 Okay, so hopefully the recording started. Right, so welcome everybody to Gauss. I just wanted to say a couple of things before we get started here, just despite graduate students sort of being in the title, attendance is pretty much open. And people's backgrounds can vary quite a lot. So just like to ask people, you know, to be respectful to your friends and colleagues here. Please do ask questions to sort of in a traditional seminar format. The primary goal here is for this to be sort of a positive learning experience for everyone. Perhaps the easiest way to ask questions is to politely interrupt and just ask out loud and meet yourself. You can also I'll try to keep an eye on chat so you can just type your question in chat and I can try to relate to the speaker. I think that's pretty much it. So with that being said, it's my great pleasure to introduce Ben Tai from the University of Chicago, will be telling us about extending differential forms across singularities. Alright, thanks. Right, so maybe I'll start off by saying like what I do. And what I research is, you know, hodge theory and singularities of, you know, spaces that you might see in the first or second course in algebraic geometry and, you know, generalizations of these things. So the top will definitely be of this flavor but I hope that at least for the first half, maybe we can just kind of get introduced to some of the terminology and get our feet wet with some of the. Yeah, the language and the theorems that you might see in my field because hodge theory can be kind of a niche subject, especially when you start work working with singularities. Throughout though, to kind of get started. We're going to let X be one of these two things we're going to either let it be a projective variety over the complex numbers, which you've taken an algebraic geometry class you know this is just going to be like some subset of projective space cut out by polynomials. And otherwise, we're going to be working with like close complex analytics subspaces of complex projective space and I make these distinctions for the moment just because I want you to be able to, you know, hopefully you know, or comfortable with one or two of these things. You're either kind of comfortable with analytic spaces in like some Euclidean standard topology, that's where the complex analytics spaces are coming in this is where complex manifolds are. You're not comfortable with complex manifolds maybe you're comfortable with smooth manifolds, and the jump isn't that big. Or you're comfortable with just like algebraic geometry that you know you've done schemes but maybe you never took a smooth manifolds class. If you've not done either. This is going to be a very fun talk for you. So let's buckle in. And as we'll see later. Let me point out here that I make these distinctions. And kind of a kind of a winky kind of way because it turns out that for all intents and purposes these things are going to be the same objects that analytically or algebraically you know we're kind of going to be studying the same things and I'll mention that a little bit later. But if you aren't familiar with these subjects you don't do algebraic geometry here kind of some of the names of the objects that you might study in this field and those include elliptic curves. Riemann surfaces or algebraic curves, kind of going back to this thing that I'm alluding to that they're the same. K three surfaces clabby have varieties fan of varieties, etc, etc. So you might have heard these names before these are kind of the objects that I play around with in hodge theory and study specifically. Maybe you've heard these names before and this is kind of like the field where these things show up. Alright, before delving into the singular world maybe I'll start off by talking about what happens when things are nice and smooth. So for the moment we're going to let x just be a smooth projective variety, or in the analytic sense we're just going to let x be a projective complex manifold. So like us, you know, complex manifolds aren't too different from smooth manifolds they just are they had their locally CN for some end with some sort of rigid analytic structure. And so, as the title suggests we're going to be talking about differentials. And so let's try to make sense of what kind of differentials we can study in this world so a holomorphic one form on x is just an object omega. Yeah, that lives in this thing capital omega one, which locally looks like, you know, fi DZ, where fi is just a holomorphic function in the Z, or Zi or just holomorphic coordinates. Okay, so on your manifold you're going to have, you know, coordinates. So maybe I'll say a little bit more about that. So if x is a complex manifold, you're going to have real coordinates on here because see, you know, complex spaces also can be considered as a real to space and then so real, you know, complex and space can be considered as real to end space. Where each of these things can be written as like the real parts of some coordinate and like the imaginary part. Maybe I'll put some eyes here with I here also being the the imaginary unit. All right, and so you can think about a complex manifold as being have locally being coordinates and holomorphic coordinate Zi, maybe I'll call it Zi here holomorphic and Zi bar which are anti holomorphic. And so I'm really just looking at differentials which are attached to differentials of these objects. Okay, and a holomorphic P form just like in the smooth case, I can just take their wedge. I can take wedges of one form support P forms and they just are locally going to look like these things. And if you don't like, you know, if you don't know what locally means. All that means is that these objects are going to glue together well with the local structure on the manifold so kind of how you form partitions of unity and the smooth you know for smooth manifolds in order to get like differentials and gluing things together in the complex sense that makes you know sense as well and algebraically or you know even analytically the local condition is just saying that this object here, a mega P or a mega one upstairs is going to be a sheaf. You don't know what a sheaf is that's fine it's just something that you're able to you know you