 So the last part will be the final conclusion. And if time permits, I will sketch the proof. Okay. So in this talk, if you are not familiar with those complex geometric terms, so for torsion-free coordinate shift, you can view it as holomorphic effect bundles with singularities. The singular set will be sub-varieties of co-dimension at least two. So you can take the ideal shift of a point, for example. So coherent just means that they behave well near the singular set. So for reflexive shift, you can use them as holomorphic effect bundles with co-dimension three singularities and a good property that it has satisfied, it's Hato's extension property. A good example can be given by this one. So you take three functions, z1, z2, z3, and then you have a map, then you take the corner. So as you can see, this map has a zero at the origin. So this E, where it has an essential singularity at the point origin, rather than that, it will just be smooth holomorphic effect bundle. So based on the definition, you know that this kind of torsion-free coordinate shift or reflexive coordinate shifts, they only appear when your base dimension is at least two for torsion-free shift and at least three for reflexive shifts. And also to go from the category of torsion-free shift to the category of reflexive shift, you can just take the home shift from this shift E to the structure shift. This will give you a reflexive shift. And a shift is the reflexive shift. Just means that if you take the home shift twice, you recover itself. So these are some elementary properties. So basically you can use them as homomorphic effect bundles with controlled singularities. So that's the meaning. Okay. So these are just some concepts. Given this, let's recall Donelson-Wollin-Baggy-Yaw theorem. It says, suppose you have a homomorphic effect bundle of a compact Taylor manifold. The theorem says that if you know this E is slope stable, then they exist a Hermitian-Yaw mass metric H on this E. So by Hermitian-Yaw mass metric, just means that you take the curvature of the chain connection determined by this H. Now you look at the contraction of the Taylor form of this curvature. It would be a multiple of the identity map. Unless mu here is a constant. So this is the equation. Let me define what is the meaning of the slope stability. So it means that if you look at all the sub bundles with singularities, now you can compute the average of the first-chain class by doing integration with respect to the Taylor form. So it will be strictly smaller than the slope of the ambient shift E. So in this case, we call the bundle to be slope stable. If you have a... Sorry. Yes. I was wondering what the first-chain class of a sheaf is. How does one define that? I know what the first-chain class of a bundle is but not the other sheaf. Yeah, so as I said, so the definition can be given as follows, actually. So this shift E is a vector bundle outside of outside of code dimension two sub variety. So you can define and determine there. And actually you can show that this determinant can be extended to be defined over the whole space. So you get a genuine line bundle. We define the first-chain class to be that one. Right, thank you. Mm-hmm. So this is the slope stability. Now in literature, the chain connection involved in this theorem is usually referred as Hermitian-Yamlos connection. And actually just by using the Taylor identity, you know that Hermitian-Yamlos actually implies a curvature to be harmonic. So it's indeed a Yamlos connection. So this is Donaldson-Wollin-Bay-Yaw theorem. Later, this was generalized by Bander on the seal to the case of stable reflector sheafs by using the so-called Hermitian-Yamlos metric. As I mentioned earlier, reflector sheaf are bundles with code dimension three singularities. So when you talk about the Hermitian-Yamlos connection, so it is just defined away from the singular set of the bundle, but at the same time, the curvature, the L2 norm of the curvature has to be finite. So this is so-called admissible Hermitian-Yamlos metric. Let me give you a genuine example of a stable reflector sheaf that is not a homomorphic of that bundle. So you consider this sheaf E over Cp3. So basically you'll take three sections of the hyperplane bundle. Now you take the quotient of this one, this will be rank two, and this E has an isolated singularity given by the zero of this one. And the one can check that this E is actually a stable reflector sheaf. By using some algebraic geometric criteria. So by bundle on the sheaf theorem, theoretically we know that they exist a singular Hermitian-Yamlos connection on this E. So this is bundle on sheaf theorem. Let me give you a few remarks here. So bundle on sheaf theorem implies theoretical existence of singular Hermitian-Yamlos connections. And for given explicitly examples, the local information of the singular set can be usually described by using resolutions like this one, like we just did with this one. So algebraically we know how to describe the singularity. And also this kind of singular connections naturally appear on the boundary if you compactify the modular space of Hermitian-Yamlos connections or given unitary bundle. And this kind of singular, this kind of as appearance of this kind of singular connections make the compatibility very difficult. So what is the meaning