 Может начать. Сейчас у вас был небольшой левтор с 2 лекциях, который я хочу объяснить. Это сравнение между Бетти и Рам. Азиоморфизм и воль-кроссинг структура в случае одной формы. Не функции, но формы. Так, что это за сетап? У меня есть обычная смуза-лжбрек-варайти из комплексных номер. У меня есть дебайсер с нормальным кроссингом в X. Больше я consider chains with boundary on D, а потом я consider closed one form in X. So, I have closed algebraic one form. И я хочу изучить бетти и рамкомолджи. И, я думаю, давайте выберем комплектификацию, в которой можно легко сказать, ажибрэк-комплектификация X-bar, которая смуза. И я сказал, что комплектификация X имеет какой-то дебайсер. Это дебайсы, которые consist of three parts. Это, опять, нормальный кроссинг дебайсер. Хоризонталога-рюфингин-вертокл, я explained it in a second, and said that the total, which is the closure of D, union with these three things, is again normal closing divisor. So, in X-bar, I have four types of boundaries, and the property as a following. Form alpha extends to kind of open part of the horizontal part. I think it's the horizontal place where 2DH open, kind of open part of the horizontal. Or equivalently, I just write that form in the same letter. Alpha will be considered as a form on X-bar minus logarithmic and vertical part. And near these things, the form is not defined. It should have the following, kind of in analytic topology. Form will look the following. Alpha will be some regular form without any poles, plus some linear combination of D log Zi, where product of Zi is equal to 0 is equation for logarithmic part of my form, of my divisor, and Zi are just some nonzero complex numbers. And plus, second part will be near vertical part, I think in terms of function. It will be product over some other variables dj to some power nj multiplied by 1 plus 1, where a product of JZJ is equal to 0 is equation for dvertical, and nj are strictly positive integer numbers. So, one can make such picture infinity. Then one can consider a coherent shift on X bar, which I think it will be vector bundle, it's omega f, which will be defined as a following. You can see the shift of forms, in which a log forms on X bar relative to all things, and vanishing on D bar. Vanishing means this complex, and such that omega, which alpha belongs to the same story, it's kind of extension of the complex, which alpha. So, this is nice coherent shift, and one can prove the theorem that it's essentially close subar result for case when device is empty, or when there's no maybe logarithmic, when you have only vertical part. In this case, if you consider a hypercomology of X bar with a risk etapology with this shift omega f, and with differential H bar plus H alpha, it's closed under both differentials, then it does not jump at all, for all H bar. And can one also put constant in front of the alpha, or not? No, no. If there's no logarithmic part, one can also put constant along the alpha, and also get it to zero. If there's no logarithmic part, it's an additional story, which I will not talk now. If the block is empty, then one can put two constants. Which are cannot be simultaneous? Could be simultaneous equal to zero. Could be. Yes, yes, yes. Equal to rank of homology of X bar with omega f as well. One can put both constants to be zero. If the logarithmic part is trivial. Yes, it's the sixth generation, which happens. But I will not talk about this story, and I will not explain you the proof of the theorem. Or maybe I will explain in some simple case what the main idea is. Let's consider this really simple case, when d total is empty. You have just closed variety and closed homomorphic one form. Why this rank of homology doesn't jump? Or maybe it should be a little bit... No, maybe I removed it. For case of one form. Let's explain me this situation. What we calculate? In this case, we have a line bundle, which is o, with flat connection d plus alpha. Let's put h bar equal to 1. We have a flat connection on line bundle. And here we calculate this homology of this line bundle, homology of x with flat connection using the RAM model. And then there is a well-known saying, that a story kind of... I should say it by Simpson and others. Yes, that there exists unique up to constant, because it's simple thing, harmonic metric on this line bundle with respect to flat connection. And then you can identify the rank homology with Higgs homology. So it means that it gives you some line bundle, another line bundle, homomorphic line bundle, e with Higgs field, phi, which belongs to sections of x and anamorphism e times one form. And then you consider different homology, consider Dalbo homology, which will be hyper-comology of x as analytic or risky. Then consider e tensoring forms with differential acting by Higgs field. And then the rank homology are equal to this thing. The reason is essentially if you consider d bar plus phi, and here you get d plus alpha, operator. And take commutant, because we get the same identity as the usual Hodge theory. That's the proof. No, that's the general fact for harmonic metric. But what is harmonic metric here for this local system? Harmonic metric is very simple. It's constant metric. Norm of one at any point is equal one. It's constant metric in this trivialization. Why it's harmonic? When consider flat trivialization, we will locally write a as differential of some function f locally. And then the metric in this trivialization matrix will be matrix exponent of f squared. Even consider logarithm of this guy. It's kind of one by one matrix, Hermitian matrix, depending on point and manifold, we can see the logarithm of these things, take d bar, we get definitely zero, so it's harmonic. No. So what is the plus in the commutator? Anticommutator. What is the condition of harmonic? There is some notion of harmonicity for matrix on flat bundles. And particularly for rank one bundle it means that if consider flat sections, consider logarithm of norm, you get bloody harmonic function. But it's also for arbitrary rank, for local system of arbitrary rank on K and manifold one can see some notion of harmonic matrix. And then there is a general thing that deramk homology, local system I equal, one can associate it by some formula, some holomorphic bundle plus Higgs field. Another bundle, yeah. You have another bundle and such a kind of Higgs homology equal to deramk homology because of this identity for Laplacians. But the identity compares things with different bundles. You can identify the sinfinity bundles. Can I? Yeah. Which shows that homology are the same, yeah. It's kind of generalization of usual Hodge theory. Yes, yes, yeah. It's a compact manifold here. But here what happens? So we get this another bundle and some Higgs field. Then we can make calculations and first of all Higgs field because it's a line bundle. It's the same as homomorphic one form because the endomorphism line bundle is constant and phi is, I think it's equal to maybe plus-minus, maybe some small constant, but form alpha maybe divided by 2, something like this, yeah. So it's