 Реколд, что в последний раз я сказал, что в последний раз что-то, которое я сказал, было полное. Если у вас есть комплексный аналитический, симпляктический манифальт, то два форма с холоморфиком, и вы получите, может, кучейн, representing some twist parameter, and which twist parameter should belong to h2 of x and take with coefficients in the group of invertible limit and cfh bar. Then this thing gives you by the Joseph construction or my deformation characterization, gives you a shift of h linear, a billion categories on x. It's in shift, which I will not denote by any specific letter. And in particular one can go to global section, get certain category of h, some kind of global category, that if we are lucky, it actually goes through shift of algebraids, but if we are lucky, say x is maybe Stein or algebraic, stations also works in analytical algebraic situation, or algebraic, and we get shift of algebras, not of algebroids, so there's no other way I think to enter into the game. Then we get some algebra over ch will be a h modules, a h is certain algebra, and it will be complete algebra, complete flat algebra over form power series. For example, if we apply to algebraic situation, again x is affine space or complex numbers with standard symplectic form, then the algebra which we get is the algebra, which equivalent to this guy. So what is my goal? You see that what does it mean? To consider form power series in each bar, in each coefficient is differential operator, but the degree of this differential operator is unbounded, it can go up. And what we want? To get kind of modify algebra and get something like this. So it will be polynomial differential operators, these coefficients are form power series in each bar, so they have kind of well defined degree. So we just want to move this form power series in the front, which drastic change in the algebra, and in principle later one can replace c of h bar in front, one can try to replace to terms of analytic functions. So it's form power series, which convergent in some disk of small radius, convergent series, or even polynomial c in each bar. Yeah, so, and similar thinking one can ask you in commutative algebraic geometry, those who get a final algebraic variety, and you consider deformation of this final algebraic variety, like changing coefficients of equations. Formal, in fact, in commutative algebraic geometry it's even more dramatic. Formal deformation theory is trivial, each one of a final variety is tangent bundle's trivial, so there's no deformation of form power series, but how to handle the story. In deformation theory to get reasonable answer compactify the situation. So the solution is compactify x. So I claim that if you embed x into some compact manifold and with the following properties. So first omega minus 1 on x is by vector field on x, extends to x bar. I assume here of course that x is, let's say, it's supposed to solve the registration, x is the risky, open the risky dense. Now obviously this thing will be Poisson bracket, because it's Poisson bracket an open part. Poisson structure on x bar, and what I assume that this x bar is a divisor, maybe with normal crossing it's normal crossing, and ideal of this divisor is Poisson ideal. I claim in this situation one can do something there, kind of, I wrote a paper about 15 years ago, quantization of algebraic variety, and in fact I look at just several two strong assumptions and one can do things better, I'll explain what is going on. So there are two general results, which were not in this paper. Suppose x is a compact algebraic variety, I say complex numbers in a field of characteristic zero, and this Poisson structure, even without Poisson structure, let's consider compact algebraic variety. This deformation quantization identified two things, you consider on x two shifts of differential graded Lie algebras, one's called T-poly, another D-poly, maybe shifted by one. T-poly is polyvector fields, you just shifted by one, so in degree minus one will be functions, and there will be vector fields, by vector fields and so on, and you get score and bracket, and D-poly also starts in degree, it's again O, but instead of T you consider differential operators. Differential operators, not only vector fields, and sort of by vector fields differential operators, and some poly-differential operators. This can be thought of as a kind of local version of homological complex for algebra functions, it gives a linear function, it's end out with some other bracket, and the main thing is, there is a kind of L-infinity, whatever it means, quasi isomorphism, which identifies shifts, using this complicate Feynman integrals and diagrams and so on, it's quasi isomorphism, which identifies some solution of Mauer-Carton equation, and there is some general story, if you get a shift of differential greatly algebra, one consider Mauer-Carton space or some function with values in maybe group poets, or whatever you want, for R gamma, this guy, gamma, gamma, and if you are interested, if you try to interpret what means this thing, it's kind of purely differential geometric equation, and here are deformations of shifts of categories. It's kind of this controllable deformations of shifts of categories. Okay. I think about this categories, which are locally equivalent to categories of modulus of some algebras, but there's no algebras. So the kind of theorem one, if x is compact, and... Actually, theorem construction, maybe. And then choose a subspace in commulger of x with complex coefficients... С such that we included to this commulgio of xc and maps to commulgio of x, this is, for compact varieties, epimorphism is rejection, yeah, so choose some splitting of some term of hodge filtration, then this choice, which is pure, you can use hodge theory, for example, to do it, it will be canonical choice, it gives quasi esomorphisms of modulus, global modulus space, not of shifts, more carton of argama, now of trunquated things, as a shift then even locally not quasi esomorphic, but globally will get certain identification modulus space, and here it has a following clean meaning, it's deformations of x and kind of zero Poisson structure, S Poisson manifold, may deform till x and also add Poisson structure in it, we start with, yeah, because if consider what it means, you have essentially do bi-vector fields, and you give Poisson structure and h1 vector fields, which are deformations, and here it will be deformations of x and o as a shift of c algebras, so consider variety with a shift of algebras, so it's not shift of algebroids, the proof, it's really one line proof, the observation that on both modulus spaces, most algebras contains constants, as a central, a