aa r X i v : . [ m a t h . AG ] A ug THE LOG PRODUCT FORMULA
LEO HERRUNIVERSITY OF UTAHUNIVERSITY OF COLORADO BOULDER*
Abstract.
We prove a formula expressing the Log Gromov-Witten Invariants of a product of logsmooth varieties V × W in terms of the invariants of V and W . This extends results of [LQ18], whichintroduced this formula analogously to [Beh97]. The proof requires notions of “log normal cone”and “log virtual fundamental class,” as well as modified versions of standard intersection-theoreticmachinery [Man08] adapted to Log Geometry. Introduction
The Log Product Formula.
The purpose of the present paper is to prove the “Product For-mula” for Log Gromov-Witten Invariants. For ordinary Gromov-Witten Invariants, the analogousformula was established by K. Behrend in [Beh97].Let V , W be log smooth, quasiprojective log schemes. Let Q be the fs fiber product M g,n ( V ) × ℓM g,n M g,n ( W ) , with maps M g,n ( V × W ) h −→ Q f ∆ −→ M g,n ( V ) × M g,n ( W ) . One can naturally endow Q with a “log virtual fundamental class” in two ways: pushing forwardthat of M g,n ( V × W ) or pulling back that of M g,n ( V ) × M g,n ( W ). The Product Formula equatesthese: Theorem 0.1 (The “Log Gromov-Witten Product Formula”) . The two log virtual fundamentalclasses are equal in A ∗ Q : h ∗ [ V × W , E ( V × W )] ℓvir = ∆ ! [ V, E ( V )] ℓvir × [ W, E ( W )] ℓvir . The symbol ∆ ! refers to a “Log Gysin Map” for which we offer a definition, along with “LogVirtual Fundamental Classes.”This theorem was formulated for ordinary virtual fundamental classes in [LQ18] and provedunder the assumption that one of V or W has trivial log structure. Like their work and [Beh97]before it, our proof centers on this cartesian diagram (Situation 5.5): M ℓg,n ( V × W ) Q M ℓg,n ( V ) × M ℓg,n ( W ) D Q ′ M g,n × M g,n M g,n M g,n × M g,n . hc p ℓ p ℓ al φ p ℓ s × s ∆ Date : August 15, 2019.*The author’s present address is the former. This article formed part of the author’s dissertation at the latter.
One applies Costello’s Formula [Cos06, Theorem 5.0.1] and commutativity of the Gysin Map tothis diagram to compare virtual fundamental classes.In the log setting, one requires this diagram to be cartesian in the 2-category of fs log algebraicstacks in order to preserve modular interpretations. The assumption of [LQ18] that V or W havetrivial log structure ensures that these squares are also cartesian as underlying algebraic stacks.These fs pullback squares in question likely aren’t cartesian on underlying algebraic stacks.Therefore, none of the standard machinery of ordinary Gysin Maps and Normal Cones is valid.This quandary forced us to prove the log analogues of Costello’s Formula and commutativity forour “Log Gysin Map.” With these modifications, the original proof of K. Behrend essentially stillworks. We pause to comment on the new technology.0.2. Log Normal Cones.
The
Log Normal Cone C ℓX/Y = C X/ L Y of a map f : X → Y of logalgebraic stacks is the central object of the present paper. Every log map factors as the compositionof a strict and an ´etale map X → L Y → Y , so the cone is determined by two properties: • It agrees with the ordinary normal cone for strict maps. • If one can factor f as X → Y ′ → Y with Y ′ → Y log ´etale, the cones are canonicallyisomorphic: C ℓX/Y ≃ C ℓX/Y ′ . This object becomes simpler in the presence of charts. Locally, we may assume the map X → Y has a chart given by a map of Artin Cones A P → A Q . The map A P → A Q is log ´etale, so we canbase change across it to get a strict map without altering the log normal cone.Because this method can lead to radical alterations of the target Y , we recall another strategythat we learned from [IKNU17, Proposition 2.3.12]. For ordinary schemes, one locally factors amap as a closed immersion composed with a smooth map to get a presentation for the normal cone[BF96]. We obtain a similar local factorization (Construction 1.1) into a strict closed immersioncomposed with a log smooth map, and the same presentation exists for the log normal cone.The above is made more precise in Remark 2.7. The charts and factorizations these techniquesrequire are only locally possible, so we need to know how log normal cones change after ´etalelocalization. We encounter a well-known subtlety noticed by W. Bauer [Ols05, § Pushforward and Gysin Pullback.
The proof of the Product Formula needs two ingre-dients: commutativity of Gysin maps and compatibility of pushforward with Gysin maps. Thecommutativity of Gysin Maps readily generalizes to the log setting in Theorem 3.12; on the otherhand, compatibility with pushforward simply fails!Nevertheless, the original proof of the product formula depends on a weak form of this compat-ibility first introduced by Costello [Cos06, Theorem 5.0.1]. We prove a log version of this theoremand will offer further complements in [AHW].We obtain another partial result towards compatibility of pushforward and Gysin Pullback. For alog blowup p : b X → X with a log smoothness assumption, we show p ∗ [ b X ] ℓvir = [ X ] ℓvir in Theorem3.10. The alternative approach of [Bar18] may extend our results by modifying the notions ofdimension, degree, pushforward, chow goups, etc. in the log setting. See also [Ran19] for aninsightful approach to Log Chow Groups.We hope the technology and the strategy of reducing statements about log normal cones to thestrict, ordinary case will be of interest.0.4. Conventions.
HE LOG PRODUCT FORMULA • We only consider fs log structures . We therefore use L , L Y to refer to Olsson’s stacks T or, T orY . • We work over the base field C . • We adhere to the convention of [Ols03] regarding the use of the term “algebraic stack”: wemean a stack in the sense of [LMB99, 3.1] such that – the diagonal is representable and of finite presentation, and – there exists a surjective, smooth morphism to it from a scheme.We do not require the diagonal morphism to be separated. • By “log algebraic stack,” we mean an algebraic stack with a map to L . Maps between themneed not lie over L . • The name “DM stack” means Deligne-Mumford stack and a morphism f : X → Y ofalgebraic stacks is (of) “DM-type” or simply “DM” if every Y -scheme T → Y pulls backto a DM stack T × f,Y X [Man08]. • The word “cone” in “log normal cone” refers to a cone stack in the sense of [BF96]. • Let P be a sharp fs monoid. Write A P = [Spec C [ P ] / Spec C [ P gp ]]for the stack quotient in the ´etale topology endowed with its natural log structure [ACWM17],[CCUW17], [Ols03]. Beware that some of these sources first take the dual monoid. Thislog stack has a notable functor of points for fs log schemes:Hom fs ( T, A P ) = Hom mon ( P, Γ( M T )) . In particular, Hom fs ( A P , A Q ) = Hom mon ( Q, P ) . We write A for A N = [ A / G m ]. Log algebraic stacks of this form are called “Artin Cones.”“Artin Fans” are log algebraic stacks which admit a strict ´etale cover by Artin Cones. The2-category of Artin Fans is equivalent to a category of “cone stacks” [CCUW17, Theorem6.11]. • The present paper concerns analogues of normal cones and pullbacks in the logarithmiccategory. We use the notation p , × , C for pullbacks and normal cones of ordinary stacks,and write p ℓ , × ℓ , C ℓ to distinguish the fs pullbacks and log normal cones. When theyhappen to coincide, we write ℓ , p ℓ , × ℓ , C ℓ to emphasize this coincidence. • Many of our citations could be made to original sources, often written by K. Kato, but wehave opted for the book [Ogu18]. We have doubled references to Costello’s Formula [Cos06,Theorem 5.0.1], [AHW] where appropriate because we will have more to say building onfuture work.0.5.
Acknowledgments.
The present article is part of the author’s Ph.D. thesis at the Univer-sity of Colorado, Boulder under the supervision of Jonathan Wise. Not a result was envisioned,obtained, or fixed without his tremendous support, guidance, and patience.The author also benefitted from email correspondence with Dhruv Ranganathan and LawrenceBarrott. The author is grateful to the NSF for partial financial support from RTG grant
Preliminaries and the Log Normal Sheaf
The present paper originated with one central construction, which we learned from [IKNU17,Lemma 2.3.12].
Construction 1.1.
