 Okay, thank you. So, before I start, I wanted to point out one small notation in consistency between my lectures and Sacha Kusintsov's lectures. So, in my lectures, so when S is a subject, is a subset of the objects of a triangle at the category, and I use these angle brackets, I mean the smallest extension closed subcategory containing that set, right, the extension closure, where extension means, I mean, if the first and the third of an exact triangle in your subcategory, then the middle one should also be, whereas in Sacha Kusintsov's lecture, it meant the smallest triangle-related subcategory containing these points. So, I mean, another thing you, yeah, let me, okay, so what I'll do today is explain for you the construction of stability condition on surfaces, and then go towards applications. And just let me start with a little bit of preparation. So, Lx will be a smooth projective surface over K, and the characteristic of K will have to be zero. And actually, that's really crucial. So, I do not know whether stability condition exists on an arbitrary surface in characteristic P. In fact, I wouldn't be surprised if they don't exist in general. And then H is a polarization, right, so a class of an, say, class of an ample line bundle. And as my, as my V, I'll, and lambda, I'll fix the following. So, so, I mean, I go from the K group into R3 by sending E to the vector rank of E, H dot churn 1 and churn 2. Right, and lambda isomorphic to Z cubed will be the image of V. And then claim, my claim is that there does not exist a stability condition thick by equal Z comma A in star lambda of X weight, where you, where you start with coherent chiefs. Right, and I, I, I'll, I mean, I, I won't, I won't prove this, but you might want to try it as an exercise yourself if you want. And this even holds, but generally, if instead of just as rank three letters, I would have chosen here the, the image of the churn character. So by taking lambda, essentially the, the algebraic homology. Right, so I cannot just use for coherent chiefs, there is my claim that there is no central chart function that has, has this upper half plane property. Right, and so instead, by the way, if you want to find the proof, it's in one of Yukinobo's papers. I forgot which one. So instead, but I'll start with, I'll start with guessing, why, so instead of guessing the harder, I'll start guessing the, the central chart function. Right, and so it will depend on two parameters, but let me start introducing them slowly. Right, so, I mean, what we did for curves, some of it did degree in this direction and rank in this direction. So this is for curves. And I mean, for surfaces, now we have three degrees. We have the rank, we have the degree of C1, and we have the degree of churn two. And some of the natural extension seems to be to go like this. Rank, h dot churn one, and churn two. Right, but now it also is about time to allow for some parameters, right? I mean, here if we use the parameter in front of the rank, we could just undo that choice of the parameter by using the gl2 action, but certainly with these three rays, we couldn't do that anymore. And so this parameter will be a, the choice of a positive real number alpha. Right, which is basically, I mean, of course this imaginary part rescaling doesn't do too much. It's really just rescaling the contribution of the rank compared to the contribution of churn two. And let me also introduce another twist here by twisting the churn character by some real number. We're here churn beta of E is E to the minus beta h churn 0 of E, right? So for beta integral, this just means you're, you're tensoring your whole construction by multiple of h, but you can numerically interpolate between these integral values. Okay, and I'll note that for alpha going to plus infinity, some of this, if you look at these terms, of course, that's dominated by churn zero and churn one. So in other words, this is asymptotically, the phase of the alpha beta is just asymptotically governed by, by the ordinary slope. So let me define this precisely. Definition for a coherent chief. I define mu h and let me also twist this by beta for convenience. Although here it doesn't, doesn't really make a big difference. It's just shifting everything by a real constant. So here I take the slope h dot churn one beta of E divided by the rank of E if that is positive and plus infinity otherwise. And you should, you should think of this as given by something like a central charge, namely by the bar equal to minus, minus h churn one beta of E plus h squared over two churn zero of E. And let me also introduce the, these factors, alpha squared and alpha here. I mean here of course this doesn't make a difference up to the gl2 action, but just to make the z bar close, more closely related to the alpha beta. I basically have just forgotten the churn two term and then rotated everything by 90 degrees. So basically the, the phase of this complex number isn't directly related to, to the slope over here. And then the definition is that E in coherent sheath is slope semi stable A into E onto B. We have, and here there's