proportion to form alpha. Line bundle is not trivial, but line bundle one can also identify because it will be line bundle of rank 0 or degree 0, so it will have unitary connections. U1 connection, flat U1 connection. And it will be a kind of homomorphic bundle associated with flat U1 connection. In flat U1 connection you take something like items maybe imaginary part of alpha and maybe again divided by 2. You get closed purely imaginary one form up to some constant. So you get some bundle with some flat connection, but the bundle is, you see it's not trivial. How we proceed. When we calculate this homology it will be in unitary apologies, it will be concentrated. The complexity will be a cyclic outside of 0s of phi locally. So it's a direct sum of contribution of neighborhoods of 0s of alpha. Call Z. It will be some union of some connected components. And this Dalbo-comologio take direct sum over B. Consider hyper-comologio of some neighborhood. What did you have in the last line? No, in the last line of the homomorphic and then somewhere of the loaded. Homomorphic bundle associated with a flat U1 connection given by trivial bundle with this carriage connection form. So now you have considered neighborhood of this ZB and you repeat the same story, you get E and some X and Y. But my form is on ZB locally is exact. It's vanishes on ZB. It can be represented as differential near ZB. It's the same with imaginary part. So you see that near ZB this bundle this flat connection this flat connection is trivial monogram is trivial. So E is identified with O this trivial bundle in some canonical way. So you see that you can replace by usual story and you get multiple deductions as forms. So you just multiply by alpha. Okay, so that's Excuse me, for the general case you also use the harmonic metric? Yeah, the same harmonic metric, but then one should use Mochizuki's analysis how one should do with irregular singularities and so on. So it's That's the thing. So you get this homology do not jump for all h-bar. And to claim it's not the RAM homology, which we are interested in. It's slightly different from when we calculate integrals we integrate something like the RAM homology depending on h-bar and alpha. So it will be something like hyper-comology of X without compactifications and you can see the forms with HD plus alpha. You can see the open part and you can see the homologon open part. You compare this close part to take dark image and you see that something will go wrong but only for problem it's different only for special values of h-bar, which are union of i gets ci. ci are this nonzero numbers and I consider ci divided by positive integers for this for such values of plant constant you don't get isomorphism gets something some small problem I want to compare this RAM homology with beta homology and for beta I just essentially already wrote you what I need I have slightly deeper yeah it's not isomorphic to the RAM homology of open variety maybe put minus maybe put things like this forms vanishing on D that's what we're interested in integrals and this is not isomorphic for this exceptional value of h-bar ok, so now we write the union of alpha is ok, just repeat z is zero of alpha so you are in the general case or we simplify assumption no, no, no in general case so now consider zero of alpha so this theorem doesn't apply sorry, yeah, it's sorry yeah, because in Sabar there's no logarithmic divisors yeah, that's yeah, it's yeah, it's yeah, it's yeah, it's zero of alpha it is a compact subset because of our picture of what in x-bar minus d log d vertical it's really compact and then we can write as a union of connected components this would be and as I just already told you near each for any b near zb in analytic topologies there exists unique function fb homomorphic function so that fb restricted maybe to reduce divisor is zero to reduce sub-scheme is zero and dfb is equal to alpha yeah, so you will locally represent as a critical point of some canonical absolutely canonical function and then I define beta realization which depends on minus this component of critical plane constant which will be yeah, I'll write you formal definition you get r gamma of zb and then you take yeah, if you don't know this notation then you can help you get shift of vanishing cycles you divide by function of beta divided by h-bar and then of what yeah, so you can see the yeah so first you can see the constant shift on x- all divisor restricted to the neighborhood then you do what you extend by factorial to d yeah, so it's responsible for chains with boundary or commodity of pair but then you should extend to steal some larger larger story x embedded to x-bar with deloc and dvertical now stand by star yeah, that's commodity yeah, so it's roughly if you don't have the things with infinity you can see commodity with shift of vanishing cycles then you take care of infinity and then I'll say conjecture, because I didn't check should be theorem I didn't check all details is the following that one should have comparison isomorphism between beta and deramkomology namely for h-bar which doesn't lie to some countable union of rays in c namely rays again by conditions that argument of h-bar is equal to argument of integral of alpha of some chain where gamma is a pass in sorry, it just was maybe little bit yeah, I was a little bit careless here just before going on I can say not just 0s of alpha but 0s of alpha and of all its restrictions to intersection of components of d-bar and d-horizontal sorry, yeah, because it has yeah, I have many, many strata and I want to assist vanishing all strata as well, yeah the same now I have some kind of bad direction for alpha for h-bar namely I consider pass in x-bar minus d-log and d-vertical connecting this ends in this z some bad points and consider arguments of such things when integrals are not 0 so get countable many how many get countable many numbers which could be everywhere dense on the plane then the plane that's for h-bar which is kind of not bad you get canonical isomorphism comparison isomorphism between what this hyper-comology of exactly these things which maybe denotes kind of h, kind of maybe modified from h-bar of x-alpha just like this this slightly modified thing so it's drum version will be isomorphic to the direct sum of all connected components of this guys beta maybe g-bar by c but here this will be some interesting point here I want to put two different plant constants here maybe I put here kind of h-bar what is h-modified h-modified h-modified is this definition this hyper-comology with this nice complex you see because the drum-comology jump and if you don't do modified for this bad values of h-bar but this commolded do not jump in h-bar they all stay the same because it looks only local contribution for critical points so I get these things and what is h-prime it should be kind of holomorphic in h-prime and h-prime should belong to the certain disk kind of open disk whose zero is in direction h-bar so it looks maybe I can say that if you go to inverse variable so maybe we can