 В последний раз я объяснял вам, что это стабильность брыжа, и показал какие-то эквестиции для стабильности в комплексном случае, и в простой версии неудобной комедийной стабильности брыжа. Теперь мы поговорим о стабильности брыжа Мамфот. Это начинка раколлессии Мамфот, геометрия Геометрии и Геометрии и Редакция Мамфот. Так, это история, которую мы придумали в Мамфоте в 1960-х годах. Так, мы можем начать с абсолютных кейов. first, suppose k is a field, and simply assumes that it has a characteristic zero. And x would be a final algebraic variety, could be singular and in principle having importance, okay. I want it to just to be one variety and g will be reductive group and it should act on x. И дефинития в том, что джейтик вошел в этот случай, будет еще другая файна варианта. Она идет от файна варианта до другая файна варианта, которая будет спектром альжебра и инверенцией. Она имеет альжебра функции. Она имеет определенную демо-демо-президенту, это contribution of trivial representations. Это альжебра, она будет другая файна варианта. Она будет спектром альжебра, и это будет файна. Но это еще-то есть. Файна, и это не то биос. Давайте definим релацию на, давайте скажем, сет of points, альжебрейт closure. Вместе, мы говорили, что 2-поинт эквивалент, Мы говорили, что 2-поинт-сайк-виллян, если вы берете 2 орбита, то они получают Common Point. Это, конечно, 포인т-эквиллян для себя, это симметричное, но почему это эффект? Зариски кложа. Зариски кложа. Это транситив. Это не то, что мы говорили. Это не то, что мы говорили. Да, например, можно представить... Один должен использовать хорошие варианты, чтобы они были хорошими. Потому что представить, что CP1, как CP1, может быть C star action. Нет, группа редактива. Но у вас 3 орбита и 1-го орбита, 2-го орбита, 2-го орбита, 2-го орбита, 0-го орбита, 3-го орбита. И что здесь действительно идёт? Если это не транситив, то они могут контрактать 1 по 1, 1 по 1, 1 по 1, и все остальное. Они получают цепи 1. Да, они, в основном, думают, что это не транситив, не содержит никакой цепи. Да, это значит, что эта цепи... Окей, это первый факт. Тогда это легко. По-моему, это никакой эквивалентной классы. Класса содержит уникальный кложный орбит. Это экономический репрессор. Это кложные орбиты. И как это связано с этим абстрактом? Тогда я предполагаю, что этот репрессор имеет значение по этой эквивалентной репрессору. Это же как уникальный кложный орбит. Уникальный кложный орбит. Это г. И это будет именно этот репрессор. Да. Так что можно... Это только один пример. Не надо знать про эту историю. Так что этот репрессор уникальный кложный орбит. И г будет г. Аккуратно действует 1 и минус 1. Так что мы получим... Что? Мы получим этот репрессор. И эквивалентной классы как следующее. Как вы можете писать, как репрессор x, y equal to constant t, который не 0. И это как уникальный кложный орбит, который consist of 3 орбит. И только 1 репрессор уникальный кложный орбит. И эквивалентная класса, конечно, это лучшая линия. Это просто спектрум. Это т. Это продукт уникального кложного орбит. И вы можете заметить, что этот репрессор, можно redefine this subset consisting of this joint union of closed orbits. It's called something like a polystable part. It's actually very bad. It's neither open nor closed. It's some kind of horrible constructable subset. It's not a scheme. Just constructable, not closed, not open, constructable subset. And that's why people don't really try to do not speak about it, because it's not part of this general story, but secretly this is this mad guy. Okay, that's absolute story, and then there is kind of relative. Ah, and maybe before continuing, I just want to say that just to understand this JT quotient, all equations reduce the equations of linear representations. Because let's take algebra of x. It's finitely generated. And choose some subspace which is generated. Choose subspace called V star, which is generate of the whole algebra, and also gene variant, because it's some fundamental representation. So you can always any set expect to find the representation. Choosing subspace, it means that you embed x into V, which is kind of linear representation of G. It will be quotient of symmetric algebra. So it means that all effects will be symmetric algebra, generated by V star functions on V, divided by some ideal. And if we take invariants, we take G by G. So it implies that x cross G is a closed subset in V cross G. So, in principle, if you want to study the JT quotient, it's enough to study linear representation, but it's kind of not obvious. And it gives us another viewpoint, kind of closed orbits in X. It will be closed orbits in V. So it's another way to see the same embedding, but it's kind of not really obvious relation between functions and closeness of these things. Okay. Now I'll go to relative case. So x will be no longer affine, x will be not necessary affine, not neither projective, but it should have ample line bundle, ample bundle, and in doubt with G action and everything. So, what I want, there's really no good name, I called it approach type situation. In this case, when we have an embedding to x to approach of some graded algebra, take some of all n greater than zero, gamma x into power n, we get graded algebra. Let's solve already the infinite dimension principle and you got to think. And I want it to be as more of approach of some graded algebra. And what is typical situation? What one have, one have x projects to some affine base and fibers are projective. So everything could be as singular as you want. So it will be family of projective things of affine base. And family will be ample bundle, which is ample on fibers. And then here is the definition of the JT quotient and here kind of make up notation using this everywhere. Yeah, suppose G z x everywhere. So this quotient, which depends on the choice of the sample bundle, will be just repeat the formula, take proj of this direct sum, blah, blah, and you take invariance. And what you see, it's because it's proj, it's kind of the same situation. You construct the same simple bundle of proj type, because it's defined as to be proj. And what happens in this situation, that in this case x cross G, we again have some line bundle, which will maps to y cross G without any line bundle. And it will be affine and projective. So we kind of repeat the same thing. And there is some kind of difference between this relative station, absent station, and absent station any point of x, give some orbit, take its closure, take closed orbits inside to get point in the quotient. But here not every point gives a point in the quotient. Only some open part of x will produce this point in the quotient, will be some part which is called semi-stable. It could be, yeah, usually it's dense open, yeah. Yeah, in principle these guys could be empty. Kind of theoretically I can say that these things are vanish for, oh no, no, in this situation they will not empty. It will be dense open in x. And what is it? How, maybe I'll just draw a picture on the side. So you can see the total space of line bundle. It contains zero section, which is x. Now I take point in x, lift it somewhere, it doesn't matter, because so rescaling, and start to x with a group. And what can happen if the orbit of group can get point x and get point x tilde, if