of that? So if you assume the base dimension is three, now if you look at all this kind of singular connections on the boundary, because it is singular, so the singular set consists of points. So basically the difficulty lies in you don't have control of the number of the points, of the singular points of the boundary. So yeah, you don't know how to compactify the singular connection in this case because topology can be very complicated. So the questions that we ask about this is that how does those singular connections behave near the essential singularities? And it only appear non-trivial when your base dimension is at least three because in dimension two we have Wollambake's removable singularity. Let me give you a few motivation for asking this question. Originally, this is motivated by Donaldson and Thun's work on singularities of a K-Li Einstein matrix. The second motivation is that it helps understand the compatibility of modular space of Hermitian-Yamlis connections that I just described. An elusive modular space usually carries rich geometry in general and has very important applications. One of the famous application is Donaldson's non-vanishing theorem. One the base dimension is two. About the compatibility of the modular space. So a good description only exists when the base is projective. And when the base dimension is two, this is due to Li-Chun. In higher dimension, it's due to Grapp, Sibley, Thoma and Richard. So in this case, the at-break geometry comes in a very essential way. Because in that case, like what I said, they can control the number of singular points. Like in the examples that I just described. Another motivation to ask this question is it provides a model case for starting gauge theory over manifolds with special hall on me, like G2 spin seven. Okay. Now let me talk about the local model that we will focus on. So first, we fix a reflexive shape, a bundle with co-dimension three singularities over the uniball. Now we fix a calor metric so that it can be written in this way. This omega zero is a standard flat metric. Now only the shift E will fix an admissible Hermitian-Yamlis metric and a connection on E. So that means that the connection, if you define at least shift E and also the Hermitian-Yamlis equation, the last one is that you want to have control of the L2 norm of the curvature. So here we can simply mention this A without E because this E is actually is exactly defined by A. To study such type of similarity, the information we get can be divided into three parts. One is analytical information that it can get from the connection A and the metric. The second part is algebraic information that it can get from just as a shift E itself. So the last part will be to use ad-break to characterize the analytical information. Let's look at how we get the analytical information. So this is a process through zooming in. What does that mean? So basically you just rescale the base, rescale the base manifold to be CN. At the same time, you can rescale your connection and take a limit. Let me make it a little bit more precise. You'll fix a number positive number. Now you can define a rescaling map of the base in this way. Through this rescaling map, you can pull back your admissible Hermitian-Yamlis connection and you get a new one. Now by letting this Namda go to zero, passing to a sub sequence is very important. Now up to some gate transforms, you can forget about the 10s. So this is used to guarantee the convergence. After you do all these things, you can take the limit of this sequence of A Namda. So the limit now leaves on CN with the standard flat metric. So a sequence of A Namda will give you a limiting pair, A infinity sigma B. I will make it more precise later, but right now you can just view them as a pair. This A infinity is a connection. Sigma B is a sub-variety of this CN and they have very good structure. So roughly this piece of information comes from the follows. The part where we have smooth convergence, the other part is we don't have smooth convergence. So this is the information we get. And the key ingredient that we used here is the Price-Motor-Nistry formula. So this formula will tell you that for this sequence of rescaled connections, you have all to control of the curvature over any compact subset of CN. So in this case, you can apply the known analytic result to get a limit. So let me make it a little bit more precise about the limiting pair. So this A infinity is an admissible Hermitian-Yamlis connection on CN. And this sigma B in literature is really called the blow-up locus. Actually, this is named by Tian. And it will help you memorize the loss of Yang-Mills energy in the following way. So as a sequence of currents, if you look at the chain two defined by A-Namta, if you look at the limit of this one, it will differ the chain two by this A infinity. The difference is given by a integer linear combination of sub varieties of CN, of pure co-dimension two. And the least MK is an integer that can be associated to each irreducible component sigma K. So it is called as an analytical multiplicity. And this kind of structure theorem is due to Tian. And also using price to modernity formula, Tian actually shows that this A infinity sigma B are pulled back from the projective space. So what does that mean? That means that from CN you