昨日のディスカッションは、バッファインバリアントフレームのディスカッションで、モジュアルスペースのステーブシップは、フィビアスリーをプロポットすることができました。私はコンジェクチャーと呼ばれました。このシーバースリーのコンプレックスナンバーフィールドは、モジュアルスペースのステーブシップのステーブシップは、フィビアスリーをプロポットすることができました。ここについては、シーバースリーのコンプレックスナンバーフィールドについて、コンプレックスナンバーフィールドが必要できます。しかし、前に品質を日に変えている暗黒の場合は、この価値としては、上下の殻の中理 boiling 技術を代表し、このような技術を調整することができます。そして私たちは、ボーファインバイアントを測定することで、しかし、ファーバーのプッシュフォードのコンスタンドシープはMxIの面状に移動されています。このファンクション、このコリノミアのYは、このように一般的に描かれています。このように、グバクマバーファインバイアントに移動されています。このファンクションは、HBPが最初に表示されているコンジェクチャーです。このコンジェクチャーは、GCでコンパクマバーファインバイアントに移動されています。G?B?BYes, this is defined from a stable pair.No, no, no.This is defined from a stable pair.So, equals to NG beta.So, this appears.Yeah.So, this one is defined by thinking aboutmodular space of stable pairsthat is something like pairs of curvesand points on them.And this is defined frommodular space of one-dimensional stable sheavesthat is modular space of curvesand vector boundaries on it.So, let me give one example.So, in the last time, I gave a competitionon the left-hand sidefor some apitical fiber carburetorthat is given byso-called wire stress modelwhich I explained yesterdayso I don't repeat it here.So, this is the wire stress model.So, this is apitic vibration.So that every fiber is eithersmooth apitic curveor nodal rational curveor cuspital rational curve.So, in this caseyesterday, we observed thatn0So, f is a fiber classn0 mfis minusvolve number of xandpn1mfp equals toes and higher genusSo, anyway, so this wasfrom the computationof the wall-crossing formulaand yesand it implementscomputation of the multiple-coverconjecture of generalizedDonald's and Thomas invariant.On the other hand, in this casethis right-hand side is much moremuch easier to compute.So, let me give someits computation.So, in this case, if youconsider, likebeta equalswellif beta is multiple offiber classin this case, the child-variety ischild-variety is nothing butthis is the modular space of cycleswith homogenous multiplied by its fiber.So, the modular space ofone cyclewith homogenousmf is nothing but the choosingsome m numberof points, m number offibers of that fibrations.So, the child-variety isnothing but thesymmetric product of s.On the other hand, if you look atif youconsider modular space ofstable sheaves, so becauseevery stable sheave should havecompact, have connected supportevery stable sheaveshould be supported onone of the fibres.And indeed, and also if youconsider modular space ofstable bundles on, for example, one offibers, then it is well known thatthe modular space of stable bundlesis isomorphic to the originalunderline empty curve.And in this case, it iseasy to see that this isindeed isomorphic to the originalx.And under this isomorphism, thishybrid trail map can bejust identified withthis diagram map.So, in order to computethis number, it is enough just tocompute thepush forward of constantsheave to here.So, if we computeyes.And alsoin this case, it iseasy to see that.In this case, a pi starofsection of complex, this isof course, constant sheaveshifted by three.This is given by sshiftedby oneplusv plussby minus one.And it breeds 13perversesheave on s.Indeed, it breeds nothing but theconstructive sheavesuch that shiftedby two.And yes, this is aconstructive sheave ondownstair.Andit is notit is notlocal constant, butif you considerthe fiber of thisconstructive sheave, then ofcourse, this is nothing buth1xsq.So, this isjustq2 ifxs is smooth.In this case, this isthe smooth apt curve.And if x isno-diverational curve.And this is 0 if x hasonly cuspidiverational curve.So, if wedo the picture of this baseI have this base spaceand there is a discriminant locusthat is a divisor on baseon whichthe fiber is singularand ofcoursein the genetic point of thisbase, the fiber issmooth apt curveand at the genetic point of thisdiscriminant locuswhere the fiber is a no-diverational curve with one nodeand atsome special pointI get some cuspidiverational curve.But in any caseyou can just write this astheq plusto minusoiler number of the fiber.So, this information is enough to computethe Copacmobuff invariant in this case.And indeedin this casewe can just computeone of the definitionthat we should computeintersection compresscsplusy plusv plusminus1csy butofcoursethis is a constant shiftso it's the order characteristicit's nothing but the order number of the baseandthe order number of vcan be computed using thiscomputationso this meansjust likeEsmy inverse plusso this becomesEsmy inverse plustoEs minusthe number of explussyand that is just written asminusoiler number of x plusoiler number ofsy plusy squareso by this expressionofcourse this isn0mfand this is n1mfso there is nohigher genus termso in this casewe have the agreement of thecomputation