21.10 The Guillemin reference metric256A3051D9FC

Proof. The main step is to invert $\nabla G_0$ near the boundary of $P$ in the coordinates of a vertex chart (Step 2); the logarithmic singularities in $H_ v$ then cancel.

Step 1: the potentials on $N_{\mathbb R}$. The function $\lambda \mapsto \lambda \log \lambda $ is convex and continuous on $[0,+\infty )$, and each $\ell _\rho $ is affine and nonnegative on $P$. Hence $G_0$ is convex, continuous and bounded on $P$, and closed and proper on $M_{\mathbb R}$. On $\operatorname{Int}P$,

The normals $u_\rho $ span $N_{\mathbb R}$, since a nonzero $y\in M_{\mathbb R}$ annihilated by all of them would give $P+\mathbb Ry\subset P$ by the facet presentation, which is impossible because $P$ is bounded; so $D^2G_0(y,y)=\sum _\rho \langle y,u_\rho \rangle ^2/\ell _\rho >0$ for $y\neq 0$, and the Hessian is positive definite.

Let $x\in N_{\mathbb R}$. The continuous function $G_0-\langle \cdot ,x\rangle $ attains its minimum on the compact set $P$, and not at a boundary point $m_0$. Indeed, let $p\in \operatorname{Int}P$ and move from $m_0$ towards $p$: for each facet with $\ell _\rho (m_0)=0$ we have $\ell _\rho (m_0+s(p-m_0))=s\ell _\rho (p)$ with $\ell _\rho (p)>0$, and the term $s\ell _\rho (p)\log (s\ell _\rho (p))$ has right derivative $-\infty $ at $s=0$, while all other terms have finite derivatives; so the function decreases initially. The minimum is therefore attained in $\operatorname{Int}P$, at a solution of $\nabla G_0(m)=x$. Conversely, every solution of $\nabla G_0(m)=x$ in $\operatorname{Int}P$ is a minimum point, by convexity, and the minimum point is unique because $G_0$ is strictly convex on $\operatorname{Int}P$. Hence $\nabla G_0$ is a smooth bijection of $\operatorname{Int}P$ onto $N_{\mathbb R}$ with invertible derivative, and so a diffeomorphism by the inverse function theorem. Writing $m(x)=(\nabla G_0)^{-1}(x)$, we have $F_0(x)=\langle m(x),x\rangle -G_0(m(x))$, so $F_0$ is smooth, and differentiating gives $\nabla F_0(x)=m(x)$ and $D^2F_0(x)=(D^2G_0(m(x)))^{-1}$, which is positive definite. Since $G_0$ is bounded on $P$, say $|G_0|\leq C$ there, we have $\chi _ P-C\leq G_0\leq \chi _ P+C$, and conjugation gives $h_ P-C\leq F_0\leq h_ P+C$.

Step 2: local inversion at a boundary point. Fix a vertex $v$, number the facets at $v$ as in lemma 21.8.2(4), and use on $M_{\mathbb R}$ the coordinates $t_ k=\ell _ k(m)$, so that $m=v+\sum _ kt_ ke_ k$, and on $N_{\mathbb R}$ the dual coordinates $x_ k=\langle e_ k,x\rangle $, so that $x=\sum _ kx_ ku_ k$ and $\langle m-v,x\rangle =\sum _ kt_ kx_ k$. If $z\in \iota (T)$ and $x=\operatorname {Trop}(z)$, then $x_ k=\log |\chi ^{e_ k}(z)|^2=\log |w_ k(z)|^2$. Let $w^0\in U_ v$, let $I=\{ i:w^0_ i=0\} $ and $J$ its complement, and let $Q=\{ m\in P:t_ i=0\text{ for }i\in I\} $ be the corresponding face (proposition 21.9.2). Put $x^0_ j=\log |w^0_ j|^2$ for $j\in J$.