first write formulas it means that argument for h-bar minus argument so no statement is not estimate and hidden construction that what is hidden construction integration over left shift symbols for all page-primes which belong to something you have isomorphism yes one can have some isomorphism which is holomorphism and h-bar should be sufficiently close argument and real part of h-bar h-prime inverse times h-bar bigger than some constant so maybe one can go better to inverse variables h-prime inverse belongs to half plane which goes in direction h-bar inverse so what are you given h I have two numbers I get h is not on not on bad rays but h-bar is some small deformation of h-bar small deformation and h-bar should be sufficiently small h-prime h-prime kind of an inverse if make inversion from this circle will get half plane inverse of numbers belonging to the circle whose boundary contains 0 get half plane yeah so it's pretty abstract story why do you need h-bar page-prime because the integrals will still converge I want to say that integrals which I defined have analytic continuation to some efficient domains yeah, actually what is really nice in this formulation that this bad exceptional sets are completely disappear from the formulation I can modify the RAMc homology a little bit using complication and global beta homology are not equal to this direct sum exactly with the same exception so this the whole thing is kind of cancel and one have very clean result and now we just show you kind of basic example one can really go to the very end and understand everything so x is c star this coordinate so x, small x form alpha will be 1 over x minus 1 times dx and this boundary divisor will be 0 so we are interested in this connection and of course the complication have only one complication cp1 and one point will be the logarithmic will be 0 because form has a logarithmic pulse at 0 and has a high-order pulse infinity and maybe in the snow horizontal pulse so we get this complication and generator of this homology of each modified whatever RAM bar x alpha is form dx over x that's called volume form it's easy to see that it's generator so set of zeros of my forms is just one point it has point x equal to 1 form vanish exactly to one point so in this case Camelja one dimensional and if each bar is positive real number then generator of it's kind of more point and you get only one left shift symbol and left shift symbol will be positive x a real part of real line sitting in x and on this line it contains it's some things which goes to point one and on this symbol we'll get a function of one of x which is equal to log x plus one minus x what is nice about this function this function at this point one which is belongs to z is equal to zero and differential of my form is form alpha functions form alpha so it's kind of normalize a function to have critical value zero at my point and now what I integrate just by definition I get e again modified in different sense in my sense of first lecture maybe we get kind of e plus of each bar will be kind of general prescription of the following we take a square root of 2 page 2 pi h bar and one is one is dimension of my manifold so it's this universal formula then we integrate from zero to plus infinity exponent of one of h bar f one of x function what the graph will be of this function we have this x from zero one to infinity and function f will have the graph like this that will be good go to minus infinity in both sense so integrate these things, multiply by volume form this is exactly volume form so you have this integral and then you can calculate it very easily i gamma s divided by square root of 2 pi s to power s minus one half exponent minus s and is equal to one plus some asymptotic series in s sorry s is denoted by h bar inverse so you get kind of stealing formula and so this function it appears in comparison isomorphism between betti and deram and what is nice it's kind of compatible with this picture this function extends to invertible function in c minus negative in h bar plane so you get function defined only for positive real number but by this comparison isomorphism should go to some bigger disk but in fact the disk will be half plane and then when I start to rotate it because my walls will be only here I get exactly the complement to negative number so I get automatically explanation why this thing is invertible from this from this picture but then one can go to have different domains now consider function e negative of h bar which is one you now consider h bar negative number again draw a symbol and calculate integral and if you calculate integral you get things related to these things it's function on complementary picture and now what are bad directions bad directions is imaginary axis asomorphism fails because you integrate alpha consider periods of my form alpha and you have only one interesting integral which is 2 pi i because residue is one so this is kind of bad directions kind of stocks race not stocks, we can imagine stocks race here I don't have things it's e r plus and e r minus and I have one function which defines on a one half what extends to larger things another function to another half another thing we consider jump if h bar belongs to e r plus then what we see that so we use the rule r plus and i i times r plus sorry if h bar belong to these things then you have one function from left hand side and as from right hand side they differ by the following things that something like e minus is equal to a plus minus of 2 pi i over h bar divided by i plus and if h bar belongs to minus i negative part of imaginary axis then i minus is equal to i plus minus exponent of minus 2 pi i by i plus and here again one can understand everything so first we get this kind of minus sign and this minus sign means the following kind of axis c star get point x is a cylinder topologically it has one zero from my point and there exists essentially one loop connecting with this other some gradient line and this multiplicity this thing means that n plus minus gamma zero is equal to minus one so we get one loop but sign so it's integer number we cannot see it's informal it's minus one but it's integer and what is exponent exponent it's integral 2 pi i what is it's 2 pi i or minus 2 pi i 2 pi i is integral of a gamma zero of alpha and minus 2 pi i is integral of minus gamma zero of alpha so one can understand all terms in this formula through this comparison isomorphism so if you consider this so my original left symbol was something like this but then start to rotate kind of you rotate the argument of h bar start to rotate and you get the same things plus union of the circle let's express in this formula so all the story explains nicely all properties of gamma functions which we know from general principles ok now I note full several simplifying notations forget about these divisors ignore that they exist just to make notation easier so what happens in general I have closed one form on some variety X then for each nonzero plan constant I have two isomorphic things I have h