center, contains constant shift, it's actually central embedding, and what does it mean, that it means that you can see the argama of this shift of the algebras, which is just graded vector space, so you see that commold of x shifted by 2, as graded abelian is a graded algebra x, on both Marko-Ton spaces, argama t-poly, n-poly. Now, these spaces, maybe I'll just, so if it acts, then the subspace V also acts, yeah, so you get action of kind of additive super space on this Marko-Ton thing, and this guys contains, because it contains sub-algebra t greater than or equal 1, and d greater than or equal 1, get inclusion of, and then what you get, you get the following, you get this V shifted by 2, and which actually, maybe because V is isomorphic, you can see commold of x by x shifted by 2, multiplied by, you can see the action on this submodular space, maps to Marko-Ton of argama t-poly, it maps to Marko-Ton of argama t-poly, because this thing sits in the larger space, and we can see the action of subgroup on this larger space, and if you check on the level of tangent space, it's at zero point, you get this quasi-isomorphism, so it's actually quasi-isomorphism, because it's quasi-isomorphism level of tangent space at initial point, this I think is quasi-isomorphic by formality, so I think that this guy is equal to this guy, action is free, and so you identify one thing with another, but when you identify, you see that you get some certain interesting things that, it will be not, once you include it, you get certain shift by commold class, so you should, you start with Poisson structure, which gives solution Marko-Ton here, and at zero here, and to transfer here, you get not shift of algebra, but shift of kind of algebra shifted by some class in the space V, so you should have some derby correction, and this correction is non-trivial, many years ago Boris Stiggen made a calculation in the case of compact symplectic manifolds, like K-3 surface, and there are not so many examples, but one can still develop a theory, and it turns out that if you want to see what an actual shift of algebras, and compare a story, you get some corrections, and corrections don't trivialize Rosanski-Witton invariance of symplectic manifold, so it's pretty non-trivial story, how we go from one to another, but as I claim, the same works for Poisson manifolds, there is some kind of final look of Rosanski-Witton story, which I don't know what is it, but this kind of simplifies the life, so we can really permit rise shifts of algebras by Poisson structures, yeah, so it's first theorem, and another theorem, it's about divisors, actually, it's some kind of unfinished manuscript by my collaborator from Cameroon, he actually proposes story Joseph Donga, about 2-3 years ago, that it's logarithmic formality. If you have, again, any compact variety, algebraic or analytic, and it contains divisor with normal closing, then one should replace polyvector fields by logarithmic polyvector fields, so, first one to have kind of logarithmic bundle, so if you have some coordinates x1 to xn on your variety, then log and divisor is given by product of x i equal to 0, then logarithmic tangent bundle is local coordinates is bundle with basis, which is denote x i d with x i, so it's become vector field tangent to this story, and that's geometric part of the story, and algebraic part of the story is diagram of functors, you get kind of shift of categories on x, and suppose its device has one компонент, you get, you quantize both x and divisor, and you get two categories, and restriction functor will give a functor between two categories, or algebraically morphism of algebras, and if devices maybe union of component, maybe let's say two component, get commutative diagram of functors, yeah, so it's kind of pretty abstract, but what to say it's, one can combine two things, if you get Poisson manifold with Poisson divisor, then when you quantize to get shift of algebras, and set it ideal of the divisor will be still, to set it ideal non-commutative algebra, you get compact Poisson variety, x divisor, maybe x, x bar, say, and Poisson divisor is normal crossing, and then you get algebras to set it ideals, after deformations, and then x bar here, so now return to the situation when x is x bar minus divisor, and assume x is a fine, and then what I claim is that by deformation you get filtered algebra, filtered algebra over, still over form power series, I still not interchange the order of series, and filtered algebra will be again h, maybe okay, will be sections of x bar over my quantized shift, and put maybe kd, allow also for the k, k01, and then we get a multiplication with the compatible filtration, this spaces are finite dimensional, then we get kind of inductive limit which will be filtered algebra, and this is essentially object of algebraic geometry, because what it says it's algebra, I think it's more or less automatic, it will be finitely generated and generated by some finite space, we get some finite dimensional space, some generators, relations, and associativity gives some algebraic constraints, so we get algebraic modular space of such filtered algebras, some modular space, and what we have here, we get a formal, we get a map from, formal spectrum of C of h to this modular space, and I told you, there exists for example, kind of one principle deformation, when we can free scale Poisson structure, when twist parameter is 0, so examples, which I showed in a minute, suggest that this formal path is extends to actually analytic drum, in fact, not analytic drum, in all examples, it's entire, some entire map, it's not algebraic map, it'll be still, or C analytic, and then we can arrange parameters, one can speak about algebraic families of these filtered algebras, yeah, so that's how one can go to formal power series into actual family of algebras, yeah, so what are basic examples, suppose X is cotangent bundle to some smooth algebraic variety, and I explained last time to kind of get usual differential operators, one should put non-trivial twist parameter, belongs to H2 X Z mod 2 Z, we can see it's kind of plus minus 1 sitting in C of h bar parameter, and it's responsible for taking square root of canonical bundle Y over 2, okay, then you get, so how we compactified, so first case, first A, A is compact itself, if Y is compact itself, so you get a vibrational Y with fibers, which are vector spaces, and you compactify each vector space to projective space, fiber-wise, compactification of cotangent bundle, cotangent bundle union projectivization to infinity, and it's very easy to see it's plus 1, the divisor if you added infinity, it's plus 1 divisor, and this procedure should give you, of