Tropical cohomology with integral coefficients for analytic spaces
aa r X i v : . [ m a t h . AG ] J un TROPICAL COHOMOLOGY WITH INTEGRAL COEFFICIENTS FORANALYTIC SPACES
PHILIPP JELL
Abstract.
We study tropical Dolbeault cohomology for Berkovich analytic spaces,as defined by Chambert-Loir and Ducros. We provide a construction that lets us pullback classes in tropical cohomology to classes in tropical Dolbeault cohomology as wellas check whether those classes are non-trivial. We further define tropical cohomologywith integral coefficients on the Berkovich space and provide some computations. Ourmain tool is extended tropicalization of toric varieties as introduced by Kajiwara andPayne.MSC: Primary 32P05; Secondary 14T05, 14G22, 14G40Keywords: Tropical Dolbeault cohomology, Tropical Cohomology, Superforms, Toricvarieties, Berkovich spaces, Tropical geometry Introduction
Real valued-differential forms and currents on Berkovich analytic spaces were intro-duced by Chambert–Loir and Ducros in their fundamental preprint [CLD12]. Theyprovide a notion of bigraded differential forms and currents on these spaces that hasstriking similarities with the complex of smooth differential forms on complex analyticspaces. The definition works by formally pulling back Lagerberg’s superforms on R n along tropicalization maps. These tropicalization maps are induced by mapping opensubsets of the analytic space to analytic tori and then composing with the tropicalizationmaps of the tori.Payne, and independently Kajiwara [Pay09, Kaj08], generalized this tropicalizationprocedure from tori to general toric varieties and Payne showed that the Berkovichanalytic space is the inverse limit over all these tropicalizations.Shortly after the preprint by Chambert–Loir and Ducros, Gubler showed that onemay, instead of considering arbitrary analytic maps to tori, restrict one’s attention toalgebraic closed embeddings if the analytic space is the Berkovich analytification of analgebraic variety [Gub16].Let K be a field that is complete with respect to a non-archimedean absolute valueand let X be a variety over K . We write Γ = log | K ∗ | for the value group of K and X an for the Berkovich analytification of X . Both the approach by Gubler and the one byChambert–Loir and Ducros provide bigraded complexes of sheaves of differential forms ( A • , • , d ′ , d ′′ ) on X an . We denote by H ∗ , ∗ (resp. H ∗ , ∗ c ) the cohomology of the complex ofglobal sections (resp. global sections with compact support) with respect to d ′′ .In this paper, we generalize Gubler’s approach, showing that one can define forms onBerkovich analytic spaces by using certain classes of embeddings of open subsets intotoric varieties. Given a fine enough family of tropicalizations S (see Section 4 for thedefinition of this notion) we obtain a bigraded complex of sheaves ( A • , •S , d ′ , d ′′ ) on X an .We show that for many useful S , our complex A • , •S is isomorphic to A • , • . The author was supported by the DFG Collaborative Research Center 1085 “Higher Invariants”.
For the rest of the introduction, we make the very mild assumption the X is normaland admits at least one closed embedding into a toric variety.The general philosophy of this paper and also the definition of forms by Chambert-Loirand Ducros and Gubler is that we can transport constructions done for tropical varietiesto Berkovich spaces by locally pulling back along tropicalizations. While Chambert-Loir,Ducros and Gubler used only tropicalization maps of tori, we will also allow tropicaliza-tion maps of general toric varieties. We will show advantages of this equivalent approachthroughout the paper.The definitions by both Chambert-Loir and Ducros as well as Gubler work with localembeddings. We show that we can also work with global embeddings. Our constructionsprovides us with the following: Let ϕ : X → Y Σ be a closed embedding into a toricvariety. Then we obtain pullback morphisms trop ∗ : H p,q (Trop( ϕ ( X ))) → H p,q ( X an ) and(1.1) trop ∗ : H p,qc (Trop( ϕ ( X ))) → H p,qc ( X an ) (1.2)in cohomology.Note that (1.1) and (1.2) were not in general available in the approaches by Chambert–Loir and Ducros resp. Gubler. This construction allows us to explicitly construct classesin tropical Dolbeault cohomology.For cohomology with compact support, we even obtain all classes this way: Theorem 1.1 (Theorem 5.9) . We have H p,qc ( X an ) = lim −→ ϕ : X → Y Σ H p,qc (Trop( ϕ ( X )) . where the limit runs over all closed embeddings of X into toric varieties. We also show that the analogous result for H p,q is not true (Remark 5.10).Further, in certain cases, we can check whether one of these classes is non-trivial onthe tropical side. Theorem 1.2 (Theorem 10.3) . Assume that
Trop( ϕ ( X )) is smooth. Then (1.1) and(1.2) are both injective. We exhibit three examples in Section 10, namely Mumford curve, curves of goodreduction and toric varieties.Another construction that we transport over from the tropical to the analytic worldis cohomology with coefficients other than the real numbers. For a subring R of R , wedefine a cohomology theory H ∗ , ∗ trop ( X an , R ) and H ∗ , ∗ trop ,c ( X an , R ) with values in R -modules. Liu introduced in [Liu17] a canonical rational subspace of H p,q ( X an ) . We show that this space agrees with H p,q ( X an , Q ) as defined in this paper(Proposition 9.3).We obtain the analogue of Theorem 1.1 where on the right hand side we have tropicalcohomology with coefficients in R (Proposition 8.4), and we provide an explicit isomor-phism dR : H p,q ( X an ) → H p,q trop ( X an , R ) . which is a version of de Rham’s theorem in this context (Theorem 8.12). Liu introducedin [Liu19] a monodromy operator M : H p,q trop ,c ( X an ) → H p − ,q +1trop ,c ( X an ) ROPICAL COHOMOLOGY WITH INTEGRAL COEFFICIENTS FOR ANALYTIC SPACES 3 that respects rational classes if log | K ∗ | ⊂ Q [Liu19, Theorem 5.5 (1)]. Mikhalkin andZharkov introduce in [MZ14] a wave operator W : H p,qc (Trop ϕ ( X ) , R ) → H p − ,q +1 c (Trop ϕ ( X an ) , R ) . Note that both these operators are also available without compact support.We show that W can be used to give an operator on H ∗ , ∗ c ( X an ) and that this operatoragrees with M up to sign in Corollary 9.1.We also obtain the following result regarding the interaction between the wave operatorand the coefficients of the cohomology groups: Theorem 1.3.
Let R be a subring of R and R [Γ] the smallest subring of R that containsboth R and Γ . Then the wave operator W restricts to a map W : H p,qc ( X an , R ) → H p − ,q +1 c ( X an , R [Γ]) . As W and M agree up to sign and Liu’s subspace of rational classes agrees with H ∗ , ∗ ( X an , Q ) , this generalizes Liu’s result for Γ ⊂ R = Q .We now sketch the organization of the paper. In Section 2 we recall backgroundon toric varieties and their tropicalizations. Section 3 contains all constructions ontropical varieties that are needed for the paper. Most of these should be known toexperts, however we still chose to list them for completeness. The main new result is theidentification of the wave and monodromy operator, which is based on Lemma 3.13. InSection 4 we consider what we call families of tropicalizations , which is what we will use todefine forms on Berkovich spaces. We give the definitions and some examples of familiesthat we will consider. In Section 5 we define for a fine enough family of tropicalizations S a bigraded complex A • , •S of sheaves of differential forms on Berkovich spaces. Wealso provide some conditions under which those complexes are isomorphic for different S and introduce tropical Dolbeault cohomology for X an . In Section 6, we prove that forso-called admissible families S , the complexes A • , •S are isomorphic to A • , • . In Section7 we discuss integration of top-dimensional differential forms with compact support. InSection 8 we introduce tropical cohomology with coefficients for X an and compare it with H ∗ , ∗ . In Section 9, we discuss the relation between the wave and monodromy operatorsand consequences thereof. Section 10 provides partial computations of H p,q ( X an ) and H p,q ( X an , R ) for curves and toric varieties, using our new approaches. In Section 11 welist open questions that one might ask as a consequence of our results. Acknowledgments
The author would like to thank Walter Gubler, Johannes Rau and Kristin Shaw forhelpful comments and suggestions. The idea for the proof of Theorem 3.13 came fromjoint work with Johannes Rau and Kristin Shaw on the paper [JRS18]. The author wouldlike to express his gratitude for being allowed to use those ideas.Parts of this work already appeared in a more a hoc and less conceptual way in theauthor’s PhD thesis [Jel16a].The author would also like to thank the referee for their detailed report and specificremarks which greatly improved the paper.
Notations and conventions
Throughout K is a field that is complete with respect to a (possibly trivial) non-archimedean absolute value. We denote its value group by Γ := log | K ∗ | . If the absolutevalue is non-trivial, we normalize it in such a way that Z ⊂ Γ . A variety X is angeometrically integral separated K -scheme of finite type. For any variety X over K , we P. JELL will throughout the paper denote by X an the analytification in the sense of Berkovich[Ber90]. 2. Toric varieties and tropicalization
Toric varieties.
Let N be a free abelian group of finite rank, M its dual and denoteby N R resp. M R the respective scalar extensions to R . Definition 2.1. A rational cone σ ∈ N R is a polyhedron defined by equations of theform ϕ ( . ) ≥ with ϕ ∈ M , that does not contain a positive dimensional linear subspace.A rational fan Σ in N R is a polyhedral complex all of whose polyhedra are rational cones.For σ ∈ Σ we define the monoid S σ := { ϕ ∈ M | ϕ ( v ) ≥ for all v ∈ σ } . We denote by U σ := Spec( K [ S σ ]) . For τ ≺ σ we obtain an open immersion U τ → U σ . Wedefine the toric variety Y Σ to be the gluing of the ( U σ ) σ ∈ Σ along these open immersions.For an introduction to toric varieties, see for example [Ful93]. Remark 2.2.
The toric variety Y Σ comes with an open immersion T → Y Σ , where T =Spec( K [ M ]) and a T -action that extends the group action of T on itself by translation. Infact any normal variety with such an immersion and action arises by the above describedprocedure ([CLS11, Corollary 3.1.8]). This was shown by Sumihiro.Choosing a basis of N gives an identification N ≃ Z r ≃ M and T ≃ G rm . Definition 2.3.
A map ψ : Y Σ → Y Σ ′ is called a morphism of toric varieties if it isequivariant with respect to the torus actions and restricts to a morphism of algebraicgroups on dense tori. It is called an affine map of toric varieties if it is a morphism oftoric varieties composed with a multiplicative torus translation. Remark 2.4.
A morphism of toric varieties ψ : Y Σ → Y Σ ′ is induced by a morphism ofcorresponding fans, meaning a linear map N → N ′ that maps cones in Σ to cones in Σ ′ .2.2. Tropical toric varieties.