which appearedfrom theroad crossing formulasin stable pair sidesoand ofcoursein this casethe conjecture comparinggobakumabuffin variants defined from stable pairsand gobakumabuffin variantsishow to definethe betawhenm issingularso the question isthatis how to definethe betawhenm issingularbutm issingularand there are some history about thisproblemand ofcourse the first approach wasgiven by Osonosaito otakahashi in the2001 paperand the argument wasto use theintersectioncohomology of the normalizationof thispossibly singular schemeequal aboutapproach by Osonosaito otakahashisoso first let's takeanormalizationso this is normalizationand let's takeyouto be dense opensubsetyes ofcourse intersectional complexcan be definedforsingular varietyand in this casethis isthe more complicatedimage of thepower sheavesanyway so this isintersectional complexthat lives in the power sheaveson m-childerand the classical resultbypermission-time domainimproved the decomposition theoremfor the hyper-cohomology of this intersectional complexwhich appearsfor when m is no singularto thisintersectional complexwhat theyconceptyeah I meanI'm assumingyeah this isyes ofcourseyesyes thank yousowhat theydid isreplacemwhen m issmoothcase bythisintersectional complexso still we getsome numberwhich I didalso on site takahashiimvariantbutunfortunatelyso thisimvariant does not matchtheimvariantcoming from stable pairs orlike ground fitting variantswell this is becausein this definitionwe only think aboutthe topological property of thismodular space and we do not take care aboutthe scheme structure and for examplearewe observed that when g equals 0genus 0 thenby the world crossingformularand the beta pmust beequal to theintegration of the variantfunction of the modular spaceof the stable sheavesand if when m is no singularthis is nothing butthe topological order number of m up to signbut when m isno singularwhen m is singular then this variantfunctionfeeds the future of the singularitiesand it may be far from theoiler number of this modular spacefor example it may happen thatwhere m might be just the topological one pointbut the scheme sort of case there might be somefat pointand in that case the variantfunction is notone andit reflectsthe length of thestructure sheave of mbut in that casethe also on site takahashi constructiononly gives just onesoin general this is notequal to n0bettergstanyway the problemis whether how toimbub variantfunctionor sheave of vanishing cyclesin this constructionyes butyeah indeed by the definition ofcopcabuff invarianceso this is for when m issmooth case so as I explainedyesterdayfor example child fend childbut it is just one point then thepolynomial which computescopcabuff invariance is nothing butthe usualfrancai polynomial andthe usual donelson-tomass invariantis something likesomething like oiler numberof the modular space of stable sheavesand soand this means that we have togive some sort of refinementofdonelson-tomass type invarianceso that it alsoimbalts likepolyncai polynomial was onso that is has to do withthe problem ofhow toconstruct so-called mochivicdonelson-tomass invarianceor gohomodical donelson-tomass invariancein some of our vigorous waybut anyway soin Wednesdaywe saw thatwe can consider modularstark of coefficient sheavesthen it is locally written ascritical locus of some functionso let's recall thatin thisso this isdivine-joicecollaborator that this modular space ofso thisproperty holdsfor any modular space ofobject in derived categorythis m iscovered by someopen-coveringeach open-setis embedded intosmooth-schemeso this is smoothand this is a regular functionsuch that ui each ui isgivenbycritical locusand in Wednesdayif we are given such datait is possible to constructsheavesthat is called the sheave of vanishing cyclesand indeedsheave of vanishing cyclesis an object ofperverse sheaveson uiso I get perverse sheaveson each uione of the problemaboutthis is thatwhether we canguage perverse sheavesso the question is thathow to glueyes this kind of problem has beenone withoutthe foundationalproblem in constructingcohomodicard-nathontomas invariantor you can also replacethis sheave of vanishing cycles byso-called mochilic-munic fibermochilic vanishing cycles and you can askwhether suchevent in the gluttonic groupof varieties glue together to getsome kind ofmochive and yesin recent years there have beenseveral worksto giverelastreatment but the gling of theseperverse sheavesso here I referthe work by Domenic Joyceusing the notion ofso-calleddecritical structureso this is after Joyceyes indeed he shows thatyeah so for this statementyeah we havestrong statement that iswe canyes in order to glue these perverse sheaveswe need to