The final section shows how our notions fit with the paper [CD12]. While we restricted to the algebraic case, the paper of Chambert-Loir and Ducros works for arbitrary analytic spaces. We assume that the reader is familiar with the theory of analytic spaces as given in [Be93] or [Te10]. For simplicity, we assume again that $K$ is algebraically closed, endowed with a non-trivial non-archimedean complete absolute value $|\phantom{a}|$ with corresponding valuation $v:=-\log|\phantom{a}|$ and that all occurring analytic spaces are strict in the sense of [Be93]. This situation can always be obtained by base change without changing the theory of differential forms and currents. As usual, we use the value group $\Gamma:=v(K^{\times})$.

The next result shows that for the construction of differential forms in the algebraic case, we may restrict our attention to algebraic moment maps.

We have the following generalization of the Bieri–Groves theorem. Working with analytic spaces, the boundary $\partial Z$ of $Z$ becomes an issue.

The next result shows that the Sturmfels–Tevelev multiplicity formula holds for analytic spaces.

Now we are ready to compare these two tropical multiplicity functions. It is clear that the degree $\deg(\varphi)$ from 4.10 appears on the algebraic side.

Let $Y^{\prime}$ be the closure of $q(Y)$ in $T^{\prime}$. Let $\omega\in N_{\Gamma}$ be a regular point of ${\rm Trop}(Y)={\varphi_{\rm trop}}(X)$, i.e. $\omega$ is contained in the relative interior of an $n$-dimensional polytope $\Delta\subset{\rm Trop}(Y)$. We may choose for $\Delta$ an integral $\Gamma$-affine polytope. We set $\tau:={\rm relint}(\Delta)$ and $\omega^{\prime}:={\rm Trop}(q)(\omega)$. We consider the affinoid subdomains $U_{\omega}:={\rm trop}^{-1}(\omega)$ in ${T^{\rm an}}$ and $U^{\prime}_{\omega^{\prime}}:={\rm trop}^{-1}(\omega^{\prime})$ in $(T^{\prime})^{\rm an}$. By finiteness of $\varphi$, the set $X_{\omega}:=(\varphi^{\rm an})^{-1}(U_{\omega})=\varphi_{\rm trop}^{-1}(\omega)$ is an affinoid subdomain of ${X^{\rm an}}$ and $\varphi$ restricts to a finite morphism $X_{\omega}\rightarrow Y_{\omega}:={Y^{\rm an}}\cap U_{\omega}$. Let ${\mathscr{X}}_{\omega},{\mathscr{Y}}_{\omega},{\mathscr{U}}_{\omega},{\mathscr{U}}^{\prime}_{\omega^{\prime}}$ be the canonical formal affine ${K^{\circ}}$-models of $X_{\omega},Y_{\omega},U_{\omega},U^{\prime}_{\omega^{\prime}}$ associated to the algebra of power bounded elements in the corresponding affinoid algebra. Moreover, let $\overline{Y_{\omega}}$ be the closure of $Y_{\omega}$ in ${\mathscr{U}}_{\omega}$. Then we have canonical morphisms

$${\mathscr{X}}_{\omega}\stackrel{{\scriptstyle\varphi}}{{\rightarrow}}{\mathscr{Y}}_{\omega}\stackrel{{\scriptstyle\iota}}{{\rightarrow}}\overline{Y_{\omega}}\stackrel{{\scriptstyle q}}{{\rightarrow}}{\mathscr{U}}^{\prime}_{\omega^{\prime}}$$

(4)

of admissible formal affine schemes over ${K^{\circ}}$ in the sense of Bosch, Lütkebohmert and Raynaud (see [BL93], §1). We claim that all these morphisms are finite and surjective. Obviously, the generic fibres of the first and second morphism are finite and surjective. To see that the generic fibre of the third morphism is finite, we note first that ${\rm Trop}(q)^{-1}(\omega^{\prime})\cap{\rm Trop}(Y)$ is finite by construction of $q$ and hence $q^{-1}(U^{\prime}_{\omega^{\prime}})\cap{Y^{\rm an}}$ is in the relative interior of an affinoid subdomain of ${T^{\rm an}}$ which is contained in $q^{-1}(U_{\omega^{\prime}}^{\prime})$. We conclude that $q^{-1}(U^{\prime}_{\omega^{\prime}})\cap{Y^{\rm an}}\rightarrow U^{\prime}_{\omega^{\prime}}$ is a proper map (see the proof of Theorem 4.31 in [BPR11] for more details about the argument). Since $q^{-1}(U^{\prime}_{\omega^{\prime}})\cap{Y^{\rm an}}$ is the disjoint union of the finitely many affinoids $U_{\rho}\cap{Y^{\rm an}}$, $\rho\in{\rm Trop}(q)^{-1}(\omega^{\prime})\cap{\rm Trop}(Y)$, we conclude that $q$ induces a proper morphism $Y_{\omega}\rightarrow U^{\prime}_{\omega^{\prime}}$ of affinoids. By Kiehl’s direct image theorem ([BGR84], Theorem 9.6.3/1), this morphism is finite and hence also surjective using dimensionality arguments. We conclude that all three morphisms in (4) are surjective and finiteness follows from [BPR11], Proposition 3.13.