The normal cone of a morphism f : B → A of finite type is constructed bychoosing a factorization B → B [ x , . . . , x r ] ։ A inducing a closed immersion into affine r -space:Spec A ֒ → A rB → Spec B. LEO HERR
The normal cone of f may then be expressed as the quotient of the ordinary normal cone of theclosed immersion by the action of the tangent bundle of A rB → Spec B .Let P → A and Q → B be morphisms from fs monoids to the multiplicative monoids of rings(“prelog rings”). A commutative square: B AQ P fθ is a chart of a map between affine log schemes. Assume f is of finite type; θ automatically is bythe fs assumption. We will obtain a factorization of the induced log schemes into a strict closedimmersion followed by a log smooth map .Start with a similar factorization B B [ x , · · · , x r , y , . . . , y s ] AQ Q s P with Q s = Q ⊕ N s mapping to B [ x , . . . x r , y , . . . y s ] by sending the generators of N s to the algebragenerators y , . . . y s . Define Q θs via the cartesian product Q s Q θs PQ gps Q gps P gp . p By definition, Q θs → P is exact, and Q s → Q θs is a “log modification:” an isomorphism on groupi-fications. Witness also that Q θs → P is surjective, so the characteristic monoid map Q θs ∼ → P is anisomorphism [Ogu18, Proposition I.4.2.1(5)] and Spec P → Spec Q θs is strict. Take Spec of bothrings and monoids [Ogu18, § II] to obtain a diagram with strict vertical arrows:
X X θ A r + sY Y Spec P Spec Q θs Spec Q s Spec Q p ℓ We’ve written Y = Spec B , X = Spec A and introduced the fs pullback X θ in the diagram. Thetop row expresses our original map Spec f as the composition of a strict closed immersion, a logmodification, and a smooth and log smooth morphism. The log modification Spec Q θs → Spec Q s and hence X θ → A r + sY may be expressed as a (strict) open immersion into a log blowup as in[Ogu18, Lemma II.1.8.2, Remark II.1.8.5]. Hence X ⊆ X θ is a strict closed immersion and X θ → Y is log smooth. Remark 1.2.
Continue in the notation of Construction 1.1. If we began with a morphism of fslog rings with f and θ both surjective, we could omit Q s → B [ x , . . . , x r , y , . . . y s ]. In that case,we obtain a factorization X ⊆ X θ → Y where X θ → Y is not only log smooth but log ´etale. HE LOG PRODUCT FORMULA As in [BF96], we will present the log normal cone locally as C ℓX/Y = [ C X/X θ /T ℓX θ /Y ] usingthese factorizations. The difficulty is then piecing together the local descriptions and checkingcompatibility. In this sense, the heavy lifting has already been done for us by [Man08]. We spendthe rest of this section collecting relevant properties of the log normal sheaf N ℓX/Y . When we definethe log normal cone C ℓX/Y ⊆ N ℓX/Y , its important properties will be locally deduced from suchfactorizations. Remark 1.3.
An algebraic stack X is DM if and only if the map X → Spec k to the base field is ofDM-type. If X → Y is a morphism of DM type and Y admits a stratification by global quotients,then so does X [Man08, Remark 3.2]. A morphism f : X → Y of algebraic stacks is of DM type ifand only if its diagonal ∆ X/Y : X → X × Y X is unramified [Sta18, 06N3]. Lemma 1.4.
Let f : X → Y be a morphism of log algebraic stacks. If the map on underlyingstacks is of DM-type, then the induced maps L X → L Y and X → L X are DM-type. Proof.
The inclusion X ⊆ L X representing strict maps is open, so it suffices to show that L X → L Y is DM-type.We will argue that the diagonal of L X → L Y is unramified [Sta18, 04YW]. The isomorphism L X × L Y L X ≃ L ( X × ℓY X ) identifies the diagonal ∆ L X/ L Y with the result of L applied to thefs diagonal ∆ ℓX/Y : X → X × ℓY X. Any diagram: S L XS ′ L ( X × ℓY X ) L ∆ ℓX/Y with S ⊆ S ′ a squarezero closed immersion of schemes is equivalent to a diagram S XS ′ X × ℓY X ∆ ℓX/Y with S ⊆ S ′ an exact closed immersion of log schemes. Composing with the fsification map X × ℓY X → X × Y X sends this square to S XS ′ X × Y X, ∆ ℓX/Y in which case the two dashed arrows have the same underlying scheme map because X → X × Y X is unramified by hypothesis. Then the maps on log structure must be the same as well, because( M X ⊕ ℓM Y M X ) | S ′ → ( M X ) | S ′ is an epimorphism. (cid:3) Recall the functor of points of the normal sheaf.
LEO HERR
Definition 1.5 (Normal Sheaf Functor of Points) . Let f : X → Y be a DM morphism of algebraicstacks. Define a stack N X/Y over X named the log normal sheaf via its functor of points: N X/Y
T X := ( T, O X | T ) X ( T, A ) Y i i is a square-zero closedimmersion with kernel O T = O Y | T a squarezero algebra0 O T A O X | T T ) An obstruction theory for f is a fully faithful functor N X/Y ⊆ E into a vector bundle stack asin [Wis11, Corollary 3.8].The notion of “square-zero closed immersion” in the definition demands elaboration, since theobjects involved are ´etale-locally ringed spaces. See [WH] for details. Remark 1.6.
Suppose we specified an obstruction theory E • → L X/Y in the sense of [BF96]. Theassociated obstruction theory according to Definition 1.5 on T -points is given by: N X/Y ( T ) = Ext( L X/Y | T , O T ) −→ E = Ext( E • | T , O T ) . See [Wis11, Corollary 4.9] and [GRR72, Chapitre VIII: Biextension de faisceaux de groupes] forcomparison and elaboration on Ext( E • , J ) = Ψ E • ( J ). In particular, our obstruction theories areall representable by obstruction theories in the sense of [BF96]. Definition 1.7 (The Log Normal Sheaf) . Let f : X → Y be a DM morphism of log algebraicstacks. Let T → X be an X -scheme. A deformation of log structures along f on T is a log structure M A → A on the ´etale site ´et( T ) of T with maps ( O Y | T , M Y | T ) → ( A , M A ) → ( O X | T , M X | T ) oflog structures such that: • The kernel ker( A → O X | T ) ≃ O T and the diagram O Y | T O T A O X | T • The diagram A ∗ O ∗ X | T M A M X | T is a pushout.The second bullet says that ( A , M A ) is a strict squarezero extension of ( O X | T , M X | T ); comparewith “deformations of log structures” [Ill97]. The square in the second bullet is also a pullback,and M A → M X | T is also a torsor under 1 + O T . HE LOG PRODUCT FORMULA Define the log normal sheaf to represent the deformations of log structures just defined: N ℓX/Y T X := { Deformations of log structures along f on T } . We show that this definition agrees with Definition 1.5 in [WH]: N ℓX/Y = N X/ L Y .To write down the functoriality of the log normal sheaf, we need to recall some of the machineryof log stacks found in [Ols05].We denote L i := L [ i ] , the stack of i -simplices of fs log structures. The j th face map d j sends( M → M → · · · → M i +1 ) ( M → M → · · · → M i +1 ) if j = 0( M → · · · M j − → M j +1 · · · → M i +1 ) if j = 0 , i + 1( M → · · · → M i ) if j = i + 1 . We write s, t : L → L = L for the “source” d and “target” d maps, respectively. We have anisomorphism L i = L × t, L ,s L × t, L ,s · · · × t, L ,s L ( i factors).Endow L i with the final tautological log structure, M i +1 in the above. All the face maps d j arestrict except j = i + 1.We continue [Ols05] to use “ (cid:3) ” to denote the category with these objects, arrows, and relations:0 12 3 ◦◦ We adopt pictorial mnemonics for fully faithful morphisms of these finite diagrams: means thefunctor [2] ⊆ (cid:3) avoiding 2, etc. Definition 1.8 (Compare [Ols05, Lemma 3.12]) . Define V := L × ℓt, L ,t L . Given a scheme T ,the points of this stack are cocartesian squares of fs log structures: V ( T ) := M M M M .ℓ y This is the “fsification” of the ordinary pullback L × t, L ,t L , endowed with the non-fs pushout M ⊕ monM M of the universal log structures.The natural embedding V → L (cid:3) exhibits the squares which are cocartesian as an open substack,as we’ll record in Lemma 1.10.For a morphism q : Y ′ → Y of log algebraic stacks, we obtain relative variants: V q := V × , L Y ′ , L (cid:3) q := L (cid:3) × , L Y ′ . LEO HERR
The fs pullback here agrees with the ordinary one because Y ′ → L is strict. The points of thesestacks over some scheme T are squares M Y | T M Y ′ | T M M , with those of V q required to be cocartesian. Lemma 1.9.
