Moss’ theorem, which relates Massey products in the

-page of the classical Adams spectral sequence to Toda brackets of homotopy groups, is one of the main tools for calculating Adams differentials. Working in an arbitrary symmetric monoidal stable simplicial model category, we prove a general version of Moss’ theorem which applies to spectral sequences that arise from filtrations compatible with the monoidal structure. This involves the study of Massey products and Toda brackets in a non-strictly associative context. The theorem has broad applications, e.g., to the computation of the motivic slice spectral sequence and other colocalization towers.

This note is devoted to refining the almost optimal regularity results of Breiner and Lamm on minimizing and stationary biharmonic maps via the powerful quantitative stratification method initiated by Cheeger and Naber and significantly developed by Naber and Valtorta for harmonic maps. In particular, we obtain an optimal regularity result for minimizing biharmonic maps.

Let X be a normal projective variety of dimension d over an algebraically closed field and f an automorphism of X. Suppose that the pullback

of f on the real Néron–Severi space

{\mathsf{N}}^1{(X)}_{\mathbf{R}}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsf{N}}^1(X)_{\textbf{R}}$$\end{document}

is unipotent and denote the index of the eigenvalue 1 by

k+1

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k+1$$\end{document}

. We establish the following upper bound for the polynomial volume growth

\text{plov}(f)

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{plov}(f)$$\end{document}

of f: This inequality is optimal in certain cases. Moreover, we prove that

k\le 2(d-1)

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\le 2(d-1)$$\end{document}

, extending a result of Dinh–Lin–Oguiso–Zhang for compact Kähler manifolds to arbitrary characteristic. By combining these two inequalities, we obtain the optimal bound that affirmatively answers the questions of Cantat–Paris-Romaskevich and Lin–Oguiso–Zhang.

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 Fujiki class

if and only if it has the bounded mass property —i.e., its Monge–Ampère volumes are bounded above—and there exists a closed Bott–Chern class with positive volume. This yields a positive answer to a conjecture of Boucksom–Demailly–Păun. To this end we extend to the Hermitian context the notion of non-pluripolar products of currents, allowing for the latter to be merely quasi-closed and quasi-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 basic results of Boucksom–Eyssidieux–Guedj–Zeriahi.

A positive integer m is called a Hall number if any finite group of order precisely divisible by m has a Hall subgroup of order m. We prove that, except for the obvious examples, the three integers 12, 24 and 60 are the only Hall numbers, solving a problem proposed by Jiping Zhang.

We give counterexamples to the evenness conjecture for homotopical equivariant cobordism. To this end, we prove a completion theorem for certain complex cobordism modules which does not involve higher derived functors. A key step in the proof is provided by a certain new relation between Mackey and Borel cohomology.

We develop non-Archimedean techniques to analyze the degeneration of a sequence of rational maps of the complex projective line. We provide an alternative to Luo’s method which was based on ultra-limits of the hyperbolic 3-space. We build hybrid spaces using Berkovich theory which enable us to prove the convergence of equilibrium measures, and to determine the asymptotics of Lyapunov exponents.

We answer a question raised by Kerzman in 1971. More precisely, we show that the canonical solution of the

-equation satisfies the

L^p

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estimate on the polydisc for

p\in [1,\infty ]

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. Moreover, the

L^p

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}

estimates for

p\in [1,\infty ]

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in [1, \infty ]$$\end{document}

of

\overline{\partial }

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\partial }}$$\end{document}

can also be obtained on the product of bounded

C^2

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planar domains by an observation based on the method developed in [Dong et al. arXiv:2006.14484].

We study the $\mathbb {F}_2$-synthetic Adams spectral sequence. We obtain new computational information about $\mathbb {C}$-motivic and classical stable homotopy groups.

We prove the $C^{1,1}$-regularity for stationary $C^{1,\alpha }$ ($\alpha \in (0,1)$) solutions to the multiple membrane problem. This regularity estimate was essentially used in our recent work on Yau’s four minimal spheres conjecture [arXiv:2305.08755].

Based on the pluripotential methods developed in Darvas and Zhang (Commun Pure Appl Math 77(12):4289–4327, 2024), we give a simplified prove for a result of Chi Li, which states that a log Fano vatiety admits a Kähler–Einstein metric if it has vanishing Futaki invariant and its reduced delta invariant is bigger than one.

In this paper, we study the ascending chain condition (ACC) conjecture for minimal log discrepancies (mlds), proposed by the third author. We show the ACC conjecture holds for singularities admitting $\epsilon $-plt blow-ups. In particular, this gives the ACC for mlds for exceptional singularities. The key ingredients in the proofs of our main results are the Birkar–Borisov–Alexeev–Borisov theorem, proved by Birkar, the boundedness of complements conjecture for arbitrary DCC coefficients, proposed by the third author and proved in this paper, and the existence of uniform $\mathbb {R}$-complementary rational polytopes.

We extend the results of [Commun. Partial. Differ. Equ. 44(12), 1431–1465 (2019)] by the third and fourth author globally in time. More precisely, we prove uniform-in-N Strichartz estimates for the solutions $\phi $, $\Lambda $ and $\Gamma $ of a coupled system of Hartree–Fock–Bogoliubov type with interaction potential $V_N(x-y)=N^{3 \beta }v(N^{\beta }(x-y))$ for $\beta <1$. The potential v satisfies some technical conditions, but is not small. The initial conditions have finite energy and the “pair correlation” part satisfies a smallness condition, but are otherwise general functions in suitable Sobolev spaces, and the expected correlations in $\Lambda $ develop dynamically in time. The estimates are expected to improve the Fock space bounds of [Ann. Henri Poincaré 23(2), 615–673 (2021)] by the first and fifth author. This will be addressed in a subsequent paper.

We prove that the Gromov–Hausdorff limit of Kähler–Ricci flow on a ${\textbf{G}}$-spherical Fano manifold X is a ${\textbf{G}}$-spherical ${\mathbb {Q}}$-Fano variety $X_{\infty }$, which admits a (singular) Kähler–Ricci soliton. Moreover, the ${\textbf{G}}$-spherical variety structure of $X_{\infty }$ can be constructed as a center of torus ${\mathbb {C}}^*$-degeneration of X induced by an element in the Lie algebra of Cartan torus of ${\textbf{G}}$.

We consider $G=Q_8,\textrm{SD}_{16},G_{24},$ and $G_{48}$ as finite subgroups of the Morava stabilizer group which acts on the height 2 Morava E-theory ${\textbf{E}}_2$ at the prime 2. We completely compute the G-homotopy fixed point spectral sequences of ${\textbf{E}}_2$. Our computation uses recently developed equivariant techniques since Hill, Hopkins, and Ravenel. We also compute the $(*-\sigma _i)$-graded $Q_8$- and $\textrm{SD}_{16}$-homotopy fixed point spectral sequences, where $\sigma _i$ is a non-trivial one-dimensional representation of $Q_8$.