Let Σ be a rational fan in N R . We write T := R ∪{−∞} .For σ ∈ Σ we define N ( σ ) := N R / h σ i R . We write N Σ = a σ ∈ Σ N ( σ ) . We call the N ( σ ) the strata of N Σ . Note that N Σ has a canonical action by N and N R and the strata are the strata of the action of N R . We endow N Σ with a topology in thefollowing way:For σ ∈ Σ write N σ = ` τ ≺ σ N ( τ ) . This is naturally identified with Hom
Monoids ( S σ , T ) .We equip T S σ with the product topology and give N σ the subspace topology. For τ ≺ σ ,the space Hom( S τ , T ) is naturally identified with the open subspace of Hom
Monoids ( S σ , T ) of maps that map τ ⊥ ∩ M to R . We define the topology of N Σ to be the one obtainedby gluing along these identifications. Definition 2.5.
We call the space N Σ a tropical toric variety .We would like to remark that the tropical toric variety N Σ can also be constructed byglueing its affine pieces along monomial maps [MR, Section 3.2]Note that N Σ contains N R as a dense open subset. For a subgroup Γ of R and eachstratum N ( σ ) we call the set N ( σ ) Γ := ( N ⊗ Γ) / h σ i Γ the set of Γ -points . ROPICAL COHOMOLOGY WITH INTEGRAL COEFFICIENTS FOR ANALYTIC SPACES 5
Let Σ and Σ ′ be fans in N R and N ′ R respectively. Let L : N → N ′ be a linear mapsuch that L R maps every cone in Σ into a cone in Σ ′ . Such a map canonically induces amap N Σ → N Σ ′ that is continuous and linear on each stratum. Definition 2.6.
A map N Σ → N Σ ′ that arises this way is called morphism of tropicaltoric varieties .An affine map of tropical toric varieties is a map that is the composition of morphismof toric varieties with an N R -translation.2.3. Tropicalization.
Let Σ be a rational fan in N R . Denote by Y Σ the associated toricvariety and by N Σ the associated tropical toric variety. Definition 2.7.
Payne defined in [Pay09] a tropicalization map trop Σ : Y anΣ → N Σ to the topological space N Σ as follows: For τ ≺ σ , the space Hom( S τ , T ) is naturallyidentified with the open subspace of Hom
Monoids ( S σ , T ) of maps which map τ ⊥ ∩ M to R . The map trop : U an σ → Trop( U σ ) is then defined by mapping | . | x ∈ U an σ to the homo-morphism u log | u | x ∈ Trop( U σ ) = Hom( S σ , T ) . We will often write Trop( Y Σ ) := N Σ .For Z a closed subvariety of Y Σ we define Trop( Z ) to be the image of Z an under trop : Y anΣ → Trop( Y Σ ) . Definition 2.8.
The construction
Trop( Y Σ ) is functorial with respect to affine maps oftoric varieties. In particular, for a morphism (resp. affine map) of toric varieties ψ : Y Σ → Y Σ ′ , we obtain a morphism (resp. affine map) of tropical toric varieties Trop( ψ ) : Trop( Y Σ ) → Trop( Y Σ ′ ) .If ψ is a closed immersion, then Trop( ψ ) is a homeomorphism onto its image. Example 2.9.
Affine space A r = Spec K [ T , . . . , T r ] is the toric variety that arises fromthe cone { x ∈ R r | x i ≥ for all i ∈ [ r ] } . By definition the tropicalization in then T r and the map trop : A r, an → T r ; | . | 7→ (log | T i | ) i ∈ [ r ] .Let σ be a cone in N R . We pick a finite generating set b , . . . , b r of the monoid S σ .Let U σ be the affine toric variety associated to a cone σ . Then we have a surjectivemap K [ T , . . . , T r ] → K [ S σ ] , which induces a toric closed embedding ϕ B : U σ → A r .By functoriality of tropicalization we also get a morphism of tropical toric varieties Trop( U σ ) → T r that is a homeomorphism onto its image.2.4. Tropical subvarieties of tropical toric varieties.
In this section we fix a sub-group Γ ⊂ R . Definition 2.10.
An integral Γ -affine polyhedron in N R is a set defined by finitely manyinequalities of the form ϕ ( . ) ≥ r for ϕ ∈ M, r ∈ Γ . An integral Γ -affine polyhedron on N Σ is the topological closure of an integral Γ -affine polyhedron in N ( σ τ ) for σ τ ∈ Σ .Let τ be an integral Γ -affine polyhedron in N Σ . For σ ≺ σ ′ we have that N ( σ ′ ) ∩ τ is apolyhedron in N ( σ ′ ) (that might be empty) and we consider this as a face of τ . Furtherwe denote by L ( τ ) = { λ ( u − u ) | u , u ∈ τ, λ ∈ R } ⊂ N ( σ τ ) the linear space of τ . If τ is integral Γ -affine, L ( τ ) contains a canonical lattice that we denote by Z ( e ) . Definition 2.11.
A tropical subvariety of a tropical toric variety is given the supportof a integral Γ -affine polyhedral complex with weights attached to its top dimensionalfaces, satisfying the balancing condition. Definition 2.12.
Let Z be a closed subvariety of a toric variety Y Σ . Then Trop( Z ) :=trop( Z an ) ⊂ N R is a tropical subvariety of Trop( Y Σ ) . For a variety X and a closed em-bedding ϕ : X → Y Σ we write Trop ϕ ( X ) := Trop( ϕ ( X )) and trop ϕ := trop ◦ ϕ an : X an → Trop ϕ ( X ) . P. JELL
We will never explicitly use the weights nor the balancing condition, so the readermay be happy with the fact that there are weights and that they satisfy the balancingcondition. If they are not happy with this, let us refer them to the excellent introduction[Gub13]. The reader who already knows the balancing condition for tropical varietiesin N R will be glad to hear that the balancing condition for X is exactly the classicalbalancing condition for X ∩ N R , there are no additional properties required at infinity.3. Constructions in Cohomology of tropical varieties
In this section, R is a ring such that Z ⊂ R ⊂ R and Γ is a subgroup of R that contains Z . Further N is a free abelian group of finite rank, M is its dual, and Σ is a rational fanin N R . Additionally X is an integral Γ -affine tropical subvariety of N Σ . Definition 3.1.
Let U ⊂ N R an open subset. A superform of bidegree ( p, q ) is anelement of A p,q ( U ) = C ∞ ( U ) ⊗ Λ p M ⊗ Λ q M Remark 3.2.
There are differential operators d ′ and d ′′ and a wedge product which areinduced by the usual differential operator and wedge product on differential forms. Definition 3.3.
For an open subset U of N Σ we write U σ := U ∩ N σ . A superform ofbidegree ( p, q ) on U is given by a collection α = ( α σ ) σ ∈ Σ such that α σ ∈ A p,q ( U σ ) andfor each σ and each x ∈ U σ there exists an open neighborhood U x of x in U such thatfor each τ ≺ σ we have π ∗ σ,τ α σ = α τ . We call this the condition of compatibility at theboundary .For a polyhedron σ in N Σ we can define the restriction of a superform α to σ . Let Ω be an open subset of | C | for a polyhedral complex C in N Σ . The space of superformsof bidegree ( p, q ) on Ω is defined as the set of pairs ( U, α ) where U is an open subsetof N Σ such that U ∩ | C | = Ω and α ∈ A p,q ( U ) . Two such pairs are identified if theirrestrictions to σ ∩ Ω agree for every σ ∈ C . Definition 3.4.
For a tropical subvariety X of a tropical toric variety we obtain adouble complex of sheaves ( A • , • , d ′ , d ′′ ) on X . We define H p,q ( X ) (resp. H p,qc ( X ) ) asthe cohomology of the complex of global sections (resp. global sections with compactsupport) with respect to d ′′ . Definition 3.5.
There exists an integration map R X : A n,nc ( X ) → R that satisfies Stokes’theorem and thus descends to cohomology. Remark 3.6.
Superforms on tropical subvarieties of tropical toric varieties are functorialwith respect to affine maps of tropical toric varieties [JSS19].
Definition 3.7.
Let X be a tropical variety and x ∈ X and denote by σ the cone of Σ such that x ∈ N ( σ ) . Then the tropical mutitangent space at x is defined to be F Rp ( τ ) = X σ ∈ X ∩ N ( σ ) ,x ∈ σ Λ p L ( σ ) ∩ Λ p R n ⊂ Λ p N ( σ ) . If ν is a face of τ , then there are transition maps ι τ,ν : F Rp ( τ ) → F Rp ( ν ) that are justinclusions if τ and ν live in the same stratum and compositions with projections tostrata otherwise.We denote by ∆ q the standard q -simplex. Definition 3.8.
A smooth stratified q -simplex is a map δ : ∆ q → X such that: ROPICAL COHOMOLOGY WITH INTEGRAL COEFFICIENTS FOR ANALYTIC SPACES 7 • If σ is a face of ∆ q then there exists a polyhedron τ in X such that ˚ σ is mappedinto ˚ τ i . • Let ∆ q = [0 , . . . , q ] . If δ ( i ) is contained in the closure of a stratum of N Σ , thenso is δ ( j ) for j ≤ i . • for each stratum X i of X the map δ : δ − ( X i ) → X i is C ∞ .We denote the free abelian group of smooth stratified q -simplices δ satisfying δ (˚∆ q ) ⊂ ˚ τ by C q ( τ ) .There is a boundary operator ∂ p,q : C p,q ( X, R ) → C p,q − ( X, R ) that is given by theusual boundary operator on the simplex side and by the maps ι τ,ν on the coefficient side,when necessary. Dually we have ∂ p,q : C p,q ( X, R ) → C p,q +1 ( X, R ) . Definition 3.9.
The groups of smooth tropical ( p, q ) -cell and cocells are respectively C p,q ( X, R ) := M τ ⊂ X F Rp ( τ ) ⊗ C q ( τ ) C p,q ( X, R ) := Hom R ( C p,q ( X, R ) , R ) We denote by C p,q ( R ) the sheafification of C p,q ( X, R ) as defined in [JSS19, Definition3.13] and by F pR := ker( ∂ : C p, ( R ) → C p, ( R )) . Note that we have F pR = F p Z ⊗ R . Definition 3.10.
We denote by H p,q trop ( X ) := H q ( C p, • ( X, R ) , ∂ ) = H q (C p, • ( X, R ) , ∂ ) andcall this the tropical cohomology of X with coefficients in R . Similarly we define tropicalcohomology of X with coefficients in R with compact support. It was shown in [JSS19] that the morphisms of complexes F p R → A p, • and F p R → C p, • , that are given by inclusion in degree are in fact quasi isomorphisms. We now want toconstruct a de Rham morphism, meaning a quasi isomorphism dR : A p, • → C p, • that iscompatible with the respective inclusions of F p R . Remark 3.11.
