the first part is really just a straightforward generalization of the proof of the result by maolik and shen-limchen the portion in red requires some some new ideas because we're working with with actually equivariant k-fee here which isn't actually a k and a comology theory in the classical sense it's really an equivariant comology theory okay so this part is kind of not a straightforward generalization and now we are almost done with describing the the proof so essentially what you do is you kind of you use the construction of this forimokai transform for one of the modulite spaces where the purpose supports are understood so for instance for v plus p and then you apply the vanishing cycles and now you get a map between sheaths of vanishing cycles if these push forwards but those you can identify just with push forwards of the sheath of k-fee respect for the actual modulite spaces you care about but the one issue is that you no longer know that the map is given by by a forimokai transform so this is the one thing that you lose now it's just some some map of sheath of spectra and then by from that by taking global sections you obtain a map between the k-fee respectra that you wanted to relate and if you repeat that process at least twice you can use this to get a map between the complex k-fee respectra of the modulite space of sln-higgs bundles this time defined with respect to the canonical device as you want to the modulite space of of pgln-higgs bundles and you probably spotted that now on this slide i'm suddenly also twisting the sln side and this is something some twist that will disappear once you've taken global sections but it's it's necessary to include it in order to describe the map between the the sheath of spectra and then you can argue just in the outline i i gave you previously so you can show that this map of spectra is an equivalence after rationalization and you have to prove that torsion banishes on both sides and from that it follows that you get an equivalence and so i'm thinking i'm already out of time so a better a better stop here um but yeah essentially if if you're going to look at my slides i'm sure that the francesco will will publish them on the slack channel you see a quick sketch of the proof of torsion freeness for the rank being a prime number for sln and pgln okay well thank you very much for your attention thank you very much are there questions yeah go on i heard belish i think oh i i just wanted to ask so the uh the the the the gerb on the sln side is that just the canonical gerb associated to the sln moduli space or is it something else yeah precisely i mean there are some you know you might have to take a power of it depending on which line bundle you choose but essentially these these moduli spaces you can represent as quotients of some variety by pgln so this means that you always have a canonical pgln torsion on these moduli spaces and that's the origin of the chirp right if you just take the classical g at construction for instance think of the the character variety it's kind of easier today you just take um just kind of you know this equation of matrices and then you quotient by conjugation so this is really a quotient by by pgln and so this defines a pgln torsion in the in the co-prime case and that's the that's the chirp up to up to a power yeah yeah is your what is your uh for removal kernel that you're using well um so we're using a rinkins kernel in the case where the divisor has you know degree at least to chi and this even and then we just take an arbitrary extension of this kernel so just any extension is a coherent chi from the fiber product will do so essentially you just need this kernel to you know give you an equivalence on the elliptic locus and then for the magic of purpose she's whatever the extension you choose it's going to be using equivalence between the complex kfc groups so there's a question on the online which is related to that so there is no chronicle choice or does this isomorphism you get in the end depend on this choice of expansion um a canonical choice for the the kernel you're saying and and there's an isomorphism in the end depend on this choice well in a in a way in a way that there should be i mean you you could definitely you know i i actually think that you can extend a rinkins kernel so big enough open subsets such that in the end you could just take some kind of push forward and it will probably be coherent so that there should be canonical choices like that if those are the the right kernels to work with in order to get a derived equivalence that's a different question um in a way you have a lot of liberty when constructing those kernels so for instance if you want the canonical choice because you have some group action in the picture then you could quite easily construct a coherent extension that is compatible with this this group action and hence will produce a a map of kind of you know Borrelic covariant kfc respect or something like this um and ask what what is the monogamy of the workism of the vanishing comology of this section um you just repeat the question because people are not yet the monogamy and the vanishing comology um so i think this is something i'm not going to to answer in public yeah sorry i can't tell from the top of my head go on davis i think yeah oh yeah so in the um in my paper with jun liang there was this kind of you know a scalar ambiguity uh which is that um i mean has that basically been rigidified in your argument then is that um i mean so remind me of various scale ambiguity coming from in your paper oh i mean i can't remember now yeah there's you know at some point we trivialized like a line bundle over a point or something like that so i mean okay you know it was that kind of thing it wasn't like okay i mean i i think so it's a trivialization like this also appears but just in order to define the map to a one right i mean yeah yeah something like that that's right yeah there's a there's a i think this just means that you know there's these maps define they depend on this choice of the regular function yeah and in order to define the regular function at some point you have to you have to trivialize some some fiber of a line bundle i guess yeah so yeah i mean there's the the maps we get i mean there are probably several of them and i think this is related to the monotomy question that was asked previously yeah um and somehow what what happened in our case i mean you can actually because you set up some kind of derived equivalents first for for certain spectrum curves right and essentially you know you just you have to choose you start with some component of the SLM or life space there's some of the line money and then you choose to turn accordingly on the picture inside such that you get at the right equivalents which then you just extends brutally to some integral occurring on the full base so yeah i have one question how does this how does that there's isomorphism supposed to behave with respect to ring structure where's there a natural ring structure on the oldest mode tonight um so i i don't think there's a nice compatibility with ring structures um like already already classically right a few you if you look at the case for n is equal to one i mean then essentially you just take getting you know the jacobian self-duality of the jacobian and you would look at the map and use by but if differing makai self-transform of the jacobian which you know on the level of derived categories intertwines tensor products with the convolution product so just because of that i don't expect that there would be there certainly shouldn't be a morphism of homology rings but it might you know it might satisfy something like some kind of head care transform compatibility like similar to what in what you see in geometric langments um so this is the kind of statement you could you could hope for but we're not claiming anything along those lines any other questions if not let us thank you again