 So, thank you for the interaction and I want to thank Ahmed Abbas for the invitation and for this organization of this conference. So today I will talk on character circle. So let me fix some notation. So small k will be a algebraic closed field over character speed. Assume this is positive and x will be a smooth of the dimension d. So this will be my variety and as a quiescent I take lambda. Lambda will be, so this will be a finite field over character scale, divided from p and k, r. So I want to write script k, so it will be like this. So this will be a construct of the complex of lambda values on the other side of x. So for such object, we want to define the character cycle. Yes, yes, yeah, exactly. You want something like what is not usually a dbc? Yeah, yeah, exactly, yeah. So, so character cycle, so this will be, this will be a linear combination of closed conic subset of the cotangent band. So this is a vector band associated to omega 1. So I take notation different from Gotendic. So, so this is a linear combination of dimension d. So it will be actually a quiescent will be the 1 over p, yeah. And this conic means this is a stable closed under multiplication. So we want to define such object and want to study property of this object. Sorry? Are they in general the grand general? No, no. Can you see them? No, like this. Okay, so on this object during, he wrote a note on an open card. So this is a written, so he write like this. So I think this is August 2nd. So, and there is a famous letter to Luke, so letter. So he wrote a famous letter to Eugene. So this is dated in, so this is written like this. But at that time he was leaving in France, so it must, this must be November 4th, yeah. So, so this note is much more detailed than this letter. But so it was already three years ago, but it took me three years to understand what he wrote. So I want to, so in this note he states some propositions and also conjectures. But under some assumption, so he states some propositions and conjectures on this characteristic cycle in this note. So the goal today is that, today and tomorrow. So I want to tell you what and how I can prove. But here I need some assumption assuming the existence of singular support. So this is, so I will explain this more in detail. So in case this is a union of finite, finite to many union of closed-colonial subsets of the tangent band. So satisfying some local acyclic condition for a family of organisms, to curves or to subsets. So this will be the goal of my series of lectures today and tomorrow. So more precisely, so this propositions and conjectures includes Milner formula. So this is a formula for the total dimension of the space of nearby cycles or vanishing cycles. And also, Euler-Pankler formula, index formula. So this is for the Euler number. So actually, this will be characterization of the character cycle. And this is the formula we want to prove. So in this sense, this is basically conditional because we need to assume the existence of singular support. But in some cases, we can get unconditional result because we have ramification theory. So ramification theory provides that we have existence of singular support, which we need to assume. But we actually have such singular support if we remove outside closed subsets of co-dimension 2, at least co-dimension 2. And if our variety is a surface, we can do better. So it's the existence of singular support on surfaces without any assumption. So this is unconditional. And in particular, we can get the formula for Euler-Pankler. So we get this Euler-Pankler formula on surfaces unconditionally. So this gives the generalization of the result of Le Mans, where he makes some assumptions on the non-fairness of the assumption. And actually, I was working on this problem when I was a student, but it was too difficult for me at that time. But now, I'm happy to do something on this. OK, so this is a free plan what I'm going to do. So let me give some table of content. So the plan is... So first, I want to define what I mean by singular support. And character cycle. So this is the first part. And then I want to explain how we can construct our character cycle. And finally, I want to tell you what property we can prove. So I'm given three words. So basically, I spend one number of which. This is my plan. OK. So now I start the first part. So the first part is singular support. And this is characterized by the local cycle. OK, so I need to give some definition. So x will be as before. So x will be smooth of dimension g. And I prepare some terminology on finite family of conic closed subset of the cotangent bundle. And so it will be of dimension d in the middle dimension. So the first definition. So we say... So we study... First, we study movement curve. So f will be... So this is a flat movement over k to see the smooth curve. OK, we say such a movement is non-characteristic with respect to this family. So we look at... So we have a conical map from the pullback of the cotangent bundle of c to that of x. The inverse image... Ah, yeah, yeah, thank you. The inverse image of s is the subset of the zero section. So this is the definition of non-characteristic. Ah, sorry. And if... So I need one more condition. So the intersection... So ti will be the intersection of the zero section. So this is the notation for the zero section of the cotangent bundle of... So for each si, we take... We have an intersection like this. And this is a subset. So we regard this as a subset of x. And so is flat over c for every i. So this is a non-characteristic condition. So in the case if k is the zero extension of some