Let L arbfine denote the stack of log structures which are fine but not necessarilysaturated. The natural monomorphism L ֒ → L arbfine is an open immersion. Proof.
Consider some scheme X and pullback diagram X fs L X L arbfine p Then X fs ֒ → X is a monomorphism, the locus where the stalks of M X are saturated. After pass-ing to an open cover of X , [Ogu18, Theorem II.2.5.4] provides us with a locally finite stratification X = F σ ∈ Σ X σ where • For each σ ∈ Σ, M X | σ is constant. • The cospecialization maps for x ∈ { ξ } ⊆ XM x → M ξ are localizations at faces.The localization of a saturated monoid remains saturated [Ogu18, Remark I.1.4.5] and a monoidis saturated if and only if its characteristic monoid is [Ogu18, Proposition I.1.3.5]. We then havethat X fs ⊆ X is locally a constructible subset which is closed under generization, and hence open[Sta18, Tag 0542]. (cid:3) We collect several results of [Ols05] adapted to the fs setting:
Lemma 1.10 ([Ols05, Theorem 2.4, Proposition 2.11, Lemma 3.12]) . These statements remaintrue in the fs context:(1) For any finite category Γ, the fibered category L Γ of diagrams of fs log structures indexedby Γ is an algebraic stack.(2) The simplicial face maps d j : L i +1 → L i are strict, ´etale, and DM-type for j ≤ i .(3) If [1] → (cid:3) avoids the initial object 0 ( or ), it induces a strict ´etale, DM-type morphism L (cid:3) → L . (4) If [2] → (cid:3) omits either 1 or 2 ( or ), it induces an ´etale, DM-type morphism L (cid:3) → L . (5) The map V ⊆ L (cid:3) is an open embedding. HE LOG PRODUCT FORMULA (6) Given an fs pullback square X ′ XY ′ Y, p ℓ q the associated square of stacks X ′ X V q L Y p is a pullback. Proof.
Facts (1) through (4) are immediate by Lemma 1.9 and the analogous facts in [Ols05]. Thelast two follow by the same arguments applied in the fs category. (cid:3)
Remark 1.11.
Apply L once more to the map L Y → Y : one gets d : L Y → L Y ( M Y → M → M ) ( M Y → M ) . The result is ´etale, so the original d : L Y → Y is log ´etale [Ols03, Theorem 4.6 (ii)]. The samereasoning concludes d i +1 : L i +1 Y → L i Y is log ´etale in general. In summary, all the face maps arelog ´etale and all but j = i + 1 are furthermore strict ´etale. Remark 1.12.
Given q : Y ′ → Y DM, the natural maps V q ⊆ L (cid:3) q → L Y ′ are ´etale. The second map is the product of the ´etale map ∗ : L (cid:3) → L over L (via ) with Y ′ . Definition 1.13.
Use Lemma 1.10, bullet (6) to turn one commutative square of DM maps intoanother: X ′ XY ′ Y. q X ′ X L (cid:3) q L Y Maps of normal sheaves ϕ : N ℓX ′ /Y ′ ≃ N X/ L (cid:3) q → N ℓX/Y arise from Remark 1.12 and the second square. We call the composite ϕ Olsson’s Morphism . Remark 1.14.
In Definition 1.13, if the first square was an fs pullback square, the second factors: X ′ X V q L (cid:3) q L Y. p Since this square is a pullback, Olsson’s morphism ϕ : N ℓX ′ /Y ′ ≃ N X ′ / L (cid:3) q ≃ N X ′ / V q ֒ → N ℓX/Y | X ′ is then a closed immersion. If q or f is also log flat, ϕ might not be an isomorphism. See Lemmas 2.15, 2.16 for the strictcase. Remark 1.15.
A commutative square of DM maps may be factored: X ′ XY ′ Y. fq X ′ X L (cid:3) q L YY ′ Y. (1)This induces a commutative square of normal sheaves: N ℓX ′ /Y ′ ≃ N X ′ / L (cid:3) q N ℓX/Y N X ′ /Y ′ N X/Y . ◦ (2)The Olsson morphisms are thereby seen to be compatible with the ordinary functoriality of thenormal sheaf via the forgetful maps N ℓX/Y → N X/Y .Now suppose the original square (1) is an fs pullback: • If q is strict, then V q ≃ L Y ′ , and our fs pullback square factors as X ′ X V q ≃ L Y ′ L YY ′ Y, pp and the functor of points witnesses that (2) is cartesian. • If instead f is strict, then X ′ → V q factors through Y ′ , and the factorization X ′ XY ′ Y V q L Y pp shows that the vertical arrows of (2) are isomorphisms and the Olsson Morphism is thesame as the ordinary functoriality of the Normal Sheaf. Remark 1.16.
Given a commutative square X ′ XY ′ Y, q HE LOG PRODUCT FORMULA of DM maps we can form two other commutative squares out of it: X ′ X L (cid:3) q L Y, X ′ L X L Y ′ L Y. They induce morphisms N ℓX ′ /Y ′ ≃ N ℓX ′ / L (cid:3) q → N ℓX/Y | X ′ ,N ℓX ′ /Y ′ → N L X/ L Y | X ′ . Form the diagram X ′ L X X L (cid:3) q L Y L Y L Y ′ s p d to see that the two morphisms of normal sheaves are compatible: N ℓX ′ /Y ′ ≃ N X ′ / L (cid:3) q → N L X/ L Y | X ′ ⊆ N ℓX/Y | X ′ . Lemma 1.17.
Suppose given a pair of commutative squares: X ′ Y ′ Z ′ X Y Z f g of DM-type maps. The diagram N ℓY/Y ′ N ℓX/X ′ N ℓZ/Z ′ commutes, where all the arrows are Olsson’s morphisms. Proof.
Introduce an algebraic X -stack W , with functor of points: W T X := M M M M X | T M Y | T M Z | T commutative diagrams offs log structures on T In other words, W := ( L (cid:3) × L L (cid:3) ) × L X . All the triangles in this diagram commute because of the definition of Olsson morphisms and thefunctor of points of N : N X/ L (cid:3) g ◦ f N ℓX ′ /X N X/ W N ℓZ ′ /Z N X ′ / L (cid:3) f N Y/ L (cid:3) g N ℓY ′ /Y ∼ ∼ ∼ Restricting the diagram to N ℓX ′ /X , N ℓY ′ /Y , and N ℓZ ′ /Z , we get the result. (cid:3) Proposition 1.18.
Given X f → Y g → Z DM-type maps of log algebraic stacks, the Olsson Mor-phisms yield a complex of stacks N ℓX/Y → N ℓX/Z → N ℓY/Z | X , in that the composite factors through the vertex.If h is smooth, N ℓY/Z = BT ℓY/Z and rotating the triangle in the derived category yields an exactsequence of cone stacks: T ℓY/Z | X → N ℓX/Y → N ℓX/Z . Proof.
The Olsson Morphisms come about from the commutative diagram
X X YY Z Z N ℓX/Y N ℓY/Y | X = XN ℓX/Z N ℓY/Z | X . The surjectivity, smoothness, and calculation of the fiber of N ℓX/Y → N ℓX/Z may all be checkedroutinely using the functor of points. (cid:3) Remark 1.19.
Suppose given a (not necessarily commutative) finite diagram of cones. If thediagram induced by taking abelian hulls is commutative, so was the original.2.
Properties of the Log Normal Cone
We are ready to define the log normal cone. We recall the essential properties of the ordinarynormal cone; the rest of the section establishes analogous properties in the log context.
Remark 2.1.
Consider a DM-type morphism f : X → Y of algebraic stacks. K. Behrend and B.Fantechi defined the Intrinsic Normal Cone [BF96] C f = C X/Y ⊆ N X/Y ;C. Manolache [Man08] removed their assumptions of smooth Y and DM X . This cone has thefollowing basic properties: HE LOG PRODUCT FORMULA (1) A commutative diagram X ′ XY ′ Y fq yields a morphism of cones ϕ : C X ′ /Y ′ → C X/Y × X X ′ . • if the square was cartesian, ϕ is a closed embedding. • if also f or q was flat, ϕ is an isomorphism.(2) For a composite X f → Y g → Z • if g is l.c.i., C X/Y = N X/Y and the sequence N X/Y → C X/Z → C Y/Z | X of cone stacks is exact. • if h is smooth, the sequence T Y/Z | X → C X/Y → C X/Z is exact.(3) Obstruction Theories and Gysin Pullbacks are obtained by placing the cone in a vectorbundle stack C X/Y ⊆ E (see [Man08], [Wis11], [Kre99]). Definition 2.2 (Log Intrinsic Normal Cone, Olsson Morphisms) . Let f : X → Y be a DM-typemorphism of log algebraic stacks. We define the Log (Intrinsic) Normal Cone C ℓX/Y := C X/ L Y ⊆ N ℓX/Y after [GS11]. Endow it with the log structure pulled back from X . Given a commutative square oflog algebraic stacks and its partner X ′ XY ′ Y q X ′ X L (cid:3) q L Y, the latter induces ϕ : C ℓX ′ /Y ′ ≃ C X ′ / L (cid:3) q → C ℓX/Y . This is again called the
Olsson Morphism . Remark 2.3.