want these things to be able to glue together locally. Okay, and so maybe let's do an example, you know, to make sure that we, you know that this object isn't too scary. Let's look at x lambda let this be the elliptic curve which just locally looks like this equation so maybe you've seen this before. I'm letting this be a subset of C squared with coordinates x and y. So first you know we can compactify this just add a point that's what I'm talking about to get this x lambda, which sits inside p square. Okay. But since you know what we're talking about our local things you know let's just look at the affine equation because it's much easier. You can check since this is a polynomial that x lambda is smooth, except that these point zero one infinity infinity being this point we add to compactify. And because you know we're only working with two variables there's just going to be you know, pretty much one form one one form on this guy. Oh man which I'll call a mega lambda, which what do we do we just take the differential of each side, and we get this object DX over why, which is going to be DX over the square root of the guy on the right side. So what you can check is that this is holomorphic away from the bad points zero one infinity. And the point I'm trying to make here is that these are just what the objects are you're doing exactly what you would do in the smooth case. And what you're wanting to check now is that instead of it like being smooth or continuous. The adjective here is holomorphic. And here's a cute picture of how maybe Omega is related to constructing the elliptic curve and I get the tourists topologically take two paths cut these things along some branch cuts in red. This is the procedure to get this tortoise here this is what the space X Omega guy is here. And as it turns out this is kind of just an aside. Once you know, in this case, what a mega lambda is you actually know what X lambda is for lambda not equal to zero one or infinity but this is a this is just an aside. So what can we do with these things. Well, since we're working with complex coordinates, you know, we can take the form we can take its conjugate. We can wedge with these things, and we can integrate, because it's a smooth complex manifold so all these things make sense in the complex world. And going a little further if you've taken a smooth manifold class, it turns out that because this guy is holomorphic. And therefore defines a class in the Durham co homology with complex coefficients and this is defined the same way as it is. In a smooth manifold class, but now you're just emphasizing that it's you know complex thing. Okay, so again, what am I trying to say here I'm just trying to say that you know these objects shouldn't be too different from what you've studied before if you've seen them. Any questions before I move on. So this is one way of describing differentials in the analytic world. Now let's go to the algebraic world. Let's get a heart shorn and make sense of algebraic differentials, which we can also define locally. So if let me kind of define this as generally as I can for the moment. So let a be a C module, a derivation is a module homomorphism from a to m to a module and satisfying these usual conditions. You want your functions to vanish when you take the differential so for W and C, those guys are going to vanish. And then we apply D to it. It's linear in addition so DF plus G is equal to DF plus DG. And then it does this. This, this the Leibniz rule type thing in three that if you take multiplication it's going to be DFG is equal to FDG plus GDF so this is just what a derivation is and going upstairs in the holomorphic sense we want these things to hold as well. And so the definition slash theorem, which you would see in heart shorn is that the module algebraic differentials is kind of just like the unique object for which there is a universal differential differential. And so I'm going to avoid describing the universal property just for brevity it's not very hard. But there is kind of a universal object for which this makes sense. And a universal differential differential, but this is going to be exactly what you think it is. So it's kind of exactly this. Once we make sense of this for projective varieties once we move from modules projective varieties. It's going to be exactly what you think it is. One quick remark. So heart shorn calls these Taylor differentials. I am myself trying to avoid using Taylor, the name Taylor as much as I can. He wasn't the best guy in the world. Nazi sympathizer for many years after World War two. Unfortunately in my field, his name comes up almost all the time so it's hard to avoid it all together, but I figured, you know, Taylor differentials are the same as just like algebraic so why. Why use it when for something silly like that, but so I will be calling them algebraic differentials. Okay, and then. So what are differentials for projective varieties what are these algebraic differentials well basically you know you take X your projective variety. And then you just take the sheaf associated to this module downstairs so locally, you know, since your variety these guys look like you know, they look like a speck of some ring and if you don't know what spec is just think about this is like being some sort of nice module of polynomials modern ideal. Okay, so locally these things look like this. And some topology. And so locally the chief of algebraic differentials is just do this operation we were just describing upstairs take this universal object, and then she fire. Okay. So I'm being purposefully vague because what I'm about to say is that the out, you know, oversee the two, the two processes that I just described, very briefly the whole morphic and the algebraic world are going to be the same. So, if you've seen this already know what it is if you don't haven't seen it before. That's okay you probably understand this just well enough without needing the technical definition. So the algebraic differentials, while kind of nuance to define outright they do have have very nice properties so, for example, you've probably seen this in hard term before that if x is a projective variety, then access smooth if and only if this object. Omega Alge is a locally free sheaf of rank the dimension of x or you know in the analytic sense this is equivalent to this being a vector bundle of rank the dimension. So, if you're not smooth basically this thing is detecting where the singularities are. It's going to jump somehow. But this is a heart shorn thing. You don't need to define this oversee all these things algebraically make sense over a field over, you know, over us just for a scheme. But what I'm about to say for the most part, as I'm stating it really is only going to make sense for smooth things overseas. So, here are some big theorems where you see these differential show up. You might have heard of these before. They're used all the time in complex algebraic geometry they're very useful tools. So, let X be a smooth projective variety over C of dimension M, then we have these like four nice things so the first one is a variation of Chow's you know a version of Chow's theorem. And this is what I was alluding to before that analytic subspaces of complex projective space are algebraic. And so what do I mean by this, well, polynomial equations are holomorphic right. As long as they're smooth. They give us smooth equations. So yeah, if if I give you a series of polynomials and their Jacobian doesn't vanish, or doesn't drop in rank. These things are going to etch out holomorphic functions so what this is basically saying is that any analytic subspace acts of CPN. So that is going to be locally given by polynomial equations. And so in particular, we can make sense of identifying these two objects together. And I put this equivalence and quotation marks because well, if you know both of these subjects the analytic topology, considering these spaces is different from like the Zariski topology, the topology would study on schemes. So basically the formal way of constructing, you know, Chow's theorem is by constructing a functor from the algebraic category to the analytic category. But basically everything that you would want to be able to say about these sheaves is just going to be the same. Once you kind of forget the topology, which is hard for sheaves but that's kind of how I think about it. So I want to share duality, which does hold in any characteristic but how I'm stating it is actually how I'm stating it also holds in any characteristic at least for smooth things so let little omega x be the top wedge. The sheaf of holomorphic differentials here. So just take the top wedge of these guys so we're looking at m forms. If we have any free sheaf F or for any vector bundle. We have this identification between these co homology groups here. Okay. And so, in particular, we have this identification between the co homology groups of the omega p's and the omega and minus p's there's this duality between these guys. Yeah, so serduality is a very important theorem. The first place you would probably see applications of serduality are in heart shorn or Griffiths and Harris's book on complex algebraic geometry. Probably the easiest one would be proving like Riemann rock for curves is the first place it comes up or anything having to do with setting the geometry of complex surfaces. And serduality is one of those big theorems that heart shorn proves and it's used all the time and for calculations, especially if you're working with co homology. Another one is codire vanishing which only holds in characteristic zero in general. And this is a heart shorn book. So L is an ample line bundle which just means that it's a nice enough. A nice enough sheaf. Then we have this vanishing of this co homology group here that the HKF x omega x times L vanishes for K greater than or equal to zero. And then we have this object showing up in codire vanishing which has to do with the differentials that I was describing so somehow there's a relationship between differentials and like this, you know, these, these, their co homology. In particular, if I rewrite things, you know, because it's smooth all these omega p's are going to be invertible. So we're able to say that the co homologies vanish in some certain range when we just replace omega x with lower degree forms. Okay, so we have nice vanishing of these, the co homologies of these differentials codire vanishing. Where would you see that. So what I think shows up I believe in the proof of the podira and read guys classification of surfaces. It's used to classify what kind of surfaces show up in algebraic geometry. Which is probably what you would see in like a first or second course on complex outbreak geometry you probably see this classification somewhere. And then the last one is the Hodgsey composition probably the least known of the three if you're definitely if you're not doing this field, which says that for each K the singular co homology of acts can be decomposed into the co homology of these differentials. In particular, and this is the nice part that the complex conjugate of one gives you the co homology of another so the HQ acts of omega p is the HP acts of omega q. And so this really does only hold in characters to zero. This was observed. I think by hodge I can't remember if his proof was right on this, but the proof, you know uses harmonic analysis. If I've stated it, this is the only way you can prove the Hodgsey composition right now. There are algebraic proofs by Dillion and the Z proving that something like this is true. Using characteristic P methods, but you can't get quite the, this nice statement here, you can't get this complex conjugate condition which basically puts a can have a restraint on what the rank of the singular co homology of a complex variety can be. Okay. And so I probably just settle if you haven't ever seen these things before I probably just, you know, you probably just heard a lot of things that are confusing. But the big point here is that I have four very important theorems in complex algebraic geometry, and they all have to do with differential somehow. And, again, the ups, you know, I want to emphasize this only holds when X is smooth. And so that's kind of what I want to focus on and the rest of the talk is, you know, what happens when we start looking at singular things. Okay, so before I move on, are there any questions. My next question is, number one, is that essentially gaga principle or is that something. Yeah, gaga