smooth c on the complement of divisor, then ti tends to be the... ti can be one of the divisive components of the divisor on the boundary. And I want at ramification, there's no vertical ramification. So this is basically such a condition. So this is the condition for one curve. But actually I want to work with a family of more smooths. So that's... Now in the second part, I assume... So I want to work with the relative version. So I work with this commutative diagram. So this is a commutative diagram of flat motions, of smooth schemes over k. So here, so I call this map f. So this b is the base space. So this is parameter space. This b plays a role of parameter space. So this is the base scheme. So I call this b. And here, so this is the family of curves. So this is the smooths of the relative dimension one. And the map here is eta. So we take this eta neighborhood, but basically... So this is the family of most parameterized by b from x to this curve. So this is what I mean, family of most two curves. So now we say... So this f is non-charstitic. If for every closed point, the fiber, the most of the fiber, this is non-charstitic with respect to pullback over. So we consider this condition in family. So this is non-charstitic. Okay. Yeah. Now we can define what I mean by a single support. So heavy as before. So then let me say... So this may be also not before. So this is the finite family of conic closed subset of... Sorry? The si are not necessarily irreducible. Yeah, actually they will be irreducible. But when you take your definition, I wonder about the following. So you assume that the intersection of si with the zero section is flat over c. And now this depends on the choice. So if you have something which is... If you have a union of things which are not irreducible and the compositional irreducible components, it's not clear that this condition is the same when you take the irreducible component or when you take... Okay. Also when you pass to etal localization, the irreducible components, something which was irreducible can have several branch. So now, so in your definition, you definitely want to fix a particular choice of index family si or does it depend in some way only on the union? Yeah, yeah. So on the level of x, I fixed... So it will be irreducible component. But after pullback, I just keep the same index. Yeah, okay. So, yeah. So now I state the condition. So, yeah. So we call such family is a single support of this k if it satisfies the condition I call SS1. I put one here because here we study map to curve. And we will get SS2 in lecture later. So for a commutative diagram as above, I mean the commutative diagram in this two, so family of motion to curve, we ask that if... Yeah, so this motion on the top is universally locally acyclic relatively to the pullback, okay. So then w is a scheme of... a total scheme of x cross b. So by first projection, we can still go further to x and we can pull it back k to w. And we... And this condition says that this motion is locally... universally locally acyclic we just do this pullback if f is... Sorry, I wanted to say this is non-characteristic over b. Yeah, if this motion... is non-characteristic over b. Yeah, I forgot to write this over b after non-char... second condition there. Maybe I... Yeah, non-characteristic. Yeah, so we require that the non-characteristic motion is universally locally acyclic. So this is the condition required for single support. So in the absolute case, we just have a flat motion to curve and we ask that there's no non-trivial vanishing cycle. So this is basically the condition, but we work in family. Okay, so we still have... Yeah, so there's some elementary properties on this. So first remark is that... So yeah, so SSK, so this will be a notation for single support. Okay, but first remark is that this is not uniquely determined. So if you add something like an extra, this condition is still preserved. So I write like this, but this is not uniquely determined by K. And I'm sorry? What did you say about it being non-uniquely determined? Because if we add something, an extra here, then the condition gets... This condition gets stronger. So this condition remain valid. If you add what did you say? To SSK, so if your SSK is already single support and if you add something extra, then the new one is still single support. Yeah, okay, and so this is compatible with the construction is compatible with smooth pullback and also finite and pushed forward by finite unified motion. So this statement is not totally trivial, but I skip this here. Sorry? Is it the case that the singular support is the minimum family of irreducible subs that satisfy this or...? Oh, yeah, you can make it like this, but yeah, I will not stick to this point there, here. Is that okay? Okay. Yeah, so I give some elementary example. So the first example is that we have our complex K and suppose we have finite closed subset. So that's the concept of finite many closed points. And such that on the complement, the cohomology sheaves are local constant. So in this case, we can take as a single support, we can just take the union of zero section. So this is the zero section and the fibers after X. So this is the consequence of local psychosity of smoothness. So this is an elementary example. And more interesting construction is by ramification theory. So ramification theory tells us that there exists some closed subset of co-dimension to codimension theory, such that if you remove this G, we have... so the ramification theory provides a single support. But today, I will not give you, I will not say how we can construct