どうもありがとうございました。最後にステーブペレーションを解説しました。このシリーズを紹介します。このシリーズはトランス・トーマスインバリアンスのシリーズについて解説しました。カウンティング・サブスキーズのカラビアンスイフォーズです。シリーズのシリーズはツルス・ロジェクトル・CY3であり、バータの字を表現しました。このシリーズは、パンデ・ハイパンデ・トーマスステーブペレーションです。このシリーズは、ラフィス・ペーキング・カウントの写真です。その数はFとS、Fは1Dシーフ、Sはこのままこのようにしています。それが、何か必要なサジェクティブではありません。しかし、サジェクティブは1Dシーフです。これは、このマークのコカネルがあるかもしれません。しかし、このコカネルは0Dシーフです。これは、1Dシーフです。広いSは、このままこのようにしています。そうすると、普段的な例目が、このデバイザーを使って、このファイナスを使ってこのデバイザーを使うことで、このようなコンポジションを取り出すことができます。FはこのカーブCのダイバンドに合わせることができます。Sはこのコンポジションです。そして、このエクゼンプルのステーブペアを使っています。ステーブペアのモジュアスペースは、このニューメイカーコンディションは、ファンタメントフォモジークラスに合わせるこのファンタメントフォモジークラスに合わせるFに合わせると、フォモジークラスのエクゼンプルのステーブペアのモジュアスペースは、このニューメイカーコンディションは、シメトリックパーフェクトブスラクションです。 sonraニューメイカーコンディションに合わせるインタdoing フォモジークラスに合わせるファンタメントフォモジークラスのインタ controlsサイコールの雪蔵を選択し、何が masks, theres,side cos mist.Generating series over N is beta X to be the generating series like this and also the full generating series is defined by 1 plus beta is positive so beta positive means that this is the homomorphic class of the effective one cycleand I take the generating series like this.So this generating series is the series that is responsible to the revisions whether in variants like the chromo-fitting variants or cobalt-mabuffin variants.So today I will talk about its revisionship to chromo-fitting variants and the sum of contextual yes and the properties for this generating series for that revisionship to make sense.So that is vector by P that was 3.3.So the relation is that this generating series of stable pairs are more positive so this is a combination of the prediction in homomorphic class of concor-fantai-pande together with the DTPT correspondence which I explained yesterday.Anyway so this is given by the potential of the generating series.So this is the, yes I might explain that this is the so called genus G and curve class beta chromo-fitting variant.So this is G and homomorphic class beta chromo-fitting variant.And this equality is whole after the variable change that is given by Q equal minus exponential i lambda.Yes so this, yes I explained that this modulate space of stable pairs is indeed a scheme and this invariant is integer.But contrary to the stable pair invariant, chromo-fitting variant is not necessary integer.But this is rational number.So I explained what is chromo-fitting variant roughly.So after speaking well, chromo-fitting variant is given by, this is defined by the integration of the virtual fundamental cycle of other modulite space of curves on carburetor world.So this is so called modulite space of stable maps.So this is modulite space of pairs that is C and F.She is a curve that has only, she is a projective curve with at worst nodal singularity and F is a map, algebraic map from C to X such that the genus of,isometric genus of this curve is G.And the push forward of this fundamental cycle coincides with beta.And the stability condition is that you consider the automohism of this map that is the automohism of this curve C that is compatible with this map is finite.Well it's the pair of C and F satisfying this property is called stable map.And it is known that this modulite space of stable maps is indeed as we know for the stark.But it may not be necessary scheme because of, well, this automohism may not be finite, but still there might be some automohism.And if there is automohism, then the modulite factor which, which is modulite factor of the family of stable maps may not be presentable by a scheme,but it is only a minimal for the stark.So because of the appearance of this automohism, the groomed hit invariant is not necessarily integer, but this is rational number.So this MNOP conjecture expresses some visions of rational invariant and integer invariant.And this gives some hidden property of the integrity of the groomed hit invariant.But yes, this equality holds up to this variable change.And the appearance of this variable change, it's not obvious that this makes sense because, well, the left hand side is some series,some series of invariant that is expansion where q is the absolute value of q is sufficiently small.On the other hand, the right hand side, this is a series where the absolute value of lambda is sufficiently small.That corresponds to that the q is, yes, lambda equal 0 means that q equal minus 1.So this means that the right hand side expands the function of the left hand side at q equal minus 1.So in order to state about this variable change, we need to know the convergence of this generating series of the left hand side.So if I draw the picture, so this is the domain where the function of the generating series of stable pairs are defined.So q equal 0, I get invariant and at q equal minus 1, so this is where lambda equal 0.The expansion near this point corresponds to groomed hit invariant.And in order to make sense about this variable change, we need to take analytic