 Okay, so thanks. So we start. So I wanted you to say a couple of comments about the lecture on Monday. So I didn't cover very many topics. There was cotangent lines and deformation theory. And then I gave a sketch of a proof of nefness of these cotangent line bundles that use the hodge index theorem. And so there's a question about references for this. I basically gave the skeleton of this argument. If you really want to look at all the details and you have to do them yourselves and as far as a reference, I tried to try to find what I mean from this argument is an old argument. It's probably goes under the rubric of well known, but I don't know whether it's there's been written down carefully somewhere if someone knows please tell me. But anyway, I gave you the structure of that argument and it's if you want to make it. If you want to completely finish it it's true you have to consider something about singularities because hodge index theorem is for smooth surfaces. But the singularities are so minor you can just resolve them and also the universal this family of universal this one dimensional family of curves over the base can have different components and you have to then take them apart and apply it to each one. Anyway, so I leave that to you but if someone knows where it's been written down somewhere carefully, please tell me. So the final topic was about these descendant integrals, and I gave you two ways as to compute them. Katie V and Versaro, and what I failed to mention is that so these are very effective and in fact, they, they, in some sense, the availability of these methods compute change the subject in the sense that one can really do computations and this is in the module space of curves rather efficiently, effectively. And the thing that I wanted to mention here is that there's some computer programs that are relevant and this the one that I recommend looking at is this program called ADM cycles and there's a bunch of co authors but you can find this on the web page of Johannes Schmidt at Bond so I leave that to you to look at. So, today we're going to talk about well so yesterday was about modular curves. And this was in some sense one dimensional lecture, and today we'll be talking about modular stable maps, and the dimension increases because we will take the target of the maps to be any non singing the project of variety. So this is now we include in some sense all dimension targets here. So we'll consider maps. So, and the map will typically be called F. And it'll have a domain and the domain is a nodal curve as, as we discussed yesterday, complete and connected and the genus is defined from the holomorphic or the characteristic of oh as usual. And the thing about this map is the domain is a nodal curve. And the target is non singular. Those are going to both be relevant for the virtual class and the deformation theory. And for the discrete invariance of the map of course there's the genus, that's the about the domain but then there's another discrete invariant which is the class represented by the map so if I take the fundamental class of the curve. So this class is a class in H two of the curve. I can push it forward to get a class in each two of the target with integer coefficients this class will typically be called beta. And it's another discrete invariance was that these are the discrete invariance here. And then the object for us today is the modular space of stable maps. So this is the terminology and there's the genus, and then there's the beta, the target, and this bar says it's stable maps. And this is a generalization of the modular space of stable curves we saw yesterday. And I just want to say a few words about the definitions not many. The first question of course is if I have such a modular point. When is this map stable what does it mean to be stable. And the answer is that it's very simple is just we have to look at the automorphisms group of this map. The automorphism group of this data, which I call just the automorphism group of F and that should be finite. And so the question is what is an automorphism and automorphism map is an automorphism of the domain which commutes with F. So in particular the automorphism group of a map lives inside the automorphism group of the curve itself. So if you if you're fortunate enough to start with a curve with a finite automorphism group, then the map is automatically automorph is automatically stable. This is a different way of talking about stability, but it does reduce to the same definitions for a stable curve. So that was rather fast. And while we're discussing the basics of stable maps and other issue which is good to be clear about is when are two stable maps isomorphic. And that is, so there exists for that for this, the data of the two maps to be isomorphic. What we mean by that isomorphism is that there is an isomorphism of the domains which commutes with the maps. And so these definitions are all parallel. Well, they're, as I said one can, if you look at these definitions and if you haven't thought a lot about the model space of stable curves and you can go back and see that they agree with the standard ways of thinking about the model space of stable curves. It's just more general. And then there's another twist is that we want to put in these mark points. And the definitions here are also parallel. There's an issue. We want now to consider automorphisms and isomorphism which would respect the markings. Okay, so that's the discussion about what it means to be a stable map and the modular space of stable maps, and it's a result which is not entirely well. It's maybe not entirely obvious, but it's true is that if under this definition of a stable map the actual modular spaces are compact, they're proper to lean on for stacks. But they could be bad from the point of view of singularities that could be reducible non reduced and very singular. So in this way they differ from the modular space of curves because we know that MG bar for example is non singular of pure dimension. And that's not true for these spaces of maps. Okay, so the first examples. The simplest example to think about our maps where this this is the beta beta is equal to zeros that's constant maps. So constant map of course is how do you define a constant map will you have to pick a point in X where the map goes. And then there's the domain of the map which is MG and bar in this case when the map is constant. The domain has to be a stable curve. So that's the very first example. And it looks like a very silly example. And in some geometric sense it is kind of silly. You have a modular space of maps to a point. It's just a product of the domain and the and x, but from the point of view of stable of the deformation theory it turns out it's not as quite as silly as it looks. And the deformation leads to a virtual class and the virtual class on this modular space here is well it's some object which is not exactly trivial so I'll mention it later. So in some sense, the next example is actually a little bit easier but it's the it's the one where I take the maybe the simplest possible target space which is not a point projective space. So I have maps that are Gina zero maps, and I map them to degree one that's the class of the line. And why this is an easy example is because such a space of maps. As you've already encountered it is the gross money and it's a gross money of lines and PN, and the virtual class here will be the same as the fundamental class so in some sense this example is actually easier. This degree one example is actually easier than this constant map example. And another example from the point which is natural from the point of classical algebraic geometry if you have that background is if I'd look at Gina zero maps with no mark points to P2. So in degree two that means the conics. And there's a lot of, there's a lot of ways one can think about conics I mean the simplest in some sense space. So the conics are given are given by equations quadratic equations and three variables