modified the RAM x bar, x alpha and it's isomorphic to this other stuff some of beta you get this comparison but this thing it's a holomorphic bundle in h bar so you can see but it doesn't have any natural flat connection no natural connection unlike for the case of function we can identify but here we get local systems if we go to beta description local systems different monodromes there's no reasons to identify homology they have just the same run but they are not asomorphic flat connection and similarly if we get the families we consider some holomorphic family x depending on some parameter u alpha depending on parameter u where u belongs to some parameter space then you get holomorphic bundle on c star h bar and your parameter space no flat connection again but still one can make a situation when you get some part of flat connection I say that my family is isomorphic if on a total space if you choose some global one form it's kind of total space of a family which fiber to u with fibers of u with u b x u get form which is closed globally and such that alpha u for any u is restriction of alpha global to the fiber in this case if you fix h bar we get kind of the same periods for one form we get isomandromic connection and so we should expect Gauss-Mannin connection in direction of u so we get a family of flat connections on parameter space which holomorphically depends on h bar so in h bar there will be still no connections so connection will be space only in this direction sorry okay this is a definition definition of isomandromic isomandromic is actually not a property it's a data it's a choice of global one form which is closed on the total family with restriction to its fibers gives our forms so the Gauss-Mannin connection is that on this on this homology depending on parameters yeah but then we have this comparison isomorphism in wall crossing structure it's some describes jumps in comparison isomorphism because I have some bad h bar when I don't have this isomorphism so it's defined on open then subset and this isomorphism jumps and in particular so isomorphism between one between sum of beta in pi not of z u zu is critical points of set and consider again this homology of this zb to u h bar c identified maybe in two different ways go through two sides of the wall with h deram just modified what the mutations I have it's from two sides of the wall here and then I get automorphism of the space of deram space and the whole thing is described using formalism of wall crossing structure which I explained last time we need some graded Lie algebra local system of graded Lie algebras on again on my parameter space so the Lie algebra will be depend on h bar and u and graded and grading lattice depend only on u it will not depend on h bar what will be the grading lattice yeah that's kind of tricky story it will be mixture of two things maybe for better notation zu will be zeros of r for u I want to put up index because low index denoted by some connected component so the grading is the following if I have a pass just before going on I have this thing which is m denoted by a number of connected components it will be something is amorphic to zm-1 this thing and it's root lattice of a series a a-1 it's like explained for the case of function you get yeah it's kind of it encodes two endpoints of the pass yeah so you have pass connecting two components you get some vectors either if connect components with b1 and b2 pass then should be vector or if it contains with itself it will be zero vector and gamma zero u it should be homology class of the pass I forget the endpoints can see the homology class of the pass yeah and yeah it looks a little bit excessive grading because if you know homology class of the pass then consider boundary will be h0 get elements here but I want to kind of to keep it separately because it will include the case of form and it will be work nicer with this situation so the central charge h bar will be 1 over h bar of the u the u maps from it's kind of trivial will be trivial on on root lattice and given by integral and on this gamma zero u is given by integral of form alpha which is well defined okay yeah so you get Lie algebra with graded by it's actually local system again some open part of u when a number of connected components stays the same and what will be this Lie algebra the corresponding this Lie algebra will be g h bar u will be block diagonal matrices kind of endomorphisms of direct sum of this h beta and correction numbers so it's kind of matrix algebra we can see there's a block matrices because we have direct sum decomposition to answer it by series in maybe it's kind of mod out by torsion here as well this group I want to mod out by torsion kind of group ring of this of this lattice yeah so this Lie algebra it's matrix valued functions on a multi-dimensional torus obviously graded because each block belongs to some grading in root lattice and by some monomial here yeah so you get this yeah so you get one can repeat this story what is wall crossing structure namely walls will be points in this parameter space when the center charge of some element of graded lattice is positive real number and then you should associate elements sitting in case of subgroup on this wall so it's like certificate condition but here this big difference we want to explain in last talk last talk Lie algebra was finite dimensional now it's infinite dimensional but each graded components is finite dimensional but algebra is infinite dimensional I lost a little bit what was the role of this definition capital ZUH bar what does it appear in this capital what something which is like a functional ah it's a functional ZUH bar I have a local system of Lie algebra and I get a local system of homomorphism of lattices to C that's it I just get grading lattice to C ah the grading lattice maps to C in some continuous way depending on yes and how do you grade the endomorphism of the direct sum endomorphism of graded sum A minus minus lattice from one guy to another there will be difference between two base vectors so it's kind of such things but you see this algebra is infinite dimensional and one should take some care about it so this wall crossing I defined some properties of wall crossing structure on specific kind of slice I consider C star multiplied by some specific point if I restrict to wall crossing structure what I get I get maybe infinitely many rays where I get should put my anterior transformations and should get elements of some Lie algebras so wall crossing on such things ah ah just before going on if considerable crossing on the C star times I think it's given by it's just collection it's a collection of elements of alpha j h bar is 0 of graded components of various components it's just a collection of these elements of all possible elements in my grading lattice such that z u h z u whatever of gamma is not equal to 0 yeah it's because in this situation I don't have any associativity property on this kind of copy of r2-0 yeah my walls will be infinitely many rays and this rays will be arguments