From complex to non-Archimedean geometry: an approach to the YTD conjecture

Abstract: 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.

• The paper develops a variational bridge from complex geometry to Berkovich geometry, interpreting non-Archimedean spaces as directions at infinity of finite-energy geodesic spaces.
• The paper proves that, when Aut(X,L) is trivial, a polarized smooth projective variety admits a cscK metric if and only if it is K-stable, with stability characterized through the non-Archimedean Mabuchi functional.
• The paper identifies major open problems, including entropy regularization, the equivalence of K-stability with ordinary K-stability, and practical methods for verifying stability beyond the Fano setting.

Overview and main result

These lecture notes by Boucksom and Jonsson, expanding on their talks at the 2025 Summer Research Institute in Algebraic Geometry, serve two purposes: they explain the relations between complex algebraic varieties, complex manifolds, and Berkovich spaces over $\mathbb{C}$ equipped with the trivial absolute value; and they use these relations to sketch a proof of a version of the Yau–Tian–Donaldson (YTD) conjecture for constant scalar curvature Kähler (cscK) metrics. The central result is Theorem A: for a polarized smooth projective complex variety $(X,L)$ with trivial identity component of the automorphism group $\mathrm{Aut}^0(X,L)$, there exists a cscK metric in $c_1(L)$ if and only if $(X,L)$ is $\hat{K}$-stable. This is a special case of Theorem A in (Boucksom et al., 18 Sep 2025), and it extends the Fano/Kähler–Einstein case proved by Chen–Donaldson–Sun and Tian [CDS15, Tia15] to general polarized varieties.

The distinctive feature of the approach is that the stability condition is formulated in non-Archimedean terms: $\hat{K}$-stability requires the non-Archimedean Mabuchi functional $M_{na}$ to be nonnegative on the space $E^1_{na}$ of finite-energy psh metrics on the Berkovich analytification $L_{an}$, with strict inequality away from constant potentials.

The finite-dimensional model: norms on a vector space

The notes open with an instructive finite-dimensional model. For a complex vector space $(X,L)$0 of dimension $(X,L)$1, the space $(X,L)$2 of Hermitian norms, additively written as $(X,L)$3, is the Riemannian symmetric space $(X,L)$4, with flats given by bases of $(X,L)$5. Equipping this space with any symmetric, permutation-invariant norm $(X,L)$6 on $(X,L)$7 yields a Finsler metric $(X,L)$8, uniquely geodesic and Busemann convex with respect to distinguished geodesics. The radial limit space $(X,L)$9 of directions of geodesic rays carries a radial metric, and the key observation is that $\mathrm{Aut}^0(X,L)$0 is canonically isometrically isomorphic to $\mathrm{Aut}^0(X,L)$1, the space of non-Archimedean norms on $\mathrm{Aut}^0(X,L)$2 over the trivially valued field $\mathrm{Aut}^0(X,L)$3, which is a union of apartments in the sense of buildings. The same picture holds for arbitrary Riemannian symmetric spaces $\mathrm{Aut}^0(X,L)$4, where the role of $\mathrm{Aut}^0(X,L)$5 is played by the conical Tits building. This model motivates the entire strategy: non-Archimedean geometry appears as the space of directions at infinity of the relevant Archimedean metric space.

Berkovich analytification over the trivial valuation

The notes then review the Berkovich analytification $\mathrm{Aut}^0(X,L)$6 of a variety over $\mathrm{Aut}^0(X,L)$7, emphasizing that it depends only on the field structure of $\mathrm{Aut}^0(X,L)$8. It contains the dense set $\mathrm{Aut}^0(X,L)$9 of divisorial valuations $c_1(L)$0, compactifies it, and admits a center map that is anticontinuous — the preimage of a Zariski closed set is open. Every divisorial valuation arises from an irreducible component of the central fiber of some test configuration, a fact used repeatedly later. The notes also discuss hybrid spaces $c_1(L)$1 fibering over $c_1(L)$2, whose fiber over $c_1(L)$3 is $c_1(L)$4 and whose fibers over $c_1(L)$5 are homeomorphic to the complex manifold $c_1(L)$6; these are presented as conceptual background rather than as tools used in the proof.

For a line bundle $c_1(L)$7, metrics on $c_1(L)$8 are defined by the same log-homogeneity condition as in the analytic setting. When $c_1(L)$9 is ample, the space $(X,L)$0 of non-Archimedean Fubini–Study metrics stands in bijection with normal ample test configurations for $(X,L)$1 via the potential function associated to a test configuration. A canonical object absent from the Archimedean side is the trivial metric $(X,L)$2, which exists on every line bundle over a proper variety; consequently Calabi's question has a trivial answer in the trivially valued case. The log discrepancy function $(X,L)$3 extends lower-semicontinuously to $(X,L)$4 and plays the role of a would-be measure density via $(X,L)$5, following Temkin's metrization theory.

Global pluripotential theory in both worlds

The technical core of the notes is the parallel development of global pluripotential theory on the two sides, based on the authors' prior work [trivval, nakstab1]. On each side one has spaces of psh metrics $(X,L)$6 and $(X,L)$7, defined identically as the smallest classes containing section metrics and closed under maxima, constants, and decreasing limits; continuity of envelopes holds in both settings. The Monge–Ampère energy $(X,L)$8 extends to the space $(X,L)$9 of finite-energy metrics, and the strong topology makes $\hat{K}$0 continuous. In the non-Archimedean case, the Monge–Ampère energy of a Fubini–Study potential reduces to the normalized top self-intersection number $\hat{K}$1, and the non-Archimedean Monge–Ampère operator produces divisorial measures supported on finitely many valuations.