is his, I think, I think chow's theorem was proved independently like before gaga stairs gaga is more statement about saying that. Not only can you not only are complex projective, or sorry, complex analytic subspaces of projective space algebraic, but actually you know their co homology theories make sense to that the co you know taking the co homology of a coherent chief analytically is the same thing as taking like the co homology of its algebraic counterpart. This doesn't say that every sheaf on a complex analytic space is algebraic, but it says that you can take the analytification functor of an algebraic thing, make it analytic and taking those co homologies are the same. I forget what kind of functor it is it's a I forget what kind of what what what the words are describing the functor between these two guys but that's basically what gaga says I believe gaga does imply chow's theorem though. I'm sorry one more thing it's maybe not super related but if you just have a, like say a scheme over see can you still make sense of the, like the canonical, canonical bundle where you wedge up to the top power. Yeah, and I'll say something about that so there is a dualizing sheaf. You can make sense of a dualizing sheaf for for schemes as long as their color Macaulay, which is a very very kind of a pretty restrictive condition. Sorry, not a restrictive condition. It's kind of the bare minimum thing you need. But the dualizing property, you know, is not exactly in terms of co homology, but that's kind of what the rest of the talk is about like what kind of what kind of how nice of a statement can we make. What kind of thing Sarah duality could I revanishing etc. Yeah, good question. All right, so let's talk about differentials with singularities. And so I was emphasizing before that everything had to be smooth. So in particular if I'm working with a projective variety, which is, which has singularities, you know has pointy edges or you know, it's et cetera. Then the then the space is no longer a complex manifold. And in particular, we can't really make sense of holomorphic differentials on all of us. So what we want to do is kind of make try to find some sort of substitute. In particular, the four theorems I mentioned above all fail for different reasons when you have singularities. The algebraic differentials still make sense, but they can be very bad, you know, they're no longer vector bundles are locally free sheaves they. They are she's but they often have torsion, which is a very bad condition like we don't want to deal with things with torsion. Sarah duality and could I are vanishing fail as stated with respect to the algebraic differentials like taking their top wedge, and then the Hodges decomposition theorem fails. So for example, I write down here the main example. Or the quickest example is probably the nodal elliptic curve. You know, take this guy here. What you can show pretty easily just by drawing some pictures of some pinched Torah I is that the rank of this guy. I believe it's equal to three. And so, like I was saying before, the Hodges decomposition if the Hodges decomposition theorem like that held, it would follow by this like conjugate property I was describing that this would have to have even rank. So some things already failing for a very easy example here. So the main question here is that can we find a replacement of the algebraic differentials, which satisfies these properties you know we want it to be nice we want it to satisfy Sarah duality and put our vanishing and then place well possibly with Hodges theory. So here's kind of. And when I, what do I mean by that I mean that there are many different ways of generalizing one of these properties, not all matching. And so that's what we're going to kind of investigate. Now. So for now, let X be a projective variety we're going to assume it's normal. So norm normality just in particular means implies that the singularities aren't too big. We're going to be having singularities in codimension to our greater, and that access co in my colleague. And so if you've gone through hard short you know what this means. If not, this basically are the bare minimum requirements that we need in order to make sense of like generalizing things to the singular world. Okay. Okay, Omega P now just for the algebraic differentials. And like I was mentioning before, Omega P has torsion, you know, basically write down any curve. And unless you're extremely unlucky, right down something with with with a singularity a curve, and just take its differentials. It's very easy to find, you know, examples where you have zero divisors and you have torsion, which is very bad in practice for deterrent, you know, taking homology of these things. But one thing we can do is we can kill the torsion, you know, when one way of doing this is taking the reflexive whole so this is an algebraic fact that extends to the to the she fee scheming world that if I take the double dual of it's torsion, it's going to be torsion free. And I'll make a remark here that this process is different than just taking Omega P X, and then modding out by the torsion subgroup that these two things are in general different. This is also a process that we could take. It's actually kind of bad. It's not going to give us some nice properties which is why I'm just describing this one. We call the chief of reflexive differentials, because we're taking the reflexive whole is what it's called, but all we're doing is just taking like the double dual in the she fee way. So if you've thought about this in a linear algebra way or done the exercise and like a commutative algebra class where you take the double dual and show it's torsion free that's the exact same thing that's happening here. Okay, and the first lemma is that this is a pretty nice object that if you is the regular locus of acts, i.e. the set of points where X is smooth, which is an open set, and J is the inclusion. So one of reflexive differentials is just taking that Omega P guy on the regular locus on the smooth thing and then pushing it forward. So we could have gone this route to start out with we could have just started with a smooth thing, look at its differentials related to acts and then push forward, turns out it's the same