course, differential operators, and each differential operators on Y, and you get a family over algebras, over, in this case, polynomially depending on H bar, if Y is not compact, is, ah, sorry, it's not algebra, it's a category of demodules, it's not, you also get a shift of categories, but I don't want to relate to all stories to speak about shifts, because after make it all things, actual families, there's no shifts whatsoever, yeah, so it's all things, just one global category, Fx is not compact, for example, if it's a fine, then you get modulus of differential approaches, but the computation is the following, first you choose compactification of Y, again, by its divisors, normal crossing divisor, and then consider cotangent bundle to Y shifted by logarithm, it's kind of dual to logarithmic tangent bundle, which I explained to you, in case of divisor normal crossing, you get total space total space of the things, and you'd, and may, may total space of practivization of the same thing, do the same story, but it's, it's again, one can easily see that divisor will have several components, it will be vertical and horizontal components, and pos will be Poisson, yeah, this, the second example is X will be C star cross C star, this coordinates X1, X2, and form is, how you can complicate, you can complicate many, many ways, this really know the surface, X bar will be any toric compactification, all, always gives Poisson surface, yeah, this should not be not necessary projective, just any toric compactification, and the algebra get analytic in each bar family of algebras, and algebras are quantum torres, so you can see the X1 hat cross X2 hat, maybe the Q is C star, and H goes to Q, Q of H is equal to exponent of H, yeah, so it's a family of algebras, algebraically depending on parameter Q, filtration is by total degree of La Ranne, X, X, X2 invertible, variable satisfies this relation, like, you get algebraic family, but when you make canonical family of star products, you get transcendental map, which is still entire map, defines for all H bar, and then next example X will be Cp2 minus smooth cubic curve, curve will be given by some equation f of X0, X1, X2 equal to 0, when X0, X2 homogeneous coordinate, and, yeah, for example, if you consider chart, when X0 is equal to 1, then Cp4 will be dx1, dx2 divided by df, dx0 equal to 1, f will be cubic polynomial, yeah, and then you get something called Sklannian algebra, Gershwin-Sklannian algebra, let me give you explicit description of this, first I consider free algebra generated by three variables, it's free tensor algebra, and then, yeah, obviously it's graded by homogeneity degree, now we take element on potential, which is element on the quotient, so it's going to consider linear combination of cyclic words of homogeneity degree 3, yeah, this space has dimension 11, these three variables, it's very easy to calculate, which is 1 plus a number of dimension of usual cubic polynomials and commutic variables, and you take potential and then consider algebra, f, modular relations, oh, so, oh, I, to say that ideal, if you get a cyclic word, then there is a well-defined derivative, which belong to algebra itself, if you draw some expression, for example, x2, dx1, all these things, what you do, you put, apply cyclic permutation, put x1 in the front and remove it, you get x2, x0, typical, yeah, so you get some quadratic relations, and this algebra, because you can act by J9, you get essentially two parameters, some canonical form of potential is the following, it will be something like x0, x1, x2, minus x0, x2, x1, plus, maybe, some, some small parameters, I consider this as a small correction, then equations of derivative is equal to 0, we'll tell you that x i, x j kind of close to 0, so you get deformation of algebra, just the things, by taking derivative, you get all commutators equal to 0, and this will be deformation of free algebras, yeah, so you get, these things, it depends on two parameters, yeah, by the way, here we get one parameter, yeah, and there is something common in these two examples, one is rank of h2 of x, of c star, of c star, and two is rank of h2 of cp2 minus cubic, yeah, so it's not really a coincidence, it's the number of parameters will be dimensional second, glomology, yeah, so this is a slanted algebra, you get, maybe, algebra of prime, then if prime is still graded, and in degree 3 it contains certain central element, which will be more or less this polynomial, which you add to the correction, you get the central element, and you consider 3, you add the inverse, and take homogeneity degrees 0 part of the story, yeah, so what is model, if you have algebra of polynomials in 3 variables, and we get some cubic polynomial, we invert, we get algebra functions on cubic space minus cubic form, and consider function of homogeneity degrees 3, we get function on p2 minus cubic curve, yeah, so this model is this glenin algebra, okay, yeah, so this is a 3 basic examples, when we have interesting algebras, what I want to say is it's kind of, there are some non-examples, non-quantizable varieties, it seems it will be a little kind of similar to this construction, for example, again you can see the sum now function in homogenous, I better denote x1, x2, x3, not x0, x1, x2, no, I will say some examples of some symplectic manifolds, not oposon, which are not quantizable, there is really no quantization, yeah, so there is first example of interesting varieties of algebras, okay, if you consider polynomial, then it gives you, ah, example will be a boson variety, you get boson structure, namely you can see the, this, I can write boson bracket like this, or this inverse things will be dx1, dx2, dx3 divided by f, and f will be, it's not homogenous polynomial, yeah, so for any homogenous polynomial, we'll get some boson structure, and you want to quantize it, and if degree of f is less than 3, then kind of naive filtration will be okay, so it's quantizable, at least on the nose, and if it's greater than 4, it's not, yeah, so that's... You mean not by algebras, by sheep, but by algebras, it's always quantizable? No, no, quantizable, I mean kind of non-pertroperty quanti, not compactifiable, maybe, not compactifiable, and the formal shift things exist here, but to get reasonable answer one should make an implication, but there are other kind of thing, which, for example, consider c2 minus 0, with the standard symplectic form, it's also not compactifiable in this sense, at infinity you just can provide a projector space, but at 0, if I add 0, it will be something wrong, it's not a divisor, and if I make a blow up, for example, at 0 of c2, then the two form will vanish, not have a pole, my form could have all these