Let v ⊗ δ be a smooth tropical ( p, q ) -cell on an open subset Ω of X .Then we define for a ( p, q ) form α ∈ A p,q (Ω) Z v ⊗ δ α = Z ∆ q δ − h α ; v i We have to argue that this integral is well defined, since δ might map parts of ∆ q toinfinity: Let X be the stratum of X to which the barycenter of ∆ q is mapped. Let ∆ q, := δ − ( X ) . Then we have supp( δ ∗ α ) ⊂ ∆ q, by the condition of compatibility atthe boundary for α , hence the integral is finite.This defines a morphism dR : A p,q (Ω) → C p,q (Ω) α (cid:18) v ⊗ δ Z v ⊗ δ α (cid:19) and one directly verifies using the classical Stokes’ theorem that this indeed induces amorphism of complexes dR : A p, • → C p, • . that respects the respective inclusions of F p R . Since both C p, • and A p, • form acyclicresolutions of F p R , the map dR is a quasi-isomorphism. P. JELL
Definition 3.12.
The monodromy operator is the unique A , -linear map such that M : A p,q → A p − ,q +1 ; d ′ x I ∧ d ′′ x J p X k =1 ( − p − k d ′ x I \ i k ∧ d ′′ x i k ∧ d ′′ x J . The wave operator W : C p,q ( R ) → C p − ,q +1 ( R ) is the sheafified version of the map dualto C p − ,q +1 ( X, R ) → C p,q ( X, R ); v ⊗ δ ( v ∧ ( ι ( δ (1)) − δ (0))) ⊗ δ | [1 ,...,q +1] . Lemma 3.13.
The diagram A p − , (cid:15) (cid:15) F p R ( − p − M ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ W ( ( PPPPPPPPPPPPPP C p − , ( R ) commutes.Proof. By the definition of the de Rham map we have to show that Z [0 , δ ∗ h M ( α ) , v i = ( − p − h α, W ( e ⊗ v ) i where δ : [0 , → X is a smooth stratified -simplex and v ∈ F p − ( τ ) , where δ ((0 , ⊂ ˚ τ and α ∈ F p . After picking bases and using multilinearity for both v and α we may assumethat α = d ′ x ∧ · · · ∧ d ′ x p and v = x ∧ · · · ∧ x p − . Then we have Z [0 , δ ∗ h M ( α ) , v i = p X i =1 ( − p − i Z [0 , δ ∗ h d ′ x ∧ . . . d d ′ x i ∧ · · · ∧ d ′ x p , x ∧ · · · ∧ x p − i ∧ d ′′ x i = Z [0 , δ ∗ h d ′ x ∧ · · · ∧ d ′′ x p − , x ∧ · · · ∧ x p − i ∧ d ′′ x p = Z [0 , δ ∗ d ′′ x p = dx p ( δ (1)) − dx p ( δ (0)) and h d ′ x ∧ · · · ∧ d ′ x p , δ (1) − δ (0) ∧ x ∧ · · · ∧ x p − i = ( − p − dx p ( δ (1)) − dx p ( δ (0)) , where dx p denotes the p -th coordinate functions with respect to x , . . . , x n . This calcu-lation holds true as long as the p -th coordinate functions is bounded on δ ([0 , . If thisis not the case however, then d ′′ x p vanishes in a neighborhood of δ (0) (resp. δ (1) ) by thecompatibility condition and we may replace [0 , by [ ε, resp. [ ε, − ε ] resp. [0 , − ε ] . (cid:3) ROPICAL COHOMOLOGY WITH INTEGRAL COEFFICIENTS FOR ANALYTIC SPACES 9
Theorem 3.14.
The wave and the monodromy operator agree on cohomology up to signby virtue of the isomorphism dR , meaning that the diagram H p,q ( X ) ( − p − M / / dR (cid:15) (cid:15) H p − ,q +1 ( X ) dR (cid:15) (cid:15) H p,q trop ( X, R ) W / / H p − ,q +1trop ( X, R ) commutes. The same is true for cohomology with compact support.Proof. The wave and monodromy operators give morphisms of complexes is a morphismof complexes W : C p, • ( R ) → C p − , • ( R )[1] and M : A p, • → A p − , • [1] , hence it is sufficient to show that A p, • dR (cid:15) (cid:15) ( − p − M / / A p − , • [1] dR (cid:15) (cid:15) C p, • ( R ) W / / C p − , • ( R )[1] commutes in the derived category. Replacing both A p, • and C p, • with the quasi-isomorphic F p R , we have to show that A p − , (cid:15) (cid:15) F p R ( − p − M ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ W ( ( PPPPPPPPPPPPPP C p − , ( R ) commutes. This follows directly from Theorem 3.13. (cid:3) Proposition 3.15.
The wave operator descends to an operator on cohomology W : H p,q trop ( X, R ) → H p − ,q +1trop ( X, R [Γ]) and W : H p,q trop ,c ( X, R ) → H p − ,q +1trop ,c ( X, R [Γ]) , where R [Γ] is the smallest subring of R that contains both R and Γ .Proof. We pick a triangulation of X with smooth stratified simplices such that all ver-tices are Γ -points. We can now compute H p,qc ( X, R ) and the wave homomorphism us-ing this triangulation by [MZ14, Section 2.2]. For a ( p, q ) -chain δ with respect to thistriangulation and with coefficients in R , we now have that W ( δ ) has coefficients in R [Γ] . Hence W restricts to a map H p,q ( X, R ) → H p − ,q +1 ( X, R [Γ]) resp. H p,qc ( X, R ) → H p − ,q +1 c ( X, R [Γ]) . (cid:3) Definition 3.16.
We denote by ∩ [ X ] R : C n,nc ( X, R ) → R the evaluation against the fundamental class , as defined in [JRS18, Definition 4.8] andalso the induced map on cohomology H n,n trop ,c ( X, R ) → R . Proposition 3.17.
The following diagram commutes A n.nc ( X ) R X / / dR (cid:15) (cid:15) R C n,nc ( X, R ) . ∩ [ X ] ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ Proof.
This is a straightforward calculation using the definitions. (cid:3) Families of tropicalizations
In this section, K is a complete non-archimedean field and X is a K -variety.4.1. Definitions.
The philosophy throughout the paper will be that we can approxi-mate non-archimedean analytic spaces through embedding them into toric varieties andtropicalizing. We will define families that approximate the analytic space well enough(fine enough families) as well as notions that tell us that two families basically containthe same amount of information (final and cofinal families).
Definition 4.1. A family of tropicalizations S of X consists of the following data:i) A class S map containing closed embeddings ϕ : U → Y Σ for open subsets U of X and toric varieties Y Σ .ii) For an element ϕ : U → Y Σ of S map a subclass S ϕ of S map that contains maps ϕ ′ : U ′ → Y Σ ′ for open subsets U ′ ⊂ U and such that there exists an affine mapof toric varieties ψ ϕ,ϕ ′ , such that U ′ ι (cid:15) (cid:15) ϕ ′ / / Y Σ ′ ψ ϕ,ϕ ′ (cid:15) (cid:15) U ϕ / / Y Σ commutes. Such a ϕ ′ is called refinement of ϕ . The map ψ ϕ,ϕ ′ induces an affinemap of toric varieties Trop( ψ ϕ,ϕ ′ ) . The restriction of Trop( ψ ϕ,ϕ ′ ) to Trop ϕ ′ ( U ′ ) depends only on ϕ and ϕ ′ , so we denote this map by Trop( ϕ, ϕ ′ ) : Trop ϕ ′ ( U ′ ) → Trop ϕ ( U ) .We further require that if ϕ ′ is a refinement of ϕ and ϕ ′′ is a refinement of ϕ ′ ,then ϕ ′′ be a refinement of ϕ .A subfamily of tropicalizations of S is a family of tropicalizations S ′ such that all em-beddings and refinements in S are also in S ′ . Further S ′ is said to be a full subfamily ifwhenever ϕ, ϕ ′ ∈ S ′ map and ϕ ′ is a refinement of ϕ in S , it is also a refinement of ϕ in S ′ .Foster, Gross and Payne study in [FGP14] so-called “Systems of toric embeddings”.They only consider the case where all of X is embedded into the toric variety, a case welater call global families of tropicalizations. Definition 4.2.
Let S be a family of tropicalizations on X . An S -tropical chart is givenby a pair ( V, ϕ ) where ϕ : U → Y Σ ∈ S map and V = trop − ϕ (Ω) is an open subset of X an which is the preimage of an open subset Ω of Trop ϕ ( U ) .Another S tropical chart ( V ′ , ϕ ′ ) is called an S -tropical subchart of ( V, ϕ ) if ϕ ′ is arefinement of ϕ and V ′ ⊂ V .Note that we have Ω = trop ϕ ( V ) . ROPICAL COHOMOLOGY WITH INTEGRAL COEFFICIENTS FOR ANALYTIC SPACES 11
Example 4.3.
The family of all tropicalizations is a family of tropicalizations in thesense of Definition 4.1. We denote it by T .In the following definition, we define the terms final and cofinal for two families oftropicalizations S and S ′ . While the definitions are a bit on the technical side, theidea is that if S ′ is either final or cofinal for S , then S -tropical charts provide the sameinformation as S ′ -tropical charts. Definition 4.4.
Let S and S ′ be families of tropicalizations. We say S ′ is cofinal for S if for every embedding ϕ : U → Y Σ in S map and every x ∈ U an there exists ϕ ′ : U ′ → Y Σ ′ in S ′ map with x ∈ U ′ an such that ϕ ′ | U ′ ∩ U restricts to a closed embedding of U ∩ U ′ intoan open torus invariant subvariety of Y Σ ′ , and that embedding is a refinement of ϕ in T . S ′ is said to be final for S for every ϕ : U → Y Σ in S map and x ∈ U an , there exists arefinement ϕ ′ : U ′ → Y Σ ′ with x ∈ U ′ an , a closed embedding m : Y Σ ′ → Y Σ ′′ , that is anaffine map of toric varieties, such that m ◦ ϕ ′ is in S ′ map and ϕ ′ is a refinement of m ◦ ϕ ′ via m in S . Definition 4.5.
A family of tropicalizations is called fine enough if the sets V such thatthere exist S -tropical charts ( V, ϕ ) form a basis of the topology of X an and for each pairof S -tropical charts ( V , ϕ ) and ( V , ϕ ) there exists S -tropical charts ( V i , ϕ i ) i ∈ I , whichare S -tropical subcharts of both ( V , ϕ ) and ( V , ϕ ) , such that V ∩ V = ∪ V i . Lemma 4.6.
