 So let me first share my note in the chat. I have put the PDF and also let me send the link again. Okay, so welcome to the third lecture. And here is the plan for today. First, deformation invariance for naive counts. Second was spines, tropical curves and the tropical modular spaces. Three, sketch of proof of the connected component theorem. Four, toric tail conditions in families. Five, gluing formula and independence on the choice of torus. Six, structure constants and associativity of mirror algebra. And seven, convex finiteness. Eight, boundary torus connection and finite generation. It's a lot of contents today. If we don't finish today, we will continue in the next lecture, the last parts. So let's start with deformation invariance of naive counts. So recall from the last lecture that given any spine H from some nodal metric tree to the essential skeleton of U, H is some piecewise affine map. So given any spine H and a curve class beta, we have the naive count of skeletal curves and H beta of a spine H and a class beta. The definition was pretty straightforward once we have the skeletal curve theorem. Question, what properties do the numbers and H beta enjoy? The most wanted property is deformation invariance. We want the invariance of the count and H beta under small deformations of H. So in fact, this property determines directly the viability of the whole project of no Archimedean curve counting. So it was the first property that we had to check before embarking on the project. Here is the rough idea for deformation invariance. We consider the space of analytical curves in U analytic and we have the map taking spines, taking associated spines of the analytical curves, and we reach the space of spines in the essential skeleton, SKU. And we are given some spine H in the space of spines. Intuitively, the count NH beta is just the cardinality of the fiber of this map SP over H. Let's ignore curve classes for the moment. So intuitively, the number of analytical curves of spine H is just the cardinality of this fiber. So for the invariance of this count NH beta under small deformation of H, we want the map SP to be somewhat atow over a neighborhood of H. And more precisely, recall that the counts NH beta are defined via evaluation of an internal market point PI. Assume for simplicity that our H is an extended spine. So let's first assume that H is an extended spine. Here is an example of H. We have five infinite one valent vertices V1, V2, V3, V4, V5. They either map to the boundary of the essential skeleton or they map to the interior. For example, V1, V2, V3, V4, they all map to the boundary and the V5 maps to the interior. The lag containing V5 shoots up vertically. It's contracted. Let P, the bold P, be the top of weight vectors at infinity. So here we have P1, P2, P3, P4, P5. They are weight vectors at infinity, meaning just the derivatives. And if V5 is an internal vertex, this derivative P5 is just zero. Recall that each PJ also lives, is also an integer point in the essential skeleton. And we have an explicit description for the set of integer points in the essential skeleton, which is just zero union positive integer multiples of essential divisorial evaluations. So recall, if the derivative PJ is zero, J is called internal. And if the derivative PJ is nonzero, J is called boundary. And in this case, we write PJ as MJ times VJ. Like here, some positive multiple of some essential divisor and assume that each, so this is grid letter nu, assume that each nu J is given by a component dJ of d. Now we have a proper modular stack. Properness will be important in the moment. We consider the modular stack m bar yp beta consisting of rational stable maps f from a nodal rational curve C with marked points PJ to y of a class beta such that each boundary marked point PJ maps to dJ with order greater or equal to MJ. So here, we say order greater or equal to MJ, because we want a proper modular stack. So we also allow some components of the curve to go completely into the boundary d. Otherwise, we cannot have a proper modular stack. And inside this proper modular stack m bar, we have the sub stack m up beta more relevant to our accounts, consisting of curves whose intersection with D is exactly given by P. In other words, each boundary marked point PJ maps to the open stratum dJ with order exactly equal to MJ, and there are no other intersections with D. So this m up beta is what we are really interested. And it leaves naturally inside this compactification m bar for any internal marked point PI. Meaning that the PI maps to the interior U, we have the natural map phi i from the proper modular stack m bar to the module, the space of domain curves times y by taking first. So to here, it's the modular space of an pointed stable and pointed rational curves. The first factor of phi i is given by stabilization of domain. Because for a stable mapping m bar, the domain curve might be unstable. So we take first factor of phi i is a stabilization of domain. And the second factor of phi i is evaluation at the ice marked point in order to obtain the eight dollars of the map of phi i. As we said that for deformation in variance, we want some sort of a darkness. And in order to obtain the eight dollars of phi i, we consider two more sub stacks. First, we consider the sub stack msd consisting of stable maps with a stable domain. In other words, there are no bubbles in the domain. And the second, we consider a further sub stack m smooth inside msd consisting of stable maps such that the pullback of the logarithmic tangent bundle is trivial. And note that these two sub stacks are in fact spaces. In other words, they are not a stack because stable pointed rational curves do not have non-trivial automorphisms. So that also implies that our accounts are positive, non-negative integer numbers as opposed to some rational numbers. Because eventually we are just working with modular spaces instead of modular stacks. And using the deformation theory of curves, we have the following smoothness theorem. So finally, I give the precise statement of the smoothness theorem which we have used already several times in the previous two lectures. We have the following smoothness theorem. First, phi i is a tau over this smooth subspace m smooth over the subspace m smooth. And the second, this subspace m smooth is sufficiently big. I mean, if this subspace m smooth is empty, then it's not very useful. So we have to show that m smooth is sufficiently big in the sense that for any fixed modulus mu inside the space of stable and pointed rational curves, there exists a risky dense open v inside u such that when we restrict our modular stack mu p beta to mu and v, it's contained in the subspace m smooth. So here, yes, yes. So here, the subscript denotes three images and usually it's clear it's pre-image by which map. So here, mu means that we consider the subspace consisting of stable maps whose domain has modulus mu and v means that we consider the subspace of stable maps whose ice marked point maps to v. And this second statement says that for any fixed