of such numbers and there's no constraints whatsoever in these things definition at wall crossing structure on this C star times u0 has support property if if there exists let's say quadratic form on what? on my grading lattice times r such that restriction of quadratic form to the kernel of my map which is typically co-dimension to subspace is strictly negative definite for any gamma such that alpha gamma is a gamma is not 0 the quadratic form of gamma is strictly positive u0 u0 yeah let's just make a break so what like imagine if grading lattice is rank 3 rank or maybe gamma 0 because in part it's really relevant rank of gamma u is 3 yeah so you get map from z3 to C and so you can imagine like integer points in r3 projects to plane and this support property means roughly the following you get 0.0 and you consider some kind of cone here which surrounds the kernel of projection z and the support of my my collection of elements are such elements of lattice which lies in this circle part of the complement is it very small gamma or oh infinitely many for all yeah so it lies in kind of this discrete set yeah why this condition yeah it's in various situations because this wall crossing story appears to various stuff that one can use using some differential geometry for example here imagine like access some Keller manifold interesting gradient lines for interesting gradient lines of of real part of one more gauge bar of you consider make its vector field real vector field consider gradient lines and then as a claim that if you bound the length of such gradient line you can bound the integral of any form any closed form because of some compactness and eventually translate you see that you cannot if you bound the length you have only finally many choices and get some uniform picture like this yeah so it's some very root some estimate holds in real life and yeah so it's one condition and the second definition wall crossing structure of the same story is analytic if it has support property and moreover norm of a gamma which is a matrix essentially yeah it's monomial it's a matrix is bounded by c1 times exponent of c2 times norm of gamma or you can replace by support property of norm of zero gamma because support property means essentially one bounds essentially equivalent that's this bound it's more difficult it's again should fall from differential geometry it's roughly say that number of gradient lines should grow exponentially that's I don't know how to prove directly but it's kind of some kind of back door one can argue that it holds and maybe if we can break it just want to go to essential part yeah so it's looks like stupid estimates support and growth that claims the following analytic wall crossing structures on the same thing can be described in kind of coordinate free language in the following ways it can be defined in kind of infinitely many ways depending on some discrete choice by the following data so we get matrix valued functions on c star to power n so let's invert c star to power n on which this function is coordinate y to some toric variety and suppose the toric variety contains a chain of cp1 kind of chain of cp1 which are toric orbits so it will be fixed points and it will be one dimensional toric orbits and then you get some chain which called c it will be some singular curve we take some neighborhood analytic neighborhood neighborhood of this chain and I consider yeah it will be a bit sloppy I consider vector bundle maybe it contains kind of have boundary divisor it still has some divisor with normal crossing around and what I consider is a vector bundle this uc is analytic neighborhood and consider vector bundle what's called e on uc essentially it relies on this divisor whose restriction d infinity it's a complement to the open orbit said that restriction to this d infinity is identified with some local system which we know a priori because it's a homotopy equivalent fundamental group of neighborhood d so we should describe what is monodrome it's we have this beta homology depending on the alpha on connected components but also depends on h bar and h bar they form a local system so this local system should be homotopically c star as I remember 2s1 neighborhood of this thing has also fundamental group so what is this analytic wall crossing structure which is collection of these elements which supports property and the growth is the same as homomorphic bundle on the neighborhood of this chain of cp1 and some toric variety with some trivialization yeah so yeah so it's kind of geometrization of this notion but this vector bundle should be identified with local system which come from this there is some story yeah no no wall crossing structure is this the algebra I have this local system of the algebras and yeah so it's all concrete story maybe I think it's time to make some small break yeah if if consider this intersection device I have my the infinity intersection with uc homotopy equivalent to my curve c which is just union of spheres kind of sphere, union sphere yeah okay and it's up to 2 cells it's homotopy equivalent to s1 which is homotopy equivalent to c star and on c star we get local system yeah up to 2 cells but maybe more complicated no no no no no I said that it's chain of toric cp1 so I got I have a wheel closed chain yeah what's the chain yeah yeah yeah yeah so it's description of local system is a bit complicated but why all this happens let's just give some root explanation so I have this a-1 lattice but have a map from whatever z u is integral of form to c which is r2 I have a map from lattice which is kind of zn to r2 I have an additive map then consider dual maps from r2 star which is again kind of r2 to what to zn dual times r which is rn I have a dual map and this can be sort of assume that's embedding so what happens in dual space I have a real plane for two sitting in rn now what I do I cover my plane also says lattice structure I cover my plane by convex polyhedral cones I say that it's r2 which lacks in the union of convex rational cones and there are some conditions but roughly the picture that it should be kind of you divide by many small sectors and you cover by small cones round and if you get such a situation you have you get a fan and if you get a fan you get a toric variety this will be my toric variety y it depends on the choice of my cone with rational cone this toric variety will contain naturally a chain of p1 because this top-degree dimensional cones will correspond to points in toric to zero dimensional orbits and co-dimensional one faces correspond to cp1 so you get automatically at infinity cp1 ok but now you do the following but this view is not real some more complicated no no no no fixed points only correspond to I don't have any it will not complete toric variety it will be some quasi affine whatever because cp1 corresponds to a race no cp1 correspond to co-dimensional one faces in toric varieties you get a fan open cones correspond to fixed points in kind of dimensions opposite and cp1 correspond to co-dimensional one faces so they touch each other