 Let's start the second lecture of the afternoon, which will be presented by Francesco Lin and the title is monopole Flair homology and invariant theta characteristics, please Okay. Well, thank you much very much for the invitation. It's a great honor and pleasure to be here Well, of course, it was a great honor and pleasure to be Tom students You know, you're always in talking to Tom. You're always in for some good time learning beautiful math and being asked as Andres was saying like very interesting questions that you know, it might take two hours two weeks Two months to solve so this is This project is inspired by a question and Tom asked me about ten years ago I think so I'm still and I'm not solving. I'm not answering the question he asked me But I'm trying to make some progress along those lines Yeah, so I want to tell you about some relation between monopole thermologies, so this is, you know, some topological invariant of three manifold defined From the cyber within equations Yeah, and this was defined by Okay, and some very classical Topic of study in the geometry of Riemann surfaces. So this is a classical Geometry of algebraic curves. Yeah, so I guess Josh told us about a theorem from 1897 I'll start from a theorem from Yacobi from 1550 I actually so it yeah, I'll I'm not sure it's his theorem like I look I try to look up where this was first proof Either him or plucker, but this is what teaching says So if you have Surface in P2 Which is a quartic smooth quartic Okay, so the smooth sets of a you know generic Degree for polynomial look at the sets. So this is a this is genius tree. Okay, so yeah, so you know you have this thing and Align because the quartic surface will intersect it at four points So you can ask yourself or you know lines that are tangent at two points and that's what it's called a bi-tangent Okay, so it's a line by tangent So the four is with multiplicity so it makes sense to ask for two points of intersection and the theorem Yacobi is that in this setup There are exactly 28 by tangents. Okay, so this is a very classical theorem 1980 Yeah, so how is this related to what we're interested in? Yeah So I don't know how they prove the exactly back in the days, but in the modern in the modern in The modern language. Okay, so this thing is a Riemann surface of genius tree No, if you pick any Riemann surface, okay, you have K the canonical bundle and Yeah The key object we will look at so key Definition so this is a holomorphic line bundle. Okay, so we can look at line bundles, which are square root of it So the cold Is a theta characteristic if It's good it's square which is another holomorphic line bundle is isomorphic as a Holomorphic line bundle to the canonical. Okay, so yeah, so let me remark So the canonical bundle has degree 2g minus 2 so this has degree g minus 1 so and as a remark Yeah, so on a surface if Sigma has genus g there are exactly 2 to the 2g Theta characteristics, okay Yeah, so that we have this all these holomorphic line bundles a lot of them so in particular on a surface of Genus 3 we have 2 to the 6 and the theorem so Jacobi the theorem follows From the fact if you have a genus of sigma equals 3, you know, there's Then so you know we have this holomorphic line bundle so we can look at a space of holomorphic sections so H zero so we look at the The this is space of holomorphic section and then the dimension of these is One or is either zero or one and it's one for exactly Okay, so somehow having holomorphic section of this square root of the canonical for response to this by tangents Okay, so I won't go into more into that. I guess the the key point is that Smooth quartic is embedded using its canonical linear system If you if you write things about that, you know at the end of the day it turns out that this theorem very geometric and Very pretty follows from the computation of the space of holomorphic sections of certain line bundles on your remote surface Any question yes, so why is this related to? Yeah, fluoromology or gauge theory at all you might ask well, this is a this is a Very beautiful observation due to a tia which I actually learned from Tom's Riemann surface class that That this is really you know So this computation here, which you know, it's something about holomorphic sections And complicated object in complex geometry. It's really a theorem in topology Yeah, so that looks like a theorem in complex analysis because about space of holomorphic section Let me tell you why it's a theorem in topology. Well, I guess depending on what you call topology exactly Something I would call topology Yeah, so I guess you know theta characteristics on sigma, you know, there's some how taking the square root of the canonical So you might expect that it's somehow related to spin structures And in fact, you know the number to the G is also the number of spin structure on your surface So indeed they are in one-to-one correspondence with spin structures, okay, and then here we can go down and You know Let's say you go to a natural number, which is you know L maps to the dimension on its L Instead of just looking at this number, you know this number in principle Depends on the Riemann surface you choose in general So if you deform the complex structure, this holomorphic bundle moves