domain modulus as long as the ice marked point goes to a risky dense open subset v, then the stable map automatically lies in this smooth locus m smooth. So it says that m smooth is very big. It's not only dense open inside this m up beta, the good modular stack. It's actually dense open over every fiber mu. So this is the smoothness theorem that we have used many times and we will be using again today. So it can be proved by the deformation theory of curves. Now we consider the following commutative diagram. We have our map phi i from the space subspace m smooth taking domain of stable map and evaluation at the ice marked point. And we also have the tropical, at the tropical level, we have the map phi i drop from the space of spines. The space of spines in the essential skeleton with infinite directions given by bold p or top bold p to this product. Similarly, the first factor is just takes domain of a spine and the second factor is given by evaluation of the ice marked point in the spine. Then we have two vertical maps, sp and rho. The left vertical map sp just takes the associated spine of stable maps inside our modular space m smooth and the right vertical map pro is the usual tropicalization map. On the first factor it sends every analytical curve, every stable and pointed rational analytical curve to an extended tropical curve with n marked points. And the right hand side is on the second factor, it's just the, the retraction map from the analytic you to to the skeleton to close the skeleton. What is the closure of skeleton here? Oh, so here, the close here, because the skeleton of you lives in the identification of you and it also lives in the identification of why. So we take a closure inside the identification of why. Why is the natural, why is the SNC compactification of you to this closed the skeleton? And yeah, so now I stated the following theorem, let's state the following theorem, the connected component theorem, which is the one main theorem of today's talk. Let S inside the space of the spines be a transverse extended span. I will give the precise definition of transverse in a moment. Just for Maxim's question, I want to say that this bar has, it's really has no importance here. I write this bar just to be able to receive a map from you analytic because it might, because it might go out of and but finally what really matters is the interior. So let's just ignore this. Okay, so the statement of the connected component to zero. We consider S a transverse extended spine inside the space of spines. I'll give the precise definition of transverse in a moment. Then there exists an open connected neighborhood. VS of S and a Zarisky dance open are inside this, which is the first I push forward VS VS is open subset here I push forward the VS. And I pull back by row. So there exists open neighborhood. VS of S and the Zarisky dance open are in this push forward pullback of VS such that when we restrict our modular space and smooth to VS and R. Meaning that we consider the sub space. So it's just a pre image by a VS and R. And the meaning is that we consider the sub space consisting of stable maps whose spine belongs to VS and whose ice marked point maps to R. So this sub space is a union of connected components of the proper modular stack M bar restricted to R. And this is the connected component theorem. It says that this space when we restricted sufficient. When this space sufficiently restricted becomes a union of connected components of this proper modular stack restricted to R. So why it's important because an immediate consequence of the connected component theorem is that by the properties of the modular stack M bar and the smoothness theorem. We deduce that the restriction of phi i to this. This space we denote temporarily by MVS R. The restriction of phi i to this over R is finite a top and whose degree is exactly the count and I S beta. Recall that we defined in the last lecture the count and I S beta using. Skeletal curves and it's just a naive kind of skeletal curves and this we defined using degree of or length of some. Zero dimensional analytic space of skeletal curves and this fiber is just one particular fiber here. So the degree of this finite data map gives exactly this count of skeletal curves we defined in the last lecture. It's the count of skeletal curves with the spine S curve class beta and evaluating at the ice marked point. Hence this connected component theorem. Especially this finite a dollar consequence shows that the count and I S beta is constant for all. For all spine S inside this connected neighborhood vs. So let me give a few remarks for this connected component theorem. First this shows the invariance of the count and S beta under small deformation for a transverse extended spine. So here we have some I but as we explained in the last lecture by the symmetry theorem. The place where we evaluate has no importance at all. So this shows the invariance of the count associated to small deformation. For any transverse extended spine S. Then we prove. For all transverse spines. By studying the Toric tail condition in families. Which I will explain. A bit later. And deformation invariance does not hold in general for non-transverse spines. Even though for the proof of the associativity of mirror algebra. And also for the proof of wall crossing homomorphism. We need a slight general generalization for non-transverse spines. Which we call almost the transverse spines. And second remark. Actually we had to prove a stronger version of the connected component theorem. We can further assume. That R intersects every fiber of the projection. From this push forward pullback of the of the open subset VS. To the modular stack of stable and point directional analytical curves. And what does it mean. It means that when restricting to R. R is a Zariski open of this space. And by further assuming that R intersects every fiber of the projection. It means that when restricting to R. We are not going to miss any modular of the domain curve. And this will be important for the proof of the gluing formula later. Where we need to consider degenerate domain modulus like this. Which contains some node. And which a priori might be missed when we restrict to a Zariski open. Because this kind of domain modulus containing nodes. They are Zariski closed subsets. They are closed conditions. So a priori if we take a Zariski open we may miss this domain modulus. And that will be bad for later proofs. We must for later proofs we must be able to degenerate domain modulus as we like. So we'd better have R big enough not to lose any interesting domain modulus. That's the second remark. Third remark. Recall that finite eta is equivalent to being proper plus eta. Finite eta is equivalent to proper plus eta. Properness is a question of compactification. While eta is a question of transversality. Compactification and transversality are two pivotal themes of enumerative geometry. Ideally we would like to treat them separately. The compactification and the transversality we want to treat them separately. However here the proofs of the two properties are intertwined for two reasons. First if we want the stronger version in remark two. Here where we do not want to miss any domain modulus. Then we must apply the smoothness theorem above with some properties conditions. And second properties prevents analytical curves in you from escaping to infinity. That is escaping to the boundary D. If we want to establish properness purely via tropicalization. We must show that tropical curves in the essential skeleton SKU do not escape to infinity. That is do not escape to the boundary of the essential skeleton. Which is just the closed essential skeleton minus the essential skeleton. So if we want to establish properness without using smoothness. Then we can try to do it purely via tropicalization. So we must have better control on tropical curves. We want to prevent it from escaping to infinity. But it's extremely complicated to consider tropical curves with components in the boundary of the essential skeleton. And especially the modular space of such tropical curves. If we think about components of tropical curves mapping to the boundary of essential skeleton. And then what is the modular space for such tropical curves how they deform. It gets very complicated at the level of combinatorics and elementary topology. And the task is greatly simplified if we use the smoothness theorem within the proof of properness. So that's the two main reasons that it's difficult to separate the two issues. The compactification and the transversality inside our proof. And they are intertwined in some way. So that is the statement of the connected component theorem. Which is the main theorem of today's talk. Now let's go to the second section. Was spines tropical curves and the tropical modular spaces. In order to better understand the behavior of non-alchimedian analytical curves. We need to study the associated tropical curves. Question, how do we take tropicalization of analytical curves inside your analytic. Recall that skeletal curves have canonical spines. Then what about non-skeletal curves. Moreover, the spine is only part of a bigger tropical curve. So how do we obtain the whole tropical curve. The idea is the following. Unlike spines of skeletal curves, tropicalizations are not canonical in general. They depend on the choice of some model. Here we will work with toric models. Recall we have our log-calabi-o containing some torus. And it's contained in some SNC compactification y. We denote by d the complement of u in y. Lemma for producing a toric model for our log-calabi-o. After replacing our pair yd by some toric blow-up. There exists a toric compactification yt dt of this torus tm. Such that the birational map pi from y to yt induces a bijection between the generic points of the strata of d. Intuitively, this bijection means that yd is simply a blow-up of some toric pair yt dt whose center does not contain any strata of dt. Here is an example. We have our toric surface yt dt associated to some fan. Then we make blow-up at two points on the boundary. And we obtain a log-calabi-o pair yd. u the complement of d in y is log-calabi-o. And the two blue curves denote the exceptional curves. Let's introduce a notation. Let e in y and e t in yt denote the complement of the isomorphism loci of the toric model map pi. In the above example, e is just the two exceptional curves. And the e t is the center of our blow-up. Oh, that's a small comment. Yeah, the whole picture shows a good direction, kind of generalized cluster varieties, sort of Lamp-Pilevsky blow-up. Yes. Devisors in a toric devices. Yes. Yeah, thank you. Thanks. Yeah, and should we talk to some kind of notion of mutation for such pictures, which you never used in your approach? Because we don't really need the mutations. The mirror algebra is built directly from the structure constants. Of course, later we can show that the structure constants are somehow invariant under mutations. Yeah, but I want to say that eventually, for counting non-archimedean curves, the assumption that it contains, the log-Halabiac contains a torus is not necessary. Although it will become much more technically sophisticated. And here, we just take advantage of this torus to make many things more transparent. And many of the geometric ideas here that we explain actually applies easily when we consider the more general situation of counting without a torus. But all things concerning tropicalization, it gets simplified in this case. Okay. Yes. So, okay, so now let's just take advantage of our torus and it's easy to tropicalize in the toric case. So recall, how do we tropicalize curves in the toric case? Given yt dt, a toric variety with a structure torus tm, m being the co-character lattice. For example, if our base field is complex numbers, then the torus tm is just m tensor with c star. We have the valuation map from the identification of the structure torus tm to mr, m tensor with r, which is isomorphic to rn. So if you take some basis, then this map is just a coordinate-wise valuation map. And this map compactifies to a map from the identification of the toric variety to a compactification of mr. So this compactification is given according to the fan of the toric variety. Now, so we are recalling how to tropicalize curves in toric varieties. And given any analytical curve c mapping into our toric variety, recall that any analytical curve is homomorphic to some infinite graph. If it's a rational curve, then it's homomorphic to some infinite tree. And we consider the composition of the map f with this tropicalization map or retraction map tau t. This composition has a natural factorization through tau c and h, where tau c from c to gamma contracts every path in c that are contracted by the composition of f with the retraction map tau t. So it's easy to think intuitively c is an infinite graph. f is an analytic map from c to the analytic space. The analytic space, it's very complicated, especially in higher dimensional case. It's difficult to imagine, but it has a nice retraction map to just Rn, some compactification of Rn. So the composition of tau t with f is a map from an infinite graph to Rn. And this factorizes, so this map may contract infinitely many, so it's a map from an infinite graph to n, and it contracts many, many things inside of this infinite graph. So we just contract them all. We factor this map through gamma, where we contract all the paths that are contracted by the composition, and it factorize like this. We make each an immersion. It's a theorem from tropical geometry that if we do this, if we contract every path like this, then gamma is just a finite metric graph, and fh is piecewise affine and balanced. So balanced meaning that the sum of weight vectors around every vertex is zero. By weight vector, we mean the derivatives. So here is a picture of a tropical curve. This green graph is gamma finite metric graph. It maps in a piecewise affine way to R2, our plane. And it's balanced, meaning that, for example, at this vertex, we take the sum of the three weight vectors, the three derivatives, they add up to zero. So this tropical curve, h, is essentially a combinatorial object, this map h, after this contraction. All the contractions, it's essentially a combinatorial object. And this is called the tropical curve associated to the analytic curve f from c to the analytic, f from c to the analytic, analytic, toric variety. Let me make a remark for this tropicalization. So if we have some marked points, pj inside c. If we have some marked points, pj inside c, let gamma s inside c denote the convex hole of all the marked points. Then we call the restriction of the composition of f with the contraction. The restriction of this composition to the convex hole gamma s, we call it the associated spine. Recall that in the last lecture, the associated spine is just f restricted to the convex hole. Because in the skeletal curve case, the spine maps automatically to the skeleton of u. Here, for a general curve, which might not be skeletal, we restrict it to the convex hole, but we need to compose with the retraction map tau t, which depends on choice of models. And in this case, when we have some marked points in the definition of tropical curve, it is natural to require that the contraction tau c, this contraction does not contract any edge of the spine. So that the spine is a subset of the tropical curve. We don't want to contract any edge of the convex hole of the marked points. So that this is a subset of gamma. In other words, we want h to be an immersion outside the spine, but on the spine we just wanted to be the spine. So this is natural to have this requirement because a priori in the definition of tropical curve, everything that are contracted by the retraction map, we just contracted them. But when we have marked points, then the convex hole of the marked points, they are the spine. The convex hole of the marked points is by definition, the domain of the spine, and we just don't want to contract anything in the spine. Otherwise, spine will no longer be part of tropical curve and it's difficult for some reasonings. So we just relax a little bit on the immersion condition for the tropical curve. But I mean, we ask in the definition of tropical curve that this should contract every pass, because otherwise, there, of course, we can contract less, but there's no canonical way of choosing what to contract. Or what not to contract. This is how to tropicalize analytical curves in the analytic torque variety. And we want to tropicalize analytical curves in our original y analytic. This is easy. We simply compose with the torque model. So we have a torque model map pi from y to y t. And we just compose with the torque model and then we tropicalize inside the torque variety. It's also possible to deal with the indeterminate locus of pi. But let's ignore this for the moment. A question, what do tropicalization of analytical curves in our modular space MUP beta look like? Recall that this modular space consists of rational stable maps in y of class beta, whose intersection numbers with D are given by the top of P. Here is an example of an analytical curve in our modular space. So it touches the boundary D in three points P1, P2, P3, and it also touches the two exceptional divisors. When we tropicalize, we get this tropical curve. The red sub tree denotes the spine and the two green rays denote the twigs. This is how tropicalization of analytical curves in our modular space look like. In observation, since our analytical curve meets D only at the marked points Pj, the twigs cannot meet the boundary of MR at arbitrary points, because most of the boundary of MR is just tropicalization of D, except the subset ET-trop inside the boundary of MR, which is of co-dimension greater or equal to one. So recall that in the lemma of Toric model, ET is the complement of the isomorphism locus of pi, like the center of the blow-up, and by the statement that the Toric model induces a bijection between the generic points of the strata of D and the generic points of the strata of DT, it implies that the tropicalization of ET is of co-dimension at least one inside the boundary of MR. So definition, let wall inside MR be the image of all possible twigs. Then, by the balancing condition, together with the above observation, we see that the subset wall in MR is a polyhedral of co-dimension greater or equal to one. And we can make it finite polyhedral by bounding the degree of twigs. Here is an example, the picture of walls for the example of blow-up of Toric surface we gave above. ET-trop, the tropicalization of the center of our blow-up is just two points at the boundary of MR. And by the balancing condition, we can see that the walls consists of two lines and then infinitely many rays in this sector. Next observation, by the balancing conditions again, a spine can only bend at wall. That is, a vertex of a spine is balanced unless it lies in the wall. Because if a vertex is not balanced, a vertex of a spine is not balanced, then there must be some twigs attached to this vertex. And by definition, the wall contains the image of all possible twigs. So a spine can only bend at wall. This gives strong constraints on the shape of spines, especially for transverse spines. Transverse meaning that transverse to wall. So since wall inside MR is of co-dimension greater or equal to one, we see that for a transverse spine H, like in this picture, this red curve, a transverse spine H, like this red curve, if we require the image h-gamma to pass through a fixed point x inside MR, then we can no longer deform the map h. In other words, it becomes rigid. We see that a priori we can translate this gamma. Not really translate, but this gamma we can deform it to something like this. It can only bend at the wall. So if we push it up, it bends here and then it bends again. It has one-dimensional deformation space. But if we fix a point x, then this spine becomes rigid. Furthermore, if we perturb a little bit this x inside MR, then the spine deforms uniquely. This is called the rigidity property of transverse spines. Let's give the precise statement. For simplicity, we just restrict to extended spines, i.e. not truncated, but rigidity holds for general spines too. It's just more complicated notations. So let's restrict to extended ones. Consider the map phi-i-trop from the space of transverse spines in MR with infinite directions given by P to the product where the first factor takes domain of the spine and the second factor takes evaluation of the ice marked point. And let S be a transverse spine inside this space. Then there exists a connected open neighborhood vS of S inside this space of transverse spines such that the restriction of phi-i-trop to vS is a homomorphism onto its image and is open. This is the precise formulation of this intuitive picture. So we just denote S bar by S bar, the image of S in the target, and vS bar, the image of vS in the target. Then the rigidity property implies that vS is a connected component of the space of spines restricted to vS bar, meaning that vS is a connected component of the sub-space where we ask the ice marked point to lie in vS bar. And