construct-singer support in this case. So in the case where X is surface, so this means that we are left with only finite many closed points. So by the same construction here, so this gives the existence of the single support if dimension is two. So in this sense, the ramification theory provides us unconditional results. Okay, so this is the definition of single support. So now I've got the tactile cycle. You need to make it non-trivial because if you take the subs to be too large, do you want them to be a certain dimension? Dimension D, yes. Yes. Definitions is also the middle dimension, yes. Yes, so the middle dimension, yes. So this is, you didn't put in the definition? Definition one. Yes, yes, yes. So sorry, please put dimension D, yes. Yeah, thank you. Okay. So now, so we are going to assume that we have single support. Then the problem is how to, problem in the definition of the tactile cycle, we just need to determine the coefficient. So the coefficient will be uniquely determined by requiring such a mineral formula. So I will tell you what formula we are going to study. Okay, so let me make one more definition. So mineral formula is a formula for isolated character points. So I'm going to define what does it mean isolated character point. So x will be as before. And so this is a flat motion. In the case where, yeah. So this is a flat motion. Ah, ah, ah, ah. Yes, so this is a motion of a key to a smooth curve. Okay, so under u, u, so this is a colloquial point. So we say u, we say, so we are, we are, so always we assume that we are given k and also s equal to s, s, k. So we say, and we, so I said that this is not unique, but we are going to fix it. So we say, so they are fixed. We say u is not isolated, u is an isolated character point. If they exist, ah, neighborhood, you such that the restriction of f to the complement here to s. So this motion is non-characteristic. This is the decision of my senior support. Yeah, so we assume that there is, so locally at u, so u is isolated point and outside u, the motion is non-characteristic. So this is my terminology for isolated character point. So under this, s is our senior support. Okay, so to state this MUNA formula, I want to define certain intersection number. So now we take omega. So this is our basis of tangent bundle below at the image of our point u. Then by this pullback, this defines, so f star of omega defines a section of tangent bundle on u. And by this condition, the intersection, so by this condition, the intersection, so this intersection, the intersection is isolated. So this in the inverse image here. So this means that we can define the intersection number in this cotangent bundle on the fiber at u. So we can define this intersection number. But our assumption that si, so this close-up set are conic, so this is independent of the choice of such basis. So by a feasible notation, we define it like this. So now we are ready to state the same. So assume senior support exists, then there exists e dink z1 over p linear combination, which we denote like this. So this is the sum that exists, such that. So this satisfies Milner formula. So the minus of the total dimension of the space of, so I will tell you more what this means, so k f. So this will be equal to that, the intersection number. So the right-hand side is defined by this remark. And here. So here you assume that si are reducible, yes? Yeah, yeah. So here f for every f defined on a time neighborhood, v of u, such that u is an isolated character point. So this is a point of wave. So we look at every map to curve c defined on a time neighborhood of v, a time neighborhood of v of u, such that u is an isolated character point. And we require that this formula holds. So we call this formula Milner formula. And this uniquely characterizes the character cycle and the claims that are actually such linear combination. And I put the star because we assume the existence of senior support. So to indicate that this is a conditional result. So let me briefly explain the notation there. So cu k, 5, yeah. So this is the stoker to u of the complex of vanishing cycle. So each cohomology, so this is the Galois representation of the local field after the image. So this is a local field of curve c at the image. So this is a classical local field. So this is the field which is close by assumption. So we have such a Galois representation and by finiteness of vanishing cycle. So we have finite many finite dimensional vector space with such Galois action. So we can define the total dimension is just the dimension plus swan conductor. So for each cohomology we can take the dimension and the swan conductor because we have this Galois representation of classical local field. But this is a complex so we just take the alternating sum. So this is the left hand side. And we require this mean formula for character cycle. So if the senior support exists, it exists after pulled back by K-tar names. So in some sense you could have just stated your theorem just for exit. Yeah, yeah, that's possible. These eyes really can split after. So it's a little bit mixture between. Yeah. If you stated just for case it's a slightly weaker. I think it's equivalent. But in case it's characterized in equilibrium so it doesn't matter so much. Yeah, so I give some example. Yes, so I have to say why I call this mean formula. So there's one X for the, in S07, X for the 16. So this is, so they are doing improve the formula in geometric case. So this is a case where K is just constant and in this case this character cycle is nothing but minus 1 to dimension D over the times the 0 section. So in this