closure of orbit intersect with zero section is empty, then it will be point in x semi-stable and you can make the orbit. Yeah, so it's, yeah, so it's formal definition. It's set of x tilde with total space minus zero section. Set that orbit intersecting with zero section is empty and up to rescaling. So it's not clear from these definitions open part, but it is open part. And this again, kind of poly-stable, I think which corresponds to closed orbits in the whole story, kind of closed orbits, but it's kind of bad, it's not scheme. But good for my point of view, that it's represent all equivalence classes, so x, p, s, g. Okay. Yeah, so there's kind of new notion semi-stable, which appears in relative situation in this thing. Howt, I know it's time to go to example. And as example, yeah, this representation quiz. In fact, this picture was really discussed not in Mumford, but by Alistar King. Yeah, somehow it took many, many years, about 20 years to this important but simple generalization. So I take finite quiver, vertices, arrows. I will be denoted vertices of the quiver and we'll have representation of certain dimension di. So the group will be product of group gl di and my space x will be actually representation. V, maybe called, it will be product of affine spaces of all arrows to the idj. So it will be entries of rectangular matrix from d-dimensional space to dj-dimensional space. So point of this space will be representation cure in coordinate spaces. Эроз. For each arrow should kind of specify arbitrary elements of rectangular matrix and group acts by conjugation. So it's actually affine space and affine variety, but line bundle will be trivial bundle all, but with non-trivial G-action. It's affine trivial, yeah, everything is affine, yeah. It's a question to get something non-affine. G-action is given by character and character will be the following. Element of the group, which is collection of square matrices, goes to product over i, determinants of gi to some power ai, v ai, some integer numbers. Ок. If sum of ei-dimension, whatever, ei is not equal to zero, then there is no semi-stable point, because kind of rescaling x in the fiber and contracts everything to point in the base. This x semi-stable is empty, and quotient is empty. If the sum is not zero, then you act by diagonal matrices, and then you see that it doesn't change, then it contracts m6 to zero, so get nothing. Yeah, so it should have this equality, and then easy kind of, but if there is some kind of evaluative criterion, my mom thought to see was point is stable not, you get one parameter subgroup, and if you translate it to the language of representational cure, you get the following. Point is semi-stable. If and only if this representation of a cure is semi-stable in the sense, which I describe in my first lecture, if you get corresponding representation of cure, that for any subrepresentation, which is not zero, sum of ai-dimension of e'i is less than zero. This is an equality. And poly-stable. By the way, semi-stable representation is given slope form abelian category, and there are simple objects in this abelian category, which is not simple with this cure, and poly-stables are exactly direct sum of simple objects in this category. Simple object means to get strict inequality. So it will be semi-simple object in some abelian category. Such simple story, and what is it going on? When you consider this quotient, this representation is neither projective nor affine. It's maps to v cross g result. I think which is affine variety, and which is algebra generated by traces of all possible cyclic loops. Modular relations is universally held in this direction, with some ideal, and at the end of the day, it will be finite dimension algebra for given dimensions, and factors are projective, and usually high-dimensional could be. So you get this thing, and how it's related to my general picture. It's kind of stupid to consider identity map, but here you endow this line bundle, and here forget line bundle. It's kind of look stupid, but even for identity map, when here fibers are point, and the quotient get high-dimensional fibers. So it's strange. So it was short introduction to general algebraic story, but now what goes on? We speak about something, maybe non-compact guys, with ample bundles. And ample bundles suggest something, that we get scalar matrix. We have scalar matrix, integer scalar class, because we get actual bundle. And the idea of Hamiltonian reduction is the following. That if on x we have scalar matrix, not just ample class, then on the quotient we also get scalar matrix. So it will be kind of way to enhance this to scalar geometry. So now if k is equal to c, it will go from just scalar class to scalar matrix. Under certain assumptions. So let's start this absolute case first. So we get a fine variety, and a fine variety can be embedded in vector space. On vector space we can see the functions like this. Some function which goes to infinity, and plurisubharmonic. So the whole story, one can speak about plurisubharmonic functions on singular varieties. And so one can fix function c to r, maybe just positive values, which will be plurisubharmonic. I don't want to spend the time, but if it will be manifold, it will be like ddbar will be positive. But in single points it's something. It defines some kind of scalar matrix on a singular variety. And it goes to infinity. It's a proper map. That's the only property we should need. And now I assume that phi is... Actually I can make a strictest harmonic. So this secondary which will be caliform, like restriction for these things from embedded space. And it can ask phi to be invariant under compact subgroup, unitary group. This is not a big deal. We take any function just average by compact group. We get invariant function. And then in this case, there is a famous campness theorem, that orbit g of x is closed if and only if this function restricted to orbit achieves a minimum. There is a function. If the orbit is closed, function goes to infinity. And that should achieve some critical point. But because of invariance, all critical points could be only minima. Let's explain why. Let's imagine that the group is c star to power n. Just for simplicity. Maximum compact group is u1 to power n. And then consider plural subharmonic function on g, which is k invariant, is the same as convex function in rn. Conditional for secondary derivative is convex function on rn, which is quotient, g mod k. And quotient function has only minima. It cannot have any other types of critical points. And the equation for minima, it's actually zero of some moment map. Because in this case, one can first construct some map from x cross le algebra of k to r. You construct a map, which will be linear in this thing. You take point, you take element of le algebra. And if you move in the direction of k, the value of function will not change. You take derivative of my function f in direction i times k. You can transfer to direction point x. You get a certain real number. So you get a map from x to le algebra. And it will be Poisson map. So it will be Poisson action. And what we are interested in, critical points will be, for bits it will be exactly zero. So it's Poisson map. And what here goes on, Funny story. In this case, I can kind of roughly write what will be the collection of, you get closed subset. It will be kind of zero, zero focus set. It will be closed subset, which intersect exactly each closed orbit ones. So this horrible set, which was kind of not convex, not closed, is contracted to closed guy. So it's kind of very beautiful construction. Is there some tropical interpretation? I don't know, but about tropical I will say very soon something. So it was story in absolute case and in relative case a little technicality, very small one. So relative case, again x maps the projected way to a fine y. And we get line bundle here, ample bundle, so we get the same situation. So on the total space of L, minus zero section. We consider, again, plural superharmonic function, which has kind of local, if you locally trivialize this bundle, which has a form, that f of kind of x t will be some function of x only, which will be potential of scalar metric, plus log t squared. There's no canonical potential on the manifold itself, but on total space of bundles, this is canonical search function. And if you change coordinate, you add here t multiplied by invertible function, you get plural harmonic function, it doesn't change scalar metric. And here we need little technical assumptions on gross property, gross property at infinity, if x is not compact. Of course, it will be typically the case. Namely choose any compactification by some variety, and extend L to x bar, this line bundle arbitrary again algebraic way, and the property is a form. If we have sequence of points, so that limit x i belongs to sec infinity, it goes to infinity, and also choose some kind of lifts to total space of line bundle, so that limit of this point will be nonzero vector in fiber x infinity. Now then, we want to ask the following property. Function of this sequence on total space divided by log of 1 over distance in any metric, because things is compacted x i to the boundary is plus infinity. So it should grow faster than logarithm. It's very, very mild condition. And this property does not depend on a choice, on choices, which I hear complication, so on. It's intrinsic property, and kind of informal meaning is that if we consider some holomorphic tube, which goes to infinity, it has infinite area. It's exactly this property. It means. So it has something to do with probably Foucaille categories at the end of the day, because... It doesn't depend on choice, it means whose choice. It shows us of x bar and extension of line bundle. You can make a different extension with these conditions all equivalent. So for the case of Quivers, this function phi will be, like in this simple case, maybe I can write what is a phi, because we have trivial bundle. Phi will be something like mod v square plus log t squared. And this is much larger than log t, log norm of v. So it definitely has this gross condition. And what goes on in general situation, you can see again, critical points of the whole things of the space, and you can mod out by, I think it will be 0 of moment map from your manifold, and you divide by compact group. So you see, you make it model like close subset, divide by compact group action, which is very nice. Not this horrible topology. So it will be definitely house door space. And x in this table is, again, set of points with subject up to representatives, subject function orbit, this function on orbit is bounded below, again by this direction, and it polystable and achieves minimum. So it's the same story. And what is the advantage of all this killer story, one can speak about not integer Camology classes of the form, because in our registration, one is obliged to do integer classes, and if I go to non-integer combinations, it's some kind of tinkering. You say that you consider all various line bundles and try to make joint lattice by p-car group model components, we get some lattice. And when it is in series of JIT quotient, there are some several walls, there are some kind of piecewise linear structure, and your rational point could line some stratum, then it says it belongs to this stratum. And here in complex geometry define it's explicitly. So this is the whole story. Now if you want to apply to representation of queers and get killer matrix, one sees that there are many, in fact maybe I'll say to be too many Pg invariant Pluris subharmonic functions. And let me just tell a few words about Pluris subharmonic functions. This Pluris subharmonic functions in general complex geometry form a sheaf of max plus algebras. Tropical. So this kind of hidden in complex geometry is this automatically tropical geometry because you take some, you take maximum. Because you can allow not necessary smooth guys positive as a flow. It's also invariant under multiplication by positive real numbers. And there are really many of them. You can make like maximum zero and log some f squared, f is holomorphic function. This thing, or it takes some of squares. Or it take meta log one plus some of squares when it's not vanish, yeah. So there are many, many, many things. And in quiver case so we want to write some kind of function depending on representation quiver is metric. Get some arrows, some metrics and get some space of metrics all representations plus metrics want to write some function. Yeah. And want to kind of write it in a uniform way. So the function which we write originally if space of collection of metrics I'll what I really write let's sum over trace t alpha cross with respect to this metrics t alpha square of norm. Yeah, it depends on the choice of metric, yeah. And class and also of all representations you can see simultaneously. And plus this standard part which will be sum over whatever mu i log bit g i mu i coming from the stability business. Some constant, yeah, but main thing is have kind of this important part which is. The first part only depends on the metric to scale. That's not the last part. Yes, yes. The last part it makes it changes, yeah. But this first part one can add many more terms. One can make trace of any expression in t and t star with arbitrary order can consider free algebra and take the same expression star this will be still even more positive. So you get huge amount of terms. Call this kind of like psi this expression. So you get many, many norms. Yeah, usually people start with this quadratic part but there's really no reason. Yeah, one can. No, no, it takes square. Trace of square. I can see the cone and where commutators come from because one can see the moment map equation for the story. It means that you start to vary Hermitian forms norms. And if you vary, then you do not change t but variation of t will be 0. But t star will be commutator of variation of h commutator was t star. And then if you write condition for fixed points for critical points you get the following things. You take sum over all arrows of commutator t alpha cross and you take this cyclic expression it will be cyclic root yeah, you take derivative of this. So in the quadratic case it will use this t, derivative will be t yeah, but in general we get some horrible expression and then it should be equal to sum over mu i projectors this will be generalized king's equation. yeah so you get huge amount of star algebras and kind of the king's equation in the special