around and the space of holomorphic section changes But yeah, they keep a very nice thing is that if you reduce mod 2 Go to Z2 and this is a topological invariant and in fact, you know spin structures on a surface have a topological invariant Which is the R pin variants. You know, this is the spin board this class You know, it's in omega to spin Omega spin 2 Okay, and well the claims that this diagram commutes So if you take a theta characteristic and take the R pin variant of the corresponding structure that gives you The dimension of the space of holomorphic section mod 2. Okay, and I guess that you know how this doesn't relate to gauge theory Well, so so the idea this is uses that you know the del bar operator Which is you know the operator whose kernel holomorphic section on a spin surface this corresponds to the deraille operator Okay, and under this correspondence, you know to tie up with the what I talked about last week, you know H0 of L Can be made to correspond to You know, it's in the corresponds to harmonic speakers. Yeah, so Okay, so yeah, and this This theorem here is somehow a manifestation of index theory of the Z2 version Okay, so maybe let me remark even more clearly that you know the dimension of H0 of L is not Deformation invariant, okay, but it's it's not to class. You know, it's a residue mod 2. It's part of these is the formation Okay, so in general if you have a family of Riemann surface with a spin structure, then the space of harmonics pin or jumps but yeah Mod 2 you will not Any question about this Yeah, so the question is like why does this imply so you can show for a surface So yeah, one thing I didn't say is that if you have genus 3 Then the the space of section kind of can be at most one dimensional So the party that determines exactly what Which which either zero one? Yeah, I guess maybe there's and you know You can I guess maybe the other thing is that the number of spin structure. Sorry, I'm Not hyper elliptic, you know, you need to use the smoothness but Yeah, I guess on a surface that is exactly 28 spin C structure with odd dark invariant. I guess that's the Ingredients you're missing. Ah, okay. So I guess more more on this side. Okay. I will talk about this later Yeah, it's a long story made very short. Yeah, but yeah, I guess yeah So if you have a known if you haven't known hyper elliptic genus 3 surface, then this this has to be It either it has one and then the party Determines whether it is zero one so you can compute the theorem follows from a topological considerations Okay Yeah, so okay, these are all about surfaces and reman's And they're all working geometry Yeah, so for today, I want to look at a very specific example So so, you know monopropyramology is a is a invariant of three manifolds I have to you know construct three manifolds in some way So I'll look at I work in a very simple setup. So consider So this is a reman's surface and you consider an automorphism. Okay, so here I'll be looking at genus You know, let's say, you know for simplicity genus. I list to and buy, you know, so this implies that Why as finite order? Okay. Yeah, so, you know, if you have a manifold like this, you know, if you have something like this you can Look at its mapping tools. So this is so I guess the goal for today is to understand how The flow of knowledge of these mapping Torah somehow relates to the geometry of these automorphism Yeah, so I guess Yeah, I said that I don't have a complete answer To Tom's question from 10 years ago, but I look at a very you know for today The big assumption for today in some sense. There's some sense this action is is is very complicated meaning that We'll assume that you know, we have this action so you can take the quotient by, you know, there's a natural complex structure. So this is We'll assume that it you get the reman's fear. Okay, so, you know, I might forget to say it at some point today But this is our work in assumption today So this implies in particular that this mapping Torah is a manifold with B1 and five Okay, so it's Yeah, so what are some examples of this? Well, for example, if you pick any reman's surface, which is hyperliptic and do the hyperliptic involution This is an example of that picture of a hyperliptic involution But in general, so if you have, you know, the reman's surface associated to Z to the D equals PW okay, it is not inside C2 So the reman's surf this will be like the reman's surface Yeah, so this has a natural action by, you know, Zeta P to the D Z mod DZ acts on this by Primitive the truth of unity. Yeah, so, you know, I'm only studying mapping Torah finite Orga automorphism which from the point of view of topology might not be that exciting, you know They're like, but yeah, I guess the point of view of geometry they turn out to be very interesting But yeah, and let me also comment that, you know, you can show that actually every example that Satisfies like this, you know is of this form. So it's Z to the D equal polynomial for some polynomial So this is really all you can think about you should think about and you know The the action can be have very complicated ramifications and stuff. So, you know, the combinatorics is pretty complicated Okay Yeah, so what's the point? Well, the point is that so in this situation, so