The map L Y → Y has a section Y ⊆ L Y which is an open immersion. This openimmersion represents strict log maps to Y .As a result, if X → Y is DM and strict, C ℓX/Y = C X/Y and N ℓX/Y = N X/Y . In addition, theOlsson Morphisms are the same as the ordinary functoriality of the normal cone (Remarks 1.19and 1.15).The Olsson Morphism of any fs pullback square is a closed immersion, because it fits into acommutative square of closed immersions from Remark 1.14: C ℓX ′ /Y ′ C ℓX/Y | X ′ N ℓX ′ /Y ′ N ℓX/Y | X ′ . Remark 2.4 (Short Exact Sequences of Cone Stacks) . Recall [BF96, Definition 1.12]. Let E be avector bundle stack and C, D cone stacks all on some base algebraic stack X . A composable pairof morphisms of cone stacks E → C → D is called a short exact sequence if • C → D is a smooth epimorphism. • The square E × C CC D, pr σ where pr is the projection and σ the action, is cartesian.These are equivalent to having C ≃ E × X D locally in X .Note that this definition is fpqc -local in the base X [Sta18, 02VL]. Another reduction we willneed applies in case there is a commutative diagram of cone stacks E C DE ′ C ′ D ′ s t with E, E ′ vector bundles. If the top sequence is exact and the arrows labeled s, t are smooth andsurjective, then the bottom is exact. To see this, pushout along E → E ′ so as to assume E = E ′ ( s, t remain smooth and surjective). The diagram on the left is the pullback along the smoothsurjection D ′ → D of the one on the right: E × C ′ C ′ C ′ D ′ p E × C CC D, and we can verify that E × C is the pullback after smooth-localizing. Proposition 2.5.
Suppose X f → Y g → Z are DM maps between log algebraic stacks, and g is logsmooth. Then T ℓY/Z | X → C ℓX/Y → C ℓX/Z is an exact sequence of cone stacks. Proof.
Encode the log structures on the maps via the top row of the diagram X L Y L Z L Y L Z L . p p Since Y → L Z is smooth, L Y → L Z is. Moreover, they have the same tangent bundle: T ℓY/Z | L Y = T Y/ L Z | L Y = T L Y/ L Z since the vertical maps are log ´etale [Ogu18, Corollary IV.3.2.4].Together with the isomorphism C ℓX/Z ≃ C X/ L Z , we obtain the exact sequence. (cid:3) HE LOG PRODUCT FORMULA Remark 2.6.
In the proof, the composite C ℓX/Y → C X/ L Z ≃ C ℓX/Z is precisely the Olsson Morphism. This is immediate from the diagram: X X L (cid:3) q L Z L Y L Z. Remark 2.7.
The introduction promised three characterizations of C ℓX/Y .The log intrinsic normal cone is characterized by the strict case of Remark 2.3 and the log ´etalecase of Proposition 2.5. This is because any map X → Y factors into the strict map X → L Y composed with the log ´etale map L Y → Y (Remark 1.11).We can unpack this definition locally using charts. Suppose a morphism has a global fs chart byArtin Cones: X YA P A Q . The morphism A P → A Q is log ´etale [Ols03, Corollary 5.23]. Let W = A P × ℓA Q Y denote thefs pullback, so that X → Y factors through a strict map to W and W is log ´etale over Y . Weimmediately get C ℓX/Y = C X/W . The reader may be reassured by working locally with this definition. If the reader wants insteadto work with charts Spec ( P → C [ P ]) in the traditional sense, then log ´etaleness is no longerimmediate and we must check Kato’s Criteria [Ogu18, Corollary IV.3.1.10].Recall Construction 1.1 – after localizing in the ´etale topology, we obtain a factorization of anymap X → Y as a strict closed immersion followed by a log smooth map X ⊆ X θ → Y. Proposition 2.5 therefore locally provides a presentation of the log normal cone: C ℓX/Y = [ C X/X θ /T ℓX θ /Y ] . Lemma 2.8.
Given a DM map f : X → Y of log algebraic stacks with X quasicompact, the map X → L Y factors through an open quasicompact subset U ⊆ L Y .Our applications require openness; otherwise the lemma is trivial. Proof.
The claim is ´etale-local in Y and X because X is quasicompact. We can thereby assume wehave a global chart X YA P A Q . The map A P × A Q Y → L Y is ´etale [Ols03, Corollary 5.25] and X factors through its open,quasicompact image. (cid:3) Remark 2.9.
This lemma ensures that any DM map X → Y of log stacks with X quasicompactfactors through X → U → Y with X → U strict, U quasicompact, and U → Y log ´etale. Example 2.10.
We provide an example of Construction 1.1 and Remark 1.2.Consider the diagonal morphism A → A . The addition map N → N gives a chart for ∆.Denote by B the log blowup of A at the ideal I ⊆ M A generated by N \ { } ⊆ N . Thepullback ∆ ∗ I is generated by the image of the composite N \ { } ⊆ N → N . The pullback is generated globally by a single element and so ∆ factors through the log blowup B .Name the generators N = N e ⊕ N f . The log blowup B is covered by two affine opens D + ( e )and D + ( f ), on which e and f are invertible.On the chart D + ( e ), the morphism A → B looks like N N e ⊕ N ( f − e ) C [ t ] C [ x, yx ] . The horizontal morphisms send f − e yx
1. Because ( f − e ) maps to 1 ∈ C [ t ], thecomposite N e ⊕ N ( f − e ) → N → C [ t ]is another chart for the same log structure on A . This means that A → D + ( e ) is strict. Thesame discussion applies to D + ( f ). In the tropical picture [CCUW17, § A at theimage of the ray corresponding to A : A A . Proposition 2.11.
Consider DM-type morphisms X f → Y g → Z between log algebraic stacks. If C ℓX/Y = N ℓX/Y , then N ℓX/Y → C ℓX/Z → C L Y/ L Z | X is an exact sequence of cone stacks. Proof.
Compare [BF96, Proposition 3.14].By Proposition 1.18 and Remark 1.16, this sequence composes to zero. Remark 2.4 allows us torepeatedly fpqc -localize in X to check exactness of such a sequence. Localizing along strict smoothcovers of Z and strict ´etale covers of X and Y ensures that the normal cones and sheaf pull back.Reduce to the case where X , Y , and Z are affine log schemes and the map Y → Z admits a globalfs chart. We are therefore in the situation of Construction 1.1. Reduction to g : Y → Z Strict
Factor Y → Z into a strict closed immersion composed with a log smooth map: Y ⊆ W ։ Z. HE LOG PRODUCT FORMULA We obtain a diagram T ℓW/Z | X T L W/ L Z | X N X/Y C ℓX/W C L Y/ L W | X N X/Y C ℓX/Z C L Y/ L Z | X . p Observe that the diagram commutes – the morphism T ℓW/Z | X → C ℓX/W in the proof of Proposition2.5 factors through an identification T ℓW/Z | L W ≃ T ℓ L W/ L Z . Because L W → W is log ´etale, the twotangent spaces are isomorphic [Ogu18, IV.3.2.4]. Thus the right square is a pullback. The verticalmaps of cones are smooth surjections, so it suffices to show the middle row is exact as in Remark2.4. We may thereby assume W = Z and g : Y → Z is a strict closed immersion. Reduction to f : X → Y Strict
Use Construction 1.1 again to factor X → Z as a strict closed immersion composed with a logsmooth map X ⊆ W ։ Z . The map X → W ′ := W × Z Y is again a strict closed immersion: X W ′ WY Z. p ℓ (3)Because the top row is strict, X → L W ′ factors through the open subset W ′ ⊆ L W ′ and C L W ′ / L W | X = C L W ′ / L W | W ′ | X = C ℓW ′ /W | X = C W ′ /W | X . The fs pullback square in (3) also induces a cartesian square of stacks: L W ′ L W L Y L Z p with L W → L Z smooth. This reveals that C L Y/ L Z | L W ′ = C L W ′ / L W . Putting this together with the above, we have computed C L Y/ L Z | X = C W ′ /W | X . The factorization (3) gives a diagram T ℓW ′ /Y | X T ℓW/Z | X N X/W ′ C ℓX/W C W ′ /W | X N X/Y C ℓX/Z C L Y/ L Z | X The composable vertical arrows are the quotients of Proposition 2.5, so the bottom row will beexact if we show the middle row is. The middle row is exact by a relative form of the original[BF96, Proposition 3.14]. (cid:3)
Remark 2.12.