case t star yeah, this is quadratic but this is huge amount of star algebras and as I told last time this is very interesting situation you get many many star algebras such as a finite dimensional representations are equivalent to something which can define pure algebraically same as polystable representation of your original guy yeah, so the conjectures the star algebras, their completions are the same so it will be some canonical star algebras depending on this choices so what you say is that you look at the critical points yeah, the critical points will be representation of these things, yeah and this will be critical with respect to the matrix yeah so it's a kind of big open question to analyze this how the star algebras are really in some sense identified yeah, but it's okay yeah, so the source was all about complex case, now we turn to non-archimedian one so what goes on in non-archimedian geometry yeah, so you get a field with a norm, a complete field with a norm and suppose you have algebraic variety then there was a story invented by Vladimir Berkovich yeah, so to this certain locally compact Hausdorff topological space it's called analytic points of analytic spectrum it's a space of complex point and I'll just give you brief introduction, what is it yeah, first if x is a fine yeah, so it doesn't depend on choice of nilpotence it's really like honest point if x is a fine, this x analytic has two descriptions as you can see the space of norms norms from algebra functions to kind of positive numbers multiplicative yeah, so to get f plus g is less than maximum fg is fg norm of 1 is equal to 1 and norm of lambda f is equal to norm of lambda k and f lambda is belongs to k yeah, so consider just such things, it's it was kind of inspired by Guilfant's theory of cistaralgebra, so yeah oh, there's another description you can see the equivalence classes of following things, you consider arbitrary extensions of your field with some norm extending norm on k and a point in x of k prime up to equivalence relation generator that you have common make, make further extensions if you extend yeah, and this is just the same story it's kind of final of Guilfant's theorem yeah, but it's pretty formal game and this third viewpoint on the story third viewpoint is another, it's actually ah, if it's not a fine you just glue open parts gives open things, but if x is projective one can do something else, one can construct things from kind of different perspective namely make all possible and suppose case has discrete valuation then they choose all possible models of your variety maybe it's even a bit smooth project, although it's not relevant, let's take models which have some kind of device with normal crossing, so the variety degenerate to some device with normal crossing then we have a model then we can construct certain finite polyhedral space it's called Clemens polytop of the model it will be kind of formal linear combinations of DI will be components of special fibers my variety can be composed in several pieces of special fiber and to take linear combinations such that sum of TI is equal to zero and intersection of D then all TI non-zero is non-empty so you take kind of vertex correspond to each device edge when they intersect triangle if it's three intersects and so on you get something of dimension no more than algebraic dimension of x you get simplicity complex for curve you get a graph you get dual polytop and now linear polyspace and if you make blow-up of the model then you add some kind of pieces and you get projective limits of this polytop you get something which doesn't change homotopy type usually you grow some kind of wing from here to here you get something contractable so it's projective limit of this polytop it's the same yeah it has nothing to looks like nothing to do but it is the same and it's actually much better viewpoint because we are not interested in functions on this guy you usually take functions which are induced for some finite model and that you reduce some kind of tropical geometry and again on this space again one get a shift of fluidity sup harmonic functions especially the good one which is induced for some finite model yeah so you get also close to the maximum plus the same story yeah so this tropical geometry is kind of in both case in sense it's kind of here more tropical than it's kind of more infinite type in complex case but it's confirm so for each varieties of an argument field you get some kind of tropical guys automatically and what does fluidity sup harmonic function I just illustrated for curves for graphs if you get a graph and get real valued function it's fluidity sup harmonic first it should be continuous function in this case and then should be convex on edges convex on edges because it has a fine structure but what happens with vertices it vertex some over kind of positive non negative function has one side derivative this is what defines the shift and if you pass to the shift of fractions then it's longer positive yeah it's differences and fluid harmonic function function if you suppose together functions minus functions are linear and balanced condition yeah this whole it's now business about geometry so get shift of fluidity sup harmonic function there is a little trouble in all this business even compare complex geometry there is no notion of strict fluidity sup harmonic function second derivative is positive namely I said that if you take an array and get strictly convex function which is the snow pieces when it kind of linear one can redefine but here there is a falling cage you have a curve but it's actually a pullback of when a contract make another model you just add some extra edge and you just contract to the point and if you consider any function which is kind of positive convex here but it's kind of constant so it's automatically making blow up and degenerate looks like a Mercedes I just wrote of course one can draw one can have other signs it means that if you consider pullback of convex function here take convex function with a fine structure it will be function which is constant here its second derivative will be zero and first derivative will be zero so there is no notion of strict convexity so one can't really speak about strict scalar matrix it's kind of always slightly degenerate and so Marco MacCulon who was a student of Bost a few years ago developed an analog of JIT so one can and all statements kind of true so you consider pullback function which is bounded below he considered only a kind of complex of action of compact varieties but with this logarithmic growth I think it's the same story extent just literally everything is true but there is only kind of catch you don't have semi-stable points those when function f is bounded below on the orbit but because you cannot speak about strict inequalities you can't say anything about this and in fact it's maybe not a bad thing as I explained just a second I think it's a good time to make a small break for 5 minutes and now continue it doesn't take the minimum it does take minimum but it's bounded below so it's this thing there is some tropical representation Hamiltonian reduction yeah tropical not but I will go to kind of photos moment map in non-archimedian story I will go just after the break yeah