thing as taking the reflexive whole. What's nice about this sheaf is that it is the perfect replacement for serduality that serduality holds as I have stated it for the most part, with respect to this dualizing sheaf here. That the reflexive differentials, the top reflexive differentials are the same as the dualizing sheaf. Omega apps. And the proof is really not that bad. And I'm just going to briefly describe what you need to check. What you need to do is just check that these guys are what are called S to sheaves, which is kind of a niche concept that you can find in heart shorn, you probably skip it. You can go through heart shorn yourself. But you know what does it mean to be S to it's basically just a generalization of this complex manifold concept of hard times theorem. And the upshot is that, because we're working with normal things and because we're going to tone my colleague things. We only need to check what's happening with these sheaves in co dimension greater than or equal to two. So this is this concept that like holomorphic functions extend across singularities and co dimension one that we can fill in holomorphic functions outside of like open on open sets to the whole thing. And then this is like I said, there are no co dimension one parts. That's the normality condition. And so we just need to check in co dimension greater than equal to two. But then it's really pretty easy to check that all these sheaves just agree with each other on the open set you, which is, you know, so that's pretty much the idea. The problem with this sheath though is that in general fails codire vanishing that if you just take, you know, there are examples where you can take a space X, a nice ample line bundle, and the sheaf and show that the co homology doesn't vanish so that this thing doesn't need to be zero for K greater than or equal to zero. And I lay out an example here by Polar. And this example is just taking like a nasty singularity and then take a nasty comb point or something it's it's a pretty easy example you just have to check that it that it feels it. Okay, so I have one sheaf. It does satisfy ser duality and doesn't in general satisfy codire vanishing. Okay, so maybe there's another method. Maybe there's another way of getting both. And the second method is similar to the lemma that we just described. Maybe we take something smooth, where we understand what the differentials look like, and we try to get that related to something on X. And that's exactly what we do here. So this, this process only works in character six zero because we're take we're going to take a resolution of singularities. So a resolution of singularities is just you know a birational model where why is smooth. And so we're going to start off with this is proved by here and not go that these things exist in characteristic zero. And so the nice thing is that the thing upstairs is torsion free right. So the omega p y is torsion free it's a it's a nice vector bundle. And so if I push forward the sheaf, I take pi and push forward the sheaf. This is also going to be torsion free. So we're already getting a nice object here. So the theorem by Brower and Reimann Schneider is that this is the object we want if we want to set you know study codire vanishing for singularities. The first thing is that with X and pi above so if I take X with a resolution of singularities, pi from y to X, then we have a relationship between these reflexive forms that the, well I write M here so maybe I should be a little more careful that this actually holds in general that the pi push forward of omega p, why upstairs injects into the thing I was just describing in the first method that the first sheaf I was describing includes this new sheaf. And for any ample line bundle X, L on X and for k greater than or equal to zero the second sheaf satisfies this codire vanishing property. Okay, so this sheaf is not too far off from the guy that satisfies ser duality and it satisfies codire vanishing. For the sake of time I will skip over the proof, because it's a little nuanced, and you can find it in like Polar and Maury's book on birational geometry but the idea is just that we know codire vanishing holds upstairs on the smooth thing of pi. That's what I was saying before. In the third theorem of the big four that I included. And what we want to do is to use codire vanishing upstairs to say something about codire vanishing downstairs. Okay, so that's basically the proof and you just have to use like higher direct images which no one really wants to play around with anyway. The first remark that I'll make, and I've already kind of alluded to it is that the second chief Omega X, this pie push forward. Omega top degree why fails ser duality in general. So in particular, this inclusion, which I was talking about of these sheaves that Brower and Reimann Schneider outlined is is not an isomorphism in general. So there's in general these two sheaves are going to be different and so we don't have a well that you know a singular sheaf necessarily that satisfies ser duality and codire vanishing. So a question you can ask and it's been a natural question that people have asked for years is, you know, when are these things the same. So when is this object. When is this inclusion morphism and isomorphism for each P. The problem is called the extension problem. So having such an isomorphism would imply that you know you have an object on your space X, which satisfies, you know, torsion free satisfies ser duality and satisfies codire vanishing so you have all of these hot ticket items to study in the singular world. And so we want to ask, when is this possible, when can we get an isomorphism here. And before I move on because I'm going to enter a new section. Are there any questions that I can answer. I'm wondering, is it expected that there would be one sheet that would ever do the job or Um, I don't think so in general. There are only really so many ways. Basically the only you know if you work with a con Macaulay thing, you have a dualizing sheaf. But if you just take a nasty enough thing you're going to show that that that thing doesn't satisfy codire vanishing. So I don't know if there's a great replacement for the dualizing sheaf that is canonically constructed and like