continuation of the function.So here, this generating series of stable pairs is defined.Here, the generating series of groomed hit invariant are defined.We have to take some analytic continuation.So it does not, it's not obvious whether it's such an analytic continuation makes sense.And in order to state about this property, it was also conjectured by MNOP.And indeed, it was proven by myself and Breachland.That is conjectured.So this is a 7.The problem is that if I fix the homogenous class beta and take the generating series of stable pair invariant.So this is a series of q.But this series of q is indeed lowland expansion of the rational function of q.And it should further satisfy the strong symmetric property that, that is invariant and taking q to 1 over q.Well, so of course, if I have this property, then this generating series has a convergence, there are values.So at least for some, and indeed, it can be shown that this generating series converges when the absolute value of q is less than 1.And it makes sense to state about this analytic continuation.Also, this symmetric property is compatible with this formula because it should consider right hand side.So this right hand side, the power of lambda is even.So this means that the right hand side is invariant and taking lambda to minus lambda.So taking lambda to minus lambda is corresponds to taking q to 1 over q.So this symmetric property is also compatible with the symmetry in the Gromhitten side.So in this picture, this means that in q equal to infinite, we can also take the analytic continuation of the generating series of stable pair invariant.And we can also go through here.And after this analytic continuation, this generating series is indeed the same value.Here I explained one example.So maybe the simplest example is separation to rational curve inside Calabria C-fold.That is, let's consider like x equal, this example is where x is not compact,but still we can make sense about the definition of stable pairs.And this is the total space of minus 1 plus minus 1.So this is the total space of length bundle on P1.And yes, let's write this c.And maybe let's take s.So this is 0 section.And let's write c to the image of this 0 section.So this is a non-compact Calabria C-fold.And the normal bundle of this curve inside x is given by, yes, of course, P1 minus 1 plus 2.So in this example, it is easy to describe the modular space of stable pairs for the curve class just for this curve class.That is, it is easy to see that the modular space of stable pairs.This is, as I said, this is given by the pure shift is c minus 1.And s is the global section of this shift that is not 0 and up to ison.And because the c is P1 and the global section of this shift is n-dimensional.So this is modular space is nothing but Pn minus 1.So as I told yesterday, if the modular space of stable pairs is non-singular,then the corresponding PT invariant is given by the particular order number of the modular space up to sine.And the sine is given by the dimension of the modular space.So in this case, the stable pair is given by Pnc equal minus 1 minus 1 multiplied by n.So in this case, it is easy to describe the generating sheets.So in this case, the generating of the sheets of stable pairs with fixed homogenous just a single curve cis given by q minus 2q square plus 3q square minus 1.And it is, of course, this is given by the rational function.And now over, this rational function has this symmetric property.This is because of course this is 1 plus q square over q.But of course if you take just formally take the substitution of q to 1 over q in this series,then of course you get series which are not true.So this symmetric property can be only seen after you take the analytic continuation of the generating sheets.But anyway, if you can once write the generating sheets of stable pairs in terms of rational function like this,then the variable change which appears in MNOP conjecture makes sense because of the variable change was q equal minus exponential i lambda.And if you substitute it to here, then it becomes 1 minus exponential i lambda square and minus exponential i lambda.And this is something like this.And of course you can expand it in terms of series of lambda.And also also make remarkable to the MNOP conjecture where it was the work by Panda-Hyperlander and Pickstone.So they put MNOP conjecture for some caribbean 3 fold given by the component intersection caribbean 3 fold into the product of project spaceby using the method of like like degenerations and localizations.But anyway, so when proving this rationality is it requires quite different method.And in this rationality statement, I'm also using the verb crossing phenomena in derived category which I'm going to explain now.Yes indeed, we have a stronger statement that has to do with the so-called gubacman-buff expansion