so now go to quivers in non-archimedian case and what was at the end of last lecture I just remind you last time we could go we have something very special we get non-archimedian field we get quiver but we consider representations which are kind of contracting so there is some norm norms such that all operators has norm less than 1 and you choose a norm on them yeah this is a choice of norms such that there exists collection of norms so that each the alpha operator norm is less than 1 or it can be say in many different ways without choosing a norm that for any cyclic word are all eigenvalues kind of spectral reduces reduces less than 1 or norm of traces less than 1 it's all the same yeah and then for such things we get kind of nice class of kind of harmonic metric but it was very boring one it's collection of norms so that each the alpha from the i, new i is as normal as in 1 or the EI EI EJ norm and then we can make reduction to the residue field which I denote reduction to these norms of E is a semi-stable representation of a residue field it was pretty boring situation so it's like defined about the integer integers yeah yes and the theorem was representation of C to semi-stable if that meets harmonic metric which is deformation of semi-stable representation but what to do if this eigenvalues are bigger than 1 2 semi-stables and yeah one can make kind of again try to steal result from complex numbers so complex numbers was trace of these things what will be non-archimedean analog of this guy and you'll see it's actually very very simple yeah for complex numbers if you have 2 vector spaces V1 and V2 and with some Hermitian norms and we got some linear map then one can always there exist a orthogonal basis V1 V2 such that matrix will be diagonal it will be maybe not rectangular matrix it will be diagonal matrix and in this diagonal terms one can also choose them to be positive and if you want to make a norm I'll just some numbers which are called singular values and the same as eigenvalues of square root of T T cross T yeah so completely yes what we can do over condition is the same as saying singular values yeah and then you get some second representation which is and now I cannot achieve it anymore so singular values will be not less than 1 yes exactly yeah actually if we assume for simplicity that this relation so all real powers then the same is true so I get notion of singular values now I claim that for example the sum of squares this will be perfectly good choice of a non-committal case as well but in fact there are many more choices choose any function g from r to r which will be like this so what will be the properties g is bigger than 0 g prime is bigger than 0 and actually double prime is bigger than 0 and limit this variable I don't know how to denote it lambda goes to minus infinity g of lambda will be 0 and limit lambda goes to plus infinity g of lambda will be plus infinity yeah for example exponent of lambda or exponent of 2 lambda which is probably good choices and the idea is to replace trace of t square trace t alpha which alpha by sum over singular singular values of the separators of t alpha you can see the g of log of the singular value this minus infinity so in particular get sum of squares because of the exponent of logarithms so we can get exactly the same function formula as in the complex case and the claim it is pulmonary subharmonic function space of operators I don't want to give you a proof but it's one can on maps from V1 to V2 In fact you can probably write it as a soup yeah maybe I think so yeah you are right because and now I can try try to apply this maculon theorem maculon theorem I can study for local critical points what are local critical points critical points of this function so as I explained your last time it will be critical points on the space of product of all vertices of matrices of my spaces and huge amount of these buildings and what I explained in your last time although it's kind of huge horrible space this tangent space is a certain cone it's correspond to real filtrations on things over the residue field so it's something nonlinear but it's space of kind of real flags and then I can make derivative spectra real flags and try to calculate what is the critical points and going to deposition of EI is at some point deposition will be real flags in finite dimensional space over a residue field G, nu, E it's a vector space for the residue field and then I can try to analyze what is going on and it's something remarkable happens we just substitute the whole things make calculation you can interpret what is going on you got the things that that certain representation of certain nuke viewers are semi-stable so these things can start to repeat themselves yeah representations of some auxiliary viewer with certain stability conditions are semi-stable yeah so I'll just explain to you in a second yeah so it's something which has something to do with this situation you get this map between two spaces which not preserving the norm so kind of unit ball goes to kind of like extend to some direction yeah something small, something bigger yeah this you get in sort of ball you get some kind of ellipsoid and some position of ellipsoid so it will be some little preparation I will need to explain this auxiliary quiver yeah I will you'll see this in a few minutes yeah suppose I got map between two vector spaces and again continuous relation to simplify my life then we know that the picture looks like this yeah actually I don't have to repeat this story but maybe what I can do I have this my log of singular values will be some zero singular values which I ignore and there will be some some finite amount of other thing which map is not zero and all these diagonal matrix I can kind of arrange in the order we kind of lambda k here and lambda 1 is the variant so I get this order and now I make in filtrations there are two filtrations I get filtrations on source and filtration on target by subspaces which are completely intrinsically determined so maybe I do not v1 bar v2 bar will be spaces over the residue field yeah spaces over the residue field and the spaces will carry these filtrations for example if you have vector in v1 bar which is not zero you can see the minimal possible log of norm of vector v underlying so it's only leading digits dv respect to second space where norm v is equal to 1 and v is a lift of v bar and we just try to minimize these things and we get one of the slog lambda i's obviously and similarly for u in u2 0 you can see the maximal possible of inverse stuff and so it's intrinsically determined filtration and and you get something v1 bar which is v1 bar less than lambda k it's because you cannot it contains part when you less than lambda k minus 1 and so on it goes to non equal to 1 to 1 and then it it contains which is equal to and then it contains the part which is minus infinity which will be reduction of the kernel which contains equal to 0 so this is the last term for kernel which we don't care and similar story with v2 bar it contains and this things we can arrange to the first things we should go to quotient naturally it's better to go to quotient this will be some non zero map to some non trivial quotient and we take u1 from and another story one get