knowsomehow visions of theselocal chartin some waywell for example the nice thingis that well iffor example if you just addsome quadratic formof this functionthe same critical locusand in that casewhere the sheave vanishing cyclesvery close to the sheave vanishing cyclesbefore addingcordatic termso his de-critical structureimproves that we can takesuchlocal critical chartin some partyeah indeed they areon the overlap they aredifferent but theyare vanishing like quadratic formslike thatyes so I expectwhat is this structureso indeed heshows that there is aanonical sheave ofshevector space issatisfying some propertiesindeed this sheavethis sheave can beexplicitly described usinglike cotangent complexbut the important propertyis that for nopen subsetand for nembedding intosmooth closed embedding intosmooth schemesuch that wherei is the ideal which definesthis closed sub-schemewe get theexact sequencethat is if you speakthis sm2you geti square tolikeso this descriptionimproves that wherefor example if you takeopen subsetclosed embedding and ifthisclosed sub-scheme isindeed defined by criticallocals of some functionthat is if f ismaking a functionsuch that whereu is justcriticallocals and i isso in this case of coursei is given bythe ideal generated by the differentialof this function fand in this casewe have a natural sectionof thissheave vector space on uthat is justif we just takethe image of f inhere gives a global sectionthis onethis isjust a huge one differentialhereso this is notmodule homomorphism so this is not justsheave vector spaceso f isno f is justf isthis is an example of the sectionnot every section islit in this wayso this is an example of the sectionthis was just one exampleyes of course where ifi is given by the differentialof the function fand if you justdifferentiate this functionthen of course by the definitionafter definitionafter taking the differentialof this f then its image iscontinued in this idealso while this function f isregarded as asection of this sheavewhen you use the smooth sheavewhen you use the smooth sheavefor examplewhen you use the smooth sheavesorry I am a bit confusingwhen you use the smoothfor example you can takejust for examplethe function f to be justconstant functionandwhereit is nothing but thecarnival of this map so this is justthe sheaveanywaythe theorem by Joyceand his collaboratoriswhere there is a certaincommunicalglobal section such thatso this is acommunical global section such thatwherethere is a certaindata like this which I do not buystarrsuch thatas restrict to eachUI givenin this wayso the good point of here is thatwhere we can takejust data so that it iscompatiblein some wayand in factif we have this conditionthen on the overlapthen eachfunction f isat firstdiffer by someadding like a quadratic formandusing that differencegiven that differencegiven by the quadratic formwe cangive acommunical isomorphism betweeneachcommunical bundlei tensor 2 restrict toUIJ restrict toUIJso this givescommunical isomorphismand this means thatthese line bundlesglued together to getline bundle on Mso this glueto giveline bundle which I do not buyM virtual so this isline bundle on Mand indeed so thisline bundle can bedescribedusing the determinant line bundleand the determinant line bundlethat isa form eand so it isthe universal shiftso this means thatif we haveby using thatglobal de-critical structureyes so that sectionis called de-critical structureusing thatde-critical structurethis meanswe have a canonical isomorphism of thisvirtual canonical bundleandthe canonical bundleof each critical charttensor by 2butandand using this datawe can describerwhat is the difference of thesheaf vanishing cycles of overlapso roughly speakingindeed the sheaf vanishing cycles of overlapnot the samebut they arethe difference is given by30rank1local systemthat is defined bythat is defined bytaking the mututosa whichpermits its local square roots of thiscanonical isomorphismso in order to cancel that differencewe needsquare root of this virtualcommunical bundle that is called orientation dataso if we choosethis square root that isso yesindeed it is notit may not be necessaryyeah indeedup here it may not beobvious that whether square root exist or notbutit is proven in the papernecrossoff and concorconcorthat is important pointand what they prove is thatindeed the first chunk class of thisvirtual canonical bundle is evenat least there is oneat least there is onesquare rootbut it's not canonicalanyway so if you chooseone of the square rootthen it is possible to groupthe local perverse sheavesso I will omit some detailbut thestatement is that there islanck1local systemon each UIsuch thatif Itest this for vanishing cyclewith fithen