this looks familiar to the connected component theorem above. For the purpose of deformation invariance, we stated the main theorem of this talk, the connected component theorem, and this is looks familiar, which the connected component theorem claimed that this restriction of M smooth to vS and R is a union of connected components of the proper modular stack M bar restricted to R. So let's recall the notations from the connected component theorem. We had the commutative diagram, the phi-i from the modular space M smooth taking domain and evaluation at PI. At the tropical level, we have phi-i-trop from the space of spines taking domain and also evaluation at the ice marked point. We have two vertical maps. The left vertical map takes the associated spine and the right vertical map is just the retraction map. And we have a transverse spine inside the space of spines and the neighborhood vS of the transverse spine. We denote by S bar the image of S and vS bar the image of vS. And R was a Zariski open subset of this push forward pullback. Assume we can extend the map SP to some SP bar from the proper modular stack to the space of the spines in the compactified skeleton MR. Then by some hypothetical continuity of this extension SP bar, the connected component statement one implies immediately that this proper modular stack restricted to vS is contained in the proper modular stack. The proper modular stack restricted to vS bar is a union of connected components. It just follows from this continuity of SP bar and this connected component statement one. And then statement two, which is the statement of the connected component theorem seem to follow from the smoothness theorem. So this seems to be a quick proof of the connected component theorem. But unfortunately, this reasoning has two major flaws. Maybe let's make a five minutes break and after the break I point out the flaws in this reasoning. If you have questions, do not hesitate to interrupt. So let's restart. This simple idea has two major flaws. Do you hear me? So this simple reasoning has two major flaws. First, the statement one was only about transverse spines. But we applied it to all spines. Non-transverse spines can be terrible. Imagine a spine lies completely in the wall. Then the bands and the weights, they are completely out of control. Because if a spine lies in the wall, then the twigs can be attached to spines anywhere. So we don't have any control on the bands or on the weights for the spine. Consequently, there is no reasonable modular space of non-transverse spines. Why am I seeing some arrows on the screen? So nonetheless, tropical curves always satisfy the balancing condition and have a nice modular space. In fact, we can prove that the space of tropical curves in MR with infinite directions given by P, whose associated spine belongs to VS, is a union of connected components of the space of tropical curves, whose ice marked point maps to VS bar. But it still falls short for remedying the flawed proof, because we must consider not only tropical curves in MR, but also in MR bar. That is, we should also allow components of the tropical curve to go to the boundary of MR. This is not theoretically impossible, but we give up because the combinatorics involved become too complicated. And secondly, spine alone, this is a more serious flaw, spine alone cannot guarantee that a stable map in the proper modular stack lies in the good modular space. That is, we cannot just use spine to impose that our stable map meets the boundary D only at the marked points, because entire components of our curve C may lie inside the boundary. So we cannot really apply the smoothness theorem to conclude. One can try to remedy this flaw by considering tropical curves in MR bar, but it's not easy. So after all these discussions, we are finally ready to sketch the proof that works. I spent some time explaining why the heuristic proofs do not work, because with Sean we also spent a long time in order to figure out a correct proof. So let me give a sketch of a proof of the connected component theorem. First, we need the theorem of continuity of tropicalization. That is, the tropicalization map from the modular space M smooth to the space of tropical curves with infinite directions P, this tropicalization map is a continuous map. Consequently, the composite map that takes the first tropical curve and then associates the spine is continuous over the transverse locus, the locus of transverse spines. We claim continuity only over the locus of transverse spines, because there is no nice topology on the set of non-transverse spines. And the continuity statement follows from a general continuity result in a previous paper of mine, or in the paper by Regnation, which is proved using formal models in my paper and using log geometry in Regnation's paper, respectively. It is also easy to give a direct proof in this special case for this specific modular space. We do not need the neither formal models or log geometry. So now let's recall the statement of the connected component theorem. We have this commutative diagram, the natural map phi i taking domain and the evaluation of the ice marked point. At the tropical level, we have phi i drop at the level of spines and the map vertical map taking associated spine factors via the space of tropical curves. Then we are given a transverse extended spine S in the space of spines. The claim is that there exists an open connected neighborhood VS of S and a connected Zariski open R of the push forward and pull back of VS such that when we restrict M smooth to VS and R, we obtain a union of connected components of the proper modular stack restricted to R. Here is the sketch of proof. Step one, the rigidity property of transverse spine implies that there is an open connected neighborhood VS of S in the space of transverse spines such that VS is a union of connected components of the sub space of transverse spines whose ice marked point maps to VS bar. This is just a follows from the rigidity property. Then up to shrinking VS, we have an analogous statement for tropical curves, but we can remove the transverse condition. So the space of tropical curves whose spine belongs to VS is a union of connected components of the space of tropical curves whose mark ice marked point belongs to VS bar. Then by the continuity theorem, the subspace of M smooths whose spine belongs to VS is a union of connected components of the subspace whose ice marked point maps to VS bar. And since the right hand side is as I risky open inside this proper modular stack M bar, we deduce that M smooth restricted to VS is as I risky open inside the proper modular stack M bar restricted to VS bar. And the connected component theorem claims that we have a union of connected components. So it remains to show that it is the risky closed after restricting to some dense open subset are in inside this pre image of VS bar. Step two, we prove the following claim. So in order to show that it's the risky to prove the risky close closeness, we use some closed analytic disk to detect to detect the risky