in the case of a single reman surface of the reman surface we had this correspondence between data characteristics and being structures Yeah, so in this situation, we also have an interesting correspondence. So we have the correspondence between so, you know, the you can So, you know by Acts on the set of data characteristics. Okay, so yeah by pullback or you know So Yeah, so and you can look at the fixed points of these action Yeah, so we have that in our setup we have In the set Of invariant data characteristic in one-to-one correspondence Spincy structures on the mapping torus So not all of them, but the ones that are conjugated to the isomorphic to their conjugate So this will call This is how you can also think of these other spincy structure coming from spin structures And this correspond also uses be one. Okay, so let me just for for clarity, let me The note, you know L Well, we're really getting spincy structure induced by spin structure Well that you you because b1 is a one you get two spin structures who induce the same spincy structure So the one-to-one correspondence is with spincy structures, which are isomorphic to its conjugate. Well, this is you know I'll take model for homology and Well, the gauge group is bigger because b1 is positive So you get z components of the gauge group if that's the so, you know You're not really losing any information, but you know, this is the the better way to phrase it for our purposes I think Yeah, so we have Yeah, the action of Here and then we have the action Yeah, so suppose we have invariance or L invariant So if I let's say you suppose that you have a characteristic, which is fine variant Well, then you can lift the action of the automorphism to the bundle on the bundle And then you can lift it You know you can look at the action on the space of holomorphic sections Hey, and the only thing to be careful is that you know, we we started with something of order D Let's say he says order D and the action the lift of the action might have double the order So that's something you know depending on which you know, there's probably some lift sometimes you might Yes, exactly. Mm-hmm. Yeah, so there's yeah, and yeah And sometimes you you you have to have order 2d. Oh, sorry The question is like the the lift is unique up to plus or minus one and the answer is yes Okay, so we have this, you know, so before we look at the space of holomorphic sections and somehow it's related to topology Yeah, so here we have this other new space of holomorphic section together with an action on it Okay, and it's a finite order action. So, you know, it's a diagonalizable action, so we have iron nice eigenvalues and Okay, so let me tell you the the the main theorem which are state in a very vague way But then I'll tell you some examples of computations Think you can do with it. So I guess the theorem of the Fleur homology groups of This pair and five with this spin C structure, which is coming from one of these Invariant with the characteristics is completely determined by The spectrum the the eigenvalues Okay, and there, you know, if you tell me the spectrum there's a recipe that Produces you the Fleur homology groups. It's a little complicated. So I won't write it out But there's a explicit formula. Well, that's in the case of a non-hyper elliptic genius tricker in general It can be arbitrarily large Yeah, in general, I'll give you I'll give you another examples of in which this thing is arbitrarily complicated. Yeah, yeah any question Yeah, what's the rate? Yes, the question is all of the groups. So really what I do is write down the chain complex Yeah, and you can also twist coefficients and yeah, I'll give you the the Fleur chain complex Without you, you know with a balance perturbation, let's say, but you others can do it. What sorry? Yeah, the great the question is I want about the grading Yeah, so in the so I can determine it when the paper worked out up to up to an overall shift It would be a grader group up to an overall shift And I think one can determine also the one works a little harder. You can determine the The actual absolute grading of everything. Yeah, so let me tell you a Basic example Um Yeah, so I got you know, we had our example the first example is hyperliptic involutions Yeah, so example one so five Relitic evolution Okay, so one nice fact about the hyperliptic involution is that it leads all spin structures Fixed. Okay. And what you can do you can use the computation that you know, I'm not really telling you To show the following so you can show that So So all the 2g structures on Sigma fixed Yeah, so and what you get by L for this for the C structure correspond to one of these Spin structures Well, I'll draw a very schematic picture. So, you know, you have the time so it's a manifold will be one positive So you have two towers So one tower will look like this You know and the other tower, you know ends up, you know the action of the the b1 action goes in this direction And you know the other tower ends a little lower. Okay, and and that's it and this quantity is let me call You know how many how much lower you go? Okay, so there's no, you know, in general, there would be some this is a stuff in the image of the You know hf infinity of hm bar In general there's reduced