Inspired by K. Fujita's algebro-geometric result that complex projective space has maximal degree among all K-semistable complex Fano varieties, we conjecture that the height of a K-semistable metrized arithmetic Fano variety $\mathcal {X}$ of relative dimension $n$ is maximal when $\mathcal {X}$ is the projective space over the integers, endowed with the Fubini–Study metric. Our main result establishes the conjecture for the canonical integral model of a toric Fano variety when $n\leq 6$ (the extension to higher dimensions is conditioned on a conjectural ‘gap hypothesis’ for the degree). Translated into toric Kähler geometry, this result yields a sharp lower bound on a toric invariant introduced by Donaldson, defined as the minimum of the toric Mabuchi functional. Furthermore, we reformulate our conjecture as an optimal lower bound on Odaka's modular height. In any dimension $n$ it is shown how to control the height of the canonical toric model $\mathcal {X},$ with respect to the Kähler–Einstein metric, by the degree of $\mathcal {X}$. In a sequel to this paper our height conjecture is established for any projective diagonal Fano hypersurface, by exploiting a more general logarithmic setup.

Kähler–Einstein currents, also known as singular Kähler–Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact Kähler spaces X and their two defining properties are the following: They are genuine Kähler–Einstein metrics on $X_{\mathrm {reg}}$, and they admit local bounded potentials near the singularities of X. In this note, we show that these currents dominate a Kähler form near the singular locus, when either X admits a global smoothing, or when X has isolated smoothable singularities. Our results apply to klt pairs and allow us to show that if X is any compact Kähler space of dimension three with log terminal singularities, then any singular Kähler–Einstein metric of nonpositive curvature dominates a Kähler form.

Divisorial stability of a polarised variety is a stronger – but conjecturally equivalent – variant of uniform K-stability introduced by Boucksom–Jonsson. Whereas uniform K-stability is defined in terms of test configurations, divisorial stability is defined in terms of convex combinations of divisorial valuations on the variety.

We consider the behaviour of divisorial stability under finite group actions and prove that equivariant divisorial stability of a polarised variety is equivalent to log divisorial stability of its quotient. We use this and an interpolation technique to give a general construction of equivariantly divisorially stable polarised varieties.

We prove that all smooth Fano threefolds in the families and are K-stable, and we also prove that smooth Fano threefolds in the family that satisfy one very explicit generality condition are K-stable.

In this paper, we define two types of strongly decomposable positivity, which serve as generalizations of (dual) Nakano positivity and are stronger than the decomposable positivity introduced by S. Finski. We provide the criteria for strongly decomposable positivity of type I and type II and prove that the Schur forms of a strongly decomposable positive vector bundle of type I are weakly positive, while the Schur forms of a strongly decomposable positive vector bundle of type II are positive. These answer a question of Griffiths affirmatively for strongly decomposably positive vector bundles. Consequently, we present an algebraic proof of the positivity of Schur forms for (dual) Nakano positive vector bundles, which was initially proven by S. Finski.

We introduce an analogue of Bridgeland’s stability conditions for polarised varieties. Much as Bridgeland stability is modelled on slope stability of coherent sheaves, our notion of Z-stability is modelled on the notion of K-stability of polarised varieties. We then introduce an analytic counterpart to stability, through the notion of a Z-critical Kähler metric, modelled on the constant scalar curvature Kähler condition. Our main result shows that a polarised variety which is analytically K-semistable and asymptotically Z-stable admits Z-critical Kähler metrics in the large volume regime. We also prove a local converse and explain how these results can be viewed in terms of local wall crossing. A special case of our framework gives a manifold analogue of the deformed Hermitian Yang–Mills equation.

Let $(M,g,J)$ be a closed Kähler manifold with negative sectional curvature and complex dimension $m := \dim _{\mathbb {C}} M \geq 2$. In this article, we study the unitary frame flow, that is, the restriction of the frame flow to the principal $\mathrm {U}(m)$-bundle $F_{\mathbb {C}}M$ of unitary frames. We show that if $m \geq 6$ is even and $m \neq 28$, there exists $\unicode{x3bb} (m) \in (0, 1)$ such that if $(M, g)$ has negative $\unicode{x3bb} (m)$-pinched holomorphic sectional curvature, then the unitary frame flow is ergodic and mixing. The constants $\unicode{x3bb} (m)$ satisfy $\unicode{x3bb} (6) = 0.9330...$, $\lim _{m \to +\infty } \unicode{x3bb} (m) = {11}/{12} = 0.9166...$, and $m \mapsto \unicode{x3bb} (m)$ is decreasing. This extends to the even-dimensional case the results of Brin and Gromov [On the ergodicity of frame flows. Invent. Math. 60(1) (1980), 1–7] who proved ergodicity of the unitary frame flow on negatively curved compact Kähler manifolds of odd complex dimension.

We prove that the alpha invariant of a quasi-smooth Fano 3-fold weighted hypersurface of index $1$ is greater than or equal to $1/2$. Combining this with the result of Stibitz and Zhuang [SZ19] on a relation between birational superrigidity and K-stability, we prove the K-stability of a birationally superrigid quasi-smooth Fano 3-fold weighted hypersurfaces of index $1$.

We study deformed Hermitian Yang–Mills (dHYM) connections on ruled surfaces explicitly, using the momentum construction. As a main application, we provide many new examples of dHYM connections coupled to a variable background Kähler metric. These are solutions of the moment map partial differential equations given by the Hamiltonian action of the extended gauge group, coupling the dHYM equation to the scalar curvature of the background. The large radius limit of these coupled equations is the Kähler–Yang–Mills system of Álvarez-Cónsul, Garcia-Fernandez, and García-Prada, and in this limit, our solutions converge smoothly to those constructed by Keller and Tønnesen-Friedman. We also discuss other aspects of our examples including conical singularities, realization as B-branes, the small radius limit, and canonical representatives of complexified Kähler classes.

