arXiv subjects
Publications and source records attributed to Sébastien Boucksom.
These notes expand on talks given by the authors at the 2025 Summer Research Institute in Algebraic Geometry in Fort Collins, Colorado. We discuss the relation between algebraic, analytic, and non-Archimedean geometry over the complex numbers, and sketch a proof of a version of the Yau--Tian--Donaldson conjecture for constant scalar curvature Kähler metrics.
Generalizing previous results of Arezzo-Pacard-Singer, Seyyedali-Székelyhidi and Hallam, we prove the invariance under smooth blowups of the class of weighted extremal Kähler manifolds, modulo a log-concavity assumption on the first weight. Through recent work of Di Nezza-Jubert-Lahdili and Han-Liu, this is obtained as a consequence of a general uniform coercivity estimate for the (relative, weighted) Mabuchi energy on the blowup, which applies more generally to any equivariant resolution of singularities of Fano type of a compact Kähler klt space whose Mabuchi energy is assumed to be coercive.
We establish a version, formulated in terms of non-Archimedean pluripotential theory, of the Yau-Tian-Donaldson conjecture for constant scalar curvature and, more generally, weighted extremal Kähler metrics with prescribed compact symmetry group on an arbitrary polarized projective manifold. This is accomplished by extending important previous work of Chi Li to the weighted case, and by establishing general slope formulas for the relevant weighted entropy and energy functionals. Among other things, our approach relies on key a priori estimates for cscK metrics due to Chen-Cheng (recently extended to the weighted case by Di Nezza-Jubert-Lahdili and Han-Liu), and on a crucial estimate for psh envelopes due to Berman-Demailly-Di Nezza-Trapani.
We study the volumes of transcendental and possibly non-closed Bott-Chern $(1,1)$-classes on an arbitrary compact complex manifold $X$. We show that the latter belongs to the class $\mathcal{C}$ of Fujiki if and only if it has the $\textit{bounded mass property}$ -- i.e., its Monge-Ampère volumes have a uniform upper-bound -- and there exists a closed Bott-Chern class with positive volume. This yields a positive answer to a conjecture of Demailly-Păun-Boucksom. To this end we extend to the hermitian context the notion of non-pluripolar products of currents, allowing for the latter to be merely ${\it quasi}$-${\it closed}$ and ${\it quasi}$-${\it positive}$. We establish a quasi-monotonicity property of Monge-Ampère masses, and moreover show the existence of solutions to degenerate complex Monge-Ampère equations in big classes, together with uniform a priori estimates. This extends to the hermitian context fundamental results of Boucksom-Eyssidieux-Guedj-Zeriahi.
For any polarized variety (X,L), we show that test configurations and, more generally, R-test configurations (defined as finitely generated filtrations of the section ring) can be analyzed in terms of Fubini-Study functions on the Berkovich analytification of X with respect to the trivial absolute value on the ground field. Building on non-Archimedean pluripotential theory, we describe the (Hausdorff) completion of the space of test configurations, with respect to two natural pseudo-metrics, in terms of plurisubharmonic functions and measures of finite energy on the Berkovich space. We also describe the Hausdorff quotient of the space of all filtrations, and establish a 1--1 correspondence between divisorial norms and divisorial measures, both being determined in terms of finitely many divisorial valuations.
We analyze the existence of Kähler-Einstein metrics of positive curvature in the neighborhood of a germ of a log terminal singularity $(X,p)$. This boils down to solve a Dirichlet problem for certain complex Monge-Ampère equations. We show that the solvability of the latter is independent of the shape of the domain and of the boundary data. We establish a Moser-Trudinger $(MT)_γ$ inequality in subcritical regimes $γ<γ_p$ and establish the existence of smooth solutions in that cases. We show that the expected critical exponent $\hatγ_p=\frac{n+1}{n} \widehat{\mathrm{vol}}(X,p)^{1/n}$ can be expressed in terms of the normalized volume, an important algebraic invariant of the singularity.