poles, and another example, if suppose x bar is, let's say, surface of general type, so it has many two forms, you pick some form, some nonzero form, and take x will be x bar minus 0 of this form, yeah, that's also bad symplectic manifold, because you can multiply by something with 0, not poles, and think, which didn't really work out, but it looks that we consider like g reductive group, and consider the quad joint orbits, and there is some kind of the worst orbit, which is kind of nilpotent orbit, it doesn't look to be quantizable, because even in the case of SL2, you get c2 minus 0, yeah, so I really lost what to do, maybe it should extend a little bit symplectic manifold, not compactifiable on the north, or at least one should change a little bit, yeah, no, no, because in my completifiable story, I just said that I should add divisor, and it's not what I'm adding here, adding a point, not a divisor, yeah, no, you can make, no, no, you can make embedded in some larger symplectic manifold, which is compactifiable, yeah, yeah, that's, yeah, no, no, but you need some kind of block, you add some points to your symplectic lift to make it, make it blow up, yeah, still, yeah, you have symplectic resolution, yeah, you should, yeah, yes, yes, yes, but at least not the orbit on the north, you should embed it as a risky open to some larger symplectic variety, yeah, I just want to add, to good example one, maybe zero example, which I forgot to mention, if X is a fine space, again with standard symplectic form, there is still another compactification X sitting in projective space, it's not the compactification of Cartesian bundle, because it's compactified in all direction, and in this case I get again differential operators in the algebra, which is differential operators of AD, but natural filtration, which I obtained here, it's called Bernstein filtration, yeah, yeah, so what actually we see here is that we can have different compactifications, giving different filtration on the same algebras, like for this affine space we can compactify, first it's kind of horizontally, interpreter is Cartesian bundle to AD, which is Y, we compactify Y horizontally, and we get one sort of configuration, one sort of filtration, or we have another configuration, another filtration of the algebra, and the same story in the second case, because one can use different toward compactification, gives different filtration, because the basis is labeled by points in Z2, in this quantum algebra, and we consider multiples of various convex polygons, it will give different filtration on the same algebra, in the third case you have only one, there's really no choice, it's just unique filtration on Jalbra, you don't have this freedom, and I want to explain, that was kind of original motivation for the conjecture, which I later made with Alexei Belov-Kanell, different ideas, that атомофизм of algebra of differential operators of AM is naturalized amorphic to athnomorphisms of affine space, is a standard symplectic form, so polynomial symplectomorphisms, so this is conjecture, which is explained, it's almost proven by now, so what was my original motivation, before I just collaborated with Alexei Belov-Kanell, it was before this, I have already the idea, on different reasons, the rough idea is the following, in all this story with compactification, it's not really clear, from couple I put the conditions that this compactification is smooth, and divisor is normal crossing smoothly, it seems to be too strong conditions, from couple, definitely one can allow singularities in good dimensions, serious stuff like this, and it hasn't been studied, but guess that compactification would be slightly non-smooth, sir? It's not even necessary, I see for example, if divisor is arbitrary singular as you like, still the claim is that the algebra will have ideal, two-sided ideal in the smooth part of this divisor, and then singularity of divisor is something for co-dimension too, so you get automatically extension of two-sided ideal everywhere by hardtox principle, yeah, so it's slightly singular, and divisor, not necessary, normal crossing, and what it will imply, suppose I have a variety and have two different Poisson compactifications, kind of, first and have two Poisson compactifications, then I can embed these things to the product, the product is still Poisson variety, and take closure of this thing, of the image, yeah, this guy will be singular, yeah, that's why I want to be singular, it's called this x people prime, but it's kind of mapped to x prime and x double prime, and this looks like kind of blow-ups at infinity, roughly, some blow-ups, in fact, after break I will speak about more detailed examples of blow-ups, yeah, it looks at the blow-ups, survive to this quantization world as well, and the algebra which we obtained, like functions which having poles are part of our devices will not change, so these things will identify quantized algebras, so the guess is that this compactification is irrelevant, so the algebra will be canonically defined, and this is kind of manifestation of this fact, it's canonically defined, yeah, so that was the original motivation, yeah, and then with Bill of Connell we have different approach using prime numbers, which gives essentially to the proof, but this proof by dot, it proves not these things, but marita after equivalence, we get some bimodules, kind of you lose algebra, you keep categories, but lose algebra, and this picture says that you really keep algebra canonically, so the big conjecture is if x omega is compactifiable, get canonical algebra, analytically, depending on each bar, I just, I forgot to say this one word about analytic dependence, in the third example, one can guess what is dependence for canonical family of this algebras, algebras have two parameters, and parameters are elliptic curve, abstract elliptic curve, and the shift of this elliptic curve, or if we had zero some element of this elliptic curve, this Poisson structure give elliptic curve plus a vector field, and then you apply by time one of each bar, you get certain shifts will be entire map, it will be like elliptic exponential map, it will be entire function of each bar, okay, so I think now I will take small break about 5 minutes, now I will kind of change the topic, but just to summarize, I will discuss formal and non-formal deformation quantizations, these are two important ingredients in my program, there will be certain ingredient Foucaille categories, which I will explain maybe in first lecture, but right now I will not go to this Foucaille category, I will explain some various versions of Riemann-Hilbert correspondence, and it will be kind of laboratory to see how