and it contains union of cp1 no no no cp1 correspond maybe just other cp1 will be for each point you get two cp1s right petal I have really two dimensional picture around just draw you get something like this correspond somehow to the intersections of two copies of cp1 like if you have an edge yeah it will correspond to cp1 yeah it's infinity will be cp1 kind of transversely yeah so roughly speaking one can think the following c start to power n maps by logarithms of norms to rn and if you consider path in c start of n which correspond to path which goes along this whole open dimension it will converge to one point in toric variety but if you go in the middle the limit will be undetermined to the point on cp1 so it gets a so it gets a toric variety and now there is some kind of easy calculation you get this curve and you consider formal neighborhood of curve it's kind of formal scheme and consider hypercommold of this formal neighborhood with coefficients of the shift of functions which vanish on the divisor at infinity of vanish of the divisor consider hypercommold consider hypercommold commold of decisions with this a shift of ideals and the claim is 0 if i is equal to 1 and if h1 has basis kind of topological basis corresponding to integer points nonzero integer points in gamma supported on something like this this complement so you can see the kind of dual cones and take commold you get exactly h1 no other commold around maybe i just give you some simple example suppose my lattice is really take care vectors in gamma satisfying support property for certain quadratic form integer points points in gamma satisfying support property yeah so it's really easy calculation with commold i can show maybe just really stupid example suppose my lattice is z and just two dimensions so i don't really have this projection but i still need a fn and my fn will be this will be my fn in z2 just four domains so the variety corresponding variety is cp1 cross cp1 kind of zero infinity support property means that it belongs to some set which may be bounded by quadratic form yeah yeah so this quadratic form was a little bit ugly thing so there exist a constant is it something like what you have there the condition yes on quadratic form on the quadratic form what is the gamma you want a point in gamma not a product each one has a basis corresponding to set of points in gamma which are possible candidates by the support property when a gamma is not zero yeah possible okay at some point you divide by one constant no no no this is not plan constant i discuss this it is a local system on s1 which is identified which depends on maybe you divide by plan constant will be an argument plan constant will be z u of gamma just like this your vector in my gamma is my gamma is vector in my lattice we consider vectors in my lattice for which which is a support property and consider no no here i don't have any no no here it seems clear okay yeah yeah one can make simple calculation for example just want to jump between two things i have a fine so variety is cp1 i consider kind of neighborhood of this view of cp1 which is not a fine surface it's not a fine surface even neighborhood and the only functions are constant and functions vanishing on these things have only first homology if one calculate using some covering check covering essentially what happens in a lattice vector so you can consider maybe integer points in this half plane this half plane and remove it will be one part of check complex and others will be octans and then we get all on zero point here yeah so something similar happens in higher dimension of course here we have a lot of choice we can make it smaller fan here and on rule space we get a larger support and eventually we can cover support property by this thing so it will be some ambiguity says it's not unique description but there are many ways to describe in this ways and you compare two different ways by some blow ups yeah so yeah so you kind of replace and this is kind of this is the basic thing from which all follows because if consider deformation theory for this question we get vector bundle in neighborhood but trivialize somewhere but now we start to deform it because deformed by something trivial on the divisor so the matrix value function vanishing on the divisor and deformation theory is given by h1 decent level of tangent space and see that you get exactly the same data for this elements of my matrices corresponding to monomials belonging to such domain and then one can make it really globally and this convergence, this gross condition that there was some analytic property implies that it can go to form from power series to actual neighborhood 21 correspondence okay so yeah so you get some kind of bizarre analytic spaces which are neighborhood of wheels of cp1 and now if we move point and want to satisfy support property we have this associativity laws for wall crossing when things go together it's all complicated business and then associativity law for wall crossing structure it's equivalent to the following thing that this bundle stays the same you see you see some e stay locally the same but what we change we change in bednik of r2 to rn when we identify the things with this wall crossing structure but analytic object stays the same but then if you go to different domain when we can use different fields we make some blow-ups but that's roughly the picture of what happens now so all this complicated wall think are embedded to some completeness kind of nice analytic language you get holomorphic bundle in some open variety kind of local system there's identification yeah yeah now I'll just be very brief now we want to jump to infinite dimensions yeah some people were not present on last talk so but you can really help with this yeah what I explained that for some infinite dimensional varieties one can try to calculate with holomorphic conformity one can try to define all the structure without definition of what a global deramkomology and so on using just this wall crossing business and the basic example was the following I have a holomorphic symplectic manifold and I have two lagrangian complex lagrangian submanifolds again holomorphic lagrangian submanifolds and my space it will be infinite dimensional it will be space of paths and points on my Lagrangian and one form is closed one form on x infinity infinity is given to integral of two form of the paths in some obvious sense yeah that's a closed one form yeah this was some issue of orientation and eventually one should we should get get homology depending on hbar which is holomorphic in hbar cstar using this walls and if I have isomonodromy deformation one should expect a flat connection we get some parameter space whole things depend on parameter we get flat connection in u holomorphically depending on hbar okay yeah so that's what one should expect naively what is capital hbar so what we use we use vanishing cycles for intersection points and then we should use wall crossing structure coming from paths in path space kind of gradient lines which we say is