The exact sequences of cone stacks in Propositions 2.5, 2.11 are natural in mor-phisms of composable pairs of arrows.There is a version of Proposition 2.11 for log cotangent complexes that we will use once later on.From any composable pair X → Y → Z , we get X → L Y → L Z and X → L Y → L Z . Bothresult in the same distinguished triangle: L L Y/ L Z | X → L ℓX/Z → L ℓX/Y → . of [Ols07, 8.10].In the next example, the log normal cone differs from the ordinary scheme-theoretic one. Example 2.13.
In Example 2.10, we considered the log blowup B of A at the origin and thediagonal map. Pull back to get the identity log blowup of A : A B A A . p ℓ Let o N , o N both be Spec C , with log structures coming from N and N , respectively. Then theinclusions of the origins o N ∈ A and o N ∈ A are strict.Take the pullback of the above diagram along the inclusion o N ∈ A : o N Do N o N . p ℓ The map D → o N is the exceptional divisor of B , which is P with log structure M x = N at theintersections with the axes and M x = N elsewhere.To see the log normal cone differ from the ordinary one, compute the normal cones of the arrowsin this square: C ℓo N /o N = o , C ℓo N /o N = C ℓo N /D = A , and C ℓD/o N = P . Although o N and o N havethe same underlying scheme, the log normal cones of o N over them are different. Remark 2.14.
A handy consequence of Proposition 2.11 is that, if Y → Z is a DM-type morphismbetween log algebraic stacks and Y ′ → Y is a strict ´etale map, then C ℓY ′ /Z ≃ C ℓY/Z | Y ′ . This is not true without the strictness assumption. This is the observation of W. Bauer precludingthe existence of a log cotangent complex with all its desiderata (see [Ols05, § C ℓY ′ /Z ≃ C L Y/ L Z | Y ′ ⊆ N L Y/ L Z | Y ′ ⊆ N ℓY/Z | Y ′ is a closed immersion which factors through C ℓY/Z | Y ′ , as in Remark 1.16.For a single example, take the log blowup B → A of the origin o ∈ A . The pullback defines astrict pullback square: D Bo A . p ℓ HE LOG PRODUCT FORMULA Because the horizontal morphisms are strict, their log normal cones coincide with the ordinaryones. Log blowups are log ´etale, so we would erroneously be led to conclude that C D/B ? = C o/ A | D . The inclusion D ⊆ B is regular, and so is o ∈ A , so the normal cones and normal sheaves agree: N D/B = O B ( D ) | B N o/ A | D = A D . The dimensions are different, so they can’t be equal.
Lemma 2.15.
Suppose given a strict pullback square X ′ XY ′ Y p ℓ q of DM-type morphisms between log algebraic stacks for which q is strict and smooth. Then theOlsson Morphism C ℓX ′ /Y ′ ∼ → C ℓX/Y | X ′ is an isomorphism. Proof.
We first note that the Olsson Morphism N ℓX ′ /Y ′ → N ℓX/Y | X ′ on log normal sheaves is anisomorphism. This is clear from the q strict pullback part of Remark 1.15 and the fact that theordinary normal sheaves are isomorphic.Now we know that the morphism of cones C ℓX ′ /Y ′ → C ℓX/Y | X ′ is a closed immersion, and itsuffices to show that it is moreover smooth and surjective. We express this map as a composite C ℓX ′ /Y ′ → C ℓX ′ /Y → C ℓX/Y | X ′ . Proposition 2.5 asserts that the first map is smooth and surjective and Proposition 2.11 says thesame for the second. (cid:3)
Lemma 2.16.
Suppose given a pair of fs pullback squares f X ′ f XX ′ XY ′ Y p ℓ z p ℓ of DM-type morphisms between log algebraic stacks for which z is strict and smooth. Then thediagram of log normal cones C ℓ g X ′ /Y ′ C ℓ f X /Y C ℓX ′ /Y ′ C ℓX/Ys ′ p s is cartesian and the arrows s, s ′ are smooth epimorphisms. Proof.
Proposition 2.11 provides a map of short exact sequences of cone stacks: BT ℓ g X ′ /X ′ C ℓ g X ′ /Y ′ C ℓX ′ /Y ′ | g X ′ BT ℓ f X /X | g X ′ C ℓ f X /Y | g X ′ C ℓX/Y | g X ′ . t ′ p e t Witness that the right square is cartesian because [Ols05] T ℓ g X ′ /X ′ = T ℓ f X /X | g X ′ and that the arrows t ′ , e t are clearly smooth epimorphisms. The arrow e t is pulled back from thesmooth epimorphism t : C ℓ f X /Y → C ℓX/Y | f X , so we have the top pullback square C ℓ g X ′ /Y ′ C ℓ f X /Y C ℓX ′ /Y ′ | g X ′ C ℓX/Y | f X f XC ℓX ′ /Y ′ C ℓX/Y X t ′ p s ′ t s p p The composite vertical rectangle of cones is the diagram we are after, and so the fact that thissquare is cartesian is clear. It remains only to note the bent arrows s, s ′ are smooth epimorphismsbecause they are the composites of t, t ′ with pullbacks of the smooth epimorphism f X → X . (cid:3) Log Intersection Theory
The Log Intersection Theory package is defined the same way as usual [Man08], mutatis mutandis.
Definition 3.1 (Log Perfect Obstruction Theory) . Define a
Log Perfect Obstruction Theory (here-after “Log POT ”) for a DM-type morphism f : X → Y to be a closed immersion of cone stacks C ℓX/Y ⊆ E (equiv. N ℓX/Y ⊆ E )of the log normal cone into a vector bundle stack E .Given an fs pullback square X ′ XY ′ Y f ′ p ℓ f and a Log POT C ℓX/Y ⊆ E for f , the Olsson Morphism C ℓX ′ /Y ′ ϕ ֒ → C ℓX/Y | X ′ ⊆ E | X ′ defines a “Pullback” Log POT .A related notion of “Pullback” Log POT arises when X ′ → X is log ´etale and f : X → Y anyDM-type map. Then Remark 2.14 shows the map C ℓX ′ /Y → C ℓX/Y | X ′ HE LOG PRODUCT FORMULA is a closed immersion, and we can compose with an obstruction theory for f to get one for thecomposite X ′ → X → Y .Given a Log POT C ℓX/Y ⊆ E for some f , suppose X has a stratification by global quotientstacks and Y is log smooth and equidimensional. Then [Kre99, Proposition 5.3.2] gives us a uniquecycle [ X, E ] ℓvir ∈ A ∗ X which pulls back to the class [ C ℓX/Y ] ∈ A ∗ E . This class is called the Log Virtual Fundamental Class (hereafter “Log VFC ”).
Remark 3.2.
When L Y is equidimensional, so is C ℓX/Y . The correct definition of the Log VFCrequires that the cone be equidimensional. If Y is log smooth, Y ⊆ L Y is dense. If Y is alsoequidimensional, we get that L Y is. This explains our assumptions in Definition 3.1. We don’tinclude these assumptions in the definition of a Log POT only because we may have Log Gysinmaps more generally. Definition 3.3 (Log Gysin Map) . Suppose a DM-type f : X → Y has a Log POT C ℓX/Y ⊆ E .Given a DM-type log map k : V → Y with V log smooth and equidimensional, form the fs pullback: W VX Y p ℓ kf The embedding C ℓW/V ⊆ C ℓX/Y | W ⊆ E | W results in a class [ C ℓW/V , E ] ∈ A ∗ W. Mimicking [Man08], we call this “map” f ! = f ! E the Log Gysin Map . Remark 3.4.
