We study the

-synthetic Adams spectral sequence. We obtain new computational information about

\mathbb{C}

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

-motivic and classical stable homotopy groups.

We study the supersingular locus of a reduction at an inert prime of the Shimura variety attached to

. More concretely, we realize irreducible components of the supersingular locus as closed subschemes of flag schemes over Deligne–Lusztig varieties defined by explicit conditions after taking perfections. Moreover, we study the intersections of the irreducible components. Stratifications of Deligne–Lusztig varieties defined using powers of Frobenius action appear in the description of the intersections.

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.

This paper mainly addresses three perspectives: potential analysis–variational calculus–convex geometry, of the potential theoretic capacities arising from mathematical physics.

We generalize H-invariant associated to TZ metric on any compact Kähler manifold.

We compute the orbifold Euler characteristics of

by applying the formalisms developed in (Wang et al., J. High Energy Phys. 2019(4):135, 2019; Zhou, arXiv:1412.1604, 2014). We take the works of Harer–Zagier (Invent. Math. 85(3):457–485, 1986) and Bini–Harer (J. Eur. Math. Soc. 13(2):487–512, 2011) as the starting point, and prove two types of recursion relations to compute

\chi ({\overline{\mathcal{M}}}_{g,n})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi (\overline{{\mathcal {M}}}_{g,n})$$\end{document}

. As applications of these recursions, we give some numerical data and derive some closed formulas, and generalize Manin’s functional equation for

\chi ({\overline{\mathcal{M}}}_{0,n})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi (\overline{{\mathcal {M}}}_{0,n})$$\end{document}

to higher genera cases. Moreover, in genus zero the results are related to Ramanujan polynomials. We also show that the generating series of

\chi ({\overline{\mathcal{M}}}_{g,n})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi ({\overline{{{\mathcal {M}}}}}_{g,n})$$\end{document}

is the logarithm of a particular tau-function of KP hierarchy evaluated at times specified by the generating series of

\chi ({\mathcal{M}}_{g,n})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi ({{\mathcal {M}}}_{g,n})$$\end{document}

.

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

-equation satisfies 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}

estimate on the polydisc 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}

. 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

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^2$$\end{document}

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

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.