which I will explain later.And indeed, I have the following.So this is the product expansion formula.And this is the theorem which implies this rationality result.So the result is that the generating series of stable pairs.I mean for the first generating series that is also after taking the sum of all the curve-curve series.So it is given by the product of the generating series of some other invariant.Which generalizes the usual notion of Donaldson-Thomas invariant in two different kind of ways.So the statement is that there are other invariant like large n and n beta.So this is a rational invariant and there also exist integer valued invariant.It has some properties that the first property is that n n beta coincides with plus omega beta beta equals to n minus betawhere omega is some arbitrary amplitude by its own x.And the property of this invariant is n beta it coincides with n minus n beta.And this is indeed 0 if the absolute value of n is sufficiently big.Such as that using these other invariant large n and large f.I can expand the generating series of stable pairs in terms of these invariants.That is px is given by product of positive exponential minus 1.Large n f beta nk beta multiplied by the generating series of large f invariant.Yes, that I will explain about these invariants.But these invariants are somewhat generalization of the usual Donaldson-Thomas invariant.And I will also let I explain that this is somehow part of the so-called Copacma buffer expansion.Yes, indeed by the property of the large n and large f invariant.This rationality statement is a consequence of this product expansion formula.This is cell A and this is cell B.Indeed cell B implies this is by the following reason.Because I am taking the exponential of the generating series.Because after taking the exponential of the generating series,after taking the exponential of the exponential,this p beta x is given by the finite number of product of the generating series of large n invariant and large n invariant.That is, it is enough to know about the rationality of the generating series.But the generating series, such generating series,it is easy to see that it is a rational function because of this property.So this is rational function invariant and taking u to bq.This is n positive.For the generating series here,indeed this is a polynomial because if you take the sufficiently big n,then both of n beta and n minus n beta are the same.Because of this symmetric property,this is lowland polynomial q that is invariant.By these properties,we can see that this product expansion formula gives the rationality.Indeed this large n and large n invariants have geometric interpretation.Indeed we can write the large n invariantin terms of the Donaldson-Thomas invariantwhich I defined yesterday.That is defined by the omega is the 13 fixed amp divisor on x.This is 0, 0, and n.So this is the Chan character.So this is h0 plus plus.And so this is the virtual number of one-dimensionalsemi-stable sheave with fundamental homogenous coincides with beta and formula characteristic of f coincides with n.But this is not precise because as I told yesterday,the usual Donaldson-Thomas invariant should be integer.But this invariant is not necessarily integer.This is because I'm going to,I want to define it so that it also imbibescontributions from stricty semi-stable sheaves.So in the talk of yesterday,I defined the Donaldson-Thomas invariantmodular stack of stable sheavesand modular stack of semi-stable sheaves are the same.That is, there is no stricty semi-stable sheaves.But in defining,so in considering this modular spaceof one-dimensional semi-stable sheaves,so there might be some stricty semi-stable sheaves.And in that case,the usual definition ofheating invariant does not make sense.And in this case,we have to give some other definitions.And yes.So the more precise definition of this invariant,which I will postpone tomorrow.So indeed,so this is the actual numberand need to allow the case that.So also another remark is thatwe have a purely in defining this invariant,we have to first pick polarizationand consider the modular spaceof omega-semi-stable sheaves.But the resulting invariant is indeedindependent of the choice of omega.So we don't have to take care ofthe choice of polarization.This is not a vice-fact,but it can be proven.So this is the meaningof this large n invariant.And by this property,by this geometric definitionof large n invariant,it is easy to see the symmetryof large n invariantand also p-o-t-o-d-c-t.For example,the p-o-t-o-d-c-tis coming from takingthe line boundaryfor the Schwarzschrank as coincides with lambda.And also,the Schemitic propertyis coming from takingthe derived dual of the one-dimensional sheaves.So today I'm going to explainabout large n invariant.Indeed,it has to do withderived category.So this might beshould