Consider a DM-type morphism f : X → Y of log algebraic stacks. The cartesiansquare L X X L Y L Y s p d from Remark 1.16 results in a closed embedding C L X/ L Y ≃ C L X/ L Y ⊆ C ℓX/Y | L X which we use to canonically extend an obstruction theory C ℓX/Y ⊆ E to a closed embedding C L X/ L Y ⊆ E | L X . Now suppose given a composable pair X f → Y g → Z as above and equip f, g with Log POT ’s: C ℓX/Y ⊆ F, C ℓY/Z ⊆ G. Define a compatibility datum for such a pair to be a traditional compatibility datum [Man08,Definition 4.5] for X f → L Y g → L Z, endowing L Y → L Z with the extended obstruction theory C L Y/ L Z ≃ C ℓY/Z | L Y ⊆ G | L Y . We offer a couple of basic remarks about our definitions before the examples and theorems.
Remark 3.5.
The map f ! just defined takes in log smooth equidimensional stacks DM over Y andproduces classes in certain Chow Groups. We do not know whether this operation may be extendedto the “Log Chow” groups of [Bar18]. Remark 3.6.
Given an fs pullback square X ′ XY ′ Y f ′ p ℓ f of DM maps where f has a Log POT C ℓX/Y ⊆ E , endow f ′ with the Pullback Log POT . Then f ! = f ′ ! when applied to log smooth, equidimensional log schemes over Y ′ . Remark 3.7. If C ℓX/Y = N ℓX/Y for a DM morphism f : X → Y , we can take E = N ℓX/Y as ourobstruction theory. If X, Y are equidimensional and Y is log smooth, unwinding definitions shows f ! ( Y ) = [ X ] , where [ X ] is the fundamental class of X . Remark 3.8.
Log Gysin Maps don’t commute with pushforward: Let X ′ XY ′ Y pf ′ p ℓ fq be an fs pullback square. Endow f : X → Y with a Log POT C ℓX/Y ⊆ E and give f ′ the pullbackobstruction theory. Then the usual equality [Man08, Theorem 4.1 (i)] can fail: f ! q ∗ = p ∗ f ′ ! . Take the square of Example 2.13 o N Do N o N p ℓ and apply both operations to [ o N ] for a counterexample. Remark 3.9.
Virtual Fundamental Classes don’t push forward along log blowups: Let X → F bethe morphism from a stack X to its Artin Fan (the reader may take a traditional chart instead of F ). Choose a finite subdivision c F → F , and form the fs pullback: c X c FX F. p p ℓ Suppose given a map f : X → Y with a Log POT C ℓX/Y ⊆ E and equip f ◦ p : c X → Y with thepullback obstruction theory C ℓ c X /Y ⊆ C ℓX/Y | c X ⊆ E | c X . HE LOG PRODUCT FORMULA Then possibly p ∗ [ c X , E ] ℓvir = [ X, E ] ℓvir . A counterexample is again given by p : D → o N , f : o N == o N as in Example 2.13: p ∗ [ P ] = 0 fordimension reasons.The rest of this section and the next should reassure the disheartened reader that commonsensefomulas of ordinary intersection theory do remain true in the log setting. We regard Remarks 3.8,3.9 as defects of the usual notion of pushforward p ∗ in the log setting. The morphisms o N → D , D → o N of Example 2.13 are monomorphisms in the fs category, and o N → o N should be a cycleof dimension one in the “two dimensional” log point o N .The paper [Bar18] introduces log chow groups to correct this defect, in particular via suitablenotions of dimension and degree. See also [Moc15]. We are eager to see which of our results maybe extended using this improved technology.For now, we content ourselves to use the observation of [Niz06, Proposition 4.3] that log blowupsare birational if the target is log smooth. We will use it to prove that weaker forms of the na¨ıveguesses of Remarks 3.8, 3.9 do hold true, as well as straightforward commutativity of the GysinMaps.We will need to use Costello’s notion of “pure degree d ” [Cos06, before Theorem 5.0.1] to makesense of pushforward on the level of cycles, given by cones embedded in vector bundles. The nexttheorem allows us to check statements about Log VFC ’s after a log blowup if the target is logsmooth. Its statement and proof are similar to [AW18]. Theorem 3.10.
Suppose given a DM-type map f : X → Y between locally noetherian algebraicstacks locally of finite type over C where Y is log smooth and equidimensional. Endow f with aLog POT E and let X → F be any DM morphism to an Artin Fan. Take the fs pullback along afinite subdivision c X c FX F. p p ℓ (4)Endow f ◦ p with the pullback Log POT C ℓ c X /Y ⊆ C ℓX/Y | c X ⊆ E | c X . Then p ∗ [ c X , E ] ℓvir = [ X, E ] ℓvir Proof.
We will actually show that the map t : C ℓ c X /Y → C ℓX/Y is of pure degree one. Then the pushforward A ∗ E | c X → A ∗ E sends the class of one cone to theother, and “intersecting with the zero section” gives the equality of VFC’s.We will reduce to the case where X → F is strict. The statement “ t is of pure degree one” maybe verified ´etale-locally in X , as we now argue.Given a strict ´etale cover X ′ → X , write c X ′ := c X × X X ′ . We have a pullback diagram C ℓ d X ′ /F C ℓX ′ /F X ′ C ℓ c X /F C ℓX/F X, t ′ p p t as in Remark 2.14. Since X ′ → X is ´etale, the other vertical arrows are as well. The property“pure degree one” is smooth-local in the target, so t has it if t ′ does.Now ´etale-localize in X so that X → F factors through a chart X → F X → F for X . Take thefs pullback along the subdivision c F → F : c X c F X c FX F X F. p ℓ p ℓ We can then replace F by F X in the proof of the theorem and assume X → F is strict.Apply the proof of Costello’s Formula [Cos06, Theorem 5.0.1] to (4) to conclude t : C ℓ c X / c F → C ℓX/F is of pure degree one, since c F → F is birational.Expanding upon (4): c X c F × Y c FX F × Y FY, p ℓ p ℓ we get a map of exact sequences of cone stacks: T ℓY | c X C ℓ c X / c F × Y C ℓ c X / c F T ℓY | X C ℓX/F × Y C ℓX/F . p b t t After pulling the bottom row back to c X , we get the identity on tangent bundles and see that theright square is a pullback. Since the property“of pure degree one” pulls back along smooth maps,the quotient maps in exact sequences of cone stacks are smooth, and t is pure degree one, b t is alsopure degree one. Because F, c F are log ´etale over a point, C ℓ c X / c F × Y = C ℓ c X / c Y and C ℓX/F × Y = C ℓX/Y ,so the claim is proven. (cid:3) Example 3.11.
One must be cautious, for Theorem 3.10 is false without the assumption that Y is log smooth. Recall the exceptional divisor D → o of the blowup of A at the origin o = Spec C from Example 2.13 and its normal cone C ℓD/o = P .For the sake of contradiction, let c X = P and X = Y = o as in the theorem. Endow C ℓo/o = o with the initial Log POT , E = o . Then[ c X , E ] ℓvir = [ D, E ] ℓvir = [ P ]and [ X, E ] ℓvir = [ o, E ] ℓvir = [ o ] , but again p ∗ [ P ] = 0 for dimension reasons. HE LOG PRODUCT FORMULA Theorem 3.12 (Commutativity of Log Gysin Map) . Given a composable pair of DM-type mapsbetween log algebraic stacks X f → Y g → Z, outfit f , g , and g ◦ f with log obstruction theories F , G , E and a compatibility datum (Remark3.4). Require X to admit stratifications by global quotients.If k : V → Z is a log smooth and equidimensional Z -stack and k is DM-type, take fs pullbacks: T U VX Y Z. p ℓ p ℓ Then the equality [ C ℓg ◦ f ⊆ E ] = [ C ℓC ℓg | X /C ℓg ⊆ F ⊕ G | X ] (5)holds on X . Proof.