take away from the origin, there is a natural map to the projective space. Now this A infinity sigma B is just the pullback of some data over this projective space. And the connection part, you need to do some elementary modifications to make things work. And the sigma B adjusts the closure of the inverse image of the sub varieties of PN. They are all pure co-dimension two. Okay, so the conclusion that we can draw is that this A infinity sigma B is at break. And we call this pair to be an analytical tangent cone of the admissible Hermitian-Yamas connection at the origin. And there are two fundamental questions about this. The first one is the uniqueness. Because as I said earlier, this depends on the choice of sub sequence if you want to get a limit tangent cone. The second question is, suppose you know uniqueness, how is it related to the initial data? How does it depend on the data E and A? So those are two fundamental questions that we are going to answer in this talk. So the main results that we got for this is as follows. So the analytical tangent cone turns out can be uniquely determined by E. So what does that mean? That means that if you have two admissible Hermitian-Yamas connection that defines the same E, the tangent cones will be the same. So that being said, we know that this tangent cone is an algebraic invariant associated to this E. Let me introduce the first theorem that we got in 2017 and 2018 in the case of Hermitian-Yamas isolated singularity. So now we assume that locally, we already know that away from the origin, the shift E is a pullback of a homomorphic effect bundle over the projective space. So in this case, the analytic tangent cone will be uniquely determined by or canonical filtration of this homomorphic effect bundle E, the so-called Hardener-Siemann-Sachadry filtration. Let me give a remark here. So when this underlying E is poly-stable, this is proved by Jacob Wapowski style using a PD or PD approach, because in this case by assumptions, since this underlying E is a direct sum of stable bundles, so you can construct a smooth Hermitian-Yamas cone type connection just by using Donaldson-Wurlmbacher theorem. Now using this a specific cone, it can be used to compare with unknown ones and you can get a higher order PD estimate. So the uniqueness is just a conclusion. But in general, such a cone with a smooth link does not exist just as our theorem indicates because the tangent cone can have essential similarities. This is why we need to work a lot harder at that time. Okay, let me introduce the term that I used here. The first one is a so-called hardener cement filtration. So this is a unique canonical filtration associated to a bundle or more generally to a torsion-free shape. So it is a filtration consists of sub-bondles with similarities. Let's look at the first factor. I will only define that one because other factors will be similar. So the first factor can be defined as the shape. That disabilizes at least E in a maximum sense. Maximum just means that both in rank and slope. So this will give you a unique sub-shape. This is how you define E1. And then similarly, you can define this E2 quotient E1 or this E quotient E1. In this case, you get E2 and keep going. So in particular, just by definition, we know that if you look at the graded factors, they are all semi-stable, torsion-free. And if you look at the slope of this graded factor, they all strictly decreases. And this filtration is unique. To understand our theorem, we need to introduce another filtration, the so-called hardener-seaman-sashadry filtration. So basically, you just need to add more factors to the first hardener-seaman filtration. Let's talk about again, talk about this first factor. So you add the first factor E1-1 to be a stable torsion-free. And it has the same slope as this E1. Then similarly, you do induction to define the remaining factors. But here, this first factor E1-1, the choice is not unique. You have more than one choice in general. But based on this hardener-seaman-sashadry filtration, we know that the graded factors are all stable and the slope of the graded factors is non-increasing. So these are properties we might need. We will need. Now, we give in the HNS filtration. You can take the graded shift associated to this one. This will be a direct sum of torsion-free shift. So it will be boundless with co-dimension two singularities. Now for this one, you can define a quotient shift that looks like this, a W quotient of the original one. This shift will be only supported on the singularities of this graded shift. So that means that supported on a co-dimension two, sub-variety. So from this, we can extract canonical algebraic data as follows. One is that W of this shift will take home twice to make it reflexive. The other one is a linear combination of this pure co-dimension two support of this tau. And there is an edge breakaway to associate a multiplicity to each irreducible component. And I will not talk about it here, but I will use an example to explain. The only information you need to know is that you can extract canonical algebraic data from the HNS filtration. So the conclusion that our first theorem basically says that the graded least of a deal of the graded shift plus the algebraic cycle you got will determine this A infinity and sigma B in an elementary