Let S ′ be a full subfamily of S and assume that S is fine enough. If S ′ isfinal or cofinal in S , then S ′ is also fine enough.Proof. Let x ∈ X an . Since S is fine enough and S ′ is a full subfamily, it is sufficient toprove that given an S -tropical chart ( V, ϕ ) with x ∈ V , there exists an S -tropical chart ( V ′ , ϕ ′ ) with x ∈ V ′ that is a S -tropical subchart of ( V, ϕ ) .If S ′ is cofinal in S , then we pick ϕ ′ as in Definition 4.4. Then ( V ∩ U ′ an , ϕ ′ ) is atropical chart.If S ′ is final, then ( V ∩ U ′ an , m ◦ ϕ ′ ) is a tropical chart. (cid:3) Examples.
In the following we will give examples of families of tropicalizations fora variety X . We will always specify the class S map and for ϕ ∈ S map simply define S ϕ to be those ϕ ′ where an affine map of toric varieties ψ ϕ,ϕ ′ as required in Definition 4.1 iii ) exists. The exception to this rule are Example 4.15, where we require the map ψ ϕ,ϕ ′ to be a coordinate projection in order for ϕ ′ to be a refinement of ϕ and Example 4.9. Example 4.7.
The family A is the family whose embeddings are closed embeddings ofaffine open subsets of X into affine space. This family is fine enough by the definitionof the topology of X an .Suppose we are given an embedding ϕ : U → Y Σ of an open subset of X into a toricvariety with x ∈ U an . Let Y σ be an open affine toric subvariety of Y Σ such that Y an σ ′ contains ϕ an ( x ) . Let ϕ ′ := ϕ | ϕ − ( Y σ ) . Now we pick a toric embedding of m : Y σ → A n asin Example 2.9. This shows that A is final in T . Example 4.8.
The family G is the family whose embeddings are closed embeddings ofvery affine open subsets of X into G nm .This family is also fine enough if the base field is non-trivially valued [Gub16, Propo-sition 4.16], but not when K is trivially valued [Jel16a, Example 3.3.1]. Example 4.9.
Assume that K is algebraically closed. Let X be a variety and U a veryaffine open subset. Then M = O ∗ ( U ) /K ∗ is a free abelian group of finite rank and thecanonical map K [ M ] → O ( U ) induces a closed embedding ϕ U : U ֒ → T for a torus T with character lattice M . The embedding ϕ U is called the canonical moment map of U . Wedenote by G can the family of tropicalizations where G can , map = { ϕ U | U ⊂ X very affine } and refinements being the maps induced by inclusions.It is easy to see that this family is cofinal in G . Definition 4.10.
A family of tropicalizations S for a variety X is called global if all ϕ ∈ S map are defined on all of X .Global families of tropicalizations will play a special role, as they will allow us toconstruct classes in tropical Dolbeault cohomology. Definition 4.11.
We say that X satisfies condition ( † ) if X is normal and every twopoints in X have a common affine neighborhood.By Włodarczyk’s Embedding Theorem, Condition ( † ) is equivalent to X being normaland admitting a closed embedding into a toric variety (cf. [Wło93]). Observe also thatit is satisfied by any quasi-projective normal variety. It is however weaker then beingnormal and quasi-projective, as there exist proper toric varieties which are not projective. Example 4.12.
Let S be a global family of tropicalizations such that X an = lim ←− ϕ ∈S map Trop ϕ ( X ) (4.1)and such that if ϕ : X → Y Σ and ϕ : X → Y Σ in S map then also ϕ × ϕ : X → Y Σ × Y Σ ∈ S map . Then S is fine enough. In fact it follows directly from (4.1) that S -tropical charts form a basis of the topology and from the product property that wecan always locally find common subcharts.Families with the properties from Example 4.12 were extensively studied in [FGP14].When X satisfies condition ( † ) , this helps us say even more: Example 4.13.
Assume that X satisfies condition ( † ) . Then we may consider the familyof tropicalizations T global , where T global , map is the class of all embeddings of X into toricvarieties.Let ϕ : U → A n be a closed embedding of an open subset U of X into an affinespace given by regular functions f , . . . , f n on U . Then by [FGP14, Theorem 4.2] thereexists an embedding ϕ : X → Y Σ such that U is the preimage of an open affine invariantsubvariety U σ and each f i is the pullback of a character that is regular on U σ . Thisexactly shows that ϕ | U is a refinement of ϕ in T , which shows that T global is cofinal in A . Example 4.14.
Let X be an affine variety and A global the family of tropicalizationswhose class of embeddings are embeddings of all of X into affine spaces.We show that this family is cofinal in A : Let U be an open subset of X and ϕ : U → A n an closed embedding given by f /g , . . . , f n /g n , where f i , g i are regular functionson X . In particular, U = D ( g , . . . , g n ) . We pick regular functions h , . . . , h k on X such that the f i , g i , h i generate O ( X ) . We consider the embedding ϕ ′ : X → A n × A k given by f , . . . , f n , g , . . . , g n , h , . . . , h k . Then ϕ ′ | U gives a closed embedding of U into A n × G nm × A k , which is clearly a refinement of ϕ in T . Example 4.15.
Assume that K is algebraically closed. Wanner and the author con-sidered in [JW18] the class of linear tropical charts for X = A . In the language of thepresent paper, they use the following family of tropicalizations, which we denote by A lin . A lin , map is the set of linear embeddings i.e. those embeddings ϕ : A → A r , where the ROPICAL COHOMOLOGY WITH INTEGRAL COEFFICIENTS FOR ANALYTIC SPACES 13 corresponding algebra homomorphism K [ T , . . . , T r ] → K [ X ] is given by mapping T i to ( X − a i ) for a , . . . , a r ∈ K . A refinement of ϕ is then a map ϕ ′ : A → A s given by ( X − b , . . . , X − b s ) where s > r and { a i } ⊂ { b j } . This family of tropicalizations isfine enough and global, for details see [JW18, Section 3.2]. By factoring the definingpolynomials of any map A → A n into linear factors, it follows that A lin is cofinal in A global . Example 4.16.
Let K be algebraically closed and X be a smooth projective Mum-ford curve. Let T Smooth be the class of embeddings of X into toric varieties such that Trop ϕ ( X ) is a smooth tropical curve. Then T Smooth is cofinal in T global by [Jel18, The-orem A]. 5. Differential forms on Berkovich spaces
In this section, K is a complete non-archimedean field and X is a variety over K .Further S is a fine enough family of tropicalizations.5.1. Sheaves of differential forms.
In this section, we define a sheaf of differentialforms A p,q S with respect to S . We will also show that for final and cofinal families, thesesheaves are isomorphic. We use the sheaves A p,q of differential forms on tropical varietieswhich are recalled in Section 3. Definition 5.1.
Let S be a fine enough family of tropicalizations. For V an open subsetof X an . An element α ∈ A p,q S ( V ) is given by a family of triples ( V i , ϕ i , α i ) i ∈ I , wherei) The V i cover V , i.e. V = S i ∈ I V i .ii) For each i ∈ I the pair ( V i , ϕ i ) is an S -tropical chart.iii) For each i ∈ I we have α i ∈ A p,q (trop ϕ ( V )) .iv) For all i, j ∈ I there exist S -tropical subcharts ( V ijl , ϕ ijl ) l ∈ L that cover V i ∩ V j such that Trop( ϕ i , ϕ ijl ) ∗ α i = Trop( ϕ j , ϕ ijl ) ∗ α j ∈ A p,q (trop ϕ ijl ( V ijl )) . Another such family ( V j , ϕ j , α j ) j ∈ J defines the same form α if their union ( V i , ϕ i , α i ) i ∈ I ∪ J still satisfies iv).For an open subset W of V we can cover W by S -tropical subcharts ( V ij , ϕ ij ) of the ( V i , ϕ i ) . Then we define α | W ∈ A p,q S ( W ) to be defined by ( V ij , ϕ ij , Trop( ϕ i , ϕ ij ) ∗ ( α i )) .The differentials d ′ and d ′′ are well defined on A p,q S and thus we obtain a complex ( A • , •S , d ′ , d ′′ ) of differential forms on X an . Lemma 5.2.
Let S be a fine enough family of tropicalization. Let α ∈ A p,q S ( V ) be givenby a single S -tropical chart ( V, ϕ, α ′ ) . Then α = 0 if and only if α ′ = 0 .Proof. This works the same as the proof in [CLD12, Lemme 3.2.2]. (cid:3)
Lemma 5.3.
Let S be a fine enough family of tropicalization. Let S ′ be a fine enoughsubfamily. Then there exists a unique morphism of sheaves Ψ S ′ , S : A S ′ → A S , such that the image of a form given by a triple ( V, ϕ, β ) is given by the same triple. Thismorphism is injective. Furthermore if S ′ is final or cofinal, then this morphism is anisomorphism. Recall that if S ′ is a full subfamily that is either final or cofinal in the fine enoughfamily S , then S ′ is itself fine enough by Lemma 4.6. Thus Lemma 5.3 implies that Ψ S ′ , S is an isomorphism. Proof.
Injectivity follows from Lemma 5.2.Assume that S ′ is cofinal in S and that we are given a form α given by ( V, ϕ, β ) .Fixing x ∈ V and picking ϕ ′ as in Definition 4.4, we define a form α ′ to be given by ( V ∩ U ′ an , ϕ ′ , Trop( ψ ϕ ′ ,ϕ ) ∗ β ) . Since ϕ ′ was a refinement of ϕ in T , we have Ψ S , T ( α ) = Ψ S ′ , T ( α ′ ) = Ψ S , T (Ψ S ′ , S ( α ′ )) which proves Ψ( S ′ , S )( α ′ ) = α locally at x by injectivity of Ψ S , T .Assume that S ′ is final in S and we are given a form α given by a tuple ( V, ϕ, β ) . Wepick m as in Definition 4.4 and define α ′ to be given by ( ϕ ◦ m , V, β ) . Note here that m induces an isomorphism of the tropicalizations of V , thus “pushing β forward along ϕ ”is possible. Then Ψ S ′ , S ( α ′ ) = α since ϕ is a refinement of ϕ ◦ m via m . (cid:3) Definition 5.4.
Let S be a fine enough family of tropicalizations. We say that S is admissible if Ψ S, T is an isomorphism. Corollary 5.5.
We have an isomorphism A A ∼ = A T . If X is affine, these are also isomorphic to A A global and if X satisfies condition ( † ) , thenthese are also isomorphic to A T global . Theorem 5.6.