The facet functions other than $t_ i$, $i\in I$, are positive on the relative interior of $Q$. Indeed, if such a function vanished at a point of the relative interior of $Q$, it would vanish on $Q$ by the observation preceding lemma 21.8.2, so its facet would contain $Q$ and $v$. It would then be one of the facets at $v$ containing $Q$, and these are the facets $i\in I$, because $v+se_ j\in Q$ for $j\in J$ and small $s>0$. Now consider the function $G_0-\sum _{j\in J}t_ jx^0_ j$ on the compact set $Q$, on which the terms of the facets $i\in I$ vanish identically. As in Step 1, applied within the affine span of $Q$, its minimum is not attained on the relative boundary of $Q$. Indeed, if all facet functions other than $t_ i$, $i\in I$, are positive at a point of $Q$, then they are positive on a neighborhood of this point in the affine span $\{ t_ I=0\} $ of $Q$, which therefore lies in $Q$. Hence at a relative boundary point $m_0$ some facet function that is positive on the relative interior vanishes, and moving from $m_0$ towards a relative interior point decreases the function initially, as in Step 1. The minimum point $q$ is unique, because on the relative interior of $Q$ the Hessian of $G_0$ in the directions $e_ j$, $j\in J$, of $Q$ dominates the positive definite matrix $\operatorname {diag}(1/t_ j)_{j\in J}$ coming from the terms $t_ j\log t_ j$.

Near $q$ all facet functions except $t_ i$, $i\in I$, are positive, so on a neighborhood $W$ of $q$ in $M_{\mathbb R}$

where $g$, the sum of the remaining terms, is smooth on $W$. Write $g_ k=\partial g/\partial t_ k$ and $g_{jk}=\partial ^2g/\partial t_ j\partial t_ k$. The point $q$ is characterized by $g_ j(q)=x_ j^0$ for $j\in J$, since on $Q$ we have $G_0=g$. At points of $\operatorname{Int}P\cap W$ the equation $\nabla G_0(t)=x$ reads

Exponentiating the first equations and putting $y_ i=\mathrm e^{x_ i}$, which equals $|w_ i|^2$ on $\iota (T)$, we obtain the system $\Psi (t)=(y_ I,x_ J)$ with

a smooth map on all of $W$. At $t=q$, where $t_ I=0$, the Jacobian matrix of $\Psi $ is block triangular: its $I\times I$ block is the diagonal matrix with entries $\exp (1+g_ i(q))>0$, its $I\times J$ block vanishes, and its $J\times J$ block $(g_{jk}(q))$ is positive definite, because $g$ contains the terms $t_ j\log t_ j$, $j\in J$, and its other terms are convex. Since $\Psi (q)=(0,x_ J^0)$, the inverse function theorem gives a smooth local inverse $t=\Psi ^{-1}(y_ I,x_ J)$ near $(0,x_ J^0)$, with values near $q$. For $y_ I\geq 0$ the solution satisfies

and its other facet functions are positive, so it lies in $P$. If $y_ I>0$, it lies in $\operatorname{Int}P$ and solves $\nabla G_0(t)=x$ with $x_ i=\log y_ i$; since $\nabla G_0$ is injective on $\operatorname{Int}P$, $t=\nabla F_0(x)$.

Step 3: smooth extension. Let $z\in \iota (T)$ be close to $w^0$, put $x=\operatorname {Trop}(z)$, and let $t$ be the coordinates of $\nabla F_0(x)$. Since $F_0(x)=\langle \nabla F_0(x),x\rangle -G_0(\nabla F_0(x))$ and $\log |\chi ^ v(z)|^2=\langle v,x\rangle $, we obtain, using the equations for $i\in I$,

The singular logarithms have canceled. By Step 2, $t=\Psi ^{-1}(|w_ I|^2,\log |w_ J|^2)$, so the right-hand side is a smooth function $h(y_ I,x_ J)$ of $(y_ I,x_ J)=(|w_ I|^2,\log |w_ J|^2)$, which in turn is a smooth function of $w$ near $w^0$, because $w^0_ j\neq 0$ for $j\in J$. Hence $H_ v$ extends smoothly to a neighborhood of $w^0$, and, $w^0\in U_ v$ being arbitrary, to $U_ v$.