way. So sigma B will be the inverse image of this tau. A infinity, if you look at the shift, it defines, it will be isomorphic to the pullback of this one. So that's our conclusion. Let me give you an example where it can be applied. Again, you'll consider a rank two reflexive shift, a rank two boundary with essential scenarios, given as follows, obviously three, we take three homomorphic functions, the one, the two, the three square. Now this section has zero out of the origin. So the quotient shift again, where has a singularity at a point origin and one can show this E is actually reflexive. So the origin is an essential singularity. We fix A to be any admissible Hermitian-Yamas connection on this E and one can show they exist a lot of some. Just based on this resolution description, we can usually see that this E is a pullback of this underline E that lies in the following exact sequence and this underline E, as we can see, it's a homomorphic vector bongo because this one has no zero over the projected space. So in this case, we can apply our first theorem. So to apply that, we need to take a hardener Seymour-Sochadry filtration. But for this one, the filtration can be easily chosen to be as follows. You can take this O2 factor. It maps to this E. Now the quotient shift, a simple calculation, which shows that it would be the ideal shift of a point and tensile O2. So that's how you get the filtration. Now, based on this filtration, you take the graded shift, it will be the first factor O2 direct sum the last factor. So this is a piece of information that you get from the filtration. Now what is the process to determine the analytical tangent comb? So to determine that, you just first take the W of this graded shift. It would be a direct sum of O2 plus O2. Now the graded W of this graded shift, we see so, as you can see, it would be just be this O2 quotient of this ideal shift P tensile O2. So in this case, we know that the support of this torsion shift of this one is given by the point P. And just based on the W of this one, the elementary modification that will be used, we tell that this A infinity is actually flat and the blow-up locus will be the line through the origin given by this point P. So in this case, the structure of the tangent comb is clear and we get a flat connection in the limit. So from this example, we actually know that the tangent comb connection A infinity can be smooth and does not capture the original similarity. And in general, they exist in abundant examples where this shift E is not pulled back from the projected space. You can look at this E to be the corner given by this section. You choose three sections and three functions given as this one. And one can show that this E is not, it's not homogeneous in this case by starting a notion of fitting ID. So given this, in order to solve the problem in general, we need to find something like at least on the line E in our first case. And indeed, this is true. So this is a theorem that we got in 2018. So how do we get the canonical algebraic geometric data? Let me briefly explain. So here's the approach. Now our shift live on this uniball B to get something over the projected space. The natural thing to do is to blow up at the origin so that you replace the origin of the B with a copy of the projected space, PN minus one, the exceptional divisor. Now we look at the space of all possible reflexive extensions of this shift of the puncture ball across this PN minus one. So that's the same, we will focus on. The point is, if you start with given reflexive extension, if originally, if the restriction of that extension to the exceptional divisor, sorry, this should be PN minus one, if the restriction to the exceptional divisor is not close to being same stable. So that means that if you think it is not good, now we can modify it along the exceptional divisor to get another extension. It turns out, if we modify it in a certain way, it will reduce the arrow of the extension from being same stable. So repeating this process, eventually you will end up with an object that is closest from being same stable. And that is the data we can use to characterize the analytic tangent cone in general. I will make it more precise later. But now the extension closest from being same stable will be called an optimal edge break tangent cone of this sheet at a point origin. And as a conclusion is that this is unique up to some special modifications that we can easily keep track of. Let me make it more precise. So first we need an arrow function that mirrors how good an extension is. So for giving extension, you take the hardener semaphiltration. Now the arrow function will just be the slope of the first factor E1 minus the slope of the last factor. So in this case, we call a reflexive extension to be optimal. That is the closeness from being same stable. If you look at the arrow function, it lies between zero and one, a strictly less than one. And of course, I should remark that if your arrow function is zero, that means that your restriction is semi-stable. And, but a semi-stable extension does not always exist in general. And there's a modification that I used in this process is called the hacky transform. It is defined as follows. If you fix a reflexive extension E hat, you can take any saturated sub-sheaf of this online E hat. There is a natural map from your E hat to this torsion free-sheaf. Like in this case, if your underlying F is zero, this is basically the restriction map. Now through this map, you can take the kernel of this E hat prime. And one can show that this is a new reflexive extension. You can usually see it is an extension just by definition because this only changed the information of this E hat. Sorry, at least we preserve the information away from the exceptional device, right? This modification is supported on PM minus one. So special hacky transform just means that for this underlying F, when you want to apply, use it to apply the modification, you choose F to be any factor in the hardener or CMAR filtration of this E hat, of this restriction. And the uniqueness of the optimal ones is up to a special hacky transform. So roughly speaking, the picture is you start with something really bad, probably, then you keep applying this kind of special hacky transform, it will reduce the error eventually after finding many steps. Now if you keep applying special hacky transform, it will be optimal always. So for that object, we call it to be optimal. And since it is, since this kind of optimal ones differ by special hacky transform, this probably will tell us that we can just extract a canonical ad-break data just as before. That is, you take an optimal ad-break tangent cone E hat. You can take a hardener or CMAR saturated filtration of this restriction. Again, you take the graded shift associated to this hardener or CMAR saturated filtration. You take the W, an ad-breakway tell you to get a linear combination of pure connection to subvarieties. So it does not depend on the choice of optimal ad-break tangent cone. It is uniquely determined by E. So that's the information that we get from E as ad-break geometry invariance. Okay, so given this, we're ready to state the final result. So that is suppose you have an admissible Hermitian-Yamlis connection on a reflector shift E over the ball B. Now the conclusion says that the analytical tangent cone of A as the origin is uniquely determined by the optimal ad-break tangent cones of E at the origin. So the way to determine that is just similar as our first theorem. So in particular, we know that the analytical tangent cone is an ad-break invariant associated to this shift E. Let me briefly sketch the proof. So the strategy to prove this is study the so-called rescaled limits of sections of this shift E. And this is motivated by Donaldson and Swince-Walk on K-line-stein matrix, where they studied the rescaled limits of homomorphic functions. The key difference here is that compared to the study of K-line-stein matrix, we don't have the so-called Hormando technique. And also we know just from our results that not all the sections of the tangent cone comes from the rescaled limits. So you would not expect that you can prove some L2 estimate for all the sections. So we don't have Hormando technique for general sections. That's the main difficulty in our case. Let me briefly explain what does that mean by studying the rescaled limits of the sections of E. If you fix a section S of this shift E through this rescaling map, you can pull back your section. Now you can normalize the L2 norm of this section over the ball to be one. Of course, this will give you a sequence of homomorphic sections that has uniform, that has L2 norm equal to one over the ball. And the standard elliptic theory tells you that you can indeed take a limit because they satisfy elliptic equations. But the technical difficulty is that because our base is just a ball. So the limit might be trivial. So that's a difficulty. So eventually the section might concentrate near the boundary and you get a zero. That does not tell you any information. So to overcome this, there is a notion of convexity similar to the Kepler-Einstein case. It says, suppose you have a section S of this shift E. If you know that the degree, that is the matching order of this section at the origin with respect to the unknown metric, it's finite. Once you know this number is finite, the rescale limit of this S will be non-trivial. And also it will be pulled back from the projective space because as I said, our connection A infinity are pulled back from the projective space. So as long as you know your number, this degree is finite, you'll know that the limit section is also pulled back from the projective space. Okay, so in order to get a non-trivial limit, we need to control this number. And there's a key observation that we got is that this degree is always finite no matter what kind of section you choose at the beginning. So indeed, you always get a non-trivial limit for any given section. Based on this, this enables us to solve the problem intrinsically compared to our first theorem in the homogeneous case. The approach that we did in this case is that here we started with some unknown Hermitian-Yamlis connections. As I said, it's degree function, it's always finite. So we have a degree function D that comes out of this metric H. Now using this D, we can do algebraic construction to get an optimal algebraic tangent cone E hat of this E at the origin. So in particular, starting with an unknown Hermitian-Yamlis connection, we can construct some intrinsic optimal algebraic tangent cone associated to the unknown