Bulletin of the London Mathematical Society最新文献

We extend the classical Burgess estimates to character sums over proper generalized arithmetic progressions (GAPs) of rank 2 in prime fields $�F_p$. The core of our proof is a sharp upper bound for the multiplicative energy of these sets, established by adapting an argument of Konyagin and leveraging tools from the geometry of numbers. A key step in our argument involves establishing new upper bounds for the sizes of Bohr sets, which may be of independent interest.

Given a family of parameterized algebraic curves over a strictly semistable pair, we show that the simultaneous tropicalization of the curves in the family forms a family of parameterized tropical curves over the skeleton of the strictly semistable pair. We show that the induced tropical moduli map satisfies a certain balancing condition, which allows us to describe properties of its image and deduce a new liftability criterion.

We prove vanishing results for Witten genera of string generalized complete intersections in homogeneous $�{\text{Spin}}^c$-manifolds and in other $�{\text{Spin}}^c$-manifolds with Lie group actions. By applying these results to Fano manifolds with second Betti number equal to one we get new evidence for a conjecture of Stolz.

We study the equivariant Kuznetsov component $��{\mathrm{Ku}}_G��\left(�X�\right)��$ of a general cubic fourfold $�X$ with a symplectic involution. We show that $��{\mathrm{Ku}}_G��\left(�X�\right)��$ is equivalent to the derived category $��D^b��\left(�S�\right)��$ of a $��K�3�$ surface $�S$, where $�S$ is given as a component of the fixed locus of the induced symplectic action on the Fano variety of lines on $�X$.

In this paper, we consider mixed curvature $�C_{�a�,�b�}$, which is a convex combination of Ricci curvