Over a non-archimedean local place, the height of a projective variety with respect to a very ample line bundle equipped with a Fubini-Study metric is related to the naive height of its Chow form. Using a non-Archimedean Kempf-Ness criteria, we characterize Fubini-Study metrics that minimize the height under the special linear action in terms of their Monge-Ampère polytopes. This polytope can be constructed either as its non-Archimedean Bergman functional or as the weight polytope for the residual action on its Chow form; it is associated with a polymatroid.
Given an ample line bundle $L$ over a projective $\mathbb{K}$-variety $X$, with $\mathbb{K}$ a non-Archimedean field, we study limits of non-Archimedean metrics on $L$ associated to submultiplicative sequences of norms on the graded pieces of the section ring $R(X,L)$. We show that in a rather general case, the corresponding asymptotic Fubini-Study operator yields a one-to-one correspondence between equivalence classes of bounded graded norms and bounded plurisubharmonic metrics that are regularizable from below. This generalizes results of Boucksom-Jonsson where this problem has been studied …
Let L be an ample line bundle on a (geometrically reduced) projective variety X over any complete valued field. Our main result describes the leading asymptotics of the determinant of cohomology of large powers of L, with respect to the supnorm of a continuous metric on the Berkovich analytification of L. As a consequence, we establish in this setting the existence of transfinite diameters and equidistribution of Fekete points, following a strategy going back Berman, Witt Nyström and the first author for complex manifolds. In the non-Archimedean case, our approach relies on a version of the Kn…
We study K-stability properties of a smooth Fano variety X using non-Archimedean geometry, specifically the Berkovich analytification of X with respect to the trivial absolute value on the ground field. More precisely, we view K-semistability and uniform K-stability as conditions on the space of plurisubharmonic (psh) metrics on the anticanonical bundle of X. Using the non-Archimedean Calabi-Yau theorem and the Legendre transform, this allows us to give a new proof that K-stability is equivalent to Ding stability. By choosing suitable psh metrics, we also recover the valuative criterion of K-s…
We present some results that complement our prequels [arXiv:1809.08425,arXiv:1907.05770] on holomorphic vector bundles. We apply the method of the Quot-scheme limit of Fubini-Study metrics developed therein to provide a generalisation to the singular case of the result originally obtained by X.W. Wang for the smooth case, which states that the existence of balanced metrics is equivalent to the Gieseker stability of the vector bundle. We also prove that the Bergman 1-parameter subgroups form subgeodesics in the space of hermitian metrics. This paper also contains a review of techniques develope…
This is a sequel of our paper [arXiv:1809.08425] on the Quot-scheme limit and variational properties of Donaldson's functional, which established its coercivity for slope stable holomorphic vector bundles over smooth projective varieties. Assuming that the coercivity is uniform in a certain sense, we provide a new proof of the Donaldson-Uhlenbeck-Yau theorem, in such a way that the analysis involved in the proof is elementary except for the asymptotic expansion of the Bergman kernel.
Let $X$ be a proper algebraic variety over a non-archimedean, non-trivially valued field. We show that the non-archimedean Monge-Ampère measure of a metric arising from a convex function on an open face of some skeleton of $X^{\text{an}}$ is equal to the real Monge-Ampère measure of that function up to multiplication by a constant. As a consequence we obtain a regularity result for solutions of the non-archimedean Monge-Ampère problem on curves.
We survey the metric aspects of the Strominger-Yau-Zaslow conjecture on the existence of special Lagrangian fibrations on Calabi-Yau manifolds near the large complex structure limit. We will discuss the diverse motivations for the conjectural picture, what the best hopes are, and a number of subtleties. The bulk of the survey highlights the role of pluripotential theory, and non-archimedean geometry in particular, with a list of open questions.
Given a polarized projective variety (X,L) over any non-Archimedean field, assuming continuity of envelopes, we show that the space of finite-energy metrics on L is a geodesic metric space, where geodesics are given as maximal psh segments. Given two continuous psh metrics, we show that the maximal segment joining them is furthermore continuous.