heart shorn. That would be my issue with that. There are other ways of constructing differentials I think this has been studied before, but I also don't think they can. These other constructions hold in general either so yeah I don't think it's certainly not a way of constructing this guy. An object which does all of these things that you wanted to. Without making some assumptions on the singularities. And so that's kind of what the last part of my talk is going to be about is how this extension problem plays with singularities that you might have heard about in algebraic geometry talks. So singularities are kind of nasty objects that can be arbitrarily bad and therefore like their theory can also be arbitrarily bad. Which is why mathematicians for years now have kind of focused on certain kinds of singularities that show up in the minimal model program. And so I thought it would be interesting just to review what these guys are, because you've probably heard their names if you've ever been to an algebraic geometry talk, or go to algebraic geometry talks, maybe you don't know what they are. So in addition, let X be a normal projective variety, let pi from why the X be a resolution of singularities with, you know, an exceptional divisor E. So what does an exceptional divisor mean. This is just, I guess this is kind of an assumption. But it's the inverse image of the singular locus of X. So I'm assuming that when you resolve these guys and here in office of this is possible. The inverse image of this guy is going to be in a divisor upstairs. And you can actually also show that it's what's called a simple normal crossing divisor, which means that the singularities of this divisor upstairs, you know they cross like this. Okay, and that is not going to come up at all but I might say it just to be technically correct. Okay, and so we can write the canonical divisors or the canonical bundles of these guys upstairs in terms of something downstairs or downstairs in terms of something upstairs so you know because I'm assuming that this is normal, we can write the canonical divisor upstairs which is just you know the divisor associated with omega y as the pullback of the thing downstairs, plus multiples of these components EI. Okay. So I'm able to do this under some sort of linear, you know, q linear equivalence tilde q. And so I won't go into what that is right now. But basically, I'm able to write it like this. And then we say that X is terminal, if all of these ai's which are just rational numbers are greater than or greater than zero. We say they're canonical if the AI are greater than or equal to zero. We say they're log terminal if the AI are greater than minus one and we say that they're log canonical if the AI are greater than or equal to minus one. Okay, these are by rational, these definitions make sense by rationally. And the term log comes from the fact that when you have a minus one here. So if you have an AI, you know, for example, are equal to minus one or approaching minus one. That means that your divisors just kind of have polls of that order along the singularities. And that's what kind of, that's what log polls tend to look like with Merrimorker functions on like complex analytics basis for example so that's kind of where that name is coming from. So the most studied in the MMP. These are probably, you know, if you go to any talk dealing with singularities. Having to do with the minimal model program these are the most common, but they're not the only ones and they're actually not the ones I typically study. I'm going to study these next objects. We say that X has rational singularities. If the push forward of the structure sheaf oh why upstairs is equal to the structure sheaf downstairs. This just means that the functions use the holomorphic functions you see upstairs pushed down to the algebraic functions you see downstairs, and that these higher direct image sheaves vanish on oh why. So this is kind of a technical condition. For example, this tells you that the co homologies upstairs are equal to the co homologies downstairs. For example this is kind of a nice technical property that you might want when studying singularities. And then the third one is actually kind of a theorem because the, you know, their definition is so very difficult to describe, but you might have also heard their name, we say that axis of is Dubois, or has Dubois singularities. If and only if this natural morphism which takes place in, you know, the morphisms in the derived category of acts is a quasi isomorphism. I'm not using this definition at all in the talk I'm just saying you're trying to describe what Dubois singularities are, just so that you have an idea about like where they show up if you go to a talk on these things. Dubois singularities are probably the most general singularities you might see in the MMP. They're the most most actively studied ones right now. Because of the following implications. So we have this nice little chart here of what singularities imply the other. So, as you can tell in the, by the definition of like terminal conical log terminal log conical going upstairs that there is an implication one, you know, going this way, clearly. The relationship between the others which is, you know, was for a while a very hard thing to show that terminal implies canonical implies log terminal implies log canonical but then these things imply each other, and then these things imply each other. For instance, so there's a relationship between these things and then kind of at the bottom are Dubois singularities, which is why I'm saying that kind of the most general singularities he might study in the MMP. Right. And so there are implications going the other way. Once you add some conditions, for example, I think if you put a certain adjective here, you have that these are the same, and you can put another adjective here and I forget which one it is but I believe that Bornstein is enough. Bornstein plus like weekly normal or something. Okay. And so these are the singularities showing up in the MMP. And so we can ask, you know, the extension problem play, you know, if I assume that, you know, these guys, you know, my, my space x has at worst to these, one of these singularities. Can I get an isomorphism