Volume 2025, Issue 824

In this article, we study the K-moduli space of Fano threefolds obtained by blowing up P 3 \mathbb{P}^{3} along ( 2 , 3 ) (2,3) -complete intersection curves. This K-moduli space is a two-step birational modification of the GIT moduli space of bidegree ( 3 , 3 ) (3,3) -curves on P 1 × P 1 \mathbb{P}^{1}\times\mathbb{P}^{1} . As an application, we show that our K-moduli space appears as one model of the Hassett–Keel program for M ̄ 4 \overline{M}_{4} . In particular, we classify all K-(semi/poly)stable members in this deformation family of Fano varieties. We follow the moduli continuity method with moduli of lattice-polarized K3 surfaces, general elephants and Sarkisov links as new ingredients.

We first introduce an appropriate family of conformally covariant boundary operators associated to the Siegel domain U n + 1 \mathcal{U}^{n+1} with the Heisenberg group H n \mathbb{H}^{n} as its boundary and the complex ball B C n + 1 \mathbb{B}_{\mathbb{C}}^{n+1} with the complex sphere S 2 ⁢ n + 1 \mathbb{S}^{2n+1} as its boundary. We provide the explicit formulas of these conformally covariant boundary operators. Second, we establish all higher order extension theorems of Caffarelli–Silvestre type for the Siegel domain and complex ball. Third, we prove all higher order CR Sobolev trace inequalities for the Siegel domain U n + 1 \mathcal{U}^{n+1} and the complex ball B C n + 1 \mathbb{B}_{\mathbb{C}}^{n+1} . In particular, we generalize the Sobolev trace inequality in the CR setting by Frank–González–Monticelli–Tan in the case γ ∈ ( 0 , 1 ) \gamma\in(0,1) to the general case for all γ ∈ ( 0 , n + 1 ) ∖ N \gamma\in(0,n+1)\setminus\mathbb{N} . The family of higher order conformally covariant boundary operators we define is naturally intrinsic to the higher order Sobolev trace inequalities on both the Siegel domain U n + 1 \mathcal{U}^{n+1} and complex ball B C n + 1 \mathbb{B}_{\mathbb{C}}^{n+1} . Finally, we give an explicit solution to the scattering problem on the complex hyperbolic ball. More precisely, we obtain an integral representation and an expansion in terms of special functions for the solution to the scattering problem.

Résumé Nous montrons que l’énergie mutuelle (ou le produit d’intersection au sens de la théorie d’Arakelov) de deux systèmes dynamiques associés à des morphismes de Lattès sur Q ̄ \mathbf{\bar{Q}} est uniformément minorée et en déduisons une preuve d’une conjecture de Bogomolov-Fu-Tschinkel : le nombre d’images communes de points de torsion de deux courbes elliptiques sur 𝐂 non isomorphes par un morphisme standard vers la droite projective est uniformément borné. La démonstration repose de façon essentielle sur la théorie des espaces de Berkovich sur 𝐙 et sur un argument original permettant d’obtenir une estimation globale à partir d’une estimation centrale (au-dessus d’un corps trivialement valué).

We present a fundamentally new proof of the dimensionless L p L^{p} boundedness of the Bakry–Riesz vector on manifolds with bounded geometry, with our proof being new even in the Euclidean setting. Our proof has the significant advantage that it allows for a much stronger conclusion than previous arguments, namely that of some new dimensionless weighted estimates with optimal exponent. Part of the importance of this task lies in the novelty of the techniques: we develop the self-similarity argument known as “sparse domination” in the setting of uniformly integrable càdlàg Hilbert space valued martingales and extend the domination to a process with infinite memory. We provide a range of optimal weighted estimates and weak type estimates for these stochastic processes. Previous geometric Riesz transform estimates relied on Bellman functions and did not provide this range of weighted estimates. The development of sparse domination in this probabilistic setting and its use for high-dimensional problems is new.