The finite-energy Calabi–Yau theorem asserts that $\hat{K}$2 is a homeomorphism in the strong topology, and the identical statement holds non-Archimedeanly. However, regularity diverges sharply: by Yau's theorem, a finite-energy metric solving $\hat{K}$3 for a smooth volume form lies in $\hat{K}$4, whereas non-Archimedeanly there is no characterization of when a solution belongs to $\hat{K}$5; the authors state plainly that such a characterization "seems currently out of reach." Another divergence concerns entropy: bounded-entropy sets are strongly compact in $\hat{K}$6 but not in $\hat{K}$7, one of the few points where the parallelism breaks down.

The Mabuchi functional is defined on both sides via the Chen–Tian formula $\hat{K}$8, where $\hat{K}$9 is the entropy term and $\hat{K}$0 the Ricci energy. In the complex case, $\hat{K}$1 is the maximal lsc extension from $\hat{K}$2 to $\hat{K}$3; in the non-Archimedean case this is not known, and its validity constitutes the entropy regularization conjecture, equivalent to density of $\hat{K}$4 within divisorial measures with entropy control. The obstruction is precisely the missing characterization of $\hat{K}$5 noted above.

Geodesic convexity and asymptotics along rays

Following Darvas, the completion of $\hat{K}$6 is identified with $\hat{K}$7, and $\hat{K}$8 is uniquely geodesic and Busemann convex with respect to psh geodesics. The radial limit space $\hat{K}$9 of directions of psh geodesic rays contains $M_{na}$0 via Berman's envelope construction applied to test configurations: the map $M_{na}$1 is injective, satisfies the slope formula $M_{na}$2, and was shown by Reboulet to be an isometry onto its image; its image consists of the maximal geodesic rays. Ricci energy obeys the analogous slope identity.

The decisive input, proved in (Boucksom et al., 18 Sep 2025) and strengthening results of C. Li [LiGeod], is the asymptotic formula for the Mabuchi functional:

$M_{na}$3

The infinite-slope alternative outside $M_{na}$4 is due to Li; the equality in the maximal case requires controlling the slope of the entropy functional, using [BHJ2] together with regularity results for psh envelopes [BD12, DNT24].

Proof of the YTD correspondence

With these ingredients, the argument proceeds through three equivalent conditions. First, a theorem of Darvas–Lu and Chen–Cheng states that when $M_{na}$5 is trivial, existence of a cscK metric is equivalent to geodesic stability (nonnegativity of the slope of $M_{na}$6 along all psh geodesic rays) and to coercivity of $M_{na}$7 on $M_{na}$8. Second, the asymptotic formula identifies geodesic stability with $M_{na}$9-stability, since rays outside $E^1_{na}$0 automatically have positive slope. Third, coercivity of $E^1_{na}$1 along maximal rays transfers to a uniform lower bound $E^1_{na}$2, giving uniform $E^1_{na}$3-stability. Thus cscK existence, $E^1_{na}$4-stability, and uniform $E^1_{na}$5-stability are all equivalent under the triviality assumption on $E^1_{na}$6 — a quantitative strengthening, since mere $E^1_{na}$7-stability already forces uniformity.

Conceptually, the theorem says that existence of a cscK metric is equivalent to the trivial metric on $E^1_{na}$8 being a "non-Archimedean cscK metric," i.e., a minimizer of $E^1_{na}$9.

Extensions, limitations, and open questions

The full results in (Boucksom et al., 18 Sep 2025) remove the triviality assumption on $L_{an}$0, replacing constants by real product test configurations ($L_{an}$1-polystability), and cover extremal and weighted extremal metrics in the sense of Lahdili. Mesquita-Piccione has developed the transcendental version for arbitrary Kähler classes, proving that uniform $L_{an}$2-stability implies cscK existence, though the converse remains unproved in that setting.

Several questions are left open. Whether (uniform) $L_{an}$3-stability is equivalent to (uniform) K-stability — the latter tested only on $L_{an}$4 rather than all of $L_{an}$5 — is unknown in general; a positive answer would follow from the entropy regularization conjecture, which itself hinges on characterizing $L_{an}$6 among divisorial measures. Even granting the valuative criterion of [nakstab2], which reduces uniform $L_{an}$7-stability to a check on divisorial measures with an explicit invariant, verification remains impractical in general, in contrast to the Fano case where Minimal Model Program techniques have made K-stability checks feasible, as in the classification of Fano threefolds [ACC+, Fuj23]. Finally, Darvas and Zhang [DZ25] have independently established a different YTD correspondence using a regularized functional $L_{an}$8 and $L_{an}$9-stability; the precise relationship between the two frameworks is not addressed in these notes.

Conclusion

These notes present a coherent variational framework in which the YTD correspondence for cscK metrics becomes a statement about the interplay between the geodesic geometry of $(X,L)$00 and the pluripotential theory of the trivially valued analytification. The equivalence between cscK existence and uniform $(X,L)$01-stability rests on the slope formula for the Mabuchi functional along maximal geodesic rays, and the remaining gaps — entropy regularization, the relation between $(X,L)$02- and K-stability, and practical verifiability of the stability conditions — are stated explicitly and reduce to identifiable analytic and algebro-geometric problems.