there's six coefficients which gives a projective space p five. That's one of the ways to think about conics that's not what happens with this modular space. The modular space actually gives the classical space of complete conics. So the complete conics are a nice smooth conic like this, also a reducible conic, but when you get to the double line the complete conics. Pick two, that's the data of two points on that double line and that, and that two points is from the point of stable maps, picking a ramified cover of that line. So this is just to say that this modular space is isomorphic the classical space of complete conics. There's some stacky issues here which I am ignoring. So the very correct way to say it is the course modular space of stable maps is isomorphic the classical space of complete conics. Okay, so those are some examples. So now the obstruction theory and it's the one of the really important parts of the definition is that the stable maps carry a deformation obstruction theory that's two term and therefore there's a virtual fundamental class and that's the somehow foundations of Gromov-Witton theory. And what the virtual dimension of this space will be is given by very simple formulas that's given here it's given by the holomorphic or the characteristics to pull back to the tangent bundle and that's given by a Riemann rock turn that's this term. And then I have to add to that the dimension of the domain, which is given by the formula for the dimension of mg and bar so this all of this together is a dimension formula, and it's a pretty simple formula. So, when one confronts this modular space of stable maps, there could be a lot of mysteries but one of them, one of the things that's not a mystery is its virtual dimension. It's given by this rather simple formula. And about the deformation obstruction theory. Again, I don't say so much actually Richard gave some lectures which involved a fair amount about the virtual class. So I hope you went to those. If not then perhaps you just take the deformation theory as given here. But the way this deformation theory works with stable maps is that there's a first the theory for a fixed domain curve. And the question is how can you deform a curve in a space and that's given by H naught of the pullback sections of the pullback to tangent bundle. And the obstructions under H one and there's no higher obstructions because it's a curve so there's no higher comology groups. And this this leads to a two term obstruction theory for a fixed domain but we're not interested in a fixed domain. We're interested in the domain possibly varying in modular. So we have a fixed domain this gives some deformation obstruction theory maybe it's better not to put bar because it's a fixed domain as a deformation obstruction theory of this virtual dimension, but we're not interested in the fixed domain and this is a wrinkle that you see on the curve side that you see in the deformation theory of sheafside so the natural way to do it is to first define this deformation theory for a fixed a fixed domain, and then view that as a relative deformation theory over the art and stack of nodal curves that's what I've written here that's why I tried to make this kind of a bold M. And you get, if you put that together, because this art and stack is non singular you get an actual deformation obstruction theory for the module space of stable maps. And this is a, in some sense the definition of the stability was proposed by can save it. And this argument for obtaining this deformation obstruction theory was given by Baron and Fantechi and slightly different way by lean Tian. And these things are now a little bit old, I think it's probably getting close to 25 years old or something like that. But that's the foundation somehow the very first underlying structure and growing up within theory is that you take any non singular projective variety X. Consider this modular space of maps for any genus any and any curve class. And this modular space is a Deline Mumford stack. And it has a good deformation obstruction theory which is given, more or less by H naught and H one of the tangent bundle up to this relative part. Using that you get a cycle on the modular space of stable maps of the virtual dimension. So now we can return to the constant map, the one that I said was the, the zero example. And, and while the modular space of constant maps, which is here is kind of trivial. Then there's the issue about what is the, what is the virtual class and that that is not exactly complete trivial statement with the virtual classes. It's not hard to compute with the virtual classes just use this definition, the obstructions come in this H one. And at the, this modular space is smooth because the modular space of curves is smooth and access smooth. And in the case the modular space is smooth, but you can see it's of the wrong dimension when it's moved the wrong dimension. The virtual class is the top turn class of a certain bundle. So in this case, the virtual class is the top turn class of a bundle and what bundle is it. It's the bundle on MG side which is the dual of the hodge bundle. And the hodge bundle is the bundle with fibers given by the vector space of holomorphic differential form so it's a ranked G bundle that's why the G is here. So that's a rather interesting bundle on the modular space of curves. And in order to get the virtual class, you must take this hodge bundle dualize it, and then tensor with the tangent bundle on the X side and take the top turn class. And so the answer is, you know, somewhat interesting thing. You have a question about this section the section question is, what is the map from deformations to obstructions of a fixed curve. And so the way that, you know, the way that obstruction theory works is that there's a, there's a space, you know, there's the obstruction space. You should view this as being cutting out inside the, you know, in some sense that there's this tangent space which is this. The obstructions cut out some modular space inside it. So when this, when this vector space is large that you have to say it's too big you view that the obstruction theory is locally given by some vector space. That's H naught and obstructions give you some equations that cut it out. I'm not sure what the question is exactly. If you want the development of this obstruction theory in general, you have to, you know, to construct this maybe the other way to answer this question is how do you construct such an obstruction theory. And this comes from looking at more or less the differential and you take this curve. And there's the modular space, and it maps to X. And that this obstruction theory comes from taking the pullback of, you know, you take F. R pi lower star of F upper star where pi is the map to M. So there's this complex and this deformation theory that the somehow the action part of deformation theory is that you have to find a map from this to the cotangent complex. Well, you take the dual of this deformation theory is some somebody that maps the cotangent complex of M. That whole map can be constructed here by total logical, you know, by total logical structures you take this curve cross the modular space maps to M to X, then just looking at the differentials and total logical structures you can make a map from the dual of this to the cotangent complex of them, but it's not completely trivial to prove that has the properties of the deferment of, you know, to to to check this is actually a two term. Perfect obstruction theory you have to check some things. I have to check that the maps and isomorphism in H zero and it's subjective in each one. And these things are not completely trivial to check here. So that, but in this case of a fixed curve. This was not done by Baron and Fantechi but it has a older. It has an older history and deformation theory. So, I can try to look up some references for that but they are you can also find them in the Baron and Fantechi paper, and also this total logical construction. Actually