Pullback via k all obstruction theories and their compatibility datum to reduce to showingthe theorem for k : V == Z . We essentially apply [Man08, Theorem 4.8] to X → L Y → L Z ,endowed with the compatible triple F, G, E by composing with an isomorphism of distinguishedtriangles: G | X F E L L Y/ L Z | X L ℓX/Z L ℓX/Y L L Y/ L Z | X L X/ L Z L X/ L Z . ∼ ∼ Use Lemma 2.8 repeatedly to obtain a strict diagram with
U, V quasicompact and ´etale over thestacks L Y, L Z : X U V L Y L Z. Endow the cone C L Y/ L Z with the pullback log structure from L Y and pull it back along thepart of the diagram above L Y : C L Y/ L Z | X = C U/V | X C U/V C L Y/ L Z . The triangle is strict and the map C U/V → C L Y/ L Z is pulled back from the ´etale U → L Y , so C ℓC L Y/ L Z | X /C L Y/ L Z = C C U/V | X /C U/V . Write i : X → U j : U → V for the maps. Then the compatibility datum pulls back and [Man08,Theorem 4.8] gives us ( j ◦ i ) ! E ([ V ]) = i ! F ◦ j ! G ([ V ]) . Unwinding definitions, this becomes[ C X/V ⊆ E ] = [ C C U/V | X /C U/V ⊆ F ⊕ G | X ] . (6) This may be rewritten as[ C ℓX/Z ⊆ E ] = [ C ℓC L Y/ L Z | X /C L Y/ L Z ⊆ F ⊕ G | X ] , the claimed equality of classes. (cid:3) Remark 3.13.
Theorem 3.12 says that ( g ◦ f ) ! = f ! g ! in the sense that any log smooth, equidimensional log stack over Z has rationally equivalent imagesunder these two operations. Remark 3.14.
Consider an fs pullback of DM-type morphisms between log algebraic stacks: X ′ XY ′ Y. pf ′ p ℓ fq Write r : X ′ → Y for the composite f ◦ p = q ◦ f ′ . If f, q are endowed with Log POT ’s C ℓX/Y ⊆ F , C ℓY ′ /Y ⊆ E , how should we give r a Log POT ?The fs pullback square induces a pullback of stacks, which may be reexpressed as a “magicsquare:” L X ′ L X L X ′ L X × L Y ′ L Y ′ L Y L Y L Y × L Y. p p The magic square induces a closed immersion C L X ′ / L Y ⊆ C L X/ L Y | L X ′ × L X ′ C L Y ′ / L Y | L X ′ which pulls back to a closed immersion C ℓX ′ /Y ⊆ C L X/ L Y | X ′ × X ′ C L Y ′ / L Y | X ′ on X ′ . As in Remark 3.4, we have closed embeddings C L X/ L Y ⊆ C ℓX/Y | L X , C L Y ′ / L Y ⊆ C ℓY ′ /Y | L Y ′ .We endow r with the Log POT given by the composite: C ℓX ′ /Y ⊆ C L X/ L Y | X ′ × X ′ C L Y ′ / L Y | X ′ ⊆ C ℓX/Y | X ′ × X ′ C ℓY ′ /Y | X ′ ⊆ F | X ′ × X ′ E | X ′ . We now construct a compatibility datum for the triangle r = q ◦ f ′ , leaving the reader to applythe same argument to the other triangle r = f ◦ p . By the definitions of the Log POT ’s, we havea commutative diagram: C ℓX ′ /Y ′ C ℓX ′ /Y C L Y ′ / L Y | X ′ F | X ′ E | X ′ × X ′ F | X ′ E | X ′ . (0 × id ) To be clear, the morphism F | X ′ → E | X ′ × X ′ F | X ′ is the vertex map times the identity. It’s clearthe bottom row comes from a distinguished triangle in the derived category and the top row comesfrom Remark 2.12. HE LOG PRODUCT FORMULA Corollary 3.15.
Suppose given an fs pullback square X ′ XY ′ Y pf ′ p ℓ fq of DM-type morphisms between log algebraic stacks which admit stratifications by quotient stacks.Outfit q with a Log POT E and f with a Log POT F ; give p, f ′ the pullback obstruction theories.Then f ′ ! ◦ q ! = p ! ◦ f ! in the sense that the operations send any log smooth equidimensional input stack to the same classin A ∗ X ′ . Proof.
Denote by r : X ′ → Y the map f ◦ p = q ◦ f ′ . Apply Theorem 3.12 to both commutativetriangles using the compatibility datum constructed in Remark 3.14 to see that p ! ◦ f ! = r ! = f ′ ! ◦ q ! . (cid:3) The Log Costello Formula
This section proves a log analogue of the Costello Formula [Cos06, Theorem 5.0.1]. We will havemore to say building on future work [AHW].
Theorem 4.1.
Consider an fs pullback square of DM-type maps between algebraic stacks: X ′ XY ′ Y. pf ′ p ℓ fq Assume • Y ′ → Y is of some pure degree d ∈ Q as in [Cos06, Theorem 5.0.1], • Y ′ , Y are both log smooth and equidimensional, • all arrows are DM-type and all stacks are locally noetherian and locally finite type over C , • X ′ , X admit stratifications by global quotient stacks [Kre99] • q is proper.Endow f with a log perfect obstruction theory E and give f ′ the pullback obstruction theory.Then p ∗ [ X ′ , E | X ′ ] ℓvir = d · [ X, E ] ℓvir in the Chow Ring of X . Remark 4.2.
Let Y ′ → Y be a map between log smooth, equidimensional stacks which is of puredegree d . Let W → Y be a smooth, log smooth, integral, and saturated morphism and f W → W alog blowup. Form the fs pullback diagram: g W ′ f WW ′ WY ′ Y. p ℓ p ℓ The property “of pure degree d ” pulls back along smooth morphisms, so it applies to W ′ → W .Then [Niz06, Proposition 4.3] shows that f W → W is birational, so g W ′ → f W is also of pure degree d . Proof of Theorem 4.1.
Consider the morphism s : C ℓX ′ /Y ′ → C ℓX/Y . We will prove that s is of pure degree d . Both “of pure degree” and the specific degree d can bechecked after pulling back s along a strict, smooth cover of C ℓX/Y . Lemmas 2.15, 2.16 show thatreplacing Y or X by a smooth cover results in such a smooth cover of cones.We may thereby assume X and Y are log schemes and the map f globally factors as in Con-struction 1.1: X → X θ → A r + sY → Y. Note A r + sY → Y is smooth, log smooth, integral, and saturated, and X θ → A r + sY is a log blowup.We are in the situation of Remark 4.2, so pulling back: X ′ XX ′ θ X θ Y ′ Y p ℓ p ℓ results in a map X ′ θ → X θ which is pure of degree d along X → X θ . The proof of Costello’s Formula[Cos06, Theorem 5.0.1] then asserts that C ℓX ′ /X ′ θ → C ℓX/X θ is of pure degree d . The short exact sequences of Proposition 2.5 T ℓX ′ θ /Y ′ C ℓX ′ /X ′ θ C ℓX ′ /Y ′ T ℓX θ /Y C ℓX/X θ C ℓX/Yt s let us conclude that s is as well. (cid:3) The Product Formula
Let V , W be log smooth, quasiprojective schemes throughout this section. We denote the stacksof prestable curves and stable curves which have n -markings and genus g by M g,n , M g,n , respectively[Sta18, 0DMG]. They are endowed with divisorial log structures coming from the locus of singularcurves [GS11, 1.5, Appendix A], [Kat99]. Definition 5.1 (Log Stable Maps) . The stack of log stable maps M ℓg,n ( V ) has fiber over an fs logscheme T the category of diagrams of fs log schemes C VT with C → T a log smooth curve [Kat99, Definition 1.2] of genus g and n marked points, such thatthe underlying diagram of schemes is a stable map of curves. HE LOG PRODUCT FORMULA Remarkably, the log algebraic stack M ℓg,n (Spec C ) of log curves without a map is isomorphic tothe ordinary stack of stable curves M g,n with log structure induced by the boundary of degeneratecurves [Kat99, Theorem 4.5]. The log structures of M ℓg,n ( V ) for a general fs target may be morecomplicated, as they have to do with the “tropical deformation space” of the curve [GS11]. Construction 5.2 ([GS11, Section 5]) . We recall the construction [GS11, Section 5] of the naturalLog POT for M ℓg,n ( V ) → M g,n to clarify differences in notation.Write U → M g,n for the universal curve. Define U V as the fs pullback, naturally equipped witha tautological map to V : V U V M ℓg,n ( V ) U M g,nπ V p ℓ This diagram induces maps between log cotangent complexes L ℓV | U V −→ L ℓ U V / U t ←− L ℓ M ℓg,n ( V ) / M g,n | U V . The map U → M g,n is integral, saturated, and log smooth according to its functor of points, so itsunderlying map of stacks is flat and the fs pullback square is also an ordinary pullback.Then t is an isomorphism [Ols05, 1.1 (iv)], and the log cotangent complex of V is [Ols05, 1.1(iii)] L ℓV = Ω ℓV [0] . We’ve written [0] to consider a coherent sheaf as a chain complex concentrated in degree 0. Viathe isomorphism t and this identification, we have obtained a mapΩ ℓV [0] | U V → L ℓ U / M g,n | U V . (7)We need the ordinary relative dualizing sheaf ω π ◦ V and the identification Lπ ! V ( · ) = ω π ◦ V L ⊗ Lπ ∗ V ( · ) . Tensor (7) by ω π ◦ V and use the adjunction:Ω ℓV [0] | U V L ⊗ ω π ◦ V −→ Lπ ! V L ℓ M ℓg,n ( V ) / M g,n ,E ( V ) := Rπ V ∗ (Ω ℓV [0] | U V L ⊗ ω π ◦ V ) −→ L ℓ M ℓg,n ( V ) / M g,n . We won’t repeat the verification [GS11, Proposition 5.1] that E ( V ) is a Log POT . Remark 5.3.