a kind of slick way to find stability here by comparing not the slope of A with E, but the slope of A with B. So I, actually I learned this from, from Yukinovo. But some of the point is that because of things that have churn zero and churn one equal to zero, DC's are properties no longer as strict. It could be that they have equality here and inequality here for the slopes, right? This could be equal and this could be strictly less than. Anyway, and I thought this is just ordinary slope stability. And so the first fact about this slope stability is that hard on cement filtrations exists for E in coax and this notion of slope stability, right? And, and the proof is, is exactly the same as the one I gave. I gave on Monday for slope stability on curves. And the other fact, and this will be really crucial, is the Bogomolov-Geseker inequality. So it says the following if, if E in coherent sheath is torsion free and slope stable or slope semi-stable. Then we have the following quadratic inequality. So then turn one of E squared minus two, turn zero of E, turn two of E is greater equal to zero. So we do have a quadratic inequality for the churn classes of semi-stable objects just as what we want for, for the support property. And as a corollary in if E in, here in sheath of axis, slope stable, and now slope semi-stable, and now here I'm allowing it to be a torsion sheath, then the following expression delta h of E is greater equal to zero, where delta h is now given by h dot churn one, we had minus two h squared from zero here. So it actually doesn't matter. This expression is invariant under changing beta. Yeah, yeah, this one here, here, it's just for convenience, right? It's just changing everything by constant. But, but, but I mean the, the more interesting thing is, right, this really depends on the polarization, but this inequality does not, right? But, I mean, somehow I'm, I'm dropping this now, I'm really, I'm changing this now into an inequality that depends on h, right? And, and, and the, the point here is just by the h-index theorem that h squared churn one squared is always less than or equal to h dot churn one squared. Okay. And, I mean, and then it's also obviously true for torsion sheath, right? If, if the rank is zero, then this is obviously greater equal to zero. Okay, any, any more questions so far? Okay, now we are basically ready for the, for the actual construction. I mean, note that z alpha beta, maybe I should also use some z alpha beta here. This is equal to minus churn two beta plus minus i times z bar alpha beta. Right? So, these are really closely related. And, right? So, so you should think of this center chart as being obtained from just first rotating this one and then adding this churn to beta term. Well, this is good news, right? Because we know what to do with rotating the center chart. We just, we just tilt the corresponding heart of the teach structure. That's, that's the, just the, this GL2 action. Right? And so, this leads to the following definitions. Maybe let me go over here. Right? So, t beta. Let me write this. And these are just all the data mu h beta stable with positive slope. And then I take the extension closure and then f beta to be the same thing. Right? So, exactly distortion pair that are associated to any stability condition and the cutoff phase. And then I let co beta to be the, to be the tilt of co here and she's at this torsion pair. Right? And so, I can either think of this as t beta, f beta shifted by one and then taking extension closures. Or I also gave you this description in terms of two term complexes. Right? So, our h minus one has to be in f and h zero has to be in, in t beta. And basically, as for stability on this on surface is the entire magic of the derived category that we're going to use is just that this is an Abellion category. Right? The heart of a bounded t structure. And so, it's an Abellion category. And almost everything we'll be doing will just be doing slope stability in this Abellion category. Right? And so, let me, let me just draw the corresponding picture here. Right? So, what I, this rotating by minus i corresponds, but by rotating by minus i means that co here and she's are now over here. Right? And what I, what's above this line here. And that's t beta, t beta, what's below this line is f beta. And so, we are replacing this by f beta shifted by one. And then in the upper half plane we have co beta effects. Right? And, and you've also seen how to do, now we are just changing the center charts by a real parameter. Right? And you've also seen how to do that in the, in my lecture yesterday when I, when I was deforming bridge sensibility conditions. Right? And it, it turns out that's almost the same. Right? And so, I mean, if I just take this minus i, the beta alpha beta co beta of x, then this is a, this is still a weak stability condition. Right? By, by, by which I mean if you do, right? I mean here we saw how to go from this kind of weak central chart. I mean, sorry, maybe let me be precise here. Right? This, what, what I mean by, this is that this is a weak central chart in the sense