be written as that.So this is a virtualnumber ofthirteen stable objects,dvx,with chunk at coincides withone-zero,better than minus n.So idea speaking,we want to takethis stable object as takingp-o-t-o-d-c-tin the derived categoryand takethe modular space ofp-o-t-o-d-c-t.But at this moment,it is quite difficult to describethe space of stability conditionson projectivecurvacy for it.But instead,there issomehowgenerated notion ofstability conditionsand using thatdegeneration ofstability conditionsgive such desiredstability conditions.And the key propertyof the stability conditionwhich appears in this invariantis that,where that isthis issafe-derivedin some sense.So of course you can also interpretthe modular space of ideashifts or modular space of stable periods.But they do not satisfythesafe-derived property.The safe-derived property is thatwhere that is preserved undertaking the deriveddual.So if you takethe two-term complex given by stable periodsand take its derived dual,then it is no longer a stable pair.But instead,we can find some stability conditionssatisfying thatsymmetric property.The invariant also has this kind ofsymmetric property thatgives the correspondingsymmetric property to the stable periodsusing thisformula.Now I'm going to explainthat this formula is a consequenceof so-called wall-crossingdenominal inTBX.So here is the ideaof the wall-crossing.If we say wall-crossing,we have to findsome family of stability conditionsand we needto investigate all themojized spaces of stableobject orassociated invariant changeand the crossing of the wall.And maybe one ofthe key idea is tomake the categoricalunderstanding of stable pairs.I have a supporting member.So this is easy tosure,but letI be some object inthe derived category.This isindeed isomorphic to two-termcomplex by stable pairand only if the followingconditions are satisfied.One is thethe thermoge of thiscomplex is isomorphicgiven by this isomorphic tothe idea of somesub-scheme.The second thing isthe firstcomplex is0Dimensionalandthat property is that there is no mapfrom 0x shifted by-1 or 0.So this is the categoryof 0Dimensionalsheaves.And I am takingthe shift by-1and the final propertyis that there is noother cohomologies except1 and 2.So 0 and 1.So this is nothingeasy exercise,butthis propertycorresponds to the fact thatI have this maps.So this iscorresponds to that.Coucane of sis most 0Dimensionaland this fact correspondsto that.This shift f is pureand of course this first factcorresponds to that.Thecomplex is just two-termthis property is thatthis is nothing but this isLand-Lanquan-Torzhoff's shift.Anyway,so I am going togenerate these properties.Soanyway,the key point is thatI have 0Dimensionalcategoryof 0Dimensional sheavesfrom here and also the co-cane ofso this is also object in the categoryof 0Dimensional sheaves.Sowe can generate these notions in the following way.So instead of consideringthe shift,let's consider0 or 1Dimensional sheaves.So this isa subcategoryof thecategoryof co-cane of the sheaves.This is a categoryofsheaves on x so thatthe dimension of the support of fis less than or equal to 1.Yesterday,I defined the notion ofsemi-stable sheaves.And the notion of semi-stable sheaveswe sticked to here.So it corresponds to thesemi-stable sheavesthat isforfix,ample diviserhighlightkega to bethe stop functionhere,given by sendinge to themultiplied by the fundamental homogenousassociated to this 1Dimensional sheaveby e.So,indeed,thestability condition,theuser stability condition imparts thatweb object in here isomega semi-stableand only iffor any non-trivialsub-object,I have the inequality like this.And then let's defineshe bigger than or equal tokey to be thethis is thesubcategoryof thecategoryof theor 1Dimensional sheavesthat isgiven by the extension closurethe semi-stablestop its bigger thant and closurethis isgiven byfor every n numberI have this categorysome equal to 1xand I alsohave some nearcategory like thisand I'm going to definesome family ofstable or semi-stableobject.So by definitionI would define objectin the derived category.So,Isay this issigma t stableif the following conditions are satisfiedso this is the analogy of the conditionwhich I wrote here.Indeedin the first condition is thatyes,this first condition is sameand the next condition is thatis contained in thesorry,stableor semi-stable.In thestable case it iscontinued in she bigger than1 equal to t.This conditionis modified thatbecause of y equal tot.Shifted byminus 1.So this is a stable caseand for the semi-stable caseit ist minus 1.And the finalcondition is also same.Yes,sothis is relative to the stabilityof stable pairs in the sense thatif you takethe limit to t toinfitthenfor t toinfit limit.So this isequivalent to that this is a stable pair.Well,this is because if you takethe limit to t toinfit,then thiscategory approaches to thecategory