For a projective variety $X$ defined over a non-Archimedean complete non-trivially valued field $k$, and a semipositive metrized line bundle $(L, ϕ)$ over it, we establish a metric extension result for sections of $L^{\otimes n}$ from a sub-variety $Y$ to $X$. We form normed section algebras from $(L, ϕ)$ and study their Berkovich spectra. To compare the supremum algebra norm and the quotient algebra norm on the restricted section algebra $V(L_{X|Y})$, two different methods are used: one exploits the holomorphic convexity of the spectrum, following an argument of Grauert; another relies on fin…
This is the second in a series of two papers studying $μ$-cscK metrics and $μ$K-stability from a new perspective, inspired by observations on $μ$-character in arXiv:2004.06393 and on Perelman's $W$-entropy in the first paper arXiv:2101.11197. This second paper is devoted to studying a non-archimedean counterpart of Perelman's $μ$-entropy. The concept originally appeared as $μ$-character of polarized family in the previous research arXiv:2004.06393, where we used it to introduce an analogue of CM line bundle adapted to $μ$K-stability. We firstly show some differential of the characteristic $μ$-…
Let $X$ be a normal projective variety over a complete discretely valued field and $L$ a line bundle on $X$. We denote by $X^\textrm{an}$ the analytification of $X$ in the sense of Berkovich and equip the analytification $L^\textrm{an}$ of $L$ with a continuous metric $\| \ \|$. We study non-archimedean volumes, a tool which allows us to control the asymptotic growth of small sections of big powers of $L$. We prove that the non-archimedean volume is differentiable at a continuous semipositive metric and that the derivative is given by integration with respect to a Monge-Ampère measure. Such a …
Let L be an ample line bundle on a smooth projective variety X over a non-archimedean field K. For a continuous metric on L, we show in the following two cases that the semipositive envelope is a continuous semipositive metric on L and that the non-archimedean Monge-Ampère equation has a solution. First, we prove it for curves using results of Thuillier. Second, we show it under the assumption that X is a surface defined geometrically over the function field of a curve over a perfect field k of positive characteristic. The second case holds in higher dimensions if we assume resolution of singu…
Given an ample line bundle $L$ on a geometrically reduced projective scheme defined over an arbitrary non-Archimedean field, we establish a differentiability property for the relative volume of two continuous metrics on the Berkovich analytification of $L$, extending previously known results in the discretely valued case. As applications, we provide fundamental solutions to certain non-Archimedean Monge--Ampère equations, and generalize an equidistribution result for Fekete points. Our main technical input comes from determinant of cohomology and Deligne pairings.
We study two kinds of push-forwards of $δ$-forms and define the pull-backs of $δ$-forms. As a generalization of Gubler-Künnemann, we prove the projection formula and the tropical Poincaré-Lelong formula. As an application, we follow the idea of Gubler-Künnemann and generalize the notion of $δ$-forms on algebraic varieties, this allows us to define the first Chern forms for any piecewise smooth metrics.
We survey some recent developments in the direction of the Yau-Tian-Donaldson conjecture, which relates the existence of constant scalar curvature Kähler metrics to the algebro-geometric notion of K-stability. The emphasis is put on the use of pluripotential theory and the interpretation of K-stability in terms of non-Archimedean geometry.
This is the first in a series of two papers studying mu-cscK metrics and muK-stability, from a new perspective evoked from observations in arXiv:2004.06393 and in this first article. The first paper is about a characterization of mu-cscK metrics in terms of Perelman's W-entropy $\check{W}^λ$. We regard Perelman's W-entropy as a functional on the tangent bundle $T \mathcal{H} (X, L)$ of the space $\mathcal{H} (X, L)$ of K"ahler metrics in a given K"ahler class $L$. The critical points of $\check{W}^λ$ turn out to be $μ^λ$-cscK metrics. When $λ\le 0$, the supremum along the fibres gives a smooth…
While the existence of a unique Kähler-Einstein metric on a canonically polarized manifold X was established by Aubin and Yau already in the 70s there are only a few explicit formulas available. In previous work a probabilistic construction of the Kähler-Einstein metric was introduced - involving canonical random point processes on X - which yields canonical approximations of the Kähler-Einstein metric, expressed as explicit period integrals over a large number of products of X. Here it is shown that the conjectural extension to the case when X is a Fano variety suggests a zero-free property o…
Zhang introduced semipositive metrics on a line bundle of a proper variety. In this paper, we generalize such metrics for a line bundle $L$ of a paracompact strictly $K$-analytic space $X$ over any non-archimedean field $K$. We prove various properties in this setting such as density of piecewise $\mathbb{Q}$-linear metrics in the space of continuous metrics on $L$. If $X$ is proper scheme, then we show that algebraic, formal and piecewise linear metrics are the same. Our main result is that on a proper scheme $X$ over an arbitrary non-archimedean field $K$, the set of semipositive model metri…
Consider a polarized complex manifold (X,L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X,L). For most of the common functionals in Kähler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as defined in our earlier paper) at the non-Archimedean metric on L defined by the test configuration. Using this asymptotic result, we show that coercivity of the Mabuchi functional implies uniform K-stability.
Given a degeneration of compact projective complex manifolds $X$ over the punctured disc, with meromorphic singularities, and a relatively ample line bundle $L$ on $X$, we study spaces of plurisubharmonic metrics on $L$, with particular focus on (relative) finite-energy conditions. We endow the space $\hat \cE^1(L)$ of relatively maximal, relative finite-energy metrics with a $d_1$-type distance given by the Lelong number at zero of the collection of fibrewise Darvas $d_1$-distances. We show that this metric structure is complete and geodesic. Seeing $X$ and $L$ as schemes $X_\K$, $L_\K$ over …
We show that, up to biholomorphism, there is at most one complete $T^n$-invariant shrinking gradient Kähler-Ricci soliton on a non-compact toric manifold $M$. We also establish uniqueness without assuming $T^n$-invariance if the Ricci curvature is bounded and if the soliton vector field lies in the Lie algebra $\mathfrak{t}$ of $T^n$. As an application, we show that, up to isometry, the unique complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on $\mathbb{CP}^{1} \times \mathbb{C}$ is the standard product metric associated to the Fubini-Study metric on $\mathbb{CP}^…