when you see it it seems like almost nothing's happening, but it is non trivial. Okay, so maybe I think that the somehow the heart of that question is probably what is the map to this cotangent complex. And as I said this comes from just looking at the differential here and pushing forward and pulling back various things. I hope that's clear. You know the cotangent complex is like the cotangent bundle so it receives a map and the differences. Okay, so. So where was the virtual class of the constant maps. So we discussed that so maps to P one so that in some sense that the last lecture on Monday was about maps to a point. That's the modular space of curves. And we discussed the cotangent lines, and up to Vera sorrow there. So the next, the target with the next complexity is maps to P one. So we're growing in dimension here and the map the modular space of maps to P one if you use that virtual dimension formula that the modular spaces, the dimension is that. And then what is a general map look like. This is a good thing to keep in mind like what does a what is a stable map to P one look like. And the answer is that, well, the domain. We know not so much about the domain except that it's a connected nodal curve so I've kind of drawn a connected nodal curve, and it must map to P one. So, you can draw that as it lying somehow lying above P one, but there's different kinds of components. So there could be components that dominate P one. But there's all it's also permitted that components are contracted. So the general map might look something like this where I have some components that that multiple that multiple cover P one. And they are attached on nodes and then there's some other components that are contracted and the contracted components can dangle like this or they could create bridges. This is roughly speaking what what a stable map looks like. So the kind of ideal map would look like actually just some curve, some irreducible non singular curve mapping to P one that somehow the ideal map, but the stable map since the compact space has to involve some types of degenerations. And so there's various types of degenerations. One is that the domain can acquire a node. And the second in some sense these contracted components can be viewed as a slightly singular behavior also. We can connect grandma Witten theory to her which is older enumerative geometry of maps to P one and this turns out to be a fruitful direction to go and I wanted to just show how how that part of the theory looks before we move on to Sarah. So her wits is older theory of enumerative geometry her wits had a theory of enumerating ramified covers with with specified branch points. This is a theory that he developed at the end of the 19th century. This is 19th century. One can wonder whether this new theory of the virtual class of maps to P one. So this is somehow a new theory that's in the developed at the end of the 20th century in some sense. What, what is it have any connection to her wits is counting of curves from the 19th century and the answer is yes and the connection is actually is pretty nice. So to get this connection there's a geometric thing one has to do so her wits. So here's P one. So what her wits, the theory her wits define is you take P one, and you pick a point on P one P, and you insist that your curve upstairs has a simple branch point of P. I mean this is the very first. The very first version of her wits is theory. So you insist that their curve upstairs has a simple branch point which I've tried to draw like this. So it's a simple branch point it means it looks like something like z goes to z squared around there, and her wits the her wits problem is then you fix lots of these points, and you ask always the upstairs curve to be simply branched, and then in the degree and if all those are fixed, you can ask how many solutions there are that's an enumerative problem, and her wits solves this problem in terms of the symmetric group and there's some story there which we're not going to go into today. But that's the her wits problem and you can say how is this related to the Gromov within Gromov within theory, and that the issue is that we have to find out how to get this critical point in Gromov within theory how to get this ramification point and there's a very nice answer because it's going to it's going to be related to the cotangent line, because the existence of this critical point has to do with some differential being zero and and what is that differential well the differential is a map from the tangent line of the curve to the tangent line of the target. Yeah, outcome is that if I look at this class which is the first churn class of the cotangent line that's the side class, and this evaluation class at the point. This is something like the imposition of a ramification condition at P. What is that. This is what I said if I have so let's say let's call this point upstairs x and the downstairs point P that that if I have a stable map, then there's a differential from the tangent space at x to the tangent space at the image of x, and such a thing can be written as a section. We have this this differential everywhere it's actually a section of the dual of the cotangent of the dual of the tangent line tensor the pullback of the pullback of the tangent bundle of P one so this is a line bundle. And the dual of the tangent line of course is the cotangent line. So this differential is giving us a section of the cotangent line tensor the pullback of the tangent bundle of P one. So if we freeze the point like her what's wants to do and we can do ourselves which is this evaluation map, we can freeze the location of that of x the image of x. If we do that, then the tangent bundle here becomes trivial because it's just the tangent tangent line at that point P in the target. So, the vanishing of DF is represented represented as a cycle class by this cotangent line. And this evaluation inverse image of a point. So this is how to get that her what's condition and grandma wouldn't theory and to do it, you have to involve cotangent line. The first result there is, this is kind of an early result about this, the most basic version of this Chrome of it and her what's correspondence. So that's an old paper which was written with the beginning of the subject so it's actually pretty easy to read I think that says that if you want to calculate her what's his number. That's this genius G degree D covers of P one with specified simple branching. Zero here is about connected covers. Oh, by the way, I, you know, Georg made this comment about whether my positivity that I was writing was for connected or disconnected I added some some slides to the last, the notes for lecture one to help I think that make that more clear, but this zero here is to say connected covers. And this is just an equality it says if you want to know her what's number. Then that's actually a specific integral over the modular space of stable maps and you have to have a lot of mark points well that's that's every one point for every branch point that in her what's his problem that's why there's that so many of these this is the branch branch number. And then for every one of these mark points, you put the cotangent line and this evaluation at P. So in some sense this formula translates old and new there's this this is the old and this is the new. But it's actually, well, we, we will we used it in various ways I don't want to say so much about it. But now you can ask have I proved this formula in this lecture, because I gave you the slide explaining how to realize a ramification condition as a intersection of a descendant here. And you could say, Well, have I proved this and the answer is kind of close but not exactly. And that what you have to do is to show that the intersections on the left. Can avoid all pathologies of modular space because in the on the on this side there's no. Her which is only counting these ideal configurations that I drew here her which is counting these ideal configurations, while