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Foundations of pluripotential theory

Mingchen Xia

These contributions appear throughout the shared chapters.

Source revision da2be417c7b3 ↗

Attribution and further reading

Original source commentary and bibliographical references accompany the corresponding Pome entries here.

1.1 Conventions

The function $\mathbb {R}\ni x \mapsto x^2$ is convex instead of concave. This is different from the convention in [CLS11].

17.1 Relative energy and Perron segments

With these equations we show in section 17.6 that, in a big class, the singularity ceiling of a potential $u$ (definition 17.6.4) coincides with its singularity envelope $P_\theta [u]$, and that $P_\theta [P_\theta [u]]=P_\theta [u]$ (theorem 17.6.5). This is [Theorem 1.2, PSWZ26]; it includes the case of zero mass, $\int _ X\langle \theta _ u^ n\rangle =0$, to which the characterization of $P_\theta [u]$ in theorem 16.22.11 does not apply.

21.1 Convex analysis and toric potentials

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). These are the foundations of convex analysis and of ample toric geometry used in [Part I, XiaSampleBook].

9.1 Complex spaces and modifications

The last section defines plurisubharmonic functions on reduced complex spaces by local extension to an ambient space, proves that they are constant on compact connected spaces, and shows that they are locally bounded above near analytic subsets of codimension at least two. Holomorphic curves through singular points, constructed in section 9.5, reduce these statements to the maximum principle in one variable. These results on reduced spaces are used in [XiaSampleBook]. Historical remarks and references for this appendix are collected in subsection 25.1.21.

9.6 Blowups and resolution

In the notation of [BM08], theorem 9.6.7 is the theorem for marked ideals with mark $1$ and $N=M$. The cosupport of $\mathfrak a_ i$ is $V(\mathfrak a_ i)$, and the transform (9.6.8) divides by a single exceptional factor at each step, even where $\mathfrak a_{i-1}$ vanishes to higher order along the center. If $M$ is compact, the sequence is finite, say with $t$ steps, and iterating (9.6.8) gives

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