We develop global pluripotential theory in the setting of Berkovich geometry over a trivially valued field. Specifically, we define and study functions and measures of finite energy and the non-Archimedean Monge-Ampere operator on any (possibly reducible) projective variety. We also investigate the topology of the space of valuations of linear growth, and the behavior of psh functions thereon.
Let $L$ be a line bundle on a proper, geometrically reduced scheme $X$ over a non-trivially valued non-Archimedean field $K$. Roughly speaking, the non-Archimedean volume of a continuous metric on the Berkovich analytification of $L$ measures the asymptotic growth of the space of small sections of tensor powers of $L$. For a continuous semipositive metric on $L$ in the sense of Zhang, we show first that the non-Archimedean volume agrees with the energy. The existence of such a semipositive metric yields that $L$ is nef. A second result is that the non-Archimedean volume is differentiable at any semipositive continuous metric. These results are known when $L$ is ample, and the purpose of this paper is to generalize them to the nef case. The method is based on a detailed study of the content and the volume of a finitely presented torsion module over the (possibly non-noetherian) valuation ring of $K$.
It was conjectured by Lang that a complex projective manifold is Kobayashi hyperbolic if and only if it is of general type together with all of its subvarieties. We verify this conjecture for projective manifolds whose universal cover carries a bounded, strictly plurisubharmonic function. This includes in particular compact free quotients of bounded domains.
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 Knudsen-Mumford expansion for the determinant of cohomology on models over the (possibly non-Noetherian) valuation ring, as a replacement for the asymptotic expansion of Bergman kernels in the complex case, and on the reduced fiber theorem, as a replacement for the Bernstein-Markov inequalities. Along the way, a systematic study of spaces of norms and the associated Fubini-Study type metrics is undertaken.
We give a variational proof of a version of the Yau-Tian-Donaldson conjecture for twisted Kähler-Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in terms of a purely algebro-geometric stability threshold. Our approach does not involve the continuity method or Cheeger-Colding-Tian theory, and uses instead pluripotential theory and valuations. Along the way, we study the relationship between geodesic rays and non-Archimedean metrics.
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 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 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-stability by Fujita and Li. Finally, we study the asymptotic Fubini-Study operator, which associates a psh metric to any graded filtration (or norm) on the anticanonical ring. Our results hold for arbitrary smooth polarized varieties, and suitable adjoint/twisted notions of K-stability and Ding stability. They do not rely on the Minimal Model Program.
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.
We study the asymptotic behavior of volume forms on a degenerating family of compact complex manifolds. Under rather general conditions, we prove that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex. In particular, this provides a measure-theoretic version of a conjecture by Kontsevich--Soibelman and Gross--Wilson, bearing on maximal degenerations of Calabi--Yau manifolds.
We prove a conjecture of Boucksom-Demailly-Păun-Peternell, namely that on a projective manifold $X$ the cone of pseudoeffective classes in $H^{1,1}_{\mathbb{R}}(X)$ is dual to the cone of movable classes in $H^{n-1,n-1}_{\mathbb{R}}(X)$ via the Poincaré pairing. This is done by establishing a conjectured transcendental Morse inequality for the volume of the difference of two nef classes on a projective manifold. As a corollary the movable cone is seen to be equal to the closure of the cone of balanced metrics. In an appendix by Boucksom it is shown that the Morse inequality also implies that the volume function is differentiable on the big cone, and one also gets a characterization of the prime divisors in the non-Kähler locus of a big class via intersection numbers.
The purpose of the present paper is to set up a formalism inspired from non-Archimedean geometry to study K-stability. We first provide a detailed analysis of Duistermaat-Heckman measures in the context of test configurations, characterizing in particular the trivial case. For any normal polarized variety (or, more generally, polarized pair in the sense of the Minimal Model Program), we introduce and study the non-Archimedean analogues of certain classical functionals in Kähler geometry. These functionals are defined on the space of test configurations, and the Donaldson-Futaki invariant is in particular interpreted as the non-Archimedean version of the Mabuchi functional, up to an explicit error term. Finally, we study in detail the relation between uniform K-stability and singularities of pairs, reproving and strengthening Y. Odaka's results in our formalism. This provides various examples of uniformly K-stable varieties.