inclusions here at the end you get map to zero but here it could be zero map and here the last guy its inclusion could be identity it's actually non equality map non zero map it's really non trivial map so spaces are different and the associated graded pieces are equal so you can see the kind of they match but they go in opposite order and now we draw the graph here finally what is this auxiliary graph just before so if we start with arbitrary quiver for example like this what will be auxiliary quiver it will have for this vertex I get just bunch of epimorphisms this vertex I get again bunch of epimorphisms and here inclusions numbers will be lengths of how many is a singular values but I mean it depends this depends on representation everyone can say this label by real numbers and almost only finitely many called set you can accept like quiver a, a, r like think a quiver we erase integers by total order set of real numbers and for this vertex you get what you get a few inclusions and a few and one chain suggestion so it just gets bunch of flags nothing else so this is this auxiliary quiver is this joint union of vertices of q of kind of star star like quivers okay all the compose on the simple pieces and now I write what is the stability structure here what is the central charge of representation of axial requiver of the residue field yeah there will be some overall kind of central there are central vertices which are the same so we put here the same spaces there will be z i e i or maybe whatever these are just numbers yeah it's the same numbers it's what we have before and then we get the following correction we take some of all inclusions when gruble was some negative number j and we take d prime of lambda j derivative of g times dimension of cokernel of this inclusion and minus sum of all epimorphism graded by lambda j g prime of lambda j dimension of cokernel yeah so what is nice g prime is positive yeah it's positive number yeah if you take some of all these things same central check because these things cancel each other because these dimensions appear in one vertex and another vertex but stability it's kind of new cure get some new if you consider subrepresentation it doesn't come from original cure so get certain condition and then the equation just really straight forward check I don't want to bother with these details it goes with exactly this semi-stable relation but that's with yeah so the theorem which we have with mycos Haydn that's Ark of Pandit and me that in fact here goes really semi-stable representations of cure exactly those which admits harmonic metric which gives this story kind of harmonic metric which gives the semi-stables so it will be this moment map equation in fact we went to this in very long way not like this I have to say that we try to imitate this flow and so on and then go for complex numbers don't commit numbers and then get fixed points but at the end of the day one can make it much simpler so this is analog of moment map equation in very strange sense so this tangent space is not linear but it's just a cone but still it's the same story the proof one have to introduce some topologies some space of flux some kind of semi-continuity it's pretty soft proof some compactness, semi-continuity and so on and there's also a natural candidate for flow for h for Harners-Siemann flow because for Harners-Siemann flow in my last talk I also explained that this is also a flow and derivative of flow should be r-filtration r-filtration was Harners-Filtration and here just use Harners-Siemann-Filtration with this axillary cure to define new dot will be Harners-Siemann-Filtration but you change to opposite for axillary cure in certain sense and conjecture these things should exist and produce the limit Harners-Siemann-Filtration naturally without choosing slope and so on so it will be interesting we kind of got lost in details it's I think it's maybe some model theoretic problems because it looks like at some point you should go kind of infinitely many kind of very bad behavior in this conical singularity so is this set? yeah point it will get axillary cure over the real small k yeah okay so this was kind of essentially one nice clean example in general we have a non-recommendian geometry and now what is a general picture? I will just go for all the story for a moment I will forget about this various fields with norms it will be pure algebraic thing about stability conditions so suppose we have triangulated category over some field which could be big k small k whatever we want maybe I denoted now small k which could be big k at the moment I don't care about norms and if I try to think kind of natural example like queues but more interesting situation of multirac geometry of okay category there is something in common very very robust so for any two objects of your category if you consider home spaces find a dimensional and and also if you consider home from e and shift by kind of there is no negative effects is 0 for n greater than 0 point to objects Max, I am sorry to ask you what does it mean? I am not familiar with this kind of rate I will explain in a moment what is it? in great categories like complexes of modules over some algebra no, it is not a billion it is additive it is definitely not a billion stability structure gives a relation with a billion category it is like all complexes and then we can see the complexity in zero not complexity which are positive it is all complex both positive and negative here the main thing is for any two objects you consider x groups and this is shift it means it is like x minus n something like this also there is a way to shift you can always shift objects and there is a notion of exact triangle that is all you can shift and so it means you consider x groups between objects it will be zero in negative degrees it is the hypothesis and something final dimensional in positive degrees for usual consider modules of final dimension also the only positive axis but you shift to get formally x in negative degrees and second thing the story is that it makes sense to speak about families of objects standardized by schemes and technically it means that you get maybe DG schemes and some cool technology you get some kind of Yes, yes everyone can say it is a shift on some topos yeah if you like but it is kind of abstraction what really all this means what it really means it is kind of basic example which is more or less efficient for things suppose you have differential graded algebra over field k which has a following form you can see the three algebras in infinitely many variables for any i differential of each variable is expression in previous variables kind of step by step you add more and more variables this is very general quasi-free resolution any countably dimensional thing you have quasi-free resolution just start to enumerate things in some way but the main property is that now we want to this degrees of these variables goes to minus infinity could be some few positive degrees but eventually it should go to minus infinity which is how do you define degrees here it's z-grade algebra and for each variable choose a degree a priori for each variable choose some integer number and the expression should be homogenous so this degree of this guy should be degree of