Proof: See [CD12], Lemme 3.2.2. $\square$

Proof: It is enough to show that for every $x\in V$, there is a non-negative smooth function $\phi$ with compact support in $V$ and with $\phi(x)>0$. Since ${X^{\rm an}}$ is a locally compact Hausdorff space which is also $\sigma$-compact, the open subset $V$ is paracompact and hence standard arguments from differential geometry yield the existence of the desired partition of unity (see [Wa83], Theorem 1.11).

Proof: Using Proposition 4.16 and shrinking the open neighbourhood $W$ of $x$, we may assume that $W$ is a tropical chart $(W,\varphi_{U})$ on which $\alpha$ is given by the superform $\alpha_{U}\in A^{p,q}({\rm trop}_{U}(W))$. By Proposition 4.14, there is a very affine open subset $U_{x}$ of $U$ and a compact neighbourhood $V_{x}$ of $x$ in $(U_{x})^{\rm an}$ such that ${\rm trop}_{U_{x}}(V_{x})$ is of dimension $d(x)$. By Proposition 4.16, there is a tropical chart $(V^{\prime},\varphi_{U^{\prime}})$ with $x\in V^{\prime}\subset V_{x}$ and $U^{\prime}\subset U_{x}$. By 4.12, there is an affine homomorphism $\psi:T_{U^{\prime}}\rightarrow T_{U}$ such that $\varphi_{U}=\varphi_{U^{\prime}}\circ\psi$. Using the same factorization for the tropicalizations, we see that the restriction of $\alpha$ to $V^{\prime}$ is given by ${\rm Trop}(\psi)^{*}(\alpha_{U})\in A^{p,q}({\rm trop}_{U^{\prime}}(V^{\prime}))$. The inclusion $U_{x}\subset U$ yields that ${\rm trop}_{U_{x}}$ factorizes through ${\rm trop}_{U}$ (use 4.12). Since $V^{\prime}\subset V_{x}$, we get $\dim({\rm trop}_{U}(V^{\prime}))\leq\dim({\rm trop}_{U_{x}}(V^{\prime}))\leq d(x)<\max(p,q)$. As ${\rm trop}_{U}(V^{\prime})={\rm Trop}(\psi)({\rm trop}_{U^{\prime}}(V^{\prime}))$, we conclude that ${\rm Trop}(\psi)^{*}(\alpha_{U})=0$. This proves $\alpha=0$. $\square$

Proof: Let $x\in W\setminus{U^{\rm an}}$. Then 4.2 shows that $d(x)\leq\dim(X\setminus U)<\max(p,q)$. By Lemma 5.11, we get $x\not\in{\rm supp}(\alpha)$ proving the claim. $\square$

Proof: By assumption, the support of $\alpha$ is a compact subset of ${X^{\rm an}}$. We conclude that there are finitely many tropical charts $(V_{i},\varphi_{U_{i}})_{i=1,\dots,s}$ covering ${\rm supp}(\alpha)$ such that $\alpha$ is given on $V_{i}$ by the superform $\alpha_{i}\in A^{p,q}({\rm trop}_{U_{i}}(V_{i}))$. Recall that $\Omega_{i}:={\rm trop}_{U_{i}}(V_{i})$ is an open subset of ${\rm Trop}(U_{i})$. By 4.13, $U:=U_{1}\cap\dots\cap U_{s}$ is a non-empty very affine open subset of $X$. We define the open subset $V$ of ${U^{\rm an}}$ by $V:={U^{\rm an}}\cap\bigcup_{i=1}^{s}V_{i}$. Since $\max(p,q)=\dim(X)$, Corollary 5.12 yields ${\rm supp}(\alpha)\subset{U^{\rm an}}$. Using 4.12, we see that ${\rm trop}_{U_{i}}={\rm Trop}(\psi_{i})\circ{\rm trop}_{U}$ for an affine homomorphism $\psi_{i}:T_{U}\rightarrow T_{U_{i}}$ of tori. Then we have

and we denote this open subset of ${\rm Trop}(U)$ by $\Omega_{i}^{\prime}$. It follows that the preimage of $\Omega:=\bigcup_{i=1}^{s}\Omega_{i}^{\prime}$ with respect to $(\varphi_{U})_{\rm trop}$ is equal to $V$. We conclude that $(V,\varphi_{U})$ is a tropical chart of ${X^{\rm an}}$. Note that $\alpha$ is given on ${U^{\rm an}}\cap V_{i}$ by $\alpha_{i}^{\prime}:={\rm Trop}(\psi_{i})^{*}(\alpha_{i})\in A^{p,q}(\Omega_{i}^{\prime})$. By Proposition 5.6, $\alpha_{i}^{\prime}$ agrees with $\alpha_{j}^{\prime}$ on $\Omega_{i}^{\prime}\cap\Omega_{j}^{\prime}$ for every $i,j\in\{1,\dots,s\}$ and hence they define a superform $\alpha_{U}\in A^{p,q}(\Omega)$. By construction, $\alpha_{U}$ gives the differential form $\alpha$ on $V$. It follows from Remark 5.7 that $\alpha_{U}$ has compact support in $\Omega$. Since $\alpha$ has compact support in $V$, we conclude that $\alpha_{U}$ is a superform on ${\rm Trop}(U)$ which defines $\alpha$ on ${U^{\rm an}}$. $\square$

Proof: By Corollary 5.12, ${\rm supp}(\alpha)\subset(U^{\prime})^{\rm an}$ and (a) follows. To prove (b), it is enough to show

for a non-empty very affine open subset $U^{\prime}$ of $U$ by using (a). The differential form $\alpha$ is given on ${U^{\rm an}}$ (resp. $(U^{\prime})^{\rm an}$) by $\alpha_{U}\in A^{n,n}_{c}({\rm Trop}(U))$ (resp. $\alpha_{U^{\prime}}\in A^{n,n}_{c}({\rm Trop}(U^{\prime}))$). By 4.12, there is an affine homomorphism $\psi:T_{U^{\prime}}\rightarrow T_{U}$ of the underlying canonical tori such that $\varphi_{U}={\rm Trop}(\psi)\circ\varphi_{U^{\prime}}$. It follows that $\alpha$ is given on $U^{\prime}$ also by ${\rm Trop}(\psi)^{*}(\alpha_{U})$. By Proposition 5.6, we have $\alpha_{U^{\prime}}={\rm Trop}(\psi)^{*}(\alpha_{U})$. The Sturmfels–Tevelev multiplicity formula shows that ${\rm Trop}(\psi)_{*}({\rm Trop}(U^{\prime}))={\rm Trop}(U)$ (see Proposition 4.11). Then Proposition 3.10 shows that (3) holds. $\square$

Proof: By Lemma 5.15, we may choose a simultaneous very affine chart of integration for both $\alpha$ and $\beta$. Then the claim follows by the corresponding property of the integration of superforms. $\square$

Proof: By Proposition 5.13, there is a very affine open subset $U$ of $X$ such that ${\rm supp}(\alpha)\subset{U^{\rm an}}$ and such that $\alpha$ is given on ${U^{\rm an}}$ by a superform $\alpha_{U}\in A^{2n-1}_{c}({\rm Trop}(U))$. Then $U$ is a very affine chart of integration for $d^{\prime}\alpha$ and $d^{\prime\prime}\alpha$ and the claim follows from Proposition 3.5. $\square$