this group to giveglobal perverse sheavesso let's writethisfor vanishing cyclesbut yesdepends on the choice of orientationyeah exactly soyesindeed this depends on the choice oforientation data so thischoices is called orientation databut if you giveone orientation datathen there isglobal perverse sheavesandwe want touse this kind of perverse sheavesto giveglobal perverse invariantfor singularmodular space mbut one of the another problemis that this sheavesvanishing cycles may not satisfythe permission of vanishing time to winthe conventional theorem this meansit is true thatthis is coming from themixed torch module that isfunctor from mixed torch moduleand it is coming fromhere andbut this may not be pureand indeedyeah this may not be pureandbut andbecause ofthis issuewellinstead of usingthis sheave vanishing cyclesin the paper ofKIM theyintroduced thebackmabuff invariantusing the associated group ofthe weight filtration so that isKIM-Duse associated groupof weight filtration ofthis sheave vanishing cyclesif you use associated groupthen each direct summand is a pureforge module and we canapply the decomposition theoremand we haveinvariantwhich I did not buyand the good thingabout this invariant is thatat least genus 0 it coincideswith thebackmabuff invariant coming fromstable pairs that ison the other handlet's consider the higher genus caseso there are some issuesyeah that's what I want to sayyeah indeed so they didn't mention aboutthe dependence of your mutation databut we find that in the higher genus caseit depends on the choice of orientation datayes yesfirst issue is that when Gis positive it depends onorientation datayes the example is easyto see the exampleindeed in thisconstruction where we may alsoassume that the modular space isjust no singular and for examplewhere if you consider super rigidapt curve inside the carbure three foldthen the modular space M is justjust empty carband in this casethere are somesquare root of the trivial line bundleand if you choose some non-trivialsquare line bundlethen we get somecontradictionabout this equality in the higher genusand another issue is thatwell in thisconstruction we are doing severalartificial things that iswe choose someorientation datatake associatedgroup of the weight filtrationwas on andwhere the definition iscomplicated and it's quitefrom this definition it's not easy tocompute the invariantfrom the definitionand indeed in the paperthey didn't mentionyes this was the paperso they didn'tgive any exampleabout the comparision withthe invariant witheither glomofit invariantwhen the modular space M isindeed singularso there is noexamplewhereng beta p equalsto the beta keyng positive and M isindeed singularnow I'm going to talk aboutyesproposal is thatyou will not understandgobakumabuff invariant I thinkwe need furtherstronger geometric statementstrong property of thegeometry of the modular space ofone dimensional stable shipso this is our proposalso this is the joint work withDanish Malikthis is still in progressyesI have tothink about the first issueso whether there is a canonicalorientation data or notindeed from the context ofMotivic to Nanson Thomas invariantin order to apply for theapplication of overcoachingwe need to choose orientation datathat is compatible withwith whole algebrabut the existence of suchorientation data compatible withwhole algebra is notwhether there is a canonical one or notbut insteadlet'sconsider this mapthat is here with the channel mapso this is somehow higherdimensional analog of the likehitching fibrations soI mean if you havemodular space for HiggsVandubes you have the hitching fibrationsand the genetic fiber isthirteenAvian variety andindeed some avian fibrationsor in particular this that iscalabia fibrations and what we thinkis that this map isinterpreted as something likecalabia fibrationsin some virtual sensein some virtual senseso in particularwe believe that if youconsider the virtual canonical bundlethen this isthis is trivialabong the fibers of this mapand this is true whenon the smith fiberwe don't know whether this is trueon shinger fibers or not butI thinkwe may expect that is truebutwe expect more strongerconjecture or I'm not surewhether this is truein general or not butin the case which we computedthis is satisfied so Ilet it as a conjecture butfor any pointingcalabia Xthere isopen subsetyou write the pullback of thisyou write the charm upso this is inverse Uthen there is acritical chart on thison this pullbackand the regular functionhere such thatthis is given bythe critical locusof this functionor most stronglythis gives the so-calleddecritical chart that isif it's the global sectionofsm restrict tomu sowhile thisproperty in