Step 4: positivity. On $\iota (T)$ we have $\partial F_0/\partial x_ k=\langle \nabla F_0(x),u_ k\rangle =\langle v,u_ k\rangle +t_ k$, hence $\partial H_ v/\partial x_ k=t_ k$. In the variables $(y_ I,x_ J)$ this says

and both identities persist at $y_ I=0$, because both sides are continuous functions of $(y_ I,x_ J)$. We now compute the complex Hessian of $H_ v$ at $w^0$, where $w_ I=0$. Since $\partial y_ i/\partial \bar w_ k=\delta _{ik}w_ i$ for $i\in I$ and $\partial x_ j/\partial \bar w_ k=\delta _{jk}/\bar w_ k$ for $j\in J$, the chain rule gives at $w^0$

all other terms contain a factor $w_ i$ or $\bar w_ i$ with $i\in I$. On $\{ y_ I=0\} $ we have $t_ I=0$ and $x_ J=g_ J(t)$, so $\partial ^2h/\partial x_ j\partial x_ k=\partial t_ j/\partial x_ k$ is the inverse of the positive definite matrix $(g_{jk})_{j,k\in J}$. Hence the complex Hessian of $H_ v$ at $w^0$, where $t=q$, is block diagonal, with the positive numbers $\exp (-1-g_ i(q))$ on the $I$ block and, on the $J$ block, a positive definite matrix congruent to $(\partial t_ j/\partial x_ k)$ by the invertible diagonal matrix $\operatorname {diag}(1/w_ j)$. It is therefore positive definite at $w^0$. Every point of $U_ v$ arises as such a $w^0$ (with $I=\varnothing $ on $\iota (T)$), so $\mathrm{dd}^{\mathrm{c}}H_ v$ is positive definite on $U_ v$.

Step 5: the metric. On $\iota (T)$, for vertices $v$ and $v'$, we have $H_ v-H_{v'}=\log |\chi ^{v'-v}|^2$ and $s_ v=\chi ^{v-v'}s_{v'}$. The function $\chi ^{v-v'}=s_ v/s_{v'}$ is holomorphic and nowhere zero on $U_ v\cap U_{v'}$, so both sides of the identity $\mathrm e^{-H_ v}=|\chi ^{v-v'}|^2\mathrm e^{-H_{v'}}$ are continuous on $U_ v\cap U_{v'}$; as they agree on the dense subset $\iota (T)$, the identity holds on $U_ v\cap U_{v'}$. Thus the squared norms $|s_ v|^2=\mathrm e^{-H_ v}$ satisfy the transition law of definition 6.14.9 and define a smooth quadratic metric on $L$, with curvature $\mathrm{dd}^{\mathrm{c}}H_ v$ on $U_ v$, which is positive definite by Step 4. For $\tau \in T_ c$ we have $|\chi ^ m(\tau )|=1$, so $\operatorname {Trop}(\tau z)=\operatorname {Trop}(z)$, the weights $H_ v$ are $T_ c$-invariant, and $T_ c$ multiplies the frames $s_ v$ by factors of modulus one; hence the metric is $T_ c$-invariant. On $\iota (T)$ the curvature is $\mathrm{dd}^{\mathrm{c}}H_ v=\mathrm{dd}^{\mathrm{c}}(F_0\circ \operatorname {Trop})$, because $\log |\chi ^ v|^2$ is pluriharmonic; since $\iota (T)$ is dense, the curvature form is the unique smooth extension of this form and represents $c_1(L)$ (section 10.6). In dimension zero the facet sum and both potentials vanish, $L$ is trivial over a point, and its curvature is zero. $\square$