holomorphic disks and all this complicated story and what is the example of isomonodromic deformation should should have should have maybe no should have a holomorphic bundle vector bundle on cstar which is constructed using the all these walls yeah which you have constructed using wall crossings should have wall crossings structures and all this business because critical points of zeros of hbar zeros of infinity are constant maps identified with its final dimensional space if we have isomonodromic deformation yeah and what is the example of isomonodromic deformation is the following if my manifold is cotangent bundle and omega is differential of leewheel form L not not exact but L1 will be L1 depending on a point will be cotangent space to the point and X and U will be X parameter space will be so we have cotangent space to X will have L0 intersect with some vertical L1X and start to move this L1 so then you should have a bundle on cstar cross U yeah I should get a bundle on cstar U on a bundle should have bundle on cstar cross X which is holomorphic here and flat here some kind of this holomorphic flow homology yeah it's all this very nice and we want to try to see how it will work and then one have a very interesting trouble because in the definition of all cross instruction consider something like gradient lines for this one form which will be holomorphic disks pseudo holomorphic disks so we are interested in general in pseudo holomorphic disk related maybe L0 and maybe L1 maybe depending on point U and we have this holomorphic disks and they are in general isolated but in one point U family get some boundary and boundary of space of disks has the disk can degenerate to sequence of two disks which is responsible in wall crossing picture to the rule if you remember something like you have kind of number of disks jump by these things you get kind of product here which correspond to product in this picture of two compositions but if you get some new phenomenon which is you never see in finite dimensions new phenomenon boundary of space of disks if you want to prove this whole associative things will not hold also contains also another type of boundaries and you could have such degenerate disks can appear from the boundary real bubbles from the boundary so it's a trouble so you get a trouble comes individually from L0 or from L1 because it means that you have holomorphic disk with the boundary on L0 it never happens in two situations for L so there's no disks with holomorphic disk with the boundary to the holomorphic disk with the boundary on L and area is strictly positive area in two situations if L is exact so in my last lecture explains the case of exact L and one form was differential of function when L is exact so it means that we express two forms differential of one form and one form on L is differential of some function M is exact yeah it means that M is exact and L is exact it's some structure it's not a property it means that M is exact and so on why the integral is 0 if consider integral of the omega of the disk is integral over d omega of any disk is integral of omega of the boundary of the disk is integral of df of the disk and 0 by stock's theorem but area is the following situation area is a real part of one over each bar of integral omega of the disk yeah so we get a contradiction for any pseudo holomorphic disk the area should be strictly positive but it's 0 yeah so it's one case but there is another case which is also funny second case when M is cotangent bundle L is a graph of closed one form then the integral is 0 but by different reasons we also get integral over disk of form omega the beginning is the same it's integral of d eta over omega over disk in the integral over eta over boundary disk it's integral to the alpha over we get projection from cotangent bundle to x itself projection of projection of dd yeah it's integral of d alpha of projection of d and this is 0 because d alpha is 0 yeah yeah so it's kind of the end is different the beginning of the argument is the same but the ending is slightly different yeah so there is a completely different story I think that for the case of one form one can go to the infinite dimension story or finite dimension story it will be the same business so I claim this is kind of things with disk and so on it will be the same as finite dimension interval in my story yeah but in general except of these two exceptions except this kind of closed one form or exact thing we should get this trouble and what is the meaning of this story yeah so what goes on yeah how to understand the wall crossing structure it looks that everything will break will be broken but it's nice explanation of what happens it's claim wall crossing structure in a larger le algebra we have this previous le algebra but we have now le larger le algebra so before we have this guy before we see the trouble we have this le algebra but after the trouble I think we should understand we have different le algebra kind of matrix new le algebra it's it's kind of old le algebra and we take semi direct product with vector fields on torus yeah this are matrix valued functions on the torus or maybe j m to power n and this are vector fields on the torus you consider also at vector fields so new gamma graded component will be something like this we take y to power gamma and you take maybe y i times du du i it will be basis of new graded components of the larger le algebra it's again graded by lattice essentially by the same lattice and yeah so it's kind of claim what is going on and I explained before you had kind of theorem how we identify wall crossing structure on c star times point as a vector bundle so before analytic wall crossing structure on c star across a point was the same as holomorphic bundle on some neighborhood over chain of cp1s plus some trivialization or identification on some divisor but now I think now one can kind of repeat the whole story this analytic wall crossing structure is largely algebra will be the same as holomorphic bundle on kind of some on some c calc not on a not a neighborhood over chain of field and toric varieties but just some complex manifold which contains the view of cp1s and have some stratification maybe some trivialization just in the first order coincide historic variety then you modify by some because vector fields give you change coordinates in your different patches and glue in new manifold yeah so it's yeah in some sense one can say that it's even before we have holomorphic bundles we have only gauge theory but now we have gravity kind of diff morphism group yeah so it's mixture of two you see modified it's a it's a germ of complex it's a complexity variety which contains view of cp1s and it's stratified like toric variety and maybe normal bundles to strata trivialized to this cp1 to keep kind of first derivative and the rest will be the same yeah so you get some kind of and if you move whole crossing structure you locally do not change this guy that's essentially what kind of discovered it was something called 4d 4d 4d 2d 4d 4d 4d 4d 4d 4d 4d 4d 