that the image is contained in the upper half plane union, the negative real line union zero minus i, the alpha beta bar of co beta x. This is contained in the similar closed upper half plane, but no, zero added. And what I mean by weak stability condition is basically just pretend that objects with central charts equal to zero have maximal phase. Right? And so pretend that if e is in co beta of x and c minus i, c bar of e equal to zero, then the phase of that is just being equal to, equal to one. Right? And by the way, what is, what does this mean? This means that this condition is equivalent to e being a zero-dimensional torsion chief. Okay? And more over it, it, it also satisfies something like the property, right? It weakly satisfies the property in the sense that, right? So we have this quadratic form that's greater equal on say my stables. And now why, why do I say weakly? Because if you look at the kernel, a kernel of z bar, then that's actually equal to zero. Right? So I, I would want, in the support property, I had that this is negative definite. And now here it turns out that the kernel is directly contained in a negative code. Right? Because it's just these vector zero, zero one. Yes. Age is still this polarization. Right? And delta age is this, this form over here. Okay? And then, I mean, what I claim is that you can adapt the arguments paragraph free to this situation to show that, right? So I mean, maybe let me attribute this due to Bridgeland for K3 surface and to Akara Bertrand, potentially for arbitrary surfaces is that co-beta x, c-alpha-beta, and maybe, maybe let me add Yukinobo for the last statement here, that co-beta x, c-alpha-beta is a stability condition satisfying the support property with respect to the quadratic form delta h. Right? So this is an actual Bridgeland stability condition. And we still have this quadratic inequality for churn classes of semi-stable objects that we had for, for slope stable sheets. Okay. Any, any questions so far? So you just look, I mean, you, you take a curve on your surface and then you look at line bundles on this curve. Right? So they, they differ, they, they, they all differ just by churn 2 and they all have to be in the upper half plane. Right? So this shows that the churn 2 has to be a real contribution. And so all the line bundles are, some are on the same height. Right? And so, I could take c, let's say in h. First you look at oc of nx. The central charge of these would all have to be on the same height because they just differ by, by some, each of them differ by z of the skyscraper sheath of a point. And the only way this can happen is if they're all horizontal. But then if you look at z of m o of n h, right? Now this depends quadratically on the, I mean, of course you can write this as a sum of z of o plus contributions from there plus things from here. And then the, the, I mean it, the imaginary part of that is, is just negative for n going to minus infinity. Okay. So maybe let me start drawing some picture, right? So in lambda r is isomorphic to r cubed and delta h is a quadratic form of signature 2 comma 1. Right? So I have this, this cone like this where the interior of the cone is the part with delta h less than zero and the outer part is where delta h is greater equal to zero. And you may wonder where this, I mean here's another justification for where these strange, maybe slightly strange looking formulas for the alphabeta come from. Well, it's just that if you take the map from r greater than zero cross r to the project, projectivization of the negative cone, right? So this is the upper half plane and this is the Poincare disk. They are of course naturally isomorphic. And here I have this map alphabeta goes to kernel of the alphabeta. And that's just a standard identification of the upper half plane with the Poincare disk. So this is an, this is an isomorphism. And in particular, I mean my, my formula is here for kernel of the alphabeta. This really precisely describes the entire component given my, given by the promise by the quadratic form. Right? I mean yesterday I told you for any central charge that's where the kernel is negative definite with respect to delta h, I can get a stability condition just by deformation. And it turns out that they are all up to the tl2rx given by the alphabeta and co-beta. So here's another thing to note. The skyscrapers these are, are stable with respect to sigma alphabeta maybe. I just, just because big, I mean they are in co-beta x of maximal phase. And so that, that's very easy to check. And I mean conversely can show that geometric stability condition by which I just mean that skyscrapers are stable of the same phase are up to the tl2rx and essentially obtained from these formulas are essentially of this form. And let, let me say what essentially means, right? So again write down the formula for the alphabeta. I mean some of the, the fact that here I was using alpha squared over two and here turned to beta. This exactly comes from the fact that directly comes from Bogomolov-Geseker inequality. And basically whenever I'm on a surface where I have