Let S be a fine enough global family of tropicalizations. Let α ∈ A p,q ( V ) be given by a finite family ( V i , ϕ i , α i ) , where ϕ i : X → Y Σ i in S map . Then α can bedefined by one S -tropical chart.Proof. Since S is fine enough and global, there exists a common refinement ϕ : X → Y Σ for all the ϕ i . Then ( V i , ϕ ) is an S -tropical subchart of ( V i , ϕ i ) for all i . Denote by α ′ i := Trop( ϕ i , ϕ ) ∗ α i and Ω i := trop ϕ ( V i ) . Then α | V i is given by both ( V i ∩ V j , ϕ, α ′ i | Ω i ∩ Ω j ) and ( V ′ i ∩ V ′ j , ϕ, α ′ j | Ω i ∩ Ω j ) . By Lemma 5.2, the forms α ′ i and α ′ j agree on Ω i ∩ Ω j , thusglue to give a form α ′ ∈ A p,q (trop ϕ ( V )) . The form α ∈ A p,q ( V ) is then defined by ( V, ϕ, α ′ ) . (cid:3) Let S be a global admissible family of tropicalizations. Let ϕ : X → Y Σ be a closedembedding in S map . We define a map trop ∗ : A p,q (Trop ϕ ( X )) → A p,q S ( X an ) by settingfor β ∈ A p,q (Trop ϕ ( X )) the image trop ∗ β to be the form given by the triple ( ϕ, X an , β ) .One immediately checks that this is well defined. We define this similarly for forms withcompact support. Theorem 5.7.
Let S be a fine enough global family of tropicalizations. Let V be an opensubset of X an such that there exists an S -tropical chart ( V, ϕ ) . Then pullbacks along thetropicalization maps induce an isomorphism lim −→ A p,qc (trop ϕ ( V )) → A p,q S ,c ( V ) where the limit runs over all S -tropical charts ( V, ϕ ) .Proof. For any S -tropical chart ( V, ϕ ) , the pullback along the proper map trop ϕ induces awell defined morphism A p,qc (trop ϕ ( V )) → A p,q S ,c ( V ) . By definition this map is compatiblewith pullback between charts. Thus the universal property of the direct limit leads toa morphism Ψ : lim −→ A p,q S ,c (trop ϕ ( V )) → A p,qc ( V ) , where the limit runs over all S -tropicalcharts of V .This map is injective by construction. For surjectivity, let α ∈ A p,qc ( V ) be given by ( V i , ϕ i , α i ) i ∈ I . Let I ′ be a finite subset of I such that the V i with i ∈ I ′ cover the supportof α . Then ( V i , ϕ i , α i ) i ∈ I ′ defines α | V ′ ∈ A p,q S ,c ( V ′ ) , where V ′ = S i ∈ I ′ V i . Since S is global ROPICAL COHOMOLOGY WITH INTEGRAL COEFFICIENTS FOR ANALYTIC SPACES 15 and fine enough, the conditions of Theorem 5.6 are satisfied and α | V ′ can be defined bya triple ( V ′ , ϕ ′ , α ′ ) . By passing to a common refinement with ( V, ϕ ) we may assume that ( V ′ , ϕ ′ ) is a subchart of ( V, ϕ ) .It follows from Lemma 5.2 that supp( α ′ ) = ϕ ′ trop (supp( α )) (cf. [CLD12, Corollaire3.2.3]). Thus supp( α ′ ) is compact. We extend α ′ by zero to ˜ α ′ ∈ A p,q Trop ϕ ′ ( X ) ,c (trop ϕ ′ ( V )) .Then α is defined by ( V, ϕ ′ , ˜ α ′ ) , which is in the image of Ψ . (cid:3) Tropical Dolbeault cohomology.
In this section we assume that S is a admissi-ble family of tropicalizations and often just write A p,q for A p,q S . Definition 5.8.
We define tropical Dolbeault cohomology to be the cohomology of thecomplex ( A p, • ( X an ) , d ′′ ) , i.e. H p,q ( X an ) := ker( d ′′ : A p,q ( X an ) → A p,q +1 ( X an )im( d ′′ : A p,q − ( X an ) → A p,q ( X an )) . Similarly we define cohomology with compact support H p,qc ( X an ) as the cohomology offorms with compact support. Theorem 5.9.
Let S be an admissible global family of tropicalizations. Then pullbacksalong tropicalization maps induce an isomorphism lim −→ ϕ ∈S H p,qc (Trop ϕ ( X )) → H p,qc ( X an ) . Proof.
The map is induced by trop ∗ . The theorem follows from the fact that takingcohomology commutes with forming direct limits and Theorem 5.7. (cid:3) Remark 5.10.
The corresponding statement for H p,q ( X an ) fails. We sketch the ar-gument. If X is an affine variety, then we may pick S = A global , the class of closedembeddings into affine space.One can show that H n,n ( Y ) = 0 for all tropical subvarieties of T n = Trop( A n ) . Thuswe have lim −→ ϕ ∈S H p,q (Trop ϕ ( X )) = 0 .Let K be algebraically closed, E be an elliptic curve of good reduction and let e be arational point of E . Let V be an open neighborhood of e that is isomorphic to an openannulus and let X = E \ e .Using the Mayer-Vietoris sequence for the cover ( X an , V ) for E an gives the followingexact sequence H , ( V \ e ) → H , ( E an ) → H , ( X an ) ⊕ H , ( V ) . It was shown in [JW18, Theorem 5.7] that both H , ( V \ e ) and H , ( V ) are finitedimensional. Further it was shown in [Jel19, Theorem B] that when the residue field of K is C , then H , ( E an ) is infinite dimensional. This implies that H , ( X an ) is infinitedimensional. In particular, it is not equal to lim −→ ϕ ∈S H p,q (Trop ϕ ( X )) = 0 .6. Comparison theorems
Comparing with Gubler’s definition.
Since Gubler’s definition only works when K is non-trivially valued and algebraically closed we assume for this subsection that thisis the case. Theorem 6.1.
The family of tropicalizations G from Example 4.8 is admissible. This theorem is not a formal consequence of the definition, since the family G doesnot see any boundary considerations. Since for algebraically closed K , Gubler defined in[Gub16] the sheaf A like we here defined A G can this theorem is crucial for us as it provesthat what we show in later sections actually applies to A . Proof.
We have to show that the map
Ψ := Ψ G , T : A G → A T is surjective. We arguelocally around a point x ∈ X an . Let α ∈ A T be locally given by ( V, ϕ, β ) , where ϕ : U → Y Σ is a closed embedding of an open subset U of X into a toric variety. Wefix coordinates on the torus stratum T of Y Σ that x is mapped to under ϕ an , identifying T with G nm for some n and we denote by Z := ϕ − ( T ) . Since very affine open subsetform a basis of the topology of X , there exists a closed embedding ϕ ′ : U ′ → G rm for x ∈ U ′ an ⊂ U an such that if π : G rm → G nm is the projection to the first n factors, then ϕ | Z = π ◦ ϕ ′ ◦ ι Z .After shrinking V we may assume that trop ϕ ( V ) is a neighborhood of trop ϕ ( x ) , onwhich β = π ∗ ( β ) holds, where π is the projection to the stratum Trop( T ) of Trop( Y ) .We define the form α ′ ∈ A G to be given by ( V ′ , ϕ ′ , Trop( ψ ) ∗ β ) . Since Ψ( α ′ ) is by defi-nition given by the same triple, we have to show that ( V, ϕ, β ) and ( V ′ , ϕ ′ , Trop( ψ ) ∗ β ) define the same form in a neighborhood of x . We do that by pulling back to a commonsubcharts, namely ϕ × ϕ ′ : U ′ → Y Σ × G rm . When pulling back on the tropical side wefind that both Trop( π ) ∗ β as well as Trop( π ) ∗ β are simply the pullback of β , hencethese forms agree indeed. (cid:3) Corollary 6.2.
The family of tropicalizations G can is admissible.Proof. This follows from Lemma 5.3, the fact that G can is cofinal in G and Theorem6.1. (cid:3) Comparing with the analytic definition by Chambert–Loir and Ducros.
We denote by A p,q an the analytically defined sheaf of differential forms by Chambert-Loirand Ducros [CLD12]. Definition 6.3.
We define a map Ψ an : A p,q A → A p,q an . Let ( V, ϕ, α ) an A -tropical chart, where ϕ is given by f , . . . , f r . Let f i , . . . , f i k be those f i that do not vanish at X . Then those induce a map ϕ ′ : V → G km and trop ϕ ′ ( V ) =trop ϕ ( V ) ∩{ z i j = −∞ for all j = 1 , . . . , k } , which is a stratum of trop ϕ ( V ) and denote by α I the restriction of α to that stratum.. Then we define Ψ( α ) to be given by ( V, ϕ G m , α I ) . Theorem 6.4.
The map Ψ an is an isomorphism.Proof. We only have to prove the statement on stalks, so we fix x ∈ X an . By constructionany form is around x given by one A -tropical chart. Then Ψ an ( α ) is also given around x by one chart. If now Ψ an ( α ) = 0 , then α I equals zero by [CLD12, Lemme 3.2.2] andthus α equals zero. This shows injectivity.To show surjectivity, we take a form α that is given locally around x by a map V → G r, an m defined by f , . . . , f r and a form α on Trop f ,...,f r ( V ) . The f r can beexpressed as Laurent series and around x we may cut them of in sufficiently high degreeto obtain Laurent polynomials f ′ . . . f ′ r such that | f i | = | f ′ i | for all i on a neighborhood V of x . Now choosing functions g , . . . , g s ∈ O X ( U ) for an open subset U of X suchthat x ∈ U an , the f ′ i and g i define a closed embedding ϕ : U → A r + s with the propertythat there exists a neighborhood V of x in V and a tropical chart ( V , ϕ ) . We mayassume that g , . . . , g t are non vanishing at x and g t +1 , . . . , g s are. Now α | V is defined ROPICAL COHOMOLOGY WITH INTEGRAL COEFFICIENTS FOR ANALYTIC SPACES 17 by a form α on Trop f ′ ,...,f ′ r ,g ,...,g t ( V ) . By construction we have Trop f ′ ,...,f ′ r ,g ,...,g t ( V ) is the stratum of Trop ϕ ( V ) , where the coordinates t + 1 -th to s -th coordinate are −∞ .Denote by π : Trop ϕ ( V ) → Trop f ′ ,...,f ′ r ,g ,...,g t ( V ) the projection Let α = π ∗ α . Thenwe define β ∈ A p,q A ( V ) by ( V , ϕ, α ) . Now Ψ an ( β ) is given by V → G r + tm given by f ′ , . . . , f ′ r , g , . . . , g t and α . Since trop f ,...,f r ,g ,...,g t = trop f ′ ,...,f ′ r ,g ,...,g t : V → R r + t , theresult follows from [CLD12, Lemme 3.1.10]. (cid:3) Lemma 6.5.
Let K be non-trivially valued and α ∈ A p,q G ( V ) given by ( V, ϕ, α ′ ) . Then Ψ an ◦ Ψ − A , T ◦ Ψ G , T ( α ) is given by ( V, ϕ an , α ′ ) .Proof. Let ϕ : U → G rm be given by invertible functions f , . . . , f r . Then the corre-sponding embedding into A r , which we denote by ϕ ± is given by f , f − , . . . , f r , f − r .Denote by π : R r → R r the projection to the odd coordinates. Then by construction Ψ an ◦ Ψ G m ( α ) is given by ( V, ϕ an ± , π ∗ ( α ′ )) . Since ϕ an ± is a refinement of ϕ an and the mapinduced on tropicalizations is precisely π , we find that ( V, ϕ an ± , π ∗ ( α )) = ( V, ϕ an , α ′ ) ∈A p,q an ( V ) , which proves the claim. (cid:3) Integration
In this section we denote by n the dimension of X and we let S be a fine enoughfamily of tropicalizations. Definition 7.1.