We classify fibrations of abstract $3$-regular GKM graphs over $2$-regular ones, and show that all fibrations satisfying the known necessary conditions for realizability are, in fact, realized as the projectivization of equivariant complex rank-$2$ vector bundles over quasitoric $4$-manifolds or $S^4$. We investigate the existence of invariant (stable) almost complex, symplectic, and Kähler structures on the total space. In this way, we obtain infinitely many Kähler manifolds with Hamiltonian non-Kähler actions in dimension $6$ with prescribed one-skeleton, in particular with a prescribed number of isolated fixed points.

We introduce $q$-stability conditions $(\sigma,s)$ on Calabi–Yau-$\mathbb {X}$ categories $\mathcal {D}_\mathbb {X}$, where $\sigma$ is a stability condition on $\mathcal {D}_\mathbb {X}$ and $s$ a complex number. We prove the corresponding deformation theorem, that $\operatorname {QStab}_s\mathcal {D}_\mathbb {X}$ is a complex manifold of dimension $n$ for fixed $s$, where $n$ is the rank of the Grothendieck group of $\mathcal {D}_\mathbb {X}$ over $\mathbb {Z}[q^{\pm 1}]$. When $s=N$ is an integer, we show that $q$-stability conditions can be identified with the stability conditions on $\mathcal {D}_N$, provided the orbit category $\mathcal {D}_N=\mathcal {D}_\mathbb {X}/[\mathbb {X}-N]$ is well defined. To attack the questions on existence and deformation along the $s$ direction, we introduce the inducing method. Sufficient and necessary conditions are given, for a stability condition on an $\mathbb {X}$-baric heart (that is, a usual triangulated category) of $\mathcal {D}_\mathbb {X}$ to induce $q$-stability conditions on $\mathcal {D}_\mathbb {X}$. As a consequence, we show that the space $\operatorname {QStab}^\oplus \mathcal {D}_\mathbb {X}$ of (induced) open $q$-stability conditions is a complex manifold of dimension $n+1$. Our motivating examples for $\mathcal {D}_\mathbb {X}$ are coming from (Keller's) Calabi–Yau-$\mathbb {X}$ completions of dg algebras. In the case of smooth projective varieties, the $\mathbb {C}^*$-equivariant coherent sheaves on canonical bundles provide the Calabi–Yau-$\mathbb {X}$ categories. Another application is that we show perfect derived categories can be realized as cluster-$\mathbb {X}$ categories for acyclic quivers.

We prove that every smooth divisor in $\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^2$ of degree $(1,1,2)$ is K-stable.

We consider the Weil–Petersson gradient vector field of renormalized volume on the deformation space of convex cocompact hyperbolic structures on (relatively) acylindrical manifolds. In this paper we prove the conjecture that the flow has a global attracting fixed point at the unique structure $M_{\rm geod}$ with minimum convex core volume.

We prove an algebraic version of the Hamilton–Tian conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a reduced uniformly Ding stable triple, which admits a Kähler–Ricci soliton when the ground field .

Since the celebrated work by Cartan, distributions with small growth vector $(2,3,5)$ have been studied extensively. In the holomorphic setting, there is a natural correspondence between holomorphic $(2,3,5)$-distributions and nondegenerate lines on holomorphic contact manifolds of dimension 5. We generalize this correspondence to higher dimensions by studying nondegenerate lines on holomorphic contact manifolds and the corresponding class of distributions of small growth vector $(2m, 3m, 3m+2)$ for any positive integer m.

We show that the singularities of the twisted Kähler–Einstein metric arising as the longtime solution of the Kähler–Ricci flow or in the collapsed limit of Ricci-flat Kähler metrics are intimately related to the holomorphic sectional curvature of reference conical geometry. This provides an alternative proof of the second-order estimate obtained by Gross, Tosatti, and Zhang (2020, Preprint, arXiv:1911.07315) with explicit constants appearing in the divisorial pole.

In this paper, we identify the five dimensional analogue of the finite energy foliations introduced by Hofer–Wysocki–Zehnder for the study of three dimensional Reeb flows, and show that these exist for the spatial circular restricted three-body problem (SCR3BP) whenever the planar dynamics is convex. We introduce the notion of a fiberwise-recurrent point, which may be thought of as a symplectic version of the leafwise intersections introduced by Moser, and show that they exist in abundance for a perturbative regime in the SCR3BP. We then use this foliation to induce a Reeb flow on the standard 3-sphere, via the use of pseudo-holomorphic curves, to be understood as the best approximation of the given dynamics that preserves the foliation. We discuss examples, further geometric structures, and speculate on possible applications.

Let A be an abelian scheme of dimension at least four over a $\mathbb {Z}$ -finitely generated integral domain R of characteristic zero, and let L be an ample line bundle on A. We prove that the set of smooth hypersurfaces D in A representing L is finite by showing that the moduli stack of such hypersurfaces has only finitely many R-points. We accomplish this by using level structures to interpolate finiteness results between this moduli stack and the stack of canonically polarized varieties.

We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kähler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) compute the stability thresholds for hypersurfaces at generalised Eckardt points and for cubic surfaces at all points, and (c) provide a new algebraic proof of Tian’s criterion for K-stability, amongst other applications.

As is well known, the holomorphic sectional curvature is just half of the sectional curvature in a holomorphic plane section on a Kähler manifold (Zheng, Complex differential geometry (2000)). In this article, we prove that if the holomorphic sectional curvature is half of the sectional curvature in a holomorphic plane section on a Hermitian manifold then the Hermitian metric is Kähler.