all this general story works, so I have Riemann-Hilbert correspondence, and essentially 6 is most clean in dimension 1, and I will start with some really basic things, so suppose y is a curve, algebraic curve of complex numbers, and I will not put here each bar parameter at all, so if I think parameter h bar is equal to 1, that's interesting in dy-modules, and d-modules contains this nice part, which is written for Riemann-Hilbert correspondence, so it's holonomic d-modules, and in the case of dimension 1, these holonomic d-modules are essentially of two types, either you get maybe some y0, which is y-minus some finite set points, and on y0 you have algebraic vector bundle plus connection, which can have regular regular singularities, and then you extend maybe to your right here, or second, I think you get a point in y, and there is some nice d-modules called delta y, it's a d-module concentrated, concentrated point y, and I recall you how it looked like, it's, if you have local coordinate t, for example, and you get point 0, then I consider, this is definitely detection of du, dt and t, on this two space, consider quotient module, and it has basis t inverse t minus 2 minus 3, and then multiplication by t acts like this, multiplication du dt gives opposite operator, and it's called delta function, because you look like delta function, it's its derivative in one variable, it will be the same d-module, okay, so there are two basic, I think, yes, so essentially vector bundles in this delta what do you say, PT trivial, and kind of this standard story about d-modules, first we want to study formal puncture disk, so the algebra function will be Laurent Philanevelle, and here delta, because it's puncture, it has no actual pointers, there will be no delta functions here, so what we are interested in vector bundles, which is the same as three finite rank CFX modules plus connection, and connection is operator, delta from module times operator satisfying labnitz, or if you realize you get just meromorphic connection, yeah, so this is some story about this connection, I want to describe in some details and see some geometry, which people usually do not describe here, first there's a class of things called regular, regular singularity, and the definition is the following, there are various definitions that M contains, the set of elements of M, said that if you apply operator X due to X, M, get some square dimensional, it is dense in M, generalize again functions, dense in M, and it's equivalent to the following thing, that there exists M0 in M, which will be three CFX module, or finite rank, said that M is equal to M0, and M0 is invariant under the same operator X dx, I think it's very easy to describe this regularity module, you can see this dense set of vector, when you get finite kind of like Eigen spaces of, generalize Eigen spaces of X over dx, and X over dx denote by lambda, so you get M finite is C lambda module, and X, if you go to this variable lambda, will be shift operator, and now equivalent under Z action, this C of lambda module will be some of torsion module, and it will be kind of infinite sum of finite dimensional modules, and how it looks like, we have lambda plane, finite dimensional module of C of lambda is some of indecomposable, indecomposable either kind of finite jets of functions some point, so if you get consider the support of this M, you get some finite collection, you get finite collection of arithmetic progressions, the support of F is finite union of Z progressions, arithmetic progressions step 1, and here you get modules of finite lengths, you can find dimensional vector space when operator act with generalized Eigen value lambda, so that's description, and in the composable regular connections over this C of X, I want one correspondence to pairs lambda, which belongs C up to C by Z, and some integer, and the module is actually has a following form X lambda times C of X plus X lambda log X times C of X, you can see this vector space, this is a sobiosection action of operators X of dx, if you go to base of monomials, you get exactly this picture and Laurent series, yeah, so you get the thing and lambda is in C mod Z, it works for any field of cryptistic zero, there's nothing, let's algebraic locals field of cryptic zero, but all complex numbers, one can say that you have, you can use exponential map by I lambda, yeah, so you kind of put these things on a cylinder by exponential map, and you get coherent shift, it gets from finitely many points, and it says that it thinks it's the same as coherent shifts, this finite support on C star, or the same as finite dimensional representation of group Z, which is fundamental group of a circle, yeah, so, yeah, so it's all equivalent ways to see the same category of regular connections regular singularities, now about irregular singularities, and there is this classical result, something called Ikehara theorem, it describes connections is irregular singularities, we don't have M zeroes, this abelian category of connections over these things, is actually infinite direct sum of some blocks, and each blocks isomorphic to this regular singularities, this is a morphic to representations of pi1 of S1, which is the same story, but blocks are labeled by some kind of irregular data, and irregular data is the following infinite set, you can see the C of X, you can see the algebraic closure, and which is what, it's a union for N goes to infinity, C over N bar, and this is called piezo series, when you get rational exponents, it contains kind of ring of integer, it's still local field, then you can see the quotient abelian group, so it means it will be serious polynomials with negative powers finite sums, and lambda belongs to rational numbers since strictly negative, C lambda X to power lambda, and consider quotient orbits by action of Gala group of this guy, which is actually Z hat, and the generator kind of X to power, topological generator will be exponent of 2 pi i. So, it means that you can see the sixth model action of this group, and how to say it more concretely, if you get any such expression, you get a kind of minimal N, N, set it all lambda, C lambda equal to 0, means that lambda belongs to 1, 0 and Z, it will be least common multiple of denominators of exponent appearing here. So, write this some of C, some of C, where e ai i, negative integers, and C i an are common 0, it now graded common devices of ai is equal to 1, you cannot, the sixth energy mod out by action of this i square root of minus 1. So, this is description of flat sections? Not flat sections, no, I didn't speak about flat sections. Now, I say what is a regular data, I said it's abstract notion, I didn't speak about