21.1 Convex analysis and toric potentials4C57CCF28620

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Let the compact torus $(S^1)^ n$ act on $(\mathbb C^*)^ n$ by separate rotations of the coordinates, $(z_1,\ldots ,z_ n)\mapsto (\mathrm e^{\mathrm i\theta _1}z_1,\ldots ,\mathrm e^{\mathrm i\theta _ n}z_ n)$. A plurisubharmonic function on $(\mathbb C^*)^ n$ that is invariant under this action depends only on the logarithmic radii $x_ j=\log |z_ j|^2$, and we show in corollary 21.2.9 that, as a function of $x\in \mathbb R^ n$, it is precisely a finite convex function. The complex Hessian accounts for the convexity; conversely, the supporting affine functions of a convex function give the submean inequality on complex lines. Questions about the growth and the Monge–Ampère mass of invariant potentials thus become questions about convex functions on a real vector space, and we answer them in terms of the slopes of supporting affine functions.

The first part of the chapter develops the convex analysis that this requires. Separation of convex sets produces supporting affine functions, and Legendre duality identifies the closure of the set of slopes of a convex function $F$ with the closure of the set where its conjugate $F^*$ is finite. An upper bound of $F$, up to a constant, by the support function of a convex body $P$ holds exactly when all slopes of $F$ lie in $P$ (proposition 21.4.2); this leads to convex analogues of the classes of potentials with bounded growth, full mass and finite energy. The real Monge–Ampère measure of $F$ assigns to a set $E$ the volume of the set of slopes of $F$ at points of $E$, multiplied by $n!$. Since the subgradient map is multivalued, we construct this measure by pushing Lebesgue measure forward under the gradient of $F^*$, which is defined almost everywhere (theorem 21.5.1). Its total mass is $n!$ times the volume of the closed set of slopes, and it depends continuously on $F$ for local uniform convergence (theorem 21.5.2); when no mass escapes to infinity, the convergence also holds against bounded continuous functions. Finally, counting lattice points in the dilates $kP$ relates these volumes to dimensions of spaces of sections.

The second part constructs, from a smooth lattice polytope $P$, a projective toric manifold $X_ P$ with a very ample line bundle $L$. Monomial charts at the vertices of $P$ make its geometry explicit, and we read off from them the correspondence between torus orbits and faces of $P$, the orders of characters along the boundary divisors, and a basis of $H^0(X_ P,L^ k)$ indexed by the lattice points of $kP$ (theorem 21.9.5). The potential of Guillemin's reference metric on $L$ is the Legendre transform of an explicit entropy function on $P$, whose gradient has logarithmic singularities along the facets. In the same charts we show that these singularities cancel after the transformation, so that the metric is smooth, with positive definite curvature, also along the boundary orbits (theorem 21.10.1).

Throughout the chapter, $N_{\mathbb R}$ is a real vector space of finite dimension $n\geq 0$ and $M_{\mathbb R}$ is its dual space, with pairing $\langle p,x\rangle $ for $p\in M_{\mathbb R}$ and $x\in N_{\mathbb R}$; we identify the dual of $M_{\mathbb R}$ with $N_{\mathbb R}$. Whenever an estimate requires a norm, we fix a Euclidean norm on $N_{\mathbb R}$ and give $M_{\mathbb R}$ the dual norm $|p|=\max _{|x|\leq 1}\langle p,x\rangle $, so that $|\langle p,x\rangle |\leq |p||x|$; the open and closed balls $B(x,r)$ and $\overline{B(x,r)}$ of section 2.5 are taken with respect to these norms. When volumes are involved, we also fix a lattice $N\subset N_{\mathbb R}$, that is, the set of integral linear combinations of a basis of $N_{\mathbb R}$, and the dual lattice $M=\{ p\in M_{\mathbb R}:\langle p,x\rangle \in \mathbb Z\text{ for all }x\in N\} $. Lebesgue measure on $M_{\mathbb R}$, written $\mathrm dp$ or $\operatorname {vol}$, and Lebesgue measure $\mathrm dx$ on $N_{\mathbb R}$ are normalized so that a fundamental parallelepiped of $M$, respectively of $N$, has volume one; equivalently, they are Lebesgue measure in the coordinates of a lattice basis. (This volume of subsets of $M_{\mathbb R}$ is not to be confused with the volume of a cohomology class in definition 13.9.1.) In dimension zero both spaces are a point and both measures are the unit point mass. A convex body is a nonempty compact convex set, possibly with empty interior.

Historical remarks and references for this chapter are collected in subsection 25.1.20.