stable maps have have all these more pathological more pathological configurations and to actually give a complete proof of this you have to show that the cycle on the left hand side is avoiding all of the pathologies and once it's in the good locus, then the argument I gave is rigorous argument. And avoid all the pathologies. There are some other geometric structures that can help so one of them is this a branch morphism. So there's actually a way to keep track of the branch points even though the domains can become complicated and this was developed in paper with Barbara Fantechi. And if you're interested in that, I urge you to go look at those. So I'm not going to say much more about that but this is an exact equality between these two different worlds. So this again brings I should say this also says that even though maybe we weren't interested in these cotangent line classes or they weren't here. They come by themselves because they just come geometrically whenever you're studying more or less modular curves, you can't avoid these cotangent line classes. And that's why the witness conjecture is so important for the subject. Or at least one of the reasons. Okay, so Yeah, how far is this formula from us be formula. Can you keep a proof to Yeah, it's not it's not it's not incredibly far and particularly in that paper. When one line this paper with Barbara which is about when I'm certainly independent as well you can just if you do this. So the LSE formula has not only D but a new here right. Right. Yeah. So this if you want to prove the ELSV formula for one to the D this basically gives you gives you the proof of that by applying localization the left hand side. I mean that this is explained you only you have to use the branch morphism. If you want to apply to prove ELSV for a new here. Then you should translate these ideas into relative grammar within theory. And this was explained by Graeber and Vakil, but yeah not not far is the answer. That's a good question. Okay, so I wanted to give some examples some some. I wanted to say a little bit about the descendant notation which we discussed a little bit for the modular space of curves. But we're going to now need to discuss them also now for this, the descendants in grandma Witten theory on the modular space of maps. And so there's a certain amount of notation here that actually nothing's happening this notation but if you haven't seen it it's a little confusing for the first time. And so the idea of this notation is to use the bracket to write for write this integral, and you could say well why do we want to do that well. You know this integral is a little bit more cum cumbersome so it's slightly more efficient, but the what the main kind of nice thing about this bracket notation is that it builds in the points. There's a SN action that acts on on these pointed spaces by renumbering the points. And for the most for the most part, unless you're dealing with some odd comology and there's some signs which we're not going to talk about now it doesn't really matter. So in this notation here, there's some ways to this notation by naming all the points and this. This descendant notation here somehow freeze us from that. So this is some maybe some small some small advertisement of efficiency of it but the way this thing works is that when you see the symbol tau k gamma. What that means is that that symbol creates a mark point, and it puts it that mark point, the case power of the descendant and the evaluation of that mark point at gamma. So maybe I didn't say what that is but I should say, maybe it's clear that if I take this modular space that for every mark point there's the evaluation map at x which just takes an element here is a map and evaluation I of f is just f evaluated with mark point. Anyway, so if I see this tau k of gamma what I'm doing is creating a mark point and putting a cotangent line to the case power that's the subscript and evaluation inverse of gamma, in particular gamma must be a comology class. So if I write down such a bracket, well x tells me what space I'm on this G and beta, tell me that I'm interested in genius G maps of class beta and is redundant because like just tells me how many tau insertions are there. So it's almost never put because it's in redundant. But it's not even you can put it if you want to emphasize. So these tau insertions and every one of these creates a mark point with a particular cotangent line power and evaluations that's the definition so there's actually nothing to that. But if you for example want to restate that relationship with her wits, you can write it like this. You're taking this insertion of a cotangent line and a point, and you're doing that to G plus 2d minus two times that you can now conveniently write just as an exponent as an exponent. That's the restatement there. Okay, so there's two immediate questions about this statement. The less even direction the different direction that's a good question but we're not going there today. Two immediate questions is there such a statement for higher descendants because we have here a only the first descendant. This is in some sense some geometric interpretation of the first descendant. And what that interpretation is it says saying the first descendant is is like imposing a branch point. So then you might think well if I put the second descendant that should be imposing so if the first descendant is like imposing branch point branch point let's say is a tube because it's either the two. So if I put a square of the cotangent line this should be like putting a Z to the third, a ramification condition that Z like Z going Z to the third. And so on this is a very naive point of view and if actually if you actually understand the geometry in this differential argument, it's very natural to believe that with some factors. And the answer is that that's kind of true almost. Here's a statement for higher descendants and then of course, is there a generalization of witness conjecture which controls the descendants for targets so these are the two immediate questions I would say at this point if we start studying general targets not just the point, and then we find ourselves bumping into these descendant integrals with the cotangent lines, then you can ask these kind of questions and the answers are actually very favorable the answers are yes in both cases. So that's what I wanted to say in this for the rest of time this lecture so there's two things I wanted to say. So the first one is about this this question about is there such statement for higher descendants and I would say roughly speaking this means that are descendants. Like tangency conditions where I interpret, like the branch point is being like some kind of vertical tangency condition, or you could say are they in. Okay. Are they like our descendants, something like the vanishing of the cotangent line and some sense the answer that has to be yes. But can one make that specific, or precise and the answer is yes, you can do that. And for P one. There's this full grom of witness correspondence is something I proved with Andre also a long time ago, and it more or less takes this very simple statement. And how to increase the one to anything you want and has a very nice answer for that. But you might not be satisfied with that, you might say well what about for any target x is there some way I can exchange these cotangent lines for some geometry, geometry, and the answer to that is yes also. You can do that, in some sense, for any x. But the answers are much more complicated and this is this goes under relative of what in theory and what's called the descendant relative correspondence. And there's, there's been some development of that. The earliest paper, I think, which took this problem in general is paper I wrote with David Malik on. I think it's called a topological view of grammar within theory. I'm not going to talk about this. The answer to this question is morally yes that that these cotangent lines can be can be thought of as some