closure. So we consider a map of five from the closed unit from a closed unit disk to M bar to the proper modular stack M bar restricted to VS bar. In other words, we consider a family of stable maps in M bar parameterized by a closed unit disk with the condition that the ice marked point maps to VS bar. Such that the image of the punctured disk lies in in the smooth locus and also has a spine in belongs to VS. Then the claim is that the central fiber belongs to M UP beta, meaning that the central fiber has correct intersection numbers with the boundary. Sorry, you said it's closed. So when put an upper index smooth and low index VS to this five zero five zero five of the punctured the disk. Yeah. So this is a condition on this family. Yeah, yeah. But if it's closed, then we want to prove that five zero also belongs to the same set. It's we want to show that it is closed after further restricting. Ah, so it's just yeah. Without further restricting. It's not true. So that's why it's just an intermediate step. Yeah. So we consider a family of stable maps. Parameterized by a closed unit disk. The fibers outside the zero. They are very nice. They lives in the smooth locus and also their spines belongs to VS. Then we claim that the fiber, the central fiber over zero. Although it might not be as nice as the general fiber, but it will still have the correct intersections with the boundary. So here we see that the central fiber is a degeneration of the generic fiber. And we can have some new bubbles appear. But they will not be too bad. They are still, they still have the correct intersection numbers with the boundary. And let's first stabilize the domain curves, contracting all the bubbles. And then so after contracting all the bubbles, we get a family of stable pointed curves. Then we take a trivial family of small caps around every marked point. So it's always possible if we shrink the disk. And we can always take a trivial family of small caps, let's say of size epsilon around every marked point. P1, P2, P3, P4 as in the example. Then we take a preimage under the stabilization map. So here the preimage is again a family of small caps. But a priori, the preimage may contain some extra bubbles. Step 2.1, we show that there exists a compact analytic domain K in U analytic such that the boundary of each cap, this orange cap of every fiber maps to the compact analytic domain. The idea is to use the rigidity property of transverse span plus the continuity zero. So first, we control the boundary of these caps. And the second, we show that the body, this white part, the body of each fiber maps to K. In particular, the body is disjoint from the boundary D analytic. Because our goal is to show that the central fibers still have correct intersection numbers with the infinite with the boundary divisor D. So we want to prevent bubbles inside the body from moving to the boundary. And we use a compact subset to prevent this. And to show that each body must lie in the compact subset, we use the affinities of U and the maximal modulus principle. So we cannot prevent the body from touching the boundary without the affinities of U. Step 2.3, we show that up to shrinking the size epsilon of the caps, the caps of the central fiber cannot contain any bubbles. For this, we need to use the fact that the set of walls has a co-dimension greater or equal to one. And we use the continuity theorem and also the stability condition of stable maps. Now combining steps 2.1 to 2.3, we see that the central fiber meets the boundary divisor D only at the marked points. And the tangency orders are also good by the curve class beta. So then the claim in step 2 follows. The really essential part is to show that in this degeneration procedure, we cannot have bubbles moving from the body to the caps and finally reach the boundary divisor. So finally, we can conclude from steps 1 and 2 and the smoothness theorem. Recall from step 1 that the modular space M smooth restricted to VS inside the proper modular stack M bar restricted to VS bar is Zariski open. We already proved Zariski openness. It remains to show that it is Zariski closed after further restricting to some dense open subset R. And for this purpose, in step 2, we try to use some family over a closed unit disk to detect Zariski closure. So the claim in step 2 says that for any family in the proper stack M bar restricted to VS bar parameterized over by a closed unit disk such that the punctured disk lies in the good locus. Then the central fiber has correct intersection numbers with the boundary. This implies that if we take Zariski closure of this M smooth restricted to VS inside the proper modular stack M bar restricted to VS bar, the Zariski closure must lie in this modular stack consisting of stable maps with correct intersections of the boundary. It's just a reformulation. And let's denote this Zariski closure temporarily by this M bar VS. And we obtain a proper map of Phi i restricted to this Zariski closure by the properness of the modular stack M bar. Here we use the properness of M bar and the claim we can take R, this dense Zariski open R to be simply the complement of the image by Phi i of this closure minus the good part. So we just, we want to get rid of the bad part and we just the bad part is the complement of the good part. And we take image of the bad part and then we take complement. And the image is Zariski closed by the properness of this map Phi i. So claim is that this R satisfies our purpose. Proof of the claim, the pre image of R lies in the good locus by definition because we have take out all the bad locus. So the Zariski closure of this M smooth restricted to VS and R inside M bar restricted to R lies in M smooth. It's just by the fact that this is Zariski closure. So this implies immediately what we want that this is a union that M smooth restricted to VS and R is a union of connected components of M bar restricted to R. But when we take a complement of the image of the bad locus, maybe the image of the bad locus is just the whole thing. And we have to apply the smoothness theorem to show that R is actually big. It intersects every fiber of the projection. So in particular, R is not empty. This completes the proof of the connected component to zero. I'm sorry, it's a bit long to explain the idea, the sketch of the proof. But that's really the essential for me the most essential statement for the non-Archimedean curve counting without this deformation invariance, nothing will work. Okay, so this is a sketch of the connected component to zero. Why am I also seeing something on the screen? Can people not annotate on the screen if you I think maybe if you annotate on the screen then everyone will see on the screen. So let's go to the next section. Tauric tail conditions in families. Recall that for counting curves with boundaries associated to truncated spines, we must impose some extra regularity conditions on the boundary so as to obtain a finite dimensional modular space. When our log-Kalabiya variety U contains an algebraic torus, Tm, we can take advantage of this torus and