xi plus 1 so and this category C which you consider one can make it's pretty very explicit I can give you some kind of definition which is abstract but it's much more concrete db and of kind of dga modules consider dga modules which have finite dimensional over your field k what does it mean it's actually something very concrete you get finitely many vector spaces kind of like v-3 whatever you understood sorry derived okay whatever it means but what really means you have you do the following like I know 5 finite dimensional vector spaces then you should then you want to introduce what is differential it's d-square equal to 0 then you want to write operator x1 it's operator of certain degree which is degree of x1 which should be commutative differential because maybe d of x1 could be constant even yeah okay you should solve equation for x1 operator of certain degree дегрессор 0 0 если x1 dg-1 dg-1 which is differential is equal to 1 then you choose something for x2 step by step you want to write d of x2 some expression x3 then it's not room for the separators no operator of degree-100 yeah, so you just stop at some point yeah, so you get only finitely many variables and finitely many equations yeah, so it's something very concrete and should be computer software for this suppose yeah and then one can define what are morphisms, homotopes and it's everything is very explicit and completely concrete and finite dimensional and and the story contains particular cases some categories which defined kind of heavy way through topos, blah blah blah derive category with compact support on separated scheme of finite type one can always find such description yeah, here in fact one can also use quivers instead of re-algebra it's all the same because one can make resolutions and in this case we want to describe what are models you really saw finitely many equations finitely many variables so you can speak of parameters things depending on parameters so it's completely clear that you got speak out families of such guys and what we want from stability structure on this thing there must be a universal family I presume also yeah, it will be inductive limit of things of finite type yeah, because you start to make these things of larger and larger dimension you shouldn't change your characteristic but you it's complex what we want from stability structures first of all the central charge should be equal to following it will be finite c linear combination of something of additive maps which is given by the following thing e goes to earlier characteristic certain functor applied to e the functor is functor to perf of k sorry, it's finite complex of finite dimension vector spaces given by e goes to our home certain p i to e p p i is a perfect complex of a it's something like algebra itself or direct sum of copies it's like finite dimensional projective module and I denote by i because in real life it's a choice you make some choice you choose some things perfect complexes your algebra replaced by quiver it's one can say that it's kind of in this abstract way more concrete way it will be not digi-algebra it will be past algebra or quiver and that will be just dimension of representation in each vertex early characteristic in each vertex so it's this abstract story about perfect complex it means digi-quiver nacycle and it's one thing which we would like to have and second that then we can see the object with certain slope and with certain class the gamma is collection of gamma i the i runs through this set essentially vertex occurs gamma i is earlier characteristic point gamma i is exponent of so the whole story will be like a quiver then this thing should be not huge infinite inductive limit it will be some kind of finite class you can describe by things of some bounded description so it will be art and stack of finite type roughly some scheme of finite type divided by finite dimensional group this space of objects or semi-stable objects which may be still in there but which may not be in still and this should be open in or should be the riskier open space of all objects and why it's art and stack there's a whole story of high stacks because when consider in this category the object can have negative x to itself have automorphism then have automorphisms in principle and then it means that you should divide not by group but then you should divide by again by not by one but by two of more space and so on but for semi-stable objects it falls from there's no negative x to itself so just divide by group and you don't have to do this high homotopy theory at all by action of a group but not divided by two group in a sense so it's kind of and I think one should have kind of course course model space when you consider something like this JT quotient which will be no longer stack but ordinary maybe singular but reduced reduced separated house dwarf scheme or maybe it's called algebraic space and the set of k bar points of this guy will be the same as poly-stable objects isomorphism classes of poly-stable objects in sense in this language you can make extension of scalars because you just consider this big categories so this all yes, yes, yes make this algebra by which I remove put here k bar that's it now so get again one can think about what I claim is by pure categorical means on this course model space I will have a kind of killer class it follows just purely from stability axiomatics just a second so the claim that this is the canonical class omega belongs to car group of this and moreover the integral of this class of any compact curve is bigger than zero this one can be used from axiomatics yes, somehow relation ordinary algebraic geometry very general framework course model space consider quotient of this stack category quotient here this is some group just terms of variable numbers now so it's kind of candidate to class of killer form and what is the formula for omega first for for any this kind of vertex of the query I have a line bundle on space of objects and a fiber at some object E is determinant of this complex FIFE or Arhomia and if you add by rescaling because it's natural things associated to object you multiply by how it will acts here it will multiply by lambda times Euler torque p-stick of this guy but now consider kind of virtual line bundle on objects of c sigma kind of semi-stable of light sigma you consider virtual bundles you should be formal product of real tensor powers which don't make sense on this geometry but namely take tensor power of Li turns going by real number which is imaginary part exponent minus theta zi and what is nice here when you calculate rescaling the whole thing here cancels yeah and I didn't check what it looks very plausible at least for polystable rescaling it's kind of simplest automorphism but there are other automorphisms actually really here on this rescaling and then one have kind of virtual line bundle on the quotient will define yeah it's kind of what does it mean one can speak about norms on this guys yeah it's kind of formal game yeah yeah yeah yeah but this guy gets some first-chain class it will be first-chain class of this L maybe L sigma called this guy L sigma you just see what is going on it's sum over i this real coefficients it will be a certain real combination of guys and I