4d 4d 4d 4d 4d 4d 4d 4d 4d 4d 4d 4d 4d 4d 4d 4d 4d и почему мы должны получать эти изменения координаты, появляющиеся в ноте. Если вы получите в ноте Омега и в ноте H-бар, то вы можете создать реальный симпляктический манифильт. Это будет М, реальный части 1 на H-бар Омега, и добавить что-то, называемое B-фильт, который будет просто прорывом, прорывом и прорывом, и это будет реальный части 1 на H-бар Омега. Это будет симпляктический манифильт плюс прорывом, прорывом и прорывом. И в такой ситуации, еще раз, мы должны рассмотреть что-то, как фукая категория. Какая фукая категория? Объекты — лагранжен-сабманифильт в М, омега-рилл, лагранжен-испект в реальной форме, плюс бандлз, эта коннекция на лагранжен-манифильт, так как корочесть коннекции будет B-фильт, как бы B-фильт, multiplied by identity operator. Да, это дефинитивная дефинитивность объекта, и хомо-сай интерсекцион-поинс. Интерсекцион-поинс может быть хомо-манифильт в бандлзе. И потом это будет а-инфинити-категория. Когда мы рассмотрим, мы рассмотрим холомофик-диск. Возьмем холомофик-диск, чтобы визуально поинтегровать омега-пласс Б. Если у нас рилл-симпляктик-манифильт, то можно сделать это в вк-категории. Найевле, это выглядит как холомофик-вомнотикеrei в каждой части. Ровный момент, но сейчас мы посмотрим, что L0 и L1 are holomorphic Lagrangians. Что это значит? Какие-то холоморфичные диски. Это contribution of some infinity terms. Это очень long story, если вы не знаете, как это происходит. Вы получите эту холоморфичную диску. Сейчас у нас два холоморфичных Lagrangian manifolds. А, перед этим, я хочу увидеть следующие. В принципе, в этой четырех категориях, L и E connection are not an object, if there is something M0, if some M0 is present M0 is responsible for holomorphic discs. If there exists a holomorphic disc for another complex structure compatible with the real, the boundary of the discs belongs to L and area is positive. Such things destroy objects in 4k category. It's a well-known thing that's holomorphic disc bounded. But see the holomorphic disc for some kind of another structure, maybe some kind of J holomorphic discs. And in this station, when we start with complex symplectic manifolds, say for simplicity it's hyperkeler manifold. The fact is, it's easy to see that such holomorphic discs, because the structure depends on each bar, don't exist for almost all each bar. For each bar argument of each bar not equal to argument or integral of two form over some kind of class of the disc belonging to relative pi2 and L holomorphic. For L holomorphic you can, this E will be flat bundle. There is no such disc. So it means that one can put his object of a category, arbitrary holomorphic Lagrangian, arbitrary local system. It will be not abstracted object of 4k category. Now if one have two guys and you try to calculate homes between, get another thing, get two objects of 4k category, when one want to calculate homes you should calculate things coming from intersection but also will be differential coming from M1 from holomorphic discs. And again, holomorphic disc will not exist except of countably many bad directions by the same reasons. Because... So it looks for generic direction of each bar you get very nice things. You get list of object of categories, you know homes are identified with local vanishing cycles. But then if you cross the wall something bad should happen. Yeah, category depends holomorphically. And nobody says that if you have your object, if you cross the wall you get the same object given by the same data. So it means that for example consider rank one local systems you should kind of change of coordinates. So it means that you introduce outside of all, you introduce coordinates in space of some domain space of objects but now you change coordinates. And this will be different morphisms which come from this story. Yeah, and eventually yeah, and also with homes you get these jumps with identification. Yeah, so it's all kind of morally follows from consideration for category where you get different morphisms. And eventually you should give kind of new definition of what is new definition of resurgent functions. I'll just try to do kind of try to explain it in most name term. Yes, so you get you see this hourly algebra is built from two pieces. You get a change of coordinates and matrices. And what happens is a function resurgent function, some function in one variable, divergent functions. Yeah, so one should there are kind of two steps in Riemann Hilbert problem. First we should kind of even imagine you have kind of finitely many rays and we get some variable y1, yn. And the first problem is the following. Find functions y over h bar where h bar doesn't belong to stocks rays. And satisfying some wall crossing along the satisfying some conditions across along the way the typical condition is the following. Suppose you get two functions y1, y2. Here it's kind of minus and here's plus. And suppose for example y2 minus is equal to y2 plus but y1 minus is equal to y1 minus times 1 plus exponent 1 over h bar y2 minus. So you have functions maybe two functions here and if you consider jumps, given by such formulas it's a typical wall crossing transformation. y1 minus it's kind of exponential decaying guy in h bar. Yeah, so you get questions like this. So y1 plus is equal to 1 over h bar. Oh, 1 plus, yeah, sorry. Yeah, this is a little bit kind of nonlinear part of jump formula but then you get jump formula for integrals. It's how we parameterize local system, but also we parameterize integrals and we get something like y, y minus is equal to yj with some different indices. You get something like yj plus maybe plus some again 1 h bar, again multiplied by exponent 1 h bar times yk plus something like this. So you get secondary wall crossing formulas for matrix part of my Lie algebra and if you consider solutions on any of these guys h bar will be functions depending only on sectors but asymptotic expansion and h bar will have kind of finite limit. So you get some formal power series expansion. This formal power series expansion doesn't depend on the stock sector because it changes by things with zero tail coefficients. So you get some universal divergent series some class of divergent series in one variable and it's I think it follows from all the geometries that you can prove as a Borel summable and if you make Laplacian form so you make something like over n factorial maybe some dual variable some kind of t to power n it will have endless analytic continuation with countable many singularities. So what people expect for resurgent functions. But this I want to expect, I think it's the project we have is young to extract just for this geometric description. We get analytic variety, which is neighborhood of chain of CP1s and analytic parallel neighborhood. And then the series will follow from nothing. Yeah, it should depend essentially this deformation of complex structure in neighborhood of a real CP1 and deformation of vector bundle in this neighborhood. Yeah, so it's with some coordinate free in this description. Okay, thank you.