particular impliesthatyes sorry yes andand yes the strong assumptionthat where this canonical bundleof j is trivial that isj equals tojyes yes exactly yesyeah I don't knowwhat this is I don't knowwhat this is mean but yesI mean calabia versionin some weak so strong senseI think we may talk aboutcalabia version that isassuming that this calabia canonical bundleis coming from the pullbackof line bundle also onindeed in this caseif this condition is satisfiedthe virtual canonical bundlemistrict to mu is indeedtrivialand alsoand if this condition is satisfiedof course we may choose theorientation data onhere just the square rootjust trivial square rootthat isanyway let's assume thatthis condition is truehere we can definethe local canonical bundle ofinvariant by theby the condition thatf is taken to gammaalsoanother good conditionof this property is that if youconsider the sheaf of vanishing cyclesyeah although this is notstill this is notthis is not coming fromthe sheaf of vanishing cyclesbut still by using thecycle sheafs andproper push forward we can see thatthis sheaf of vanishing cyclessatisfies thethe composition theoremand it isand of course this is given by like thisand of course theyes and the globaland the point is thatif we take this chart as long asit gives thethe critical structurethisof invariant gives awell defined constructive functionthat is indeed independentof the choice of orientation dataso this is well definedyes right but as long asif we takeyes whenever j iscolorbl and whenever thisfunction gives the critical chartthis gives a well defined constructive functionso it is independent ofthe choice of orientation dataandthe globalcobacmabuff invariant isjust given byintegration ofthis functionso this givesas long as this conjecture is truethen this gives a well definedcobacmabuff invariantbut of course where this conjectureI am not surethis seems to be not so easybut of courseanother conjecture is thatof coursep equals tong betaunder conjecture Aso here isthe main theoremthis is withthat X bethe total space ofcanonical bundle onsmooth projective surfaceand let's takebetter to bebetter is not written as effective curve classwith some of effective curve classsorry and withfirst batch number 0so this is still working progress and I thinkit should benot difficult to improvethe result but anywaysthe conclusion is thatin this casethe both conjectures are truewith first batch number 0only thiswe canafford that this is general typebut the strong assumption is thatbetter is eligibleand I think thatI think we can improve thatbetter is primitive curve classbut yeah maybewith some effort I thinkI think we have alreadyalmost proved that butanyway so the point is thatin this case it impartslike moremany examplesthat m is singularhere yes I'm givingan example solet s to beenke surfacewhich admitseptic vibration such thatlet's take one of the double fiberso thathso that h inversep is twice of cand assume that c isnot a rational curveso there are some examplesof any surfacelike thisso in this case let's takebetter to be thethis curve classbecause this is double fiberthe curve c is rigid inside sso the shell varietyinside this surface is justone pointbecause this curve c is rigidbut if you consider3 port xthen it is notone point I meanindeed if you restrictit'sc toshso this isnormalizationone dimensional this means thatwell on surface I have thisnot a rational curve and insideon 3 port so there are somesomething like this happensand this is xso indeed in this casethe shell variety isby a1and yes and in thiscase the modular spaceof one dimensionalstable sheaves isindeed height sothis is eitherrank one torsion freesheaves on this rational curveorline boundary onp1 so this iscomposing to c or a1and c goes to 0and a1 to a1 is justidentity map so if youdivorce the picture this lookslike this so this is a modularspace and here so there isa similarity so this similarity isgiven by critical locus ofsome function like thisand the hybrid shell map isgiven by like thisso we can write it asthe calvarythe critical locus thatislet's takethis is theversal deformation of c so thisis ofcourse calvary and if wetake product a1tothe product a1to a1and this function isxy to xy and wedo it with this onethen this modular space ofstable sheaves is justgiven bydfe f of 0and in this case over here thisis calvary and thisgivesexample of modular spaceof stable sheaves withsingualities and indeed inthis case the theorem can beproven like using the resultofshending and thecombrutals showingsome shimia result for the burstreformations ofifdshable planar curvesand just applying thesheaf of vanishing cycles andproving the conjecture awe get this resultso thank you very much