The map (7) comes from the map on normal cones C ℓ M ℓg,n ( V ) / M g,n | U V ∼ ←− C ℓ U V / U −→ BT ℓV | U V . We needed duality, so we opted for the other perspective.
Remark 5.4 (Variants) . The reader may choose to work in the relative setting of a log smoothand quasiprojective map V → S . Obstruction Theories are obtained in the same way.We can naturally impose “contact order” conditions [ACWM17] in the log setting, but we onlyfix genus and number of markings to be consistent with [LQ18]. The reader may readily vary thenumerical type conditions in our formulas.We need one more stack, D : Points of D over T are diagrams ( C ′ ← C → C ′′ ) of genus g , n -pointed prestable curves over T whose maps are partial stabilizations (they lie over the identitiesin M g,n ) that don’t both contract any component. In other words, C → C ′ × C ′′ itself is a stablemap. This stack is only necessary to form an fs pullback square: Situation 5.5 ([LQ18, Section 2]) . Recall the fs pullback square: M ℓg,n ( V × W ) M ℓg,n ( V ) × M ℓg,n ( W ) D M g,n × M g,nc p ℓ a f ∆ (8)Let C → V × W be a log stable map over a base T . The maps ( C → V ), ( C → W ) needn’t bestable; denote their stabilizations by ( C ′ → V ), ( C ′′ → W ), respectively.The top horizontal arrow in (8) sends ( C → V × W ) to the induced log stable maps ( C ′ → V, C ′′ → W ). The vertical arrow c sends ( C → V × W ) to the partial stabilizations ( C ′ ← C → C ′′ ). Themap f ∆ sends a diagram ( C ′ ← C → C ′′ ) to the pair of prestable curves C ′ , C ′′ . Finally, a sends apair of log stable maps ( C ′ → V, C ′′ → W ) to the prestable curves ( C ′ , C ′′ ).This square has a factorization: M ℓg,n ( V × W ) Q M ℓg,n ( V ) × M ℓg,n ( W ) D Q ′ M g,n × M g,n M g,n M g,n × M g,n , hc p ℓ p ℓ al φ p ℓ s × s ∆ (9)where s : M g,n → M g,n stabilizes a prestable curve.To be clear, Q = M ℓg,n ( V ) × ℓM g,n M ℓg,n ( W ) and Q ′ = M g,n × ℓM g,n M g,n are the analogues of[LQ18]’s P , P , etc. Theorem 5.6 (The “Log Gromov-Witten Product Formula”) . With V , W log smooth, quasipro-jective schemes, h ∗ [ M ℓg,n ( V × W ) , E ( V × W )] ℓvir = ∆ ! ([ M ℓg,n ( V ) , E ( V )] ℓvir × [ M ℓg,n ( W ) , E ( W )] ℓvir ) . Our proof will be the same as K. Behrend’s [Beh97]: we compute the log normal cone of the map Q → Q ′ in two different ways. Remark 5.7 (On Diagram (9)) . We equip a with the product E ( V ) ⊞ E ( W ) of the natural LogPOT ’s of Construction 5.2, adopting the notation E ⊞ E ′ := E | V × W ⊕ E ′ | V × W . The cotangent complex L ℓ ∆ is of perfect amplitude in [-1, 0] because its source and target are logsmooth. Therefore C ℓ ∆ = N ℓ ∆ serves as a natural Log POT for itself. We equip φ with the pullbackobstruction theory, resulting in ∆ ! = φ ! by Remark 3.6. We endow the square bounded by φ and a with the natural compatibility datumafforded all such squares as in Remark 3.14.All of the arrows in Diagrams (8) and (9) are of DM-type. Lemma 5.8.
The stabilization map s : M g,n → M g,n is log smooth. Proof.
The cover F m M g,n + m → M g,n given by forgetting marked points and not stabilizing is strictsmooth [LQ18, 1.2.1]. This map is in particular kummer and surjective, and [INT13, Theorem 0.2]applies with P = “log smooth” once we argue that the composite F m M g,n + m → M g,n is logsmooth. HE LOG PRODUCT FORMULA The forgetful map M g,n +1 → M g,n is the universal curve, so it is tautologically log smooth. Wesee the map M g,n + m → M g,n is log smooth by iterating this forgetfulness, and this completes theargument. (cid:3) Remark 5.9.
The map D → M g,n which records the initial curve is log ´etale since the originalmap was ´etale [Beh97, Lemma 4] and ours is the fsification thereof. The stack Q ′ is log smoothbecause the map Q ′ → M g,n is pulled back from s × s .Given a log ´etale map X ′ → X of log smooth log algebraic stacks with X equidimensional, weclaim X ′ must be as well. The maps X ′ ⊆ L X ′ , X ⊆ L X are dense because of the log smoothnessassumption and the map L X ′ → L X is ´etale. Thus L X and L X ′ are equidimensional, as wellas X ′ ⊆ L X ′ . This argument shows that fsification preserves equidimensionality of log smoothstacks, so our fs versions of D , Q ′ are equidimensional because the original versions [Beh97] were. Lemma 5.10.
The obstruction theories E ( V ), E ( W ), E ( V × W ) are compatible in the sense that f ∆ ∗ ( E ( V ) ⊞ E ( W )) ≃ E ( V × W ) . Proof.
We completely echo the proof of [Beh97, Proposition 6].Consider the diagram of universal log curves and tautological maps with the notation:
V V × W U V e U V U V × W M ℓg,n ( V ) M ℓg,n ( V × W ) . f V π V s V e π V ℓ q q V f V × W π V × W r V We claim F → Rq V ∗ q ∗ V F is an isomorphism for any vector bundle F on U V . The map q V represents partial stabilization. We make the argument for contracting one P at a time.We first compute that R p q V ∗ q ∗ V F = 0 for p = 0. This claim is local in U V , so assume F istrivial. The fiber of R p q V ∗ q ∗ V F at a point x is H p ( q − V ( x ) , q ∗ V F ). Hence the fibers q − V ( x ) are eithera point or P . On each fiber, the cohomology of the trivial vector bundle is concentrated in degree0 [Sta18, 01XS]. Not only are F and q V ∗ q ∗ V F abstractly isomorphic in that case, but the naturalmap is an isomorphism [FGS +
05, Exercise 9.3.11].The universal curve π V is tautologically flat, integral, and saturated. The fs pullback squareit belongs to is therefore also an ordinary flat pullback, subject to cohomology and base change[Sta18, Tag 08IB]. This gives: Lr ∗ V Rπ V ∗ Lf ∗ V Ω V = R e π V ∗ Ls ∗ V Lf ∗ V Ω V = R e π V ∗ Rq V ∗ q ∗ V Ls ∗ V Lf ∗ V Ω V = Rπ V × W ∗ Lf ∗ V × W (Ω V | V × W ) . All the same goes for W . Add the two together to get Lr ∗ V Rπ V ∗ Lf ∗ V Ω V ⊞ Lr ∗ W Rπ W ∗ Lf ∗ W Ω W = Rπ V × W ∗ Lf ∗ V × W (Ω V ⊞ Ω W ) . This is dual to the compatibility we set out to prove, so we are through. (cid:3)
Proof of Theorem 5.6.
Compute the log virtual fundamental class [
Q, E ( V ) ⊞ E ( W )] vir in two dif-ferent ways: [ Q, E ( V ) ⊞ E ( W )] vir := [ C ℓQ/Q ′ ⊆ E ( V ) ⊞ E ( W )]= a ! ( Q ′ )= a ! φ ! ( M g,n × M g,n )= φ ! a ! ( M g,n × M g,n )= ∆ ! [ M ℓg,n ( V ) × M ℓg,n ( W ) , E ( V ) ⊞ E ( W )] vir . On the other hand,[
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