Hermitian-Yamlis connection. Because this is intrinsically associated to the metric H, the shape E, the A, we can then use at least one to characterize the analytical tangent cone. So this is, yes? So is E hat generated by those limit sections? Is that? E hat is, E hat lives on this B hat. So it's really just an extension of E. It's not in the limit. It's just in the place at the beginning before you take limit. Okay. So it's an optimal tangent cone. So the upshot is that this degree function is a well-defined function. This part we already know based on the observation. The non-trim part is that it is satisfied enough good algebraic properties so that we can construct some optimal ones. I will explain the construction a little bit more. So for simplicity in the following, I will always assume that the degree function is a fixed number modulo is the integer z. Actually, at least correspond to the assumptions that the optimal tangent cone is actually same stable. The general case can be dealt with similarly. Step one, let's construct a shape over this projective space. How do we do the construction? Now we define this mk to be the space of sections that has degree strictly bigger equal to mu plus k. So let's give us a space. Now we can take the gradient module as follows. It's mk quotient mk plus one, then you take direct sum. And the technique point is to show that this gradient shift, sorry, this gradient module is a finitely generated torsion-free gradient module over the ring of homogeneous polynomials. And in particular, it defines a torsion-free coefficient of the projective space. And actually, this finite generated property really comes from the fact that we already know the tangent cone is at break plus some technical difficulty involved with the degree function. We can show that. And this is the technique point that we overcame in the paper. So after you get this, you can get a, you can define torsion-free shape. It's on the line E hat. And at this point, we don't know, we don't even know what is the rank of this shape, actually during this process, there is an analytical technique point that I want to point out. So, because just imagine if you start with two sections and do this rescaling process, it might happen that at least two sections, if you do convergence, if they converge to the same thing, when you do rescaling. So the angle become zero, you get one section. So in order to overcome this, we need to do this Grunge-Mid Osomalization so that during this process, you want to Osomalize the two sections. But again, the difficulty is that you don't know whether you can get a non-trivial limits after you do this Osomalization. So that's the difficulty we overcame. Also, even you know that you get a non-trivial limit, you don't know the degree of the limit section. So this is a technical point, a really technical point. That we need to deal with. So after that, the next technical point is that since we already constructed something of the exceptional divisor, the next step is to construct an extension of E hat so that it really restricted to the shape that you construct on the exceptional divisor. So in particular, we know that the shape we construct has a good rank equal to the rank of the original shape. And during this process, again, the reflective property of the extension is very technical. So after you have done the construction, we basically get an extension of this E. And based on our assumption, there is one thing we need to show. We need to show it is optimal. So that means in this case, because our assumption, we need to show that it is semi-stable. And to solve this, in this case, it really comes from the fact that not just this E, we can also do a construction related to the dual of this shape E. Like if you look at the admissible Hermitian-Yamets connection associated to the dual shape, like the dual of this bundle, you can get a construction. Now you do pairing of that one. It will tell you the information about the semi-stability. So again, this is another technical point. And of course, without assuming that the degree is a fixed number modulated as an integer, this process can only give us an optimal at break tangent cone. So that's a construction. Eventually we'll end up with something optimal, intrinsic associated to this shape E. Okay, now we can finish the proof using this intrinsic optimal at break tangent cone. So by the construction, if you start with a shape S of this shape E, it can be naturally identified with section S hat of this optimal at break tangent cone, but has certain vanishing order along the exceptional divisor determined by the degree. So indeed, if you start with section S, you get something at break over the exceptional divisor. So that rough idea is that when you take the analytical limits of this section S, you can also take the algebraic limit of this S hat at the same time. Since in this case, our E hat, our restriction is semi-stable. Let's look at the two pieces of information that we get. So analytical side, you'll have limits of the riskier sections that homogeneous pullback from the project space and help you detect as an analytical shape defined by this A infinity on the algebraic side. So when you do this Grunge-Meet of summarization, at the same time, you did something to the corresponding algebraic sections when you do