it, it's not flat section. But in this logarithmic case it's flat sections, right? No, no, no, it has nothing to do, it's regular singularity, it corresponds to this sum of C lambda, lambda is equal to 0, it's kind of the most degenerate case. And what is the corresponding block? It is the following, you go to coordinate text to power 1. So, this quotient that you wrote, this is the index set? Index set, index set, yeah, it's not a group, it's a set of orbits of the group by, yeah, if you fix one of this guy, we get, let's introduce variable w, so we get we get n-fold covering of punctured disk, where will w, and what we do here is, we consider the following, we consider the exponent of sum of C i n w to power a i, negative numbers will be something like exponent of whatever, x to minus 1 half plus blah, blah, blah. But we consider this guy, tensing by regular connection in w, we get demodule upstairs and take direct image. Yeah, this will be block corresponding to each regular term. Okay, so that's the theorem and when I start to learn the subject, I kind of, I never really learned the proof, I kind of was eager to believe, yeah, it's some technical results by induction, blah, blah, blah. But there is actually interesting algebraic geometry behind this statement, which was not clarified at the time. What is geometry behind this? Just geometry of this, of this decomposition, block decomposition. It should go kind of, maybe I first consider kind of classical analog. Let's consider formal spectrum of at zero points, so we're not Laurent series, but Taylor series, and I can see the Quartangian bundle here, this vertical coordinate y. And when I consider Quartangian bundle, the algebra functions will be t of x, y will be vertical coordinate. It's to form. Now, now I compactify these things at infinity. Yeah, so let's try to imagine what does it mean to compactuate infinity. We get not a fine variety, not a fine formal scheme. We get formal neighborhood, it will be kind of like p1 multiplied by, in y-direction multiplied by spectrum of c of x. And when we compactify, we can cover by kind of two, in a sense, open spaces in the intersection. We have open part, which, we get this algebra. We have some piece at infinity, which will be c of x y inverse, formal power series in two variable at infinity. And the intersection will be a spectrum of what I think Laurent series in actually was the wrong direction here. Yeah, so I get through rings describing this complement in this completion to the section formal neighborhood of this word, dormant infinity of the section and of formal equal to this point and what would be intersection. And if you want to describe the shifts, you can see the models of this guy, models of this guy with identifications of localization of this guy. Yeah, so that's some gadget and what we have here, we get, it's kind of small, but we get a device here, where to form has pole of order two. So it has pole of order two. And now we want to make blow-ups at this point. And let's consider Poisson blow-ups. If you have a device, where the form has pole of order two, if you make a blow-up, then you get, you add some exceptional devices. When the form will have pole of order one, okay. Now we can make this point. Then we can make blow-up again this point. We get a new device here, where the form will have pole of order two again. What are the rules? If you have just two numbers, let's say A and B. And form has pole of order A and B in these two x's. For example, if it will be some coordinates, x1, x2. I say that omega is something like dx1 dx2 divided by x2 to power A by x1 to power B. If you make blow-up at this point, then we get things of order A plus B minus 1. Yeah, they're not intersecting anymore. Yeah, and then we can continue the game. One can make blow-up maybe at this point of order one. You get again some device of pole of order one. Or you can blow-up at this point. You get 1, 2, 3, 2. They can make blow-up here. It gets pole of order 2. Yeah, so it can make many, many trees. It gets... Temporarily, let's not allow to blow-up at points, where at smooth points of devices of pole, where order of pole is 1. In principle, one can make blow-up at the point of order one. And what I get, I get a new device and I get pole for the zero. So it means you can make a larger symplectic leaf. And I want to keep my symplectic leaf the same. Yeah, yeah, so you get this tower of blown-up surfaces and the claim that all these irregular terms, this quotient set, active limits of set of devices on this blow-up things, where omega has first order pole. Yeah, that's hidden algebraic geometry in this quotient set. And there are many reasons why it's a good notion. Yeah, for example, suppose we get a germ of curve, germ of curve, which lands at y equal to infinity. It does like completely land infinity here. Yeah, this germ of curve, it intersects devices that have pole of order 2 in original picture. Then we make a blow-up. When we make a blow-up, what will happen? If we have this germ, several things can happen. As it goes here and we stop it this way. Or it goes here to intersection points. And if it goes to intersection point, we start to make blow-up, intersection will decrease. Eventually we reach the situation when it goes to device when we have pole of order 1. And we stop at this moment. Yeah, so what can achieve? Achieve that the following thing. That germ of curve in, let's say in this cotangent space, which goes to infinity. After some blobs, after this boson blobs meets smooth point of device when we have pole of order 1. Actually these devices when we have pole of order 1. Yeah, so it's really, how they correspond to this irregular terms. If you get, I didn't describe this response. If you get some of c lambda x to power lambda, lambda is rational. If you get this series, you can see the graph of df. Graph of df will be certain curve and it will pick this divisor at geometrically. There is some small detail here. For example, if you look here in this description, it works if f is not equal to zero. For if not to zero, it doesn't really goes to infinity. So it's something is slightly wrong. And actually the right way is the following. This picture was, one should a little bit modify this picture. One have this device and we have pole of order 0. It maps to this my spec of cx. We get like p1 times cx. Now we make first blow up. Yeah, it has a vertical fiber. We make a blow up, we get something of order 1. And all the rest we do at this point. But this vertical fiber after this blow up became minus 1 curve and you can contract it. And here is a place for all future. You can contract it and you get something different. It's again, this is the