Let α ∈ A n,n S ,c ( X an ) be an ( n, n ) -form with compact support. A tropicalchart of integration for α is a S -tropical chart ( U an , ϕ ) where U is an open subset of X and ϕ : U → Y Σ is a closed embedding of U into a toric variety, such that α | U an is givenby ( U an , ϕ, β ) for β U ∈ A n,nc (Trop ϕ ( U )) . Lemma 7.2.
There always exists an A -tropical chart of integration. If K is non-triviallyvalued and algebraically closed, there always exist G -tropical charts of integration. If S is global, then there always exist S -tropical charts of integration.Proof. The existence of A -tropical charts of integration is proved in [Jel16a, Lemma3.2.57] and the existence of G -tropical charts of integration is proved in [Gub16, Propo-sition 5.13]. Note that [Jel16a] assumes that K is algebraically closed, but the proof goesthrough here.In the case of global S , let α be given by ( V i , ϕ i , α i ) i ∈ I . Then we may pick finitelymany i such that supp( α ) is covered by the V i and apply Theorem 5.6 to obtain an S -tropical chart of integration. (cid:3) Theorem 7.3.
Let α ∈ A n,n S ,c ( X an ) and ( U an , ϕ, α U ) an S -tropical chart of integrationfor α . Then the value Z S X an α := Z Trop ϕ ( U ) α U depends only on α , in the sense that it is independent of the triple ( U, ϕ, α U ) representing α . Further it is also independent of S , in the sense that Z S X an α = Z T X an Ψ S , T ( α ) . Proof.
Assume first that K is non-trivially valued and algebraically closed. Then Gublershowed in [Gub16, Lemma 5.15] that as a consequence of the Sturmfels-Tevelev-formula[ST08, BPR16] the first part of the statement holds for S = G . Let ϕ : U → Y Σ . We may assume that ϕ ( U ) meets the dense torus T . Denote by ˚ U = ϕ − ( T ) and ˚ ϕ := ϕ | ˚ U . We have supp( α ) ⊂ ˚ U and supp( α U ) ⊂ Trop ϕ | ˚ U (˚ U ) by [Jel16a, Proposition 3.2.56], which implies that (˚ U an , ϕ, α U | Trop ϕ (˚ U ) ) is a G -tropicalchart of integration for α . Thus we get Z S X an α = Z Trop ϕ ( U ) α U = Z Trop ϕ (˚ U ) α U | Trop ϕ (˚ U ) = Z G X an Ψ S , T ◦ Ψ − G , T α (7.1)The right hand side of equation (7.1) does not depend on U, ϕ or α U by Gubler’s result,hence neither does the left hand side. So we proved the first part of the theorem when K is non-trivially valued. Then second part follows also from (7.1) applied once to S and once to T .We reduce to the non-trivially valued case by picking a non-archimedean, complete,algebraically closed non-trivially valued extension L of K . Let p : X L → X be thecanonical map. Since tropicalization is invariant under base field extension (cf. [Pay09,Section 6 Appendix]), we can define α ∈ A n,nc ( X an L ) to be given by ( U an L , ϕ L , α U ) . Nowwe have Z Trop ϕ ( U ) α U = Z Trop ϕL ( U L ) α U and the right hand side depends only on α L , which depends only on α . The last partfollows because the maps Ψ are also compatible with base change. (cid:3) Definition 7.4.
We define Z X an α = Z T X an α U . Lemma 7.5. R X an does not change when extending the base field.Proof. This follows from the last part of the proof of Theorem 7.3. (cid:3)
Chambert-Loir and Ducros also define an integration for ( n, n ) -forms with compactsupport, which we denote by R CLD . Lemma 7.6. R CLD does not change when extending the base field.Proof.
The base changes of any atlas of integration in the sense of [CLD12] is still an atlasof integration for the base changed form. Then since tropicalizations and the multiplicity d D from their definition does not change, we obtain the result. (cid:3) Theorem 7.7.
Let α ∈ A n,nc ( X an ) . Then Z CLD X an Ψ an ( α ) = Z X an α. Proof.
Let L be a non-trivially valued non-archimedean extension of K . After replacing K by L , X by X L and α be α L , we may, since neither integral changes by Lemmas 7.5and 7.6, assume that K is non-trivially valued.Then, by Corollary 6.2, G can is an admissible family of tropicalizations and thus byTheorem 7.3, we may take α ∈ A n,n G can ,c , say given by ( V i , ϕ i , α i ) i ∈ I . Then by Lemma 6.5we have to show Z CLD X an ( V i , ϕ an i , α i ) i ∈ I = Z X an ( V i , ϕ i , α i ) i ∈ I . This was precisely shown in [Gub16, Section 7]. (cid:3)
ROPICAL COHOMOLOGY WITH INTEGRAL COEFFICIENTS FOR ANALYTIC SPACES 19 Tropical cohomology
In this section we assume that S is a fine enough family of tropicalizations for X thatis cofinal in T . We let R be a subring of R . We will use the sheaf of tropical cochains C p,q ( R ) and the constructions from Section 3. Definition 8.1.
Let V be a open subset of X an . An element of C p,q S ( V, R ) is given by afamily ( V i , ϕ i , η i ) i ∈ I such that:i) The V i cover V , i.e. V = S i ∈ I V i .ii) For each i ∈ I the pair ( V i , ϕ i ) is an S -tropical chart.iii) For each i ∈ I we have η i ∈ C p,q (trop ϕ i ( V i ) , R ) .iv) For all i, j ∈ I there exist S -tropical subcharts ( V ijl , ϕ ijl ) l ∈ L that cover V i ∩ V j such that Trop( ϕ i , ϕ ijl ) ∗ η i = Trop( ϕ j , ϕ ijl ) ∗ η j ∈ C p,q (trop ϕ ijl ( V ijl ) , R ) . Another such family ( V j , ϕ j , η j ) j ∈ J defines the same form if and only if their union ( V i , ϕ i , η i ) i ∈ I ∪ J still satisfies iv).For each p we obtain a complex of sheaves (C p, •S ( R ) , ∂ ) on X an . If S = T we will dropthe subscript and write C p,q ( R ) := C p,q T ( R ) . Remark 8.2.
It follows the same way as in Section 5 for A p,q that C p,q S is isomorphicto C p,q S ′ when S ′ is final or cofinal for S .The author does not know whether one gets isomorphic sheaves when one appliesDefinition 8.1 with for example S = G . The missing piece here is that for differentialforms on tropical toric varieties, the condition of compatibility requires every forms tolocally be a pullback of a forms from a tropical torus (i.e. R n ). The same is not true fortropical cocycles, hence the arguments form Theorem 6.1 do not work. Lemma 8.3.
Let η ∈ C p,q ( V, R ) be given by ( V, ϕ, η ′ ) . Then η = 0 if and only if η ′ = 0 .Proof. The proof for forms [Jel16a, Lemma 3.2.12] works word for word. (cid:3)
Proposition 8.4.
Comparing with tropical cohomology, we obtain C p,qc ( X an , R ) = lim −→ ϕ ∈S C p,q trop ,c (Trop ϕ ( X ) , R ) and H p,q trop ,c ( X an , R ) = lim −→ ϕ ∈S H p,q trop ,c (Trop ϕ ( X ) , R ) . Proof.
This follows from Lemma 8.3 in the same way as for forms in Theorems 5.7 and5.9 follow from Lemma 5.2. (cid:3)
Definition 8.5.
The maps dR defined in Section 3 define maps dR : A p,q → C p,q ( R ) the induces a morphism of complexes of sheaves. Definition 8.6.
The cohomology H p,q trop ( X an , R ) := H q (C p, • ( X an , R ) , ∂ ) is called tropical cohomology with coefficients in R of X an . Similarly H p,q trop ,c ( X an , R ) := H q (C p, • c ( X an , R ) , ∂ ) is called tropical cohomology with coefficients in R with compact support of X an . Definition 8.7.
We denote by F pR := ker( ∂ : C p, ( R ) → C p, ( R )) . Lemma 8.8.
The complex → F pR → C p, ( R ) → C p, ( R ) → · · · → C p,n ( R ) → is exact.Proof. Exactness on the tropical side is true by [JSS19, Proposition 3.11 & and Lemma3.14] (with real coefficients, but the proof goes through here). It is then automaticallytrue on the analytic side using the definitions (cf. the proof for forms [Jel16b, Theorem4.5]). (cid:3)
Remark 8.9.
The sheaves A p,q S admit partitions of unity, which can be shown the sameway as it was shown by Gubler for S = G in [Gub16, Proposition 5.10]. This proofhowever uses the R -structure of those sheaves.The sheaves C p,q ( R ) on a tropical variety (as defined in Section 3) are flasque sheaves[JSS19, Lemma 3.14], hence in particular acyclic.However, it is not clear whether this property also holds for C p,q ( R ) on the analyticspace X an in general. We will prove some partial results in the next Lemma.Recall condition ( † ) from Definition 4.11. Lemma 8.10.
Assume that R = R or that X satisfies condition ( † ) . Then the sheaves C p,q ( R ) are acyclic with respect to the functor of global sections as well as global sectionswith compact support.Proof. Using the map dR : A , → C , ( R ) , we see that C , ( R ) admits partitions ofunity. Hence C , ( R ) is a fine sheaf and since C p,q ( R ) is a C , ( R ) -module (via the capproduct) we see that C p,q ( R ) is also a fine sheaf.In general, if X satisfied condition ( † ) then T global is a global family of tropicalizationsthat is cofinal in A , which is final in T . hence C p,q T global ( R ) ∼ = C p,q ( R ) . Any section of C p,q T global ( R ) that is defined by finitely many charts ( V i , ϕ i , η i ) can be defined by a singlechart. This can be shown the same way as for forms in Theorem 5.6.Since any section over a compact subset of X an is defined by finitely many charts,each such section can be defined by one chart ( X an , ϕ, η ) . Now since the sheaf C p,q ( R ) on Trop ϕ ( X ) is flasque, this section can be extended to a global section. This showsthat the sheaf C p,q ( R ) on X an is c-soft in the sense of [KS94, Definition 2.5.5], whichimplies that it is acyclic for the functor of global sections with compact support [KS94,Proposition 2.58 & Corollary 2.5.9].Since C p,q ( R ) is c-soft, to show that it is acyclic for the functor of global sections, wehave to show that X an admits a countable cover by compact sets [KS94, Proposition2.5.10]. Since A n, an is covered by countably many discs, this holds if X is affine. Sincegeneral X is covered by finitely many affine varieties, the claim follows. (cid:3) Corollary 8.11. If R = R or if X satisfies condition ( † ) we have H p,q trop ( X an , R ) = H q ( X an , F pR ) and H p,q trop ,c ( X an , R ) = H qc ( X an , F pR ) . Further, we have H p,q trop ( X an , R ) = H p,q trop ( X an , Z ) ⊗ R. and H p,q trop ,c ( X an , R ) = H p,q trop ,c ( X an , Z ) ⊗ R. Proof.