impose a simple boundary regularity condition called Tauric tail condition. We have introduced the Tauric tail condition for skeletal curves in the two previous lectures. For the deformation invariance of counts associated to transvert to truncated spines, we must study Tauric conditions in families. So first, let us extend the Tauric tail condition to non-skeletal curves. Question, how to specify a tail inside a rational curve? Recall from the symmetry theorem that adding or removing internal marked points does not affect the counts. So we can specify tails by adding as many extra internal marked points as we want. For example, consider a rational analytic curve C with marked points P1 to P4, P1, P2, P3, P4. The red subtree is their convex hull. If we want to specify a tail containing some boundary marked point P1, for example E equals 2 containing this boundary marked point P2, we add an internal marked point PS here and consider the preimage by R of the path connecting PS and PE. Where R denotes the retraction map from our curve C to the convex hull of all the marked points. The red subgraph plus the pink one. If we want T to be a genuine tail containing the boundary marked point PE, we should choose PS sufficiently close to PE so that T does not contain other boundary marked points. If this is anywhere else, it may be T may contain other boundary marked points, then it's not a tail. It's many tails. And we denote by T star to be T minus the boundary marked point PE, the punctured tail. Recall that the toric tail condition asks the punctured tail to map to the torus. The example shows that in general if we want to impose tail conditions for stable maps in our modular space M smooth, we can specify a tail by picking an internal marked point PS and a boundary marked point PE and we should restrict to the subset theta inside the space of domain modular where the preimage T that is preimage by R of this path does not contain other boundary marked points. And then once we specify a tail, we can impose the toric tail condition. We have the following lemma that of equivalent formulations of toric tail condition. For any stable map in our modular space M smooth, whose suppose that its domain lies in the subset where tail makes sense, then the following are equivalent. First, the punctured tail lies in the torus. Second, the whole tail lies in the isomorphism locus of the toric model pi. Third, the tail does not intersect the exceptional locus of the toric model. And fourth, there are no tweaks of the tropical curve, F attached to the path to the path PSPE inside the spine. So this is a picture of the tropical curve associated to F. The red sub tree denotes the spine and we have some tweaks. Then the toric tail condition equivalently says that there are no green tweaks attached to the path connecting PSNP. This part is three of tweaks. And now we have the proposition for toric tail condition outside the wall. Consider let N in M smooth with a good domain modular be a subspace such that F of the marked point PS does not lie in wall for all F in N. Then the subspace satisfying the toric tail condition and tail inside the N is a union of connected components. So I recall that for deformation invariance, we need to show something is a union of connected components in another so as to obtain finite a tallness. And here we are doing it in two steps. First, for the extended spine, we have the connected component zero. And the second, we have to show that when we impose the toric tail condition, we further cut out union of connected components. Yeah, so since the toric tail condition is the second equivalent formulation here is an open condition. So the openness of the inclusion follows. For closeness, we use the equivalent formulation for and pick any sequence F lambda in the subspace and tail conversion to some F in N. More precisely, we do better pick any net instead of sequence because the underlying topological space may not be first accountable. For any F lambda consider the associated tropical curve like here by Toric tail condition for there are no tweaks associated to the path PS to PE for any F lambda. In particular, the leg LS this leg containing PS must be contracted. Then by the continuity of tropicalization for the limit F, the leg LS this leg must also be contracted. And moreover, if any tweak moves into the this path connecting PS and PE under the limit, then the tweak must attach to the vertex PS bar, which is the intersection of the leg. L E and the leg LS. We see that this is impossible because by our assumption, the image of PS does not lie in wall and the leg LS is contracted. So if it's not it's not lying wall, there can be no tweaks attached. So F must satisfy Toric tail condition for this shows closeness. And this proposition plus the connected component theorem implies the invariance of the count NS beta under small deformations of any transverse spine. Here the spine can be truncated and the transverse means in particular that the finite one-valent vertices do not map to wall. Second remark for the proof of the associativity of the mirror algebra and the proof of the wall crossing homomorphism. We will need a deformation invariance for not only transverse spines, but also certain non-transverse spines. For example, for the associativity of the mirror algebra, we need to show that the spines like the following SL, SMSR, they all have the same counts. Imagine that we deform the spine SL to SM, we move it a little bit to the right, and then to SR. In the middle, we get a non-transverse spine SM. And similarly, for the proof of wall crossing homomorphism, I will explain wall crossing in the next lecture. When we need to verify that the ring homomorphism, we want to verify that the wall crossing transformation is a ring homomorphism. If we multiply by a theta function that is parallel to a wall, we will need to show some deformation invariance as follows. So we want to show the deformation invariance associated to such a spine when we move it across the wall. This is important for showing ring homomorphism, showing that wall crossing transformation is a ring homomorphism. And this edge can just be parallel to a wall if we are multiplying by a theta function that is in the same direction as the wall. So eventually what we are interested in are just the equality between the counts associated to SL and SR. But to prove the equality between the counts associated to SL and SR, if we deform SL into SR, it's inevitable that we encounter some non-transverse spine SM. In both cases, when a finite end maps to a wall, we observe that in both cases, when a finite end maps to a wall, the derivative at the end is contained in the wall. So although they are non-transverse, but they all have this special condition, they all satisfy this special condition we call such a spine almost transverse. Question, do we have deformation invariance for almost a transverse spine? The answer is yes, but the above proposition does not apply. We must restrict to skeletal curves. So that's another application of the theory of skeletal curves. And