same place when it makes all future blobs. You make a formal scheme, which is again, kind of like y, looks again like product of pp1 times spectrum of c of x. But here to the new coordinates y may be prime or y tilde. And simplectic form is dx over x times dy tilde. So, this is the right variety. And incidentally, it's compactified logarithmic cotangent bundle to the punctured disk. To the disk, which was how I explained what should do for open variety. It's compactified in vertical direction, t, this simplectic form is dx over x times dy tilde. t, this spec c of x, logarithmic divisor at 0. Okay. And now the whole thing will get essentially the same picture. But then any curve, any section will go either to this divisor with regular similarity or to some blob irregular similarity to this point in this picture. Okay. So, it was kind of classical picture, what it has to do with differential operators. The story here is the following. We get this formal scheme, which is a thickening of p1. We have this formal scheme, but it's thickening of some of formal sub-scheme, which will be reduced schemes, which will be divisors of, maybe I'll go to reduced schemes of p1, union spec kx, multiply by this point y tilde equal to infinity. You can see, just do some more guys. And I claim that this small thing carries a shift of non-community falgebras. Actually, not a shift, I'll try to think. The claim is a following. If you make kind of some finite blob in this picture, finite kind of boson blob, one can get certain nice abelian category, or maybe, better to say, triangulated category, which maps to perfect complexes on non-reduced formal scheme, which forms schemes of zeros of boson structure. So, if we get multiplicity 1, we get just divisors multiplicity 1. If multiplicity 2, we get sincvisors derivative. So, what do we do? But we include the ones. Include the ones, yeah. What do we do for original object? Again, I cover by two kind of open in the formal sense subsets. And this I cover by kind of U1, U2 in the intersection. This triangulated category will be perfect complexes, glued from category of perfect complexes on three different algebras. From U1, you do the following. And you do models of this quantized ring, and there is no H bar here at all. From U2, you can see the perfect complexes of C. Yes, yes. Yeah, that's really bizarre rings, which enters the game. And U1 intersect in Q2, we get therefore complexes of the intersection. Then there is a kind of easy calculation that if you do blow up in classical geometry, then you also can put things in quantum algebras, and you can just go step by step. If you make blow up at one point, you go to local standard normal form, and one can check that the sincvisors point one to another in some canonical way. Then it gives a way to kind of make parallel between the question about curves, which goes to some device number with multiplicity 1, and the question for quantum representation of quantum algebra, kind of flat connections. So, the correspondence goes roughly like this. If you have a polynomial model, which is bundled with flat connection, in this case. If you have a polynomial module, you can sort something which lives on an open part of this variety morally, then you kind of extend it non-canonically way, non-canonically to some coherent object on this compactified non-commutative space. And the model is holonomic, if and only if, if this object has, if you restrict to the boundary, get pretty classical by this restriction functor, gets something with zero-dimensional support. And this is analogous to interior demodel of kind of choice of good filtration, it's in similar situation, when concrete quotangent bundle, you choose a good filtration, or in Benstein filtration also good filtration, it's a similar story. So, extent to this category and holonomic means that you get in its restriction to the classical to commutative boundary, infinity has zero-dimensional support. So, it looks like several curves, maybe at this point at infinity. And then, by the same reasons as in commutative geometry, start to make blob and want to move this support to multiplicity of one pieces. And here the story is pretty interesting. On pieces, when order of zero of pole is at least two in the intersections, get actual shift of algebras. And, but when get devices of pole of order one, you don't get a shift of algebras, shift of algebras order one, you get different geometry. I think it's what you get, this devices is canonically identified with C, it has some canonical coordinate, white has canonical coordinate. First, it has canonical one form, because two form has pole, first of the pole it has residue, which is one form. Residue of omega gives one form, which is closed in dimension one. But, but omega, in fact, is differential of form y dx origin, differential of some Louisville form, and automatically this one form will be differential of a function. So, these things will be canonically coordinate. And then the ring, which we consider here is the following. It's a, it will be ring of form of power series, that's lambda will be coordinate, canonical coordinate, which is roughly correspond to x dot dx. And the ring, which we consider is a following. We consider form of power series in kind of operator, maybe t. T is a shift operator, and consider shifts only in positive directions. Yeah, because lambda will correspond to x dot dx, and t correspond to x. So, t, commutator of t in lambda will equal to lambda. No, t, commutator of t in lambda is equal to t. I think that this is actually, this thing should generalize to higher dimensions, that we can make some make blow ups, and eventually we get certain pieces, when we get automorphism of my variety. So, what goes on? If you consider module over this algebra, module over this algebra, c of lambda and t, which is finite rank, maybe over c of t. This module, we can make, this algebra maps epimorphically, where t goes to zero with c of lambda. We get finite dimension of c of lambda, just multiply module times c of t by c, and t x goes to zero. We get finite dimension module of c of lambda. So, we get kind of like finitely many points, as a support and some important operators with the second values. But the same module, we can in about t kind of make a lattice, a different lattice, then it will have a different support at this line. So, it means that different good filtrations gives different point on this special fiber. Different good filtrations gives a different support lambda goes to some constant. Where is the parameter lambda? Where is the coordinate lambda? Lambda is vertical. It's here, for example, it's here. It will be vertical coordinate on this. Each one. Yeah, on