geometric condition about the differential vanishing in some sense. But I will tell you the theory of the formula here because it's kind of nice and simple. So that's the grom of witness her what's correspondence that I proved with Andre, and it's best here to use disconnected domains because the formulas are a little bit better. And it tells you how to interpret for for P one or in general curves but anyway for P one how to interpret this. Descendant insertion at a point and I said we already know how to do it when K is one that's a simple ramification condition, but how can we do it for hire K and the answer is that. It's always the case that this thing translates into a certain insertion and her what's this theory meaning the theory that her what's defined. But the translation is interesting. You have to translate into something kind of slightly new that I certainly heard it didn't know about, which is called this completed cycle. And this is a theory this is an object in the theory of symmetric functions and this has some development by many people and it's not my plan here to give you the. Development of this subject from the point of view of symmetric functions, but rather I'll just tell you what this thing is explicit formula for this thing this is a completed cycle so it starts with the usual cycle. This was the intuition saying that the cotangent line corresponds to some ramification condition meaning that particular if you had K equals one that's one cotangent line. It's two cycle. That's a simple ramification ramification condition or I said if you have a two of the cotangent line to cotangent lines of course would correspond to this three points coming together. So this completed cycle starts out as the usual cycle and has a correction. And so what are the examples well this is explained in this paper I wrote with Andre about Gromovic theory her what's theory and completed cycles. And you can look up some examples of for example, the first case. This was the first case we thought about this. If I have a two, the completed two cycle just happens to be just a two cycle itself so it's kind of a small miracle here and that's the reason why this correspondence was so simple in the original Hurwitz case. But in general, these completed cycles have corrections, and these corrections are given by a nice formula. It tells you how to what more you have to put in the Hurwitz theory. And this formula depends on this function which is this cinch function. That cycle then has a formula in terms of normal cycles, and the formula has coefficients and the coefficient is given by this blue means you look at this product of cinches and you extract this particular z coefficient. And that's the formula for this. So this rule tells you how this this rule gives like a completely explicit solution to how to interpret this such a descendant insertion. And the theory of curves this is for P one or for curves I should say correspondence P one gives an explicit rule for how to interpret this descendant insertion into the language of Hurwitz covers and that that translation has no complexity in that simple case but in the general it has a certain complexity it tells you that if you want to if you want to know what this completed cycle is it has corrections for basically all partitions of this bounded size and the coefficient of that correction has to do with a certain coefficient in a product of cinches. Anyway that's the explicit answer. And this this is the precise dictionary between the grammar written theory of these type of descendants insertions and Hurwitz's theory from the 19th century. Okay, and then then you could say why do you want to do this and that's a different line that leads the total equations and in that direction but I'm not going that direction I want to go for the part be now. I'm going to go over these two questions here. Is there such a statement for hard descendants and the answer is yes. And as a practical matter I must say to get it to get a feeling for what these descendant insertions are like. This language of translating them into geometric conditions is very useful because you already in your head have some idea of how those geometric conditions work. All right so then the second question I wanted to address is is there a generalization of witness conjecture, which controls the descendants for all x. So this is a great, this is a great question. And so somehow by this point in the lecture series you're supposed to be already interested in descendant insertions excited motivated them in the geometry of the deformation three of curves and also the cotangent lines of witness conjecture which had to do with, you know shallow water waves and now also descendants for one dimensional targets have to do with Hurwitz's theory. So by now you're supposed to already be interested in the problem of descendants. But now for arbitrary targets. That means you're supposed to be interested in this. In this question of computing such integrals, how to do it. Or if not how to do it. Did they satisfy some properties that will help you do it. Okay, so. And the thing that's amazing is that there is a proposal for various our constraints for arbitrary targets x. And this proposal was the first part of it was in this paper by Gucci Hori and Chong and it's also a long time ago probably 25 years ago. And that that proposal. In some sense was valid for when x only had pp comology. And, and for the general case was a modification by Sheldon cats. The outcome of all of this is a precise proposal for various our constraints for an arbitrary target x. And by arbitrary here should be arbitrary dimension, and the comology could be can be arbitrary Hodgkin composition the comology. But for the formulas I'm going to write today, we're going to assume x has only even comology this is helpful because odd comology gives a headache of signs. Also, I'm going to assume it has only type pp comology so I don't really interact with this other grading. This is the simple this is the easier case. But the general case is important for example even if you're considering access dimension one, you're going to quickly encounter algebraic varieties that don't just have pp comology so I don't mean the other cases and important it's rather important very interesting. It's just as a first pass it's maybe better not to do it. So this is kind of fun. And, and as I said that this was these these these things are written as a proposal. Let x be announcing a projective variety, and we'll need to know the dimension of x. That's our, and then I want to pick a basis of the comology of x nothing's going to matter too much about this basis. And the essential thing is basis independent but to write it I'm going to pick a basis. And this basis has index a and I want my very first element be index zero and it's the identity class. And for every element of this basis. It's going to have a grading. And since the comology is all pp it's all even comology, and I keep the p basically. To write these various our constraints. We use this bracket language for the descendants. Okay. So just as with the discussion of witness conjecture and the various our constraints for a point. We're going to write down some generating functions. And the easiest way to write down a generating function is to write down, first of all this formal linear combination of all the descendant insertion so here's the descendant insertion. I shouldn't damage the formula. Here is a descendant insertion. What are all the descendant insertions you get to pick any number of cotangent lines and any basis element of the comology. The basis elements of the comology. That's a finite list. So this is in some sense this is a finite index. We've just used an a there but it's it's conventional to use gamma a. This is a finite somehow index, but you can put any number of cotangent lines that's an infinite index. Now, this shouldn't scare us because already with a point, we've seen the infinite index K here. So we're not really changing much