sense this formula is the definition of classical mean formula. Now I give some example. The first example is the time case. So this is, so U is a complement of the device of the simple number constant. And they are irresistible component. And let's take K to be just 0 X station. And F is the smallest shift on U, which is the time identified along D. So in this elementary case, and in this case, single support is computed by Luke in this appendix to determine the finitive in this one and a half. So, and in this case, so this gives us an unconditional result. So in this case the characteristic cycle is a little bit similar, but a little bit is a little more complicated. So here are some of the takeover i. So here X i is the intersection. So this capital i runs, so maybe I'll make it more precise. Every subset denotes the intersection and this is the notation for the conormal one. So this is more or less a well-known formula, but our character cycle is characterized by this Milner formula. So we need to check Milner formula. And this is actually done recently by my Chinese student. So I wanted this Milner formula in this case is checked by Indianian. And here almost completed writing, but not completely yet. So where I put the... Okay. The second example is even more close, huh? Can it be reduced to the one, the rank one case? Yeah, yes, yes. Yes. Yeah, we can do this to rank one case, but still we have to work a little bit. Okay. The second example is a case where X is... X is the curve. Then in this case, character cycle is minus... So first we have... Yeah, first we have rank times zero section. Yeah. Ah, okay. So I need to introduce Z. So Z is as before. So this H... The cohomology shifts local constant. Then... So we have finitely many close points and I need to put the arting contactors there. So in this case, we just have two cases. So this is zero section and five lines. And this is the household, F is this K. So this arting conductor is rank of K minus dimension of stock at K plus this one conductor of K. It's integer. Yeah, it's a half-ass cell. Yeah. Yeah. Okay, so... Yeah, and in this case, this middle formula is classical induction formula. Okay, so that's it for part one. So, yeah, so thank you. Any questions? Yeah. Actually, you will show examples then. What is the simplest approach? There's no Lagrangian, so wait. I'm not integer. Ah, so... Yeah, up to here, no problem. But if you go higher dimension... Yeah, so there's an example of non-Lagrangian case in dimension two. We expect to have integer anyway. Yeah. So you have cases where the coefficients are not integers? No, we can expect that the coefficient is always integer. So in the same case, your theorem also applies, because in the same case, the signal support exists. Yes. So in some sense, what is not precise here is the precise description of the characteristic. Exactly, yeah. Yeah, so we need to determine this question, yes. So is it the case that in some sense you give a compatible family of ellatic sheaves, the characteristic cycles or the signal support are all depend... So... Yeah, as long as the lamification still works, then we can sense that. But out of that, I have no control, so I don't know. I have a question about your definition of non-characteristic... Yeah. Yeah. Because you ask that the map F to the curve should be flat on each piece of eye. Yeah, yeah. Okay, so how can piece of eye can be a close point for instance? Can it be a close point? It's excluded. But the mineral formula is for isolated characteristic point. So in this case, we can also include this. Because in the definition of an isolated characteristic point, we remove this point. We ask non-characteristic, outside this point. Yeah. So outside the TTI, that will never fit in the definition of non-characteristic, but it's always outside the character point. Sometimes the condition is not in the characteristic case, the mineral formula is true. But here we go a little bit beyond the characteristic. So we assume we have an isolated non-characteristic point. So if you have an isolated point, you can just remove it. You can include it in the center. Yeah. So non-characteristic case, both sides are zero. But you said that the first... Yeah. Yeah? You have no map which is non-characteristic. Yeah, yeah, yeah. Before you remove that point. Yes. Yes, because... Yeah, so we... So this is... We are working on curves. Yeah. So we cover the dimension one, and you put this one on the shoulder of minus one, then you get minus one. But on the right-hand side, you get the middle number or minus the usual middle number? No, no, just the intersection number. But is it the classical middle number? Thanks to the quotient of all divided by the partials. Yeah, yeah. But then there... And you put also a minus one. Not just a minus one, to be mean. There was a minus one. Yeah. I think it's consistent. Yeah. If you have a family of smooth varieties, let's say over... So a smooth family with a smooth base and you have a ship on the total space, so you can consider its characteristic cycle or singular support on the total space and on various fibres. So is it true that it is compatible on a dense, open set of the bed? It's compatible in some sense with restriction to fibres? If your muscle is non-characteristic, then it's like that. I think you need some assumption. So I was thinking about the case of the generic point. So you have a map... Yeah. My closure of the generic point, you have a new variety of energetically closed fibres. Well, yes. Yeah. Even on the generic fiber, you need some assumption. Yeah. So no location to restart at 11.45.