each device it gets its own local coordinate lambda. Yeah, so that's, first it's kind of explanation of this level to routine story. I think one can make it really rigorous proof, but that explains the geometry. But also this picture gives explanation of the following classical fact. If consider on a curve, in any dimension, but on a curve, to consider two holonomic demodules, then X groups are finite dimensional. This is again kind of standard fact and proved by some induction but what is geometric reason? And yeah, you see, it looks kind of paradoxical because this quantum geometry, it's quantization of commutative geometry and consider shifts, which supported on curves and cotangent bundle. This is a non-compact variety. And X groups are, of course, intersection point, but if it's two curves, let's say coincide, then X groups are infinite dimensional. How they became finite dimensional after quantization. So the explanation is the following. Suppose you have two modules, demodules over D C modules, C is a curve, Y is a curve. And suppose both are holonomic. I want to make, to claim that X dimension of X group between 1 and 2 is finite. I want to explain why it's finite also to write kind of topological form of earlier characteristic. First of all, I have, for each of the demodules, I have, yeah, first of all, I put my curve to the compactification and consider total space of cotangent, logarithm cotangent bundle, total space and union, this particularization of the same thing. I get my Poisson surface, which could have some vertical and and the Poisson structure will have pole of order 1 at, at, at punctures at infinity and pole of order 2 to infinity. Now my demodule have some irregular singularities, maybe at infinity, maybe at finite points. And it means that I can make some bunch of blow-ups somewhere and I can construct Poisson surface, which are responsible for both modules. It's sufficiently large blow-up. This is possible for both modules, so I get global compact Poisson surface. It will have a classical boundary at infinity, boundary at infinity and for these modules I get kind of, like, make a good filtration of modules, gives extensions to objects of the corresponding non-commutative space, such as in commutative boundaries, they have finite, finite support at smooth points when I get to objects, which at classical boundary have finite supports, which is the smooth part of this pole of order 1 devices. Yeah, because we do this kind of compact guy, the x-groups of this, between these compact couplings will be finite dimensional. I don't understand the story, because this is a local system outside of the finite set, if you take the local x. No, it's a regular singularity. No, it's irregular, but outside of the finite set it's a local system, so it's a local problem, the finite one. Find it, I see. Yes, yes. If you look at x, it's a shift, and it's a global x. Yeah, no, no, I can't see the global x. Okay, but the shift, it's a… Yes, yes, no, my perspective is not to use projection to the base curve, to speak simplectically. Yeah, so I don't want to do kind of projection, yeah. Do you think, because the same argument should work for like models of quantum torus, or in other situations, yeah. Yeah, so the philosophy is the following. We have this extensions. Mi tilde. And because this thing is compactness implies its dimension of extension of this m until this doesn't infinity. But what it has to do with this x-group of original modules? This looks like a different problem, because… And I claim that inappropriate things are the same. So it's already used to kind of local question. We have this lambda and this shift operator t. Yeah, we get this, the same algebra which… When t lambda is equal to t. I get this algebra, which is not shift like at all on this lambda space. And kind of simple modules are simple this… Holonomic modules are characterized by point lambda, which is roughly… try to think it's kind of small sections. It's called m lambda, m mu, where mu is in C. And if you go to this more usual picture and t is equal to x, this modules will be x mu times c of x. Yeah, so it's close under x over the x and multiplication by x. These are simple modules. And let's calculate x-group between m mu and m mu. Now, for this m mu, we should have this algebra. To calculate the x-group, we should make a kind of resolution of module and it has two-step resolution. And resolution is multiplication by x dx minus mu on the right. Yeah, so you see that it's given to the commodulture of this following things. You can see that this x mu… you can see the commodulture of this complex. You can see the whole from algebra to module itself. And this… so x-group is given to the commodulture of this two-step complex. And this thing has a basis x mu plus n, n is integer. And only n equals 0, for n, 0 get 0 map, for n equals 0 get non-zero map. So, it's commodulture, this is c plus c, it's in one degree, it's like commodulture of s1. And this is x-group. Okay, but now one can kind of consider modules over open part. It will be… now consider m mu times rank c of x, open. Now let's calculate the same thing for open part. And we get the same story here. But complex will be… instead of this complex, we get this complex. If you interested in equation x of kind of open variety. And you see the symbedic is quasi-izomorphism, because we get pieces when commodulture do not add. Yeah, so it shows that this… going to this good filtration, it doesn't change x, at least for this guy to itself. And also it doesn't change x when, for example, a real part of lambda i is between 0 and 1. Yeah, that's what usually kind of like deline extension does. You choose… you have various way to extend your model to the complication. And usually kind of deline says, let's do extension such that this variable lambda will state some strip. There will be some representative of this shift guy. And this is really essential. So it means that the same will happen after localization. And if you don't do this extension with various strip, you, for example, get morphism from lambda mu x to mu minus 1 to x to mu, whatever. You consider x from mu mu to mu mu minus n. It will be again commod of s1, because you still get the same eigenvalue appearing here. Yeah, so it means that this kind of intricate picture, if you allow shifts by one, but if you put… pick some representative model shifts, you get exactly equivalence of categories x. So this extension to complication doesn't change homology. And I don't think these things should be finite dimensional. Yeah, so it's kind of, I don't know, very detailed understanding of finite dimensionality in a very classical situation, but I think it's useful.