The δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}-invariant is a central algebro-geometric quantity that reflects the stability properties of a Fano variety and plays an important role in detecting the existence of Kähler–Einstein metrics. In recent years, explicit computations of δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}-invariants have become essential in the study of K-stability of Fano varieties, particularly in applications of the Abban–Zhuang method, which reduces higher-dimensional problems to the study of lower-dimensional subvarieties. Del Pezzo surfaces with Du Val singularities are natural objects in this context. In this article, we compute the δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}-invariants of all Du Val del Pezzo surfaces of degree 2. We show that, for such a surface, the δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}-invariant is uniquely determined by three geometric features: its degree, the number of lines it contains, and the types of its singularities, and we compute the precise values explicitly. As part of the argument, we also give an explicit description of the Mori cones of Du Val del Pezzo surfaces of degree 2. These computations are particularly useful in applications of the Abban–Zhuang method to the K-stability of higher-dimensional Fano varieties. In particular, they provide the necessary computations of δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}-invariants for proofs of K-stability for Fano threefolds in the families 1.12, 2.2, and 2.3 of the Mori–Mukai classification.
The δ -invariant is a central algebro-geometric quantity that reflects the stability properties of a Fano variety and plays an important role in detecting the existence of Kähler–Einstein metrics. In recent years, explicit computations of δ -invariants have become essential in the study of K-stability of Fano varieties, particularly in applications of the Abban–Zhuang method, which reduces higher-dimensional problems to the study of lower-dimensional subvarieties. Del Pezzo surfaces with Du Val singularities are natural objects in this context. In this article, we compute the δ -invariants of all Du Val del Pezzo surfaces of degree 2. We show that, for such a surface, the δ -invariant is uniquely determined by three geometric features: its degree, the number of lines it contains, and the types of its singularities, and we compute the precise values explicitly. As part of the argument, we also give an explicit description of the Mori cones of Du Val del Pezzo surfaces of degree 2. These computations are particularly useful in applications of the Abban–Zhuang method to the K-stability of higher-dimensional Fano varieties. In particular, they provide the necessary computations of δ -invariants for proofs of K-stability for Fano threefolds in the families 1.12, 2.2, and 2.3 of the Mori–Mukai classification.
Let 𝒜 be a Banach ideal of bounded multilinear operators. If A∈𝒜( X_1,… ,X_n;Y) we consider the composition operator Θ _A:L( Y,Z) →𝒜 ( X_1,… ,X_n;Z) defined by Θ _A( U) =U∘A . Similarly, if B∈𝒜( Z_1,… ,Z_n;W) let Ψ _B:L( Y_1,Z_1) ×⋯×L( Y_n,Z_n) →𝒜( Y_1,… ,Y_n;W) be the composition operator defined by Ψ _B( U_1,… ,U_n) =B∘( U_1,… ,U_n), and if A_1:X_1→ Y_1 , … , A_n:X_n→ Y_n are bounded linear operators let Ψ _A,B:L( Y_1,Z_1) ×⋯×L( Y_n,Z_n) →𝒜( X_1,… ,X_n;W) be the composition operator defined by Ψ _A,B( U_1,… ,U_n) =B∘( U_1,… ,U_n) ∘( A_1,… ,A_n) . We study the summability properties of these compositions of the operators.
A T^* -presymplectic Lie algebra (𝔤,[·,·], ω ) is a Lie algebra equipped with a non-degenerate skew-symmetric bilinear form ω such that ω ([u,v], w) = ω (v, [u, w]) for all u, v, w ∈𝔤 . In this paper, we first introduce the Levi-Civita product on anti-commutative algebras equipped with a non-degenerate skew-symmetric bilinear form, in analogy to the construction in pseudo-Riemannian Lie algebras. A T^* -presymplectic Lie algebra is shown to be a particular case of this product, where the bilinear form satisfies the compatibility condition with the Lie bracket. We then prove that every T^* -presymplectic Lie algebra is 2-step nilpotent. Moreover, we introduce the concept of the T T^* -extension, and we prove that any T^* -presymplectic Lie algebra is a TT^* -extension of the abelian Lie algebra. Additionally, we define the notion of a T^* -presymplectic double extension and show that all T^* -presymplectic Lie algebras can be constructed through a sequence of T^* -presymplectic double extensions starting from the trivial algebra {0} . Finally, we prove that every T^* -presymplectic Lie algebra admits a quadratic structure.
We study tensor-product maximal operators generated by weighted Walsh means on dyadic product Hardy spaces. Under suitable conditions on the weights, we prove that the two-dimensional critical exponent is the larger of the corresponding one-dimensional exponents, and that this result is sharp. Applications are given to Fejér, Cesàro, Riesz, and regularly varying means. We also obtain a weak- L_1 estimate and almost everywhere convergence for every H_1^#(𝕀^2) .