We show that every regular domain 𝒟 in Minkowski space R n , 1 \mathbb{R}^{n,1} which is not a wedge admits an entire hypersurface whose domain of dependence is 𝒟 and whose scalar curvature is a prescribed constant (or function, under suitable hypotheses) in ( − ∞ , 0 ) (-\infty,0) . Under rather general assumptions, these hypersurfaces are unique and provide foliations of 𝒟. As an application, we show that every maximal globally hyperbolic Cauchy compact flat spacetime admits a foliation by hypersurfaces of constant scalar curvature, generalizing to any dimension previous results of Barbot–Béguin–Zeghib (for n = 2 n=2 ) and Smith (for n = 3 n=3 ).

The 𝑘-th secant variety of a projective variety X ⊂ P N X\subset\mathbb{P}^{N} , denoted by σ k ⁢ ( X ) \sigma_{k}(X) , is defined to be the closure of the union of ( k − 1 ) (k-1) -planes spanned by 𝑘 points on 𝑋. In this paper, we examine the 𝑘-th secant variety σ k ⁢ ( v d ⁢ ( P n ) ) ⊂ P N \sigma_{k}(v_{d}(\mathbb{P}^{n}))\subset\mathbb{P}^{N} of the image of the 𝑑-uple Veronese embedding v d v_{d} of P n \mathbb{P}^{n} to P N \mathbb{P}^{N} with N = ( n + d d ) − 1 N=\binom{n+d}{d}-1 , and focus on the singular locus of σ k ⁢ ( v d ⁢ ( P n ) ) \sigma_{k}(v_{d}(\mathbb{P}^{n})) , which is only known for k ≤ 3 k\leq 3 . To study the singularity for arbitrary k , d , n k,d,n , we define the 𝑚-subsecant locus of σ k ⁢ ( v d ⁢ ( P n ) ) \sigma_{k}(v_{d}(\mathbb{P}^{n})) to be the union of σ k ⁢ ( v d ⁢ ( P m ) ) \sigma_{k}(v_{d}(\mathbb{P}^{m})) with any 𝑚-plane P m ⊂ P n \mathbb{P}^{m}\subset\mathbb{P}^{n} . By investigating the projective geometry of moving embedded tangent spaces along subvarieties and using known results on the secant defectivity and the identifiability of symmetric tensors, we determine whether the 𝑚-subsecant locus is contained in the singular locus of σ k ⁢ ( v d ⁢ ( P n ) ) \sigma_{k}(v_{d}(\mathbb{P}^{n})) or not. Depending on the value of 𝑘, these subsecant loci show an interesting trichotomy between generic smoothness, non-trivial singularity, and trivial singularity. In many cases, they can be used as a new source for the singularity of the 𝑘-th secant variety of v d ⁢ ( P n ) v_{d}(\mathbb{P}^{n}) other than the trivial one, the ( k − 1 ) (k-1) -th secant variety of v d ⁢ ( P n ) v_{d}(\mathbb{P}^{n}) . We also consider the case of the fourth secant variety of v d ⁢ ( P n ) v_{d}(\mathbb{P}^{n}) by applying main results and computing conormal space via a certain type of Young flattening. Finally, we present some generalizations and discussions for further developments.

We provide an alternative proof of Vishik’s nonuniqueness theorem for the forced 2D Euler equation in the vorticity class L 1 ∩ L p L^{1}\cap L^{p} with 2 < p < ∞ 2<p<\infty . The main variation is the construction of a smooth and compactly supported unstable vortex, which is split into two steps: firstly, we construct a piecewise constant unstable vortex, and secondly, we find a regularization through a fixed point argument. This structure of the unstable vortex yields a simplification of the other parts of Vishik’s proof.

We develop a Morse–Lusternik–Schnirelmann theory for the distance between two points of a smoothly embedded circle in a complete Riemannian manifold. This theory suggests very naturally a definition of width that generalises the classical definition of the width of plane curves. Pairs of points of the circle realising the width bound one or more minimising geodesics that intersect the curve in special configurations. When the circle bounds a totally convex disc, we classify the possible configurations under a further geometric condition. We also investigate properties and characterisations of curves that can be regarded as the Riemannian analogues of plane curves of constant width.