complexity from a point from, you know, some from looking afar. We've just had some finally we have finally many more values for this comology. And then we have one variable for every one of these. This is an infinite, the set of variables infinite, because we have this infinite possibility of the cotangent line so this is the finite index. Okay, and then I do what was done before, if I want to write this generating series of descendant integrals F at the sum overall possible genera, and then after some overall possible curve classes this is new we didn't have that in the line because there were no curve classes. And there's an extra variable for this. And one can try to treat this in very fancy ways but it does just be naive about it is a variable that keeps track of the curve class. And then there's this exponential generating series of all the senate integrals. So if you think about this expression, you'll find that it can, it contains all the descendants integrals. In some sense contains them exactly once up to some combinatorial coefficients. And as in the case of the point, we want to take this exponential of this, the partition function, and the Versaille conjecture should be that we find some differential operators that annihilate this function. And these operators are going to be for K grid and equal to minus one just as we had before, and they're supposed to annihilate this function so this is supposed to happen in all cases so it's kind of an amazing thing. It means that if I take any non-singular projective variety. I'm going to have to of course know a little bit about this variety to write these operators I'm going to give you the formulas for the operators. But I'm going to of course have to know a little bit about the right to write the operators but not much. I can write these operators and then this proposal, this Versaille conjecture says that these operators annihilate this partition function of descendants for any target X so it's kind of an amazing thing. All right, so now the exciting part is what's the formula for this. Let's say before, even for a point there was already two special cases there was L minus one and L zero this was what was called the string equation this is the dilaton equation. And it is the case again that these equations L minus one and L zero are geometrically known and for the same reason. So that's the part of the first lecture I skipped I didn't give the geometric derivation of the string equation. Raoul says that in the problem session that he might cover that if people ask I'm not sure exactly. But once you do that exercise, you can also do this exercise it turns out that the L minus one and L zero, the string and dilaton equations are not a mystery and are in some sense from the point of view of the complexity of the subject elementary. So the issue here is all about what happens for the question is what happens for K grade and equal to one that's the interesting part, just as it was before. Okay, so then the exciting thing is what is the formula for these. And that's what was the proposal in the Gucci Hori shong paper, and it's a formula. There it is. That's kind of small formula. It's only half a page. So we look at it so it's good formula. So this is a formula for this differential operator L, and you can see it's operating here. There's the partial here it's linear on this part and quadratic on this part. And here is just multiple multiplication by some constant. So this is partial of course is partial differentiation and that's the, this is partial K. Oh, sorry. A K is, is this partial partial T a K, so that the first thing is the homology index a second is the descendant index. Okay, so what is the. So first of all, there's these lists of these differential operators that's good. And then the main, the main things that you have one has to explain is that what are these green terms and those are just some combinatorial coefficients. I'll write the formulas. And so in some sense, the interesting part of this formula is what's going on with these C's and this is some matrix. Actually, this is also pretty interesting. There's a lot of things to say about this formula, and I will try to say it slowly here. Okay, so anyway, that's the formula and I just have to kind of explain how to unwind it. So what do I need to write this formula, why need to know something about x. So the first thing I need to know is the intersection pairing that's this matrix GB. And this matrix as usual, but will be used to raise and lower indices. And then there's this matrix C matrix and the C matrix is defined by the matrix of cut product with the tangent with the first class of the tangent bundle. So I've kind of explained to you already the C so the C is a matrix of multiplication with the first term class of the tangent bundle, and it appears here with some raised and lower indices and I raise and lower them using the metric. And that's already explaining these things. So the combinatorial coefficients are given by this. Where is it. So there's the be the be is given by the P the grading plus some dimension shift. And then finally there's this symbol here. There's some integer here and there's some integers out there. Actually they're not necessarily just bees can be half integers. There's a to their anyway there's some number here. And then there's the K and I and they're given by taking some evaluation of this elementary symmetric function. We saw some of these coefficients they were they came as double factorials in the in the point case and the reason they're double factorials are zero there and there's a half and anyway, they eventually become double factorials there. So I've now explained the green the black and maybe the only thing left to explain is this tilde. This tilde is a is what's called a built on shift and it just shifts these T variables except in the case 01. So anyway, oh I even wrote that so this has been written kind of carefully here. I tried to get this formula completely right but I had long thing. I think I had some people look at it also. Of course one can copy blindly but even when you copy blindly often you make mistakes. This is a very interesting term here what is this doing. And this has to do with a certain this has to do with constant contributions and Gina zero and one. Constant contributions, but they're put in by hand so there's some churn classes of churn classes of the tangent bundle on X. And this this stuff is pretty interesting and what the what the meaning of it is. And I am not going to say so much about it but roughly speaking, you know you see the top turn class and top turn class is of course given by the Betty numbers. And so that can be obtained. If you know the dimension gradings here, and this thing is a more interesting term. And so it's not the case that C one CR minus one of X can be determined by the Betty numbers, but it can be determined by the hodge numbers so when you write this whole thing in terms of the hodge grading it's important that this characteristic class can be turned by the hodge numbers. So even even to write this is is pretty interesting and not not clear one could do it, but it is true you can write it and I did I write it here but the point is this thing satisfies the virus our bracket. These, these operators satisfy the bracket, the same virus our bracket that I had in the last lecture. Okay, so that's the, that's actually a very precise statement of this your sorry constraints you take any variety X. And in the case I've said here, I made the small assumption that has only even come all just maybe a large assumption but anyway, only even come all do type pp but you have to believe me that all of this can be written for in general and it's actually not much different. I just didn't want to worry about it. You take that variety and then you, then you use in some sense aspects of its classical comology to build these operators LK. And that's the formula. And then magically this is supposed to annihilate this partition function which which