restriction to the exceptional divisor. So basically you get a sequence of elements in a certain code scheme given by this restriction shape. Now on the algebraic side, because this leaves in a code scheme, by passing to a sub sequence, you can actually take a limit, take an algebraic limit of this QI. That will give you an algebraic limit Q infinity. But in general, the limits you get in the code scheme is can be very bad, like you don't know what it is. So the naive idea is that since you get an analytical limit, you also get an algebraic limit. So just you want to connect them. So you want to identify them. Indeed, this is true, but very technical because we are working with different base. One is the exceptional divisor, one is four. So we want to connect these two limits. It turns out, even though it is technical, but it is true, the identification of the algebraic and the analytical sections can help you realize the following. You can realize this algebraic limit, you'll get as a torsion-free coherent sub-sheaf of this analytic one. So indeed, your algebraic limit can detect a lot of information of the analytic one. And furthermore, this algebraic is actually a sub-bondle of this E-infinity analytic outside the code dimension two set. So based on this, we can conclude that our E-infinity analytic is actually the W of some algebraic ones. At the same time, because of this relation, we know that this algebraic limit is not so bad. It's a torsion-free and same stable sheaf. So the limit we get in the code scheme is well-behaved, that's a point. So after this point, we transform our problem into a algebraic geometrical problem for which the conclusion is already known. Okay, so on the algebraic side, the limits can be actually rid off from Graben-Tomac's results on compatibility of modular space of semi-stable sheafs. So they construct a modular space and mu SS, this compactifies some space of semi-stable sheafs over projective manifolds, which is given in your market class and fixed determinants. The point is you have a map from the semi-stable sheafs in the code scheme to this modular space. Using this map, in our case, if we look at the sequence of elements in a code scheme, through this map, actually the image inside this modular space, it will be a fixed point. So in particular, we know that this q-infinity and qi are the same point in the modular space. The key property is that we already know this q-infinity is semi-stable, that comes from the analytical information. So q-infinity and qi are the same point in the modular space, in this modular space. And from the geometry of this modular space, we can conclude that this q-infinity and qi gives the same w of the graded sheaf, same cycle. So in particular, we know that the analytical sheaf we get is a w of this graded sheaf, and so does the cycle, they also coincide. So yeah, so that's how we solve the problem. And after this, after you identify the limited sheaf, the identification of this blow-up locals follows from a single blockchain formula by Sibley and Richard, but since here, we assume that the restriction is semi-stable. In general, this can be dealt with inductively because we can actually construct m such a hat, where this m, we denote the number of the possible images of this d. And actually, this kind of, all those optimal tangent cones are related by special hacky transformers, so. And you can use them to find other factors. Okay, so that's all what I want to say. Thank you. All right, thank you very much. Show me how. Are there any questions? So the analytic tangent cone never sees the different hecking modifications. I mean, the different, the optimal, you say you have different optimal algebraic tangent cones, but those never get distinguished by the analytic one, right? Okay. So you mentioned that you have different... Yes, you're right. So the point is this optimal ones, they all give you the same upgraded sheaf in a cycle. Yes, it does not, yeah. So do they somehow get distinguished by, again, by the limits of these sections? I mean, in principle, is that what's going on? You get, I mean, so in other words, you get different sub-sheafs of the analytic limit, but they have the same quotient of a torsion sheaf basically. Like in that, if you, after you pass to the limit, you don't really see the difference in this case. Like you said, you mean, if you look at a sub-sheaf generated by this limiting sheaf, by this limiting sections, then corresponding, you look at the difference. So the limiting sheaf, and it's our original sheaf, they have the same graded, higher than our CMOS filtration, this one. If you forget about the extension property, forget, like use them as direct sum, I don't see why it can make a difference. If you don't have a problem set, you always get direct sum in this case. So originally, you can get different extension. So it might not be able to tell the difference in this case. Anything else? So if not, then let's take it from now again for the beautiful talk. Thank you. Yeah, and really nice, really nice results as well. All right, good. So I guess see you guys, when is the next talk in two weeks? Jacob. I think so. Is that the Conor Moone or it could be? It could be next in two weeks anyway. Okay, well, see you guys. You, thank you, Shamiya. Hey, thanks. Thank you very much, Shamiya.