The degree $[X_{\omega}:U^{\prime}_{\omega^{\prime}}]$ of $X_{\omega}$ over the affinoid torus $U^{\prime}_{\omega^{\prime}}$ is well-defined as $U^{\prime}_{\omega^{\prime}}$ is irreducible (see [BPR11], Section 3, for a discussion of degrees). Since the degree does not change by passing to an affinoid subdomain of $U^{\prime}_{\omega^{\prime}}$ (see [BPR11], Proposition 3.30), we get

$$[\varphi_{\rm trop}^{-1}(\tau):{\rm Trop}(q)(\tau)]=[X_{\omega}:U^{\prime}_{\omega^{\prime}}].$$

(5)

The projection formula ([BPR11], Proposition 3.32) shows

$$[X_{\omega}:U^{\prime}_{\omega^{\prime}}]=\sum_{B}[B:({\mathscr{U}}^{\prime}_{\omega^{\prime}})_{s}]=\sum_{B}[B:({\mathbb{G}}_{m}^{n})_{{\tilde{K}}}],$$

(6)

where $B$ ranges over all irreducible components of $({\mathscr{X}}_{\omega})_{s}$. We conclude from (5) and (6) that

$$[\varphi_{\rm trop}^{-1}(\tau):{\rm Trop}(q)(\tau)]=\sum_{C}\sum_{\text{$B$ over $C$}}[B:C]\cdot[C:({\mathbb{G}}_{m}^{n})_{{\tilde{K}}}],$$

(7)

where $C$ ranges over all irreducible components of $(\overline{Y_{\omega}})_{s}$ and $B$ ranges over all irreducible components of $({\mathscr{X}}_{\omega})_{s}$ mapping onto $C$. Since the special fibre of $\overline{Y_{\omega}}$ is isomorphic to the initial degeneration ${\rm in}_{\omega}(Y)$, all irreducible components $C$ are isomorphic to the torus ${\rm Spec}({\tilde{K}}[M_{\Delta}])$ (see [BPR11], Theorem 4.29) proving

$$[C:({\mathbb{G}}_{m}^{n})_{{\tilde{K}}}]=[M_{\Delta}:M^{\prime}].$$

(8)

Using (7) and (8), we get

$$m_{\rm an}(\omega)=[\varphi_{\rm trop}^{-1}(\tau):{\rm Trop}(q)(\tau)]\cdot[M_{\Delta}:M^{\prime}]^{-1}=\sum_{C}\sum_{\text{$B$ over $C$}}[B:C].$$

(9)

Since $X_{\omega}$ is the preimage of the affinoid subdomain $Y_{\omega}$ of ${T^{\rm an}}$, we deduce from [BPR11], Proposition 3.30, that $X_{\omega}$ is of pure degree $\deg(\varphi)$ over $Y_{\omega}$ and hence the projection formula again shows the equality

$$\deg(\varphi){\rm cyc}((\overline{Y_{\omega}})_{s})=(\iota\circ\varphi)_{*}({\rm cyc}(({\mathscr{X}}_{\omega})_{s})$$

(10)

of cycles in $({\mathscr{U}}_{\omega})_{s}$. Inserting (10) in (9) by using that the special fibre of ${\mathscr{X}}_{\omega}$ is reduced, we get

$$m_{\rm an}(\omega)=\deg(\varphi)\sum_{C}m(C,(\overline{Y_{\omega}})_{s}),$$

where $m(C,(\overline{Y_{\omega}})_{s})$ is the multiplicity of the irreducible component $C$ in the special fibre of $\overline{Y_{\omega}}$. By definition, the right hand side is equal to $m_{\rm alg}(\omega)$ which proves the claim for $\Gamma$-rational points $\omega$ in ${\rm Trop}(Y)$. An obvious density argument finishes the proof. $\square$