between the sheets I was describing above these reflexive differentials or you know these push forward from the resolution. And so the first theorem is that if X has rational singularities so I'm in this situation right here, then the top degree guys. The push forward of the dualizing sheaf upstairs is isomorphic to the push forward of the top degree differentials on the regular locus. Okay, so the extension problem holds so in particular for rational singularities. Say our duality and codire vanishing hold, and you can tell this is a pretty general space rational singularities are probably the second nicest kind of singularities you can play around with. In my mind and they're kind of the singularities you want to do in wants to play around with if you're doing hodge theory. So proof of this you can see in Clara Maurice book book it's not very difficult but it's a I feel like a little technical for to give during the talk. Okay, here's another theorem this is by grab. This covac and Petter now in 2011 and in their like big paper that they all kind of collaborated on it. MSRI. If X has log terminal singularities then this isomorphism holds for each P. So in particular, not only do podire vanishing and serduality hold for spaces with log terminal singularities, but this isomorphism holds for every sequence of variables, you know, for P lesser equal to the top dimension. Okay. So this was, this is kind of an older result at this point, about 10 years. But this doesn't say, you know, so I had these two results one about rational singularities and one about log terminal singularities. And remember a lot of terminal singularities imply rational but not you know, the theorem by grab Kevin Kish covac and Petter now don't say anything about rational singularities but this was resolved by Kevin Kish and Chanel in 2018 that if X has rational singularities, then be this isomorphism holds for each P lesser equal to the dimension. Okay. The proof is pretty cute. You know, it's the, it's the entire pretty much the entirety of their 40 to 50 page paper, but they basically use cytos theory of mixed hard modules to show that it's enough to reduce to this case, in which it's already known. Yeah, it's a, it's a bonkers cool paper, using a very technical result to prove some, you know, reduce it to a very easy problem. And kind of reading the past couple of months which is why I'm giving the stock. Okay, but you know let's look at the MMP. So what have we covered we've said we've shown that the X, you know, we've discussed that the extension problem basically holds for these guys here, which means it holds for these guys here. What are we left with where we're left with like log canonical and the boss singularities. And I've already said that there are examples where code that you know the extension problem fails that they're that these two sheaves aren't isomorphic to each other. And that actually takes place. That's the level of the boss singularities so for the boss singularities, we're not going to get such a nice theorem, but we can say something slightly related. And so this is a theorem of Kovac and Schwede and Smith from 2007 that if X is a column Akali space and X has the X has the boss singularities if and only if this version of the extension problem with logarithmic singularities. And that injection is an isomorphism. So, again, I didn't talk about what like this log E thing is here, and I won't really say it right now. But basically I'm saying that there are versions of the extension problem with which hold for the boss singularities, except that this she here doesn't really satisfy codire vanishing or anything so it's independent in its own right you know it's it's interesting in its own right to consider when this thing is an isomorphism but it doesn't quite do what we want. So the boss singularities yeah it's a little up in the air how nice of a theory we can get. And Kevin kitchen show also showed that in general if we have one implication, you know, if the top degree thing holds, so we're in the DuBois setting that it actually holds for all kinds, every kind of differential for all p so for all these logarithmic p differentials and then these are holomorphic p differentials on the regular local so the similar statement that I was staying up above. So let's get to this next part and end with just some applications of the extension theorem in the literature. The extension theorem has been used a bunch of times to prove, you know, in recent years to prove some pretty interesting results. The advisor and his co author Christian Chanel use the extension problem the fact that this guy can be an isomorphism for each p for rational to singularities to study the hodge theory deformation theory and some local and global theorems for singular p varieties. So this is their papers from. I think they're put on the archive in 2016 and 2018. I've also this is kind of something I'm wrapping up on right now I'm also doing something similar for some plectic varieties and the extension theorem. Studying the intersection co homology of these objects. So I hope to have that up on the archive in the next month or two. And you know the extension problems also been used to study the bowville the gamma love decomposition for K trivial varieties. Which is an interesting theorem, it's you know this is an older result in the smooth manifold K or the complex manifold case which says that anything with like trivial canonical bundle can be covered by collabia is Torah or plectic varieties. And this has been analogs of this have been proven in the singular case by various authors. I think my, for example, my my advisor and his co authors have kind of put the finishing touches on versions of this theorem for non projective things which I think for the most part has wrapped up the entire problem. I'm extending it to certain singularities. And then finally there's this litman czariski conjecture which is actually very easy to state, which uses the extension theorem, and it's still an open conjecture in general. And this has been for about 50 years. That if the tangent chief, which is just the dual of the chief of differentials. If this guy's locally free the next of smooth. And so this is kind of you know an old conjecture has been studying