in fact, for other applications of skeletal curves such that for the symmetry theorem, it's possible to do it without skeletal curves. We can have other proofs, but for this almost a transverse spine, I don't know how to do it without using skeletal curves. And that's really important, these almost transverse spines because that's for the proof, that's for the associativity and also for wall crossing homomorphism. Without associativity, we have nothing. So let's explain how to get a deformation invariance for almost a transverse spines. Recall, we have our map phi i from the modular space m smooth, taking domain and evaluation of the iso-marked point. And we denote by isk the pre-image by phi i of the skeletal. Tony, what does let i mean since it's notation isk? i means we are evaluating at the iso-marked point. Ah, it's a skeletal isk here. Bigot, it means inverse of skelet. But it's just a notation. Yeah, inverse image of skeletal. Yes, so it's just notation, inverse image of skeletal, it consists of skeletal curves by the previous lecture. So now here is the proposition. Toric tail for almost a transverse spine. Let n be a subspace of m smooth with good domain modulus intersect isk. That is, we only consider skeletal curves. Let n be an open subspace such that for any f in n, f-trop of p-s lies in some polyhedral cell sigma of wall. Then the linear span of sigma contains this derivative p-e. So this is the condition for almost a transverse. And once the condition of, once f is almost a transverse for all n, for all f in n, then the subspace satisfying the toric tail condition is a union of connected components. For me, that's really the next, after the connected component theorem, that's the second most essential technical result for the whole theory to work. Now, again, openness follows from the toric tail condition too, because it's an open condition. For closeness, let f in n be a point in the closure. Since n in isk is open, the restriction of phi i to this n tail is also open by the smoothness theorem. Because we have this open inclusion and by the smoothness theorem, we have a dial map, a dial map is open. So we have restriction of phi i to n tail is open. So we can find a net f lambda in n tail convergent to f such that the associated tropical curve f-trop has a transverse span for all lambda. So this is this blue, two blue rays are part of our wall, and this red tree is the spine associated to f lambda, and we have some tweak. So let i lambda denote the interval in the spine of f lambda connecting the point p s bar and the nearest branching point. Toric tail condition 4 implies that the associated tropical curve f lambda trop contracts the leg ls containing p s. And moreover, it is balanced on this i lambda union, the leg le with derivative pe. So the Toric tail condition says that there are no tweaks on this path connecting p s to pe. But since the branch nearest branching point is a bit away from p s bar, by the balancing condition at p s bar, we see that moreover this part i lambda is also good. It's balanced everywhere, also containing i lambda with the good derivative pe. So if in the limit f-trop of p s lies in some cell, sigma for example in this cell, our assumption says that the linear span of sigma must contain this direction pe. So we imagine that when we move this tropical curve to this cell sigma in the wall, we see that our assumption is satisfied. The direction of this leg lives inside the linear span of the wall. Then the transversality of the span of f lambda implies that the image of all this part, this lower part, does not meet any such cell of wall. Because it's out of wall and also it's parallel to the wall, so it cannot meet any such cell. Therefore, by continuity of tropicalization for lambda sufficiently big, the distance between p s bar and the pre-image of all walls has a positive lower bound. Since when it's out of the wall, it's just parallel to the wall, therefore when we compute the pre-image, we throw away all this sigma. And then we obtain a positive lower bound between the wall and this point. So this prevents, this will prevent that any twig, this green twig slides through this interval i lambda and reaches p s bar in the limit. So the limit f still satisfies the toric tail condition, completing the proof. So here the main idea is to have a lower bound between this pass and the twigs. We want to prevent any twig from sliding into our pass, so we try to find a lower bound of distance between them. And this is only possible if we restrict to skeletal curves. A remark, this proposition implies the deformation invariance for almost a transverse span. It is necessary for the proof of the associativity of the mirror algebra and the proof of the wall crossing homomorphism. That's all I want to say for toric tail conditions. And now in the remaining five minutes, I'll give a quick introduction to the gluing formula. Recall that we have defined the counts of curves with boundaries associated to truncated spines by imposing an extra regularity condition on the boundaries, namely the toric tail condition. In this way, the counts of open curves are translated into special counts of closed curves. So it is natural to have some skepticism at this point. Do the counts we define really reflect open curve counting? Or is it just some false advertising? We would like to relieve this skepticism by establishing the next important property of our counts, the gluing formula. The idea is the following. Roughly, the gluing formula states that we can glue two open curves along two opposite boundary components, like here, and form a bigger concatenated curve. Then the counts of the concatenated curve should be the product of the counts of the two initial curves. Here we have two open curves, and we glue them at the boundary. We obtain a concatenated curve, and the gluing formula states that the counts of the concatenated curve should be the product of the number of this sort of counts for the first one and times the counts of the second one. So this is a more convincing evidence that our counts really reflect open curve counting. And it is also an essential ingredient in the proof of the associativity of the mirror algebra. Maybe let me stop here for today, and next on Thursday, this Thursday, I will continue from here. I will give a quick proof idea of proof for the gluing formula. Although in the two dimensional case, the gluing formula was in some previous paper of mine, but here I'm giving a more conceptual proof. And then I will talk about the idea of proof for associativity. But the main topic of the last lecture will be wall crossing transformations. How do we get wall crossing transformations from our counts of skeletal curves? And how do we compare the resulting wall crossing structure with the wall crossing structure of gross-hacking keel condensates in the case of cluster algebras? So if you think that some technical details in today's proof are difficult to follow, in the next lecture, there will be some more statements and with less proofs. So thank you very much for your attention.