the stronger inequality than Bogomolov-Geseker inequality, then I have more freedom with these parameters, right? So a stronger, if my surface admits a stronger Bogomolov-Geseker inequality, then I have more freedom with the, with the parameters of the central chart. And, and in particular I mean solving the problem which central chart is precisely are allowed for geometric stability condition is basically equivalent to proving sharp bounds for Bogomolov-Geseker type bounds for germ classes of slope stable vector bounds. Any more questions at this point? Well, well I mean what I mean is that I, that I'm not, so I don't mean to varying alpha and beta more, but say, you know, say using a bigger number than one here for example. So for example replacing this here by minus turn to beta plus some constant times to rank something like this. Okay, so then I should say a bit about small crossing and I'll first say some things in the abstract and then make this more concrete for surfaces and, and, and use it in examples. Right, so let's say I fix V in lambda and let's say for simplicity that it's primitive and then for, for sigma in star lambda of x. Let me look at the m sigma of v, which is the set or of course you could hope that it's a stack or an algebraic space, maybe a more moduli space, cross moduli space of sigma stable objects d v of v equal to, v of v equal to v and now I can, I can vary sigma and what does wall crossing mean? Wall crossing means that this behavior is controlled by a, by a set of walls, a proposition that there is a locally finite set of walls, which I mean real typically, typically co-dimension one sub manifolds so that if I let, if I let chambers to be the components of the, of the, of the complement and the following holes, so let me first draw a picture in, in, in, in the space of stability with this. Okay, so as I said typically there are co-dimension one sub manifolds or I could also have some theoretical, I could also, sometimes I could also phenomena like this, like in isolated co-dimension two point or wall set stop. So that the first of all m sigma of v is unchanged as sigma varies in a chamber and sorry here I should have, should have insisted on semi-stable objects. Secondly that m sigma of v contains strictly semi-stable objects, objects if and only if sigma is in the wall. So in other words there's nothing happening as long as sigma varies within the chamber. Once we cross a wall the model there are some strictly semi-stable objects on the wall and the modulized space or will be different on the other side of the wall, or will parametrize different objects on the other side of the wall. And so mean let me sketch a proof. If e is strictly semi-stable, right then it has a Jordan-Helder filtration and let's say e i are the Jordan-Helder factors, the stable filtration quotients, then, right then the picture must look something like this. Here we have c of e. All the z of e i's are somehow on the same ray but of strictly smaller, strictly smaller absolute value, right. So you have z of e i are less than z of e which is the same as z of e. But you also have that I mean in my notation from yesterday by the support property you have the norm of p of e i is less than or equal to the norm of z of e i less than z of e. Right and so this shows that there are there are only finitely many such classes because I might remember as soon as I bound z of the class and np of the class then there are only finitely many candidates. Right so only finitely many possibilities, finitely many choices for v of e i as long as v of v is bounded. So in other words as long as I'm varying within a complex set there are only finitely many such choices and then locally the wall I mean the wall associated to this strictly semi-stable objects is given by z of e i and parallel to z of e for all i and that defines the real co-dimension one sub manifold. I mean in typical it's a real co-dimension one sub manifold if these say if they're just two general factors or if they're all contained in a rank two lattice of the of lambda. And I mean how does this look like in this picture I mean as a or the indicated I mean I always like to think of the varying a stability when it's in terms of varying the kernel of its central charge right so here how does it right if this is the kernel of of z then how does this this kind of condition look like on z right I mean it's actually the same thing this condition is equivalent to saying that the that the kernel of z is contained in the rank two lattice span by v of e i and v right so in other words say if somewhere here I have v then this means that the kernel is and here I have v of e i then I have this a rank two plane span by those and the wall is exactly given by the set where the kernel is contained in this rank two lattice so in other words if I draw a cross section and I'm all interested in the walls for my fixed class v then they all look like line segments of lines going through going through v right and so in particular for example in the situation I've been in for surfaces with either becoming one or with fixing this rank three lattice long time then the walls can never intersect okay and so and the second thing I should say is about this wall crossing is that there's always this geesecker chamber at a point that's sometimes also called large volume limit right and it's actually nice that I still have the central charge over here right so remember for for alpha sufficiently large the phase of this complex number is essentially determined by turn one but and also if I mean the some of the slopes of two objects are the same then some are turned to access it tiebreaker and that's exactly the same that happens for for geesecker stability right and so um and and so this leads to the to the following proposition by the gain fix v in lambda and let's say that um better such that mu h of better of v is positive right in other words that's the same as better being smaller than the ordinary slope of of v then um for alpha sufficiently big we have that this m sigma alpha beta of v that's exactly the modular space of geesecker stable sheaths a which m v of v equal to v right and and maybe if you don't know geesecker stability just assume that the rank and the m c one of v are co prime and then it just means you're looking at slope stable sheaths of this class right and so in particular the third here is really it's a it's a it's a projected variety that can be constructed via g it and um but but but i i i mean more than that i just i don't just mean an identification of the spaces i really mean that um so e weight v of e and so if you have e and co beta v of e equal to v then e is sigma alpha beta semi stable if and only if e is a geesecker stable sheath i'm really precisely identifying the objects on both sides right and so so in other words in this in this picture over there there's a of all these chambers there's a geesecker chamber so somewhere here there is zero zero one which corresponds to alpha going to plus infinity and somewhere here you have maybe let's say here you have v here there is possibly a wall where um beta is equal to u h of v and then there's always a geesecker chamber so that along this entire chamber um this idea this identification holds and maybe let me also say um what's happens here so this here is geesecker derived dual so um here m sigma alpha beta is parameterizing them derived duals of geesecker stable geese right so if you take a vector bundle then the dual is and it's um a zero zero one so I mean when alpha is going to infinity then the then the kernel is just given by then the kernel of the alpha beta is just given by um vectors of the form zero zero one right and then there are there might be more of course there might be more walls down here okay and so in particular I mean it's it's an interesting question to ask you have this classical modular space of geesecker stable sheath and there's a wall crossing here what's happening okay so what I'll do in the remaining time today and tomorrow is basically give a survey of applications and um I mean I'll I'll touch on a few different things and so my I mean now my talk will be a little bit more um a survey start and the the first thing I'll I'll want to to say is to relate to something that came up in Sasha Kuznetsov lecture yesterday namely relating um geesecker stability um with um with quiver g i t Sasha explained is that an example but it turns out that stability condition also tell you um quite systematically what choices you can make and so let's let's say x is equal to p2 and then yesterday you saw that by um right by Baylor's on steam this is the derived category of p2 is isomorphic to the representations of this quiver with three vertices and three arrows like this from the first to the second from the second to the third modular some relations and maybe maybe let me make this relation a little bit more precise so right for the um for the I mean you're of course there are many such equivalences essentially depending on the choice of your exception collections um but right here there are always these um simple objects as one as two and three by which I mean the sample representations just given by one dimensional vector space at the corresponding vertex and zero everywhere here in sum of 40 standard exception collection you can say precisely what they correspond to they correspond to o of minus one shifted by two into omega one shifted by one and by all right and for every other exception collection okay it's easy to work out what these objects are they're always just given by the dual exception collection so for example right you could replace these by any um for example by tensoring all of them by the same line model but you can also apply these mutations and so on right and so we would so of course people knew since the 80s that you can describe um vector bundles on p2 using I mean even though they weren't saying it in this language at the time maybe using representations of these quivers but if say we want to know precisely which choices of exception collection and so which choices of derived equivalence like this would work for a given churn character and um and Gisika stability then the answer is that