bar, but in this case, I'm saying that there's no reduced and you know here I'm doing the computation without twisting coefficient, but you can do a twisting coefficient. So and what is an L? So an L. Yeah, so now I think the cool thing here is that now this is a theorem that relates You know topological invariance to objects in algebraic geometry that people have been studying for 150 years So, you know, you can use computations in algebraic geometry to get computations in In fluoromology. So what you get here is that, you know, so I guess it's an exercise in Albarello-Cornalba-Griffith-Aritz book in Appendix B implies that so this an L is zero for 2 to the g plus 1 over g, the true g, data characteristics and then for every w between 1 and g and w odd that exists exactly 2g plus 2 g minus w data characteristics Such that an L is w plus 1 over 2 Okay, so this is the whole computation of the sub conjugate spin C structures on all the fluoromology for the mapping tools over the hyperlipidic evolution. Okay Any question? I'm not sure I know how to solve the exercise, but that's fine Okay, yeah Yeah, this yeah, it turns out that so the question is I use the dimension Yeah, and yeah, it turns out that in this case, this would be just the dimension of the space of linear systems. Yeah Yes, exactly. Yeah, but yeah, and yeah, that might lead you to ask Oh, is it always this kind of boring example, which is the two towers and the answer is no So you can get actually arbitrary complicated examples. So let me just so you can get examples Yeah, so how many torsion species structure which I'm not going to have I think they're all in that case. There should be all I think no Yes, there should be all but you know, I'm I have to think about it a little more But I think that they're all self-conjugating that Yeah, so some other fun examples, so this is example example two Okay, so let me tell you so can get I can get you know, this is a little boring because you know There's only these two towers, but you can get as much a lot of reduced stuff if you want to so so can get You know something that looks like this. So you have the two towers And then something that goes down a little bit the other tower by let's say And and then you might have something at the bottom that some reduced part of the bottom that goes up And you can get this for and And this you just need to look at order five Automorph say it turns out if you look at order three You still get just this boring thing without extra reduced if you look at order five you start looking at You start getting extra interesting stuff and in general Yeah, in general Yeah, any question about the statement. Yes, the question is like how hard would it be? Yeah, I'm not sure I guess It might be doable. I don't know how Yes, that was also for fiber, but they have be one positive, right? So you need to be careful about reduce the reducible of Yeah, yeah, so you can use the cypher vibration Yeah, yeah, but yeah, but you have the torus of reducibles, right? I have to be careful how to perturb. So actually, yeah, maybe I say a word later Really the core of the theorem is a yeah, the core of the theorem is a transversality and results. So yeah It's a you should be able. Yeah, the hard part of the theorem is being able to perturb the equation while keeping understanding what's going on but Somehow being able to achieve transversality. So that's where I use the Assumption that the quotient is P1. Otherwise. Yeah, that that I don't know how to do Yeah, the question is like what about the non-spin structures? Yeah, I think this the method I can Talk about I talk about can generalize to some other spin C structures But in general that they won't work for all spin C structures. Yeah I don't really use the cypher vibration in the in the in the computation. Yeah, so yeah Yeah, I guess I realized that after I wrote the papers. Oh, this is cypher fiber, but yeah Yeah, so you don't need yeah, so yeah, okay, and in general you can you can write you can write you can okay So this is you know, okay, this is like useful computation if you know how to compute This action, okay, and then you know, there's a formula very complicated formula. That's how to do this But yeah, of course, this is not an easy thing to do right you need to find out invariance spin C structure Spin structure and compute the space of the spectrum of the action of the automorph is on it So it's a pretty complicated thing, but you can actually do you can compute Automorphies of order p. Okay, in the case of non-prime, I don't know, you know, you might get stuck But you know, I tell you a way to compute for p prime or every case And yeah in particular, you know one thing that in the case p prime there exists unique Yes, so the question is like how you know, I have a recipe that's from the action tells you the groups so the question is like how do you compute the action and Yeah, so so when p is prime so in general you can use that he about G-spin theorem. Okay to get info about Action H0l in terms of Of fixed-point locus data Okay, so if you know everything about the fixed-point locus, which is very easy to compute for example from representation z to the d equals p w You just look at the local singularities of the polynomial You can get