25.1.20 Convex duality and real Monge–Ampère measures482A6169B3A6

For the separation of convex sets and for the local regularity and subgradients of convex functions (lemma 21.2.3, proposition 21.2.4) compare [Theorem 2.1.10 and Corollaries 2.1.11–2.1.13, pp. 44–45; Theorem 2.1.22, pp. 55–56, Hor07]. Closed convex functions and Legendre conjugation are treated in [Propositions 2.2.1–2.2.2 and Theorem 2.2.4, pp. 66–68; Theorem 2.2.5, pp. 68–69; Theorem 2.2.8, p. 70, Hor07]. The corresponding duality statements in this book are theorem 21.3.2, proposition 21.3.3; in the formula for a sum in which the infimum is attained, both functions are assumed to be finite on the whole space.

The classes of convex functions, the formulas involving slope bodies and the real Monge–Ampère measures follow the appendix on convex analysis of [XiaSampleBook]; as there, the factor $n!$ is kept, with Lebesgue measure normalized by the lattice. In theorem 21.5.1, measurability with respect to the completed Lebesgue measure and countable additivity are proved using the fact that the inverse of the subgradient map is single-valued almost everywhere. Theorem 21.5.2 concerns vague convergence, and example 21.5.3 shows that narrow convergence requires an additional condition that prevents mass from escaping to infinity. The mixed measures of proposition 21.5.4 are obtained from the positivity of matrices, convolution and polarization. We keep the convention of the source for full mass when a slope body has empty interior, and example 21.4.5 shows that the hypotheses on slope bodies in the formula for sums cannot be omitted.

The lemma on the finiteness of rotation-invariant plurisubharmonic functions, [Lemma 1.2.1, XiaSampleBook], is proved there by the logarithmic convexity of circle means, and it provides the finiteness step in [Lemma 5.2.1, XiaSampleBook]. Proposition 21.2.8, Corollary 21.2.9 give a direct proof, by smoothing and the fact that the complex Hessian of a function invariant under imaginary translations is one quarter of its real Hessian; this proof does not use Kiselman's minimum principle.

For the conjugate of the maximum of finitely many convex functions, the convex appendix of [XiaSampleBook] uses [Theorem 16.5, Roc70]. Lemma 21.6.1 proves the case of functions whose domains lie in a compact set, which is the case of the finite suprema in the chapter on ample toric varieties, [Proposition 5.2.4, XiaSampleBook]; compactness takes care of points whose weights tend to zero. The computation of the energy in [Proposition 5.2.9, Step 1, XiaSampleBook] passes to the limit in integrals of a bounded function on a noncompact space against real Monge–Ampère measures; lemma 21.6.2 justifies this passage to the limit by a cutoff and the convergence of total masses.

The asymptotic count of lattice points used in [Section 5.2, XiaSampleBook] follows from the case of saturated cones in [Theorem 1.15, p. 939, KK12], where Kaveh and Khovanskii reduce the assertion to Riemann sums; their introduction states the more general theorem on the growth of semigroups on p. 929. Theorem 21.7.3 gives a complete proof for convex bodies by means of fundamental cells, allowing arbitrary translations, bodies of smaller dimension and sublinear errors at the boundary. The continuity of volume needed in its proof is proposition 21.7.2, and corollary 21.7.4 isolates the two-sided bound used in the toric computation of dimensions.

The chapter on ample toric varieties in [XiaSampleBook] relies on the theory of toric varieties developed in [CLS11]. Here the smooth projective toric variety of a Delzant lattice polytope is constructed directly from its characters in theorem 21.8.3; its fan and its boundary divisors are identified in proposition 21.9.1, proposition 21.9.2, and the basis of global sections given by characters, theorem 21.9.5, is obtained from Laurent expansions in the charts at the vertices. No general classification of toric varieties is needed.

The potential of theorem 21.10.1, built from the function $G_0=\sum _\rho \ell _\rho \log \ell _\rho $, is Guillemin's reference potential. For the Legendre computation compare [Proposition 2, pp. 7–8; Theorem 1, pp. 9–10, CDG03v1] (first arXiv version). That source uses $G=\frac12\sum _\rho \ell _\rho \log \ell _\rho $ and holomorphic logarithmic coordinates $y+\mathrm i\theta $; with $x=2y=\log |w|^2$ we have $G_0=2G$ and $F_0(x)=2F(x/2)$, up to an additive constant. Its convention $\mathrm d^{\mathrm c}f=-\mathrm df\circ J$ gives $\mathrm d\mathrm d^{\mathrm c}=2\mathrm i\partial \bar\partial $; with the curvature convention of this book the form is divided by $2\pi $. Smooth extension and strict positivity in the monomial charts are proved by the inverse function theorem at every face, including the vertices.