this is precisely described by by a bridge and stability right namely I mean we have no seen we have two ways to construct stability conditions on db of x right so first via this sigma alpha beta or co beta c alpha beta right just coming from this side as they derive category of coherent chiefs but and also by what I did in my first lecture um via quiver of representations of course the natural condition is when do they coincide and um and the simple lemma here is that if s1 s2 and s3 are sigma alpha beta stable with so that their faces are contained in a interval of length one for some phi in r then sigma alpha beta is isomorphic up to the gl2r action to a quiver stability condition right and I mean let me let me sketch the proof here right so I mean up to the up to the gl2r action we may of course assume that s1 s2 and s3 um have faces between 0 and 1 and what does this mean this means that they're contained in the heart associated to our associated to our stability condition right and so this means that now if you take the extension closure since this is an extension closed subcategory the extension closure of these three vertices is contained in a okay but what is this extension closure well we are looking at the quiver without without cycles so loops and so this means all representations can be written as extensions of the simple representations right so this is just isomorphic to my category of quiver representations and in particular this is this is I mean this is the heart of a bounded t structure and this is also the heart of a bounded t structure and I mean if it's a general fact if you have two bounded t structures and one heart is contained in the other then they have to be the same and then it just follows from the fact that the stability condition is uniquely determined by its heart and the central chart okay and now I mean of course um how does this help you well the point is that I mean these are all slope stable sheets in some particular they are they're stable at the large volume limit but you can also actually say a lot more you can you can really compute explicitly where the first wall is and I mean without going into the details of this computation the picture that you get is now the following so here you are what it turns out is that you can cover the entire boundary of this cone by regions that are that are described by quivery stability condition right so each such um segment that I'm drawing here there's the quiver stability conditions for a fixed quiver and the fixed exceptional collection right take I mean what I mean by this start with the balance on quiver mutated in some way you get this derived to a quiver lens between db of p2 and db of some quiver and then the region where this the assumptions of this lemma hold will be this given by this by a region like this in particular you can cover the entire boundary and in fact you can cover the entire boundary just by twists of the of the standard exception collection right and now let's look again at the picture we had the wall crossing picture we had here we have we and here we have some more the first wall and here you have the geyser got chamber right and then what this lemma is saying take any of the quivers occurring in this chamber and you'll you'll get that m h of v is equal to this g it modular space of quiver representations right and then it precisely it tells you which choices you can make just take any any choice so that this region of quivery stability condition intersects your intersects your geyser got chamber right so if you want to know exactly which exception collections you can use to describe one geyser can modular space then the answer is just take any exception collection so that the associated quivery region intersects the geyser got chamber of your given class right so here this would be the first one among all the set that that route that you could use and then you can use all the ones coming up here right and right and so you could use some up here or you could use some down here yeah exactly right so here from here you have lots of chambers quivery chambers that are somewhat connected volume mutations and I mean if one region is like this then the next one might look like this and like this and like this and I mean the statement is you can easily show that you can cover the entire boundary and so this in particular this means that any I mean any geyser got chamber will intersect many of these regions and so you have many possible choices for your exception collection so that your modular space of geyser can stable she is becomes just an ordinary modular space of quiver representations and your corresponding yeah I mean up up to this up to this yield to our twist right we need the the quivery heart will never be isomorphic to co beta but it will be isomorphic to a tilt of co beta okay and also note that I mean any wall crossing any so any right so I said that any MH of we is described by quiver representations but really any m sigma alpha beta of we and any wall crossing can be described by a j quiver GIT okay I'm out of time so let me stop here for today thanks