controls all of the descendant invariance. So you get these relations in some sense for free if you know this is true is the question is maybe is this true. The kind of list of when that's true cases known is this is more or less all the, all the things that I know, maybe there's something missing but anyway, when next to the point. This is true that's the story of witness conjecture, which I explained last time in the case, the next case is when X is not a point but a curve dimension one. This is true and this is a series. This was the last in a series of papers on dimension one targets with Andrea Concoff. And this proof. I mean, this, this argument uses that from of written her what's correspondence and the LSV and lots of things about the LSV since people mentioned that. So it uses these kind of things that uses a lot of techniques about one dimensional targets. So that gives us one. And then there's a different direction. So I should say that there, I mentioned, somehow three different approaches to witness conjecture there was the original approach by can save it to the matrix model. Then there's the Hurwitz approach the Hurwitz approach actually in some sense also works in dimension one. And then there's the approach that Mirza Connie used for hyperbolic using hyperbolic volumes. That's kind of three different ways and there's more maybe. Most of those don't really generalize in any way to higher dimension at the moment. Okay, so after dimension one. So it's a higher dimension you need some new ideas that one of them was this from a written Hurwitz and LSV to get dimension one in. There's another new idea that can be used which is actually a very powerful idea. So what it has to do with studying the quantum comology and I, and the statement is that if the quantum comology of access semi simple. Then the entire come up with in theory in the form of its coft can be classified using this given total classification. So this is the whole subject which is not where I'm going. If you were the comology semi simple, then in fact one can reduce the various our constraints for these targets to the various our constraints for a point. And that this is not a trivial reduction is quite a complicated reduction which uses the classification of comological field theories. But the outcome of that is that one gets a lot of examples. The examples tend to it's very it's a difficult condition to be semi simple. Trivials tend to be things like P energy might be your flag varieties. But not only there are some phonons also. You might say what about Columbia threefolds and it turns out that the the various our constraints don't tell you anything interesting. Maybe trivial is kind of a strong word but they just they don't contribute at all to the subject of Columbia threefolds they're true but they don't they don't say anything interesting. And the reaction is you could say well maybe I take this is too ambitious for all genus and everything I can take a target X and I'm just interested in Gina zero. And that's also known the various our constraints hold in Gina zero for any target. That's a kind of old paper by Leo and Tianan I think other people have also. So that's the, that's more or less the state of affairs of the cases that are known. Which is in some sense. A fair number of cases that cover a lot of classical examples, like very often in classical geometry you're considering maps to projector space or its cousins. Or you're considering Gina zero, or you're considering maps of dimension that to other curves, and all of those cases are covered here, of course but in the full richness and landscape of algebraic varieties. This is a rather small slice. So the other point of view of another point of view on this data is that it's known in rather few cases and that's also I think a legitimate point of view. Even if you take hyper surfaces surface of general type finals. There are not general results there. So you can depending on how you look at the glass you can see, see what you want in it. But in practice there are some examples. So I wanted to say, the next thing I want to do is to explain how you use this in practice, but maybe I'll do that next time. But maybe, maybe I stopped with the last slide to be something more philosophical. I wrote it here. Oh yeah so I wanted to just point out. If you tell me if you asked me somehow what's the, the one of the richest geometries and also somewhat simple that's not in the list of known cases with a very very sorry constraints are proven I would say the example then Rika surface that's really a beautiful example. And the very sorry constraints are not even known in genus one. And I wrote a paper a long time ago with David where we get some nice predictions of what, what would be true if the various our constraints are known and then they lead to a complete solution of the in Rika surface in genus one. There's very nice formula so I've always hoped that there'd be some. Some idea that would prove the various our conjecture for the Rika surface but it has not come yet or at least has not come to me yet. So what we've got then Rika surface is that it's quantum homology courses not send me some. And now maybe on the other side is that there's some different view of the these various our constraints which says that, which is that. And this is true that come up with in theory, maybe I should have said at the very beginning is a theory, not an algebraic geometry really although I've developed an algebraic geometry. It's a theory it's a theory of symplectic geometry, meaning that I always started out here in these lectures with x being a non-singular projective variety. But you can also type the x to be a symplectic manifold. And for a symplectic manifold. There's a way to define also grimoire invariance using pseudo whole holomorphic maps you have to take a compatible almost complex structure. There is a development of the full theory in that line in that generality which is strictly bigger generality than an algebraic geometry. Meaning that there's more symplectic manifolds in their algebraic varieties. Okay that's good. So in particular we can define these full generating series. Sorry. And then you could ask well what about this and the mysterious thing is that they at the at the moment there is no way to define the operators in a symplectic case and you could say well why didn't I didn't I define them here. And the answer is because in full generality in algebraic geometry you need to know the hodge decomposition to write down these various our constraints. I mean the whole thing is a conjecture but anyway to write them so they're sensible you need to know the hodge decomposition and a symplectic variety does not have a hodge decomposition. So, it is a fact that we don't know how to write. Maybe a sad fact, but we don't know how to formulate these various our constraints and symplectic geometry. And this is a very peculiar state of affairs. And this peculiarity has led to some doubts about the constraints. So it could be the case the constraints are not true. They're only true with some other assumption. It would be somehow a miracle if they were true in algebraic geometry but somehow false and some type of geometry. So it's rather puzzling state of affairs. So maybe one of you will be able to contribute to this. All right, so I stop here. Any questions. We have a question. Is it possible to generalize the data sort of constraints for any homological field theory. I think that the answer is probably going to be no but if it's semi simple then the answer is yes. Now that you can kind of conjugate. I mean if it's semi simple that you can use this this classification theorem, more or less reduce you to the point and then you can kind of conjugate the various our constraints that's how it's proven. But if it's an arbitrary homological field theory, I think the answer is going to be, well, I don't know. I'm not so optimistic about that. You have another question on, can you say something more about why the data sort of