claim omega if I want to have specificity the claim has this inequality let's make plane this this it belongs to p-car group turns r it's real numbers it's elements of p-car group yeah this is now it's kind of makes essentially perfect says so y integral of omega over some curve y compact curve y is positive yeah so what what does means you have a curve now a little mental it will be for a while so I get a curve so I have a map from curve to this course model space which was contracting of this of actual stack yeah so it's I think one can choose at least yeah it's some kind of proper map and you lift for proper curve it's subjective map yeah why not to it's not unique choose a lift yeah at least in a real life situation no problem at all choose a lift it's not canonical and what does it mean before we have kind of isomorphism class of representation we have actual familiar representations of whatever we were okay so we get a family of object family of objects parameterized by points on a curve of semi-stable objects okay now with kind of quantum lift imagine like you have a representation of curves depending on point on a curve one take our gamma at each vertex and get again representation of curve, again finite dimensional yeah and it works at least in this concrete example which I I think erased it's some object of category C there are many tricks how to do it algebraically but at least it's pretty clear what want to do in this concrete example we can multiply by ample line bundles take sections and blah blah blah there are many things what will be the central charge of this guy, let's calculate it yeah first of all needs by definition it's sum over I this number is the I times earlier curve D stick of V I E dot and what is electric T stick of V I E dot it was some space of homes or you can think more concrete in terms of cures it's electric of R gamma of coherent shifts on curve Y where fiber is F I applied to E Y we get some kind of complex of vector bounds on a curve we want to use Riemann-Roch's theorem then we get 2 terms it's integral Y over degree of this bundle first chain class of this thing complex of bundles which is the same as integral of Y firm chain class of L I determines yeah and plus 1 minus genus of the curve multiply by rank at any point which was essentially my number gamma I now substitute this thing cancels yeah and what you get you get this integral C1 it was definition so what you get get up to minus sign because I define minus sign it's integral of my class omega this not central charge exponent of this visible part of central charge is equal to this real number because central charge is complex number ok now we want to prove that this number whatever is negative why it's so here this argument I claim that this object E total when you make this equation has slopes between C2 and C2 minus pi only what is intuition let's consider simplest possible case vector spaces it will be one dimensional complex vector space when you consider commode it gets commode in degree 0 and 1 in one in this language goes shift by minus pi yeah this argument it was kind of minus pi times degree so the claim is that all terms in harmonic infiltration of E total have slopes in interval C2 minus pi theta so this is some kind of maximal and minimal we should prove both of them we have kind of maximal which maps on zero map it's kind of morally embedding to E total and then it's projected to E sigma minimum that's the two extreme points of my harmonic infiltration are like this and I want to prove that sigma maximum sigma minimum will license this interval yeah it goes by contradiction so I assume that sigma maximum is bigger than sigma so I go along this interval so what I get I get H0 of R home of E sigma max to E total is on zero but this R home E total is R gamma of my curve and what I should put here I get point wise R home and this point wise my axiom of stability I see it in degree greater than 1 and the total commode also in degree greater than 1 so I can't have anything degree zero so I get contradiction and similar for but negative other case but it's a bit tricky actually for other case now I assume that T minimum is less than sigma inspired I also want to get contradiction so I get zero R equal to R home E total sigma minimum and at least in concrete case it's true and I don't know with that general standards H H0 multiply by canonical class shift by 1 it's some kind of serduality here hidden R home point wise now this guy sits in degree 2 here should be in degree 1 and here it's again in degree 1 gain contradiction yeah so what is the conclusion so in this hardness infiltration we have object only in some sector so if it rises sector sigma it will be sigma it will be sigma minus pi and here we get kind of contribution of vectors only in this side so the center yeah so the center of charge stay here and you see immediately that integral part of exponent minus i theta the center of charge if you rotate here it goes down that's in zero exponential minus i theta that's in terms of computation no I always multiply by exponent minus theta was my slope which I fixed and here I do I always rotate things by this point my theta yeah so I get one six and one can analyze if it's equal to zero then the things will be degenerate curves so it looks like you really want to have a killer class yeah so it was I discovered this argument many years ago and for me it was kind of very good science it's all bridged lexiomatics which kind of very very formal things somehow interacts with algebraic geometry but very simple one, one dimensional it doesn't need high dimensional algebraic geometry only for curves but it's about something more commutative the whole story basically ample class yeah so and then there's a kind of general idea is the following that this this bridged stability gives a lot of varieties with ample classes this space of polystable objects which you like quivers polystables guys in such and such dimensions and GIT series says that if you fix some kind of formula for killer metric you produce natural killer metrics in complex or non-archimedian case and there's ideas should be some kind of underlying notion of non commutative killer geometry some triangulated category with something such as on each modular space you get naturally killer metrics not just killer classes and then maybe flaw and all this Donaldson machinery so it's very general theme of this thought and we have this nice examples with quivers which are more or less clean and in the last two lectures I will show completely different class of examples kind of more differential geometric nature when you get the same just get naturally again this killer metrics and so on or non-archimedian guys but have nothing to do with quivers and more be special Lagrangian manifolds and so on and will be essentially no series afterwards it will be speculative but all the story with flaws and so on will have completely parallel just to incarnations quiver like and some differential geometry okay thank you so all these evolutions were compact curves so if you have like for parabolic bundles if you just have some puncture no no it's compact curve in more less space of objects it's not the curve on which the bundle is it's something in more less space of objects