of zero-dimensional shifts.And if you just replacethis categoryby the category of zero-dimensionalshifts,then you will think about thestability conditions of stable pairs.And also for generalitymati-stableis equivalent to thatmati-semi-stable.Sogeneral means that for exampleif you like not containingrational number,then I have this.So anyway,so I have the notionso here indeed it can be alsoshown that this notioncan be expressed as t structureand some generalized centercharge.Butfor this purposeit is more efficient to describethe stability condition in this art of clay.Anyway,we havesimilar to the modular space of stable pairs.We can consider the modular space of stable objectsatisfying these properties.Butcontrary to the modular space of stable pairs,I don't know whether the modular space of satisfying this propertyis indeed a projective scheme.But at least it is realizedas algebraic space of finite type.So let's writemati-semi-stableand tdata.So this ismati-spaceofsigma-t stable object.Chan character coincides withone zero minus beta minus n asbefore.Soit is shown that this isargibike spacefinite type.Butfor the application to theanimative geometry,it is enough to knowargibike space.Becausewe have the very defined notion ofvariant function on this modular space.And by taking the integration of thevariant function,I get some invariant.Yes,for theindeed,I didn't give the definitionofvariant function,but that willbe postponed tomorrow.Butanyway,when beta is givenbythe integrationof thevariantconstructivefunctionand this isinteger-variant invariant.Andwe also set the geneticsheets likesigma-beton-nmetathe.So thebub crossing phenomenaask that how this generatingsheets varies underchange of stability condition.So thechange of stability condition is given bychanging the parameter t.Soas expected by thefinite formula which Iwrote before,this isthebub crossing formulathat isfor each fixed real numberif we consider theto be meeting generatingsheetsminus f.And the difference isgiven by thefornation of the generatingsheetscounting one-dimensionalsheets.This isgiven by nature,beton,andt-beton.So this isgiven by the formula which appears in theblossom-domas and hypandetomas correspondence.Andyes.So Ihave this picture.So t is theparameter space of stability condition.And yes,as Iwrote before,if t isselection to big,then thepresponding modular space correspondsto stable pairs.But there mightbe some rules on thisstability parameter.Andso this is the formula whichexpresses the difference of thegeneratingsheets at which non-specialpoint.And in thebub crossing,where there areat the wall,there mightbe some destabilizingsheets.And each destabilizingsheets,in eachdestabilizing sequence,thereappear some contributionsfrom one-dimensionalsemi-stable sheets.Sofor this reason,I havethis contribution ofpim-variance countingone-dimensional sheets.Sonow thethe previous productexpansion formula isgivenby taking becauset to infinite,etx equals to thegeneratingsheets of stable pairs.Andif I set eventbetter to bet to plus 0,eventbetter.Andapplying thiswall-coaching formula from t to 0to be sufficientlyclose to 0andt to infinitegives thelnb.Sofinally,I will explain a bitabout the symmetry of thislarge variable invariant.Soby definition,large variable invariant isgivenby countingobject,whichfears sufficiently small to 0.Yes,indeed,we can alsoapply thiswall-coaching formulaat t to 0 equal to 0.Sothis thing,and if t to 0 equal to 0,then this means thatwhether it's a correspondingthird-channel character small n,so this isalways 0.So this wall-coaching factor punishes.Soindeed,in this case,theplus equals tof to the minus x,eventbetter,theplus equals f to theminus.Andon the other hand,one can show thatby applyingderived-dual,so an objectin the derived category is,so this issigma-t-stableif and only ifit'sderived-dual isminus-t-stableand if this haschan-character10-bet-n,then thischan-character haschan-character10-bet-n.So thismeans that this schematic propertyalso implies that,where this thingis also equals tol-nbetter-minus.Sothis implies thatevenbetter equals to n-better.Soyes.Soby consideringthe schematic property,yes,I meanby,so thismeans that by taking the deriveddual,if I have stability conditionfor example,some machine like here,thenby duality,I get stability conditionlike this.So,and of courseif you take stable pair,thestable pair is that where the t,theformat t is essentially big.Andfor this stability parameter,sothis is very far from thesafe duality.But if you takethe stability parameterthat is essentially close to0,then I get the stability conditionthat is very close to thesafe duality.Andif you take t,thenexactly 0,then this is thesafe duality.But anyway,soin this way,the wall crossingphenomena and the symmetry ofthe derived category is importantin thinking about the generatingsheets of stable pairs.Sotoday,that's it.