constraints are uninteresting for a club your three fold. Yeah, well, it's just that if you have a club your three fold. Yeah, so the way you look at a club your three fold, from the point of view of grandma wouldn't theory is that had had some kind of so you're in genus G, then it has some genus G curve in it has some maybe finally many alright. And the interesting thing is how many genus G curves that's the interesting question. GN beta is how many genus G curves. It's an interesting number, but what this thing is going to be is about integration over the space, and this is a there's no integration that's interesting over them only enter only integral is interested in this number. But you can say well what about the modular space of maps with mark points because they modular space of maps. This is the picture of the modular space with no mark points. If you now put mark points. What you get is you get some space that's not zero dimensional, because now I can put have all these mark points running around here. But they're not really doing anything really interesting they're creating some configuration spaces every time I have this genus G curve. These mark points will just be creating some configuration space of that genus G curve, whatever your name for this configuration spaces. I don't know. It's like endpoints on this genus G curve, but, but all you're going to get is this many copies of that configuration space. So what the virus are constraints will be something about the geometry of this configuration space, which has nothing to do with the Columbia geometry and roughly speaking the virus are constraints will be tell you a little bit about that. And you might say okay that's a little maybe a little interesting but that already came up earlier in lower dimension. It's really not that interesting and it's just doing that this many times. This is the interesting part and the various are constraints are not interacting with it. I guess it's probably like we should say that for a lot of your three folk. I get that theory is not more interesting than the case for point right. No, it's a no the case of a point is much more interesting, because this is this is a case of a. Alright, if you want me to say that it's a like, like the various are constraints for Columbia three fold. Are as interesting in some sense as the descendant theory of a fixed curve in modular. You want it means like you want to take a curve, and you want to take the cotangent line you know if your curve is zero dimension just one thing, but then I can take basically products of that curve. That's what the points are. Of course it's not true at some blow ups of products of that it's some, you know some cotangent lines on this. It's not totally uninteresting but it's not you know this is not the difficult thing. And so you're basically integrating on products of a curve. Sorry, is that enough. But as I said it's not that the code the growing up in theory of the club I was incredibly interesting it's just that this is this is just missing it missing the interesting part we're going to get to that later in these lectures. It's just that the various are constraints tells you lots of things about a lot of a lot of things about a lot of geometric objects but not about club you have to have everything in life. I have two quick questions. The first one is actually related to the first lecture. Yesterday. So you gave us a formula for the intersection of side classes in genius zero. Yeah, yeah, yeah, you know, any other close formula in higher genre. Yeah, I mean there's various formulas. There's a subject about this. So, in genus one. It's a little bit harder but the way that the way the subject has been developing is you can ask for another thing which is that was for all integrals in genus zero we could ask for something else which is that I say that I only want to descend and say, you know, tau, or I could just say one descendant, but I want it in all genus. You can ask for this is this is called the one point function you're asked for one descendant all DG in this and that's a very simple answer it's something like one over 24 to the G over G factorial something like that. I meant for generic and for generic number of insertions. Yeah, so then you can try yeah so but I'll get there is a you can then say well what about the two point this is the one point function. And then there's a two point function also has a closed form, and the endpoint function has kind of, I mean, it's not as simple as this but there are ways to write then point five like there's a paper by Andre, who Andre Kunkow on the endpoint function where it gives it as a is some kind of integrals. I don't know exactly how you want to answer the question but there, this is a this is a way the theory is gone to write better and better formulas for endpoint functions. The way you're saying is to find higher and higher I think for genus one there is something one can write. Usually it has some differentiation but in genus two and higher. The way you're saying it. And it's not. I mean I think anyone. So maybe the other way you can say it is that any particular genius, you can write down some formula. And the reason you can do that is that maybe I should have said this, maybe this is what you're asking. But suppose I'm interested in something like genus 17. All right, and I want for some reason to know every single cotangular line integral, all of these, for any number of points but genus 17. Okay, then how do I do it. Well first I tell you that what's the dimension of the space. This dimension of the space is was three times 17 minus three plus and. And so this thing the dimension of the space is kind of growing linearly and right. But when I look at these cotangent lines remember we know how to handle the tau zero. I'll use the other line we know how to handle tau zero. This is fine that's a string equation. We know how to handle tau one. Okay, so the only the the things that we're interested in are the things that that we have to solve or tau two tau three these insertions. So if you look at this dimension formula and you think about it for a moment you realize that if I only allow these and there's only finally many, finally many tau insertions to think about. So if you have 17. The way you can think about this problem is that there's finally many integrals that have only have no tau zero and Taiwan if you have tau zero and Taiwan you can have infinitely many integrals. But each one of these eat some dimension so they can only have finally many of these the number is given by a partition. And one way to do this is you can just solve these by hand however you want by the very circumstance you get one number for each one of these, and then you can extend to tau zero and Taiwan by some formula. So you can write some kind of rather messy formula that has a partition number of some ends in a particular genus. You know, it's not the most beautiful thing in the world but if you really want to prove something sometimes that can be useful. And the other quick thing, if I may, is for the week in percentage case so engine bar and classes. We have to be a sorrow. We also have the KTV. Yeah, so I was wondering this time you spoke about your serve for a victory acts. Oh yeah that's a really good question and the answer is I don't really know but the one the case that we do know is when it's P one. There's this this very sorrow that I wrote here. But there's also a replacement for the KTV which is this Tota hierarchies this was proven by Andre Kunkow and myself and this is somehow the end result of following the Hurwitz line. But for P one we have both. This was also this this total was predicted by a Gucci and his co authors. Also, so for P one is so for a point. So this is the this very sorrow formulation and this KTV. And for P one we have your sorrow and Tota, and the higher ones. I mean, I don't really know how to generalize this. This for the higher general acts you just have the very sorry. Thank you. Thank you very much. Okay.