many different cases. And just because I'm running out of time, I'll just state a version of the theorem which is a little bit different from what we have to do with the extension problem. Suppose that the natural morphism between these two objects, omega push forward, sorry pipe push forward of omega one into J push forward of omega one x rag is an isomorphism so the extension problem holds for for one forms. The czariski litman conjecture holds that if tx is locally free then x is smooth. And so I'll skip over the proof for now maybe I can return to it if people are interested, but the upshot here is that the czariski litman conjecture is, I believe completely solved for KLT singularities, and rational singularities by this paper I was talking about of Greg Kevakesh, Kovac and Fennel, and rational singularities as well by Kevakesh and Schnell. And then it's also been studied in, you know, by Graf and Kovac for log canonical singularities which it turns out is equivalent to the problem for Duvall singularities as well so singularities showing up in the MMP problem the czariski litman conjecture this very easy thing the state has been wrapped up at this point, but it's still an open problem in general. All right. Thank you guys for enter, you know, listening. Sorry. Take that for a wonderful talk. Does anybody have any questions? I have a vague question, which like I barely understand myself, the question so like I'm not expecting an answer necessarily but like, is there a kind of approach to all this kind of stuff or instead of using like the complex differential forms you use the whole like cotangent complex like this whole like derived algebraic geometry approach that's you know so fancy these days like is there some similar story that happens there where this can detect smoothness does this like solve any problems, or is it totally yeah maybe I'll lightly point you to maybe I won't scroll up maybe I'll just write under here. So the definition of Duvall singularities. The construction of Duvall singularities was done in like the 80s. We generalize the Durham complex for for smooth and complex manifolds. And the reason that I didn't bring up its definition is because it depends on a hyper resolution of some official schemes of a space, and with like 10 conditions or something on like the graded components of this complex, which is why I just brought up this this theorem of Karl Schwede because it's very easy to stay and it doesn't depend on a hyper resolution. So yeah the Duvall complex, you know, I forget what it how people write it, I think it's just saying something like this in the literature. So this is how you know what it was and is how people studied, you know, singularities, kind of in this derived complex way. Another way you can study singularities in this derived category world is the intersection complex, which maybe you know about. This is kind of how Kedakish and Schnell prove their extension theorems. They turn the problem of extending these differentials across singularities to studying the Durham complex of like the underlying D module on the intersection complex ICX. You know, for a complex space, you know you kind of look at the graded pieces of this guy, and what you can do and what you can show is that there are some other pieces if you know what the decomposition theorem is. We have some other pieces which are supported on the singular locus, and that these are isomorphic to the higher direct push forward sheaves of on a resolution. And the upshot here is that if I take you know a certain she fee co homology of this thing, then I can relate these push forward guys to certain she fee co homologies of these graded pieces in this complex. And as it'll turn out, we, that's it. We don't get these guys supported on the singular locus and so what they do is that they choose this decomposition they use this identification to show that extending singularities this way is equivalent to looking at the dim support of this guy and showing that this actually holds her rational singularities. It's a pretty cute way, but it does definitely use like you know there's a derived way of studying these guys using the decomposition theorem for hodge modules. Okay, that's great. Thanks. Maybe this is something, maybe you mentioned it earlier but in this situation, do you have analogs of the hodge decomposition and filtration on that. Some parts to go through. Yeah, so there are ways of extending the hodge decomposition theorem. I just when I was writing this up I just decided to avoid it because it was just another kind of list of theorems to bring up but basically hodge theory makes sense. Orbefolds. The things which are locally quotients. They also kind of make sense for rational singularities, like, as I've stated, the hodge decomposition theorem. He also makes sense for co homology in rank. And they're equal to which is actually a very nice thing to have because the h2 kind of contains the line bundles and the ample bundles and stuff so having a decomposition like that is extremely useful. But then also just in general, there's this idea of mixed hodge structures. And this was Delene's thesis, I believe. Where he showed that any the co homology of any space, any any projective space I think after you already but basically anything that emits a resolution of singularities emits a mixed hodge structure. So it's not as nice as a pure hodge structure, you're not going to get as nice of a decomposition. But basically it's saying that there's like two filtrations you have the hodge filtration going one way and you have the weight filtration going the other way. You can form like graded components on the weight filtration on the on the co homology, and these things form pure hodge structures so it just says that like, you take certain graded components of the co homology, then this can this has a decomposition this has a this is a pure hodge structure. Yeah. But a lot of the time you know there is a pure hodge structure on these guys so for example if you don't want to study co homology you can study intersection co homology and it turns out that for projective guys this is these things are always pure hodge structures. Any, any other questions. So if not, let's thank them once more. I'll stop recording.