info about here and let me tell you this now I'm not just using a t about here But because I t about give you a formula up to science which is not very useful in concrete places I use luckily well Danny and his collaborators work out the sign that exactly in the cases I needed so this is Yeah, so if I give me a polynomial Z to the d equals p w a curve From there you can compute some Partial information about the faces of this action for example and that happens that when the piece prime That's all you need to compute the action So when p prime the whole and you know one piece most prime You know there's things that you know don't don't work perfectly, but you can still use this machinery To you know you can reverse the machinery and use computations in topology to compute for example the dimension of spaces of holomorphic sections Yes, so for example, we all know how to compute the dimension of zero surgery on the Figure on the torus not you know that has a dimension how to compute and that's you know the mapping towards our finite order automorphism And that gives you you know using the computation fluorology you get the dimension of this for a holomorphic section for this bundleology compute So for example the Surgery on So that's the mapping torus on a surface of genus p minus one times q minus one over Okay. Yeah, so let me just comment a little briefly about the idea of this this computation. So the idea I Guess is that you know on surface times a circle? So, you know these manifolds are mapping to our finite order things So they always have a finite cover with this surface times a circle Yeah, so for the trivial spin C structures Has no the cyber written equations Have no irreducible solution. Okay. Yeah, and this is Yeah, well in fact But yeah, you know, this doesn't allow you to compute the fluoromology because the equations are extremely degenerate. So equations are Extremely degenerate. Okay. In particular, you know, there's that there's a torus of that, you know of you want connections on on this manifold And the singularities are a copy of the theta divisor singular locus So the Hessian of the equations are extremely singular along this theta divisor, which is a co-dimensional three thing And it's a extremely complicated. So this is very complicated Yeah, a good portion of this book is about studying how complicated the theta divisor is Yeah, so I guess that the point though is that you know What you can do You know, so this is the this is the harder You know, this is a simplest case in some sense because this is a mapping torus of the identity By some sense, it's the hardest because you know transversality is impossible here Yeah, and the idea is to look at the like more come the most complicated case which is the case of Then, you know, the torus of that, you know, this has be one and five one and You know, so the this torus Circle and you know, the singularities can be that bad. They're just points Yeah, the the question is like can you perturb? You know, the hard part now is to Figure out, you know, if you perturb properly, you know, you don't introduce irreducible solutions and then figure out how the singularities affect the homology and how they're related to the Space of holomorphic section of your curve Yeah, and that's That's unfortunately, I don't have time to talk about it. So I'll end up here. Yes, exactly Yeah, so the the question is yeah in general H zero well Depends on the space of this theorem The theorem because this is a pathological invariant this implies that the dimension of this is Doesn't depend on the complex structure if you if you're the form within fine variant complex So the so the the so you can go perfectly from one way to the other but not the other so yeah The eigenvalues I don't yeah in principle it could depend but I don't know but it might not depend Yeah, the question is like can you do this for instant towns? Yeah, I don't know that this is Because a very linear theorem, so that's why you matches perfectly with You know special sections and linear algebra on them. Yeah, I'm not sure what happens in the instant on case But definitely hard surely hard Yes, the question is like how do you prove this? Yeah, so this this theorem here. It's proved by essentially it's some kind of dimensional reduction Yeah, so you show that there is somehow invariant under the circle and then boil them down to equation on the circle and the question is there so you You analyze the singularities of the derac operator the family of the rock operators And you can show that somehow each eigenspaces will correspond to a point at a different angle somehow identify the singularities with the eigenspaces with of that and Yeah, and then it's about under you know spectral flows So the I each dimension I give you some shift between towers and then So that allows you to write down actually explicitly the the third chain complex for these examples. Oh, yeah, the question Can you see pin to homology? Yes, this is the perturbation I use are into a covariance So you can you compute pin 2 for homology using the same approach? Yeah The question can you interpret it this on the statistics under or default? Maybe yeah, I've always been very confused by line bundles on or before so I guess But yeah, probably yeah