This follows from Lemmas 8.8 and 8.10, the fact that since R is torsion free, thusa flat Z -module and F pR = F p Z ⊗ Z . (cid:3) ROPICAL COHOMOLOGY WITH INTEGRAL COEFFICIENTS FOR ANALYTIC SPACES 21
Theorem 8.12 (Tropical analytic de Rham theorem) . There exist isomorphisms H p,q ( X an ) ∼ = H p,q trop ( X an , R ) and H p,qc ( X an ) ∼ = H p,q trop ,c ( X an , R ) . that are induced by the de Rham morphism on the tropical level, as defined in Remark3.11.Proof. We have a map dR : A p,q → C p,q ( R ) that is locally given by using the de Rham map on the tropical side constructed inRemark 3.11. This makes the following diagram commutative: A p, • dR (cid:15) (cid:15) F p ♥♥♥♥♥♥♥♥♥♥♥♥♥ ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ C p, • . This is now a commutative diagram of acyclic resolutions of F p , which proves the theo-rem. (cid:3) Wave and monodromy operators
Since both the monodromy operator M on superforms and the wave operator definedin Section 3 on tropical cochains commute with pullbacks along affine maps on tropicaltoric varieties, we obtain maps M : H p,q ( X an ) → H p − ,q +1 ( X an ) and W : H p,q trop ( X an , R ) → H p − ,q +1trop ( X an , R ) . Theorem 9.1.
The wave and the monodromy operator agree on cohomology up to signby virtue of the isomorphism dR , meaning that the diagram H p,q ( X an ) ( − p − M / / dR (cid:15) (cid:15) H p − ,q +1 ( X an ) dR (cid:15) (cid:15) H p,q trop ( X an , R ) W / / H p − ,q +1trop ( X an , R ) commutes. The same is true for cohomology with compact support.Proof. The proof of Theorem 3.14 works word for word. (cid:3)
In [Liu19], Liu defined a Q -subsheaf of F p R and defined rational classes in tropicalDolbeault cohomology. Definition 9.2 (Liu) . Denote by J p the Q -subsheaf of F p generated by sections ofthe form ( V, ϕ, α ) , where ϕ : U → T and α ∈ Λ p M . The classes in H p,q ( X an , J p ) ⊂ H p,q ( X an , R ) are called rational classes . Proposition 9.3.
Assume that X satisfies condition ( † ) . We have isomorphisms F p Q = J p and H p,q ( X an , Q ) = H p,q ( X an , J p ) .Proof. The explicit computation [JSS19, Proposition 3.11] of F p R works also for rationalcoefficients. Then this follows directly from the definitions and Corollary 8.11. (cid:3) The following statement in particular shows that H p,q ( X an , Z ) ( H p,q ( X an , Q ) . Theorem 9.4.
Assume that X satisfies condition ( † ) . Then is a non-trivial R -linearmap ∩ [ X an ] R : H n,n trop ,c ( X an , R ) → R. If R = R , then this agrees with the map induced by integration via dR .Proof. The maps [Trop ϕ ( X )] R as defined in Definition 3.16 are compatible with pullbackalong refinements, so by Proposition 8.4 we get a well defined map on H n,nc ( X an , R ) .The last part of the statement follows from Proposition 3.17. (cid:3) Liu showed that if the value group of K is equal to Q , then his monodromy map M maps rational classes to rational classes [Liu19, Theorem 5.5 (1)]. We generalize to thefollowing statement: Theorem 9.5.
Assume that X satisfies condition ( † ) . The wave operator W (and byvirtue of Corollary 9.1 also the monodromy map M ) restricts to a map W : H p,q trop ,c ( X an , R ) → H p − ,q +1trop ,c ( X an , R [Γ]) . Proof.
By Proposition 8.4, it is sufficient to prove this theorem for
Trop ϕ ( X ) . Since Trop ϕ ( X ) is an integral Γ -affine tropical variety, this follows from Proposition 3.15. (cid:3) Mikhalkin and Zharkov conjectured that for a smooth tropical variety X , the iteratedwave operator W p − q : H p,q trop ( X, R ) → H q,p trop ( X, R ) is an isomorphism for all p ≥ q [MZ14, Conjecture 5.3].Liu conjectured that if K is such that the residue field ˜ K is the algebraic closure of afinite field and X is smooth and proper, then the iterated monodromy operator M p − q : H p,q ( X ) → H q,p ( X ) is an isomorphism for all p ≥ q [Liu19, Conjecture 5.2].As a consequence of Theorem 9.1 we can tie together both of these conjectures. Proposition 9.6.
Let X be a proper variety. Assume there exists a global admissiblefamily of tropicalizations S for X such that for all ϕ ∈ S map the tropical variety Trop ϕ ( X ) satisfies Mikhalkin’s and Zharkov’s conjecture. Then X satisfies Liu’s conjecture.Proof. This follows directly from Theorem 9.1 and Theorem 5.9. (cid:3)
Non-trivial classes
In this section we (partially) compute tropical cohomology with coefficients in threeexamples: Curves of good reduction, toric varieties and Mumford curves.For the first theorem assume that the value group Γ of K is a subring of R . We denoteby log |O × X | the sheaf of real valued functions on X an that are locally of the form log | f | for an invertible function f on X . Theorem 10.1.
Let K be algebraically closed and X be a smooth projective curve ofgood reduction. Then there exists an injective morphism Pic ( ˜ X ) → H , ( X, Z ) , where Pic ( ˜ X ) denotes the group of degree line bundels on the reduction ˜ X of X . ROPICAL COHOMOLOGY WITH INTEGRAL COEFFICIENTS FOR ANALYTIC SPACES 23
Proof.
We have the following exact sequence → Γ → log |O × X | → F Z → , (10.1)which is a non-archimedean version of a well-known exponential sequence from tropicalgeometry [MZ08, Definition 4.1]. This induces the following exact sequence in cohomol-ogy groups → H , ( X an , Z ) → H , ( X an , Γ) → H ( X an , log |O × X | ) → H , ( X an , Z ) → . (10.2)In particular, since X is a curve of good reduction, X an is contractible and H , ( X an , Γ) =H ( X an , Γ) = 0 . Hence we have that H ( X an , log |O × X | ) → H , ( X an , Z ) . is an isomorphism. Therefore it is sufficient to prove that there exists an injectivemorphism Pic ( ˜ X ) → H ( X an , log |O × X | ) . We have an exact sequence → log |O × X | → H Z → ι ∗ Pic ( ˜ X ) → (10.3)where H Z is the sheaf of real valued functions on X an that locally factor as the retractionto a skeleton composed with a piecewise linear function with integer slopes and valuesin Γ on the edges of said skeleton. This is sequence is the integral version of [Thu05,Lemme 2.3.22]. Further, Thuillier showed that every harmonic function on a compactBerkovich analytic space is constant [Thu05, Proposition 2.3.13]. Thus we obtain thefollowing long exact sequence: → Γ → Γ → Pic ( ˜ X ) → H ( X an , log |O × X | ) . (10.4)This shows the existence of an injective morphism Pic ( ˜ X ) → H ( X an , log |O × X | ) . (cid:3) Remark 10.2.
Since the Picard group of a smooth projective curve of positive genus overan algebraically closed field contains torsion, Theorem 10.1 implies that H , ( X an , Z ) cancontain torsion. In other word the map H p,q ( X an , Z ) → H p,q ( X an , R ) = H p,q ( X an , Z ) ⊗ R need not be injective.It is very possible that one can drop the assumption for K to be algebraically closedin Theorem 10.1. Theorem 10.3.
Let ϕ : X → Y be a closed embedding of X into a toric variety Y .Assume that Trop ϕ ( X ) is a smooth tropical variety. Then trop ∗ : H p,q (Trop ϕ ( X )) → H p,q ( X an ) and trop ∗ : H p,qc (Trop ϕ ( X )) → H p,qc ( X an ) are injective.Proof. By [JSS19, Theorem 4.33], since
Trop ϕ ( X ) is smooth there is a perfect pairing H p,q (Trop ϕ ( X )) × H n − p,n − qc (Trop ϕ ( X )) → R induced by the wedge product and integration of superforms. Thus given a d ′′ -closed α in A p,q (Trop ϕ ( X )) whose class [ α ] ∈ H p,q (Trop ϕ ( X )) is non-trivial, there exists [ β ] ∈ H n − p,n − qc (Trop ϕ ( X )) such that R Trop ϕ ( X ) α ∧ β = 0 . Thus we have Z X an trop ∗ ϕ α ∧ trop ∗ ϕ β = 0 . Since integration and the wedge product are well defined on cohomology this meansthat [trop ∗ ϕ α ∧ trop ∗ ϕ β ] and consequently [trop ∗ ϕ α ] is not trivial. The argument for α ∈ H p,qc (Trop ϕ ( X )) works the same except [ β ] ∈ H n − p,n − q (Trop ϕ ( X )) . (cid:3) Example 10.4.
Let Y Σ be a smooth toric variety. Then Y Σ is locally isomorphic to A n and hence Trop( Y ) is locally isomorphic to Trop( A n ) and hence is a smooth tropicalvariety. Thus trop ∗ : H p,q (Trop( Y )) → H p,q ( Y an ) is injective by Theorem 10.3. Let Y Σ ( C ) be the complex toric variety associated with Σ .Then H p,q Hodge ( Y Σ ( C )) ∼ = H p,q (Trop( Y Σ ) , C ) [IKMZ19, Corollary 2]. In particular we have dim R H p,q ( Y Σ ) ≥ dim C H p,q Hodge ( Y Σ ( C )) . One may figure out the latter in terms of Σ using[Ful93, Section 5.2] or with the help of a computer and in terms of the polytope of Y using the package cellularSheaves for polymake [KSW17]. Note that H p,q Hodge ( Y ( C )) = 0 if p = q by [Dan78, Corollary 12.7]. Example 10.5.
Let K be algebraically closed, X be a smooth projective curve of genus g and ϕ : X → Y be a closed embedding of X into a toric variety such that Trop ϕ ( X ) is a smooth tropical variety (this exists if and only if X is a Mumford curve by [Jel18]).Then H p,q trop (Trop ϕ ( X ) , R ) → H p,q trop ( X an , R ) is an isomorphism. In particular we have H , ( X an , R ) ∼ = H , ( X an , R ) ∼ = R and H , ( X an , R ) ∼ = H , ( X an , R ) ∼ = R g . Proof.
Assume that
Trop ϕ ( X ) is smooth. Then trop ϕ is a homeomorphism from askeleton of X an onto Trop ϕ ( X ) [Jel16b, Theorem 5.7]. Using comparison with singularcohomology we obtain H , (Trop ϕ ( X ) , R ) = R and H , (Trop ϕ ( X ) , R ) = R g . Usingduality with coefficients in R as proven in [JRS18, Theorem 5.3] and comparison withsingular homology, we also obtain H , ( X an , R ) = R and H , (Trop ϕ ( X ) , R ) = R g .One immediately verifies all transition maps induced by refinements in the family T Smooth defined in Example 4.16 are isomorphisms and hence the claim follows from Theorem5.9. (cid:3)
Open questions
In this section, we let X be a variety over K .When X is smooth, Liu constructed cycles class maps, meaning maps cyc k : CH( X ) k → H k,k ( X an ) that are compatible with the product structure on both sides and have theexpected integration property [Liu17]. Question 11.1.
What is the image of cyc k ?In light of the tropical Hodge conjecture and Corollary 9.1, one might conjecturethat the image of CH( X ) Q is H k,k ( X an , Q ) ∩ ker( M ) . One might start with the case k = dim X − . Here one knows the answer tropically [JRS18], but the non-archimedeananalogue is not a direct consequence.The following question was asked by a referee and the author thinks it is worthwhileto include it here along with a partial answer. Question 11.2.
Is there an analogue of Theorem 10.1 when X has semistable reduction?Let X be a smooth projective curve, let X s be the special fiber of a strictly semistablemodel of X and let C , . . . , C n be the irreducible components of X s . Then we can ROPICAL COHOMOLOGY WITH INTEGRAL COEFFICIENTS FOR ANALYTIC SPACES 25 construct a map
Pic ( X s ) → H , ( X an , Z ) by the composition Pic ( X s ) → n M i =1 Pic ( C i ) → H , ( X an , log |O × X | ) → H , ( X an , Z ) . Here the first map is pullback along normalization, and the second and third mapsare induced by (10.4) and (10.2), which remain valid when replacing
Pic ( ˜ X ) with L ni =1 Pic ( C i ) . It is well known that the first map need not be injective, hence thecomposition need not be injective. Whether the map n M i =1 Pic ( C i ) → H , ( X an , log |O × X | ) → H , ( X an , Z ) is injective is unclear to the author. The map H , ( X an , log |O × X | ) → H , ( X an , Z ) willnot be injective when X an is not contractible, since the map H , ( X an , Z ) → H , ( X an , Γ) from (10 . will not be surjective. But that of course does not imply that the compositioncan not be injective. Question 11.3.
Does there exists a toric variety Y and a closed embedding ϕ : X → Y such that trop ∗ : H p,q (Trop ϕ ( X )) → H p,q ( X an ) and trop ∗ : H p,qc (Trop ϕ ( X )) → H p,qc ( X an ) are isomorphisms?The statement for H p,qc ( X an ) is implied by the finite dimensionality of H p,qc ( X an ) viaTheorem 5.9. It is in fact equivalent to the finite dimensionality of H p,q ( X an ) if one knewthat H p,qc (Trop ϕ ( X )) is always finite dimensional, though the author is not aware of sucha result (without regularity assumptions on Trop ϕ ( X ) ).Other questions related to this concern smoothness of the tropical variety. Question 11.4.
Let ϕ : X → Y be a closed embedding of X into a toric variety Y suchthat Trop ϕ ( X ) is smooth. Are then trop ∗ : H p,q (Trop ϕ ( X )) → H p,q ( X an ) and trop ∗ : H p,qc (Trop ϕ ( X )) → H p,qc ( X an ) isomorphisms?This is certainly a natural question and “optimistically expected” to be true by Shaw[Sha17, p.3]. We now know it holds for curves, as we showed in Example 10.5, but eventhe case X = Y is open in dimension ≥ . Question 11.5.
Let ϕ : X → Y be a closed embedding of X into a toric variety Y suchthat Trop ϕ ( X ) is smooth. Does the diagram Z k ( X ) trop (cid:15) (cid:15) / / CH( X ) cyc k / / H k,k ( X an ) Z k (Trop( ϕ ( X )) Z R Z / / H n − k,n − k (Trop ϕ ( X )) ∗ PD − / / H k,k (Trop ϕ ( X )) trop ∗ O O commute? Here PD denote the Poincaré duality isomorphism on tropical varieties[JSS19] and cyc k denotes Liu’s cycles class map [Liu17].Let us finish with the remark that the author does not know of any variety X with dim( X ) ≥ and any < p ≤ dim( X ) and < q ≤ dim( X ) with ( p, q ) = (1 , where weknow dim R H p,q ( X an ) . (No, not even H , ( P , an ) or H , ( A , an ) .) References [Ber90] Vladimir G. Berkovich.
Spectral theory and analytic geometry over non-Archimedean fields ,volume 33 of
Mathematical Surveys and Monographs . American Mathematical Society, Prov-idence, RI, 1990.[BPR16] Matthew Baker, Sam Payne, and Joseph Rabinoff. Nonarchimedean geometry, tropicalization,and metrics on curves.
Algebr. Geom. , 3(1):63–105, 2016.[CLD12] Antoine Chambert-Loir and Antoine Ducros. Formes différentielles réelles et courants sur lesespaces de Berkovich. 2012. http://arxiv.org/abs/1204.6277 .[CLS11] David A. Cox, John B. Little, and Henry K. Schenck.
Toric varieties , volume 124 of
GraduateStudies in Mathematics . American Mathematical Society, Providence, RI, 2011.[Dan78] V. I. Danilov. The geometry of toric varieties.
Uspekhi Mat. Nauk , 33(2(200)):85–134, 247,1978.[FGP14] Tyler Foster, Philipp Gross, and Sam Payne. Limits of tropicalizations.
Israel J. Math. ,201(2):835–846, 2014.[Ful93] William Fulton.
Introduction to toric varieties , volume 131 of
Annals of Mathematics Stud-ies . Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures inGeometry.[Gub13] Walter Gubler. A guide to tropicalizations. In
Algebraic and combinatorial aspects of tropicalgeometry , volume 589 of
Contemp. Math. , pages 125–189. Amer. Math. Soc., Providence, RI,2013.[Gub16] Walter Gubler. Forms and currents on the analytification of an algebraic variety (afterChambert-Loir and Ducros). In Matthew Baker and Sam Payne, editors,
Nonarchimedeanand Tropical Geometry , Simons Symposia, pages 1–30, Switzerland, 2016. Springer.[IKMZ19] Ilia Itenberg, Ludmil Katzarkov, Grigory Mikhalkin, and Ilia Zharkov. Tropical homology.
Math. Ann. , 374(1-2):963–1006, 2019.[Jel16a] Philipp Jell. Differential forms on Berkovich analytic spaces and their cohomology. 2016. PhDThesis, availible at http://epub.uni-regensburg.de/34788/1/ThesisJell.pdf .[Jel16b] Philipp Jell. A Poincaré lemma for real-valued differential forms on Berkovich spaces.
Math.Z. , 282(3-4):1149–1167, 2016.[Jel18] Philipp Jell. Constructing smooth and fully faithful tropicalizations for Mumford curves. 2018. https://arxiv.org/abs/1805.11594 .[Jel19] Philipp Jell. Tropical Hodge numbers of non-archimedean curves.
Israel J. Math. , 229(1):287–305, 2019.[JRS18] Philipp Jell, Johannes Rau, and Kristin Shaw. Lefschetz (1,1)-theorem in tropical geometry.
Épijournal Geom. Algébrique , 2:Art. 11, 2018.[JSS19] Philipp Jell, Kristin Shaw, and Jascha Smacka. Superforms, tropical cohomology, andPoincaré duality.
Adv. Geom. , 19(1):101–130, 2019.[JW18] Philipp Jell and Veronika Wanner. Poincaré duality for the tropical Dolbeault cohomology ofnon-archimedean Mumford curves.
J. Number Theory , 187:344–371, 2018.[Kaj08] Takeshi Kajiwara. Tropical toric geometry. In
Toric topology , volume 460 of
Contemp. Math. ,pages 197–207. Amer. Math. Soc., Providence, RI, 2008.[KS94] Masaki Kashiwara and Pierre Schapira.
Sheaves on manifolds , volume 292 of
Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] .Springer-Verlag, Berlin, 1994. With a chapter in French by Christian Houzel, Correctedreprint of the 1990 original.[KSW17] Lars Kastner, Kristin Shaw, and Anna-Lena Winz. Cellular sheaf cohomology of polymake . In
Combinatorial algebraic geometry , volume 80 of
Fields Inst. Commun. , pages 369–385. FieldsInst. Res. Math. Sci., Toronto, ON, 2017.[Liu17] Yifeng Liu. Tropical cycle classes for non-archimedean spaces and weight decomposition ofde Rham cohomology sheaves. 2017. https://users.math.yale.edu/~yl2269/deRham.pdf ,to appear in
Ann. Sci. Éc. Norm. Supér. [Liu19] Yifeng Liu. Monodromy map for tropical Dolbeault cohomology.
Algebr. Geom.
ROPICAL COHOMOLOGY WITH INTEGRAL COEFFICIENTS FOR ANALYTIC SPACES 27 [MZ08] Grigory Mikhalkin and Ilia Zharkov. Tropical curves, their Jacobians and theta functions. In
Curves and abelian varieties , volume 465 of
Contemp. Math. , pages 203–230. Amer. Math.Soc., Providence, RI, 2008.[MZ14] Grigory Mikhalkin and Ilia Zharkov. Tropical eigenwave and intermediate Jacobians. In
Ho-mological mirror symmetry and tropical geometry , volume 15 of
Lect. Notes Unione Mat. Ital. ,pages 309–349. Springer, Cham, 2014.[Pay09] Sam Payne. Analytification is the limit of all tropicalizations.
Math. Res. Lett. , 16(3):543–556,2009.[Sha17] Kristin Shaw. Superforms and tropical cohomology. 2017.[ST08] Bernd Sturmfels and Jenia Tevelev. Elimination theory for tropical varieties.
Math. Res. Lett. ,15(3):543–562, 2008.[Thu05] Amaury Thuillier. Théorie du potentiel sur les courbes en géométrie analytique non archimé-dienne. Applications à la théorie d’Arakelov. 2005. .[Wło93] Jarosław Włodarczyk. Embeddings in toric varieties and prevarieties.
J. Algebraic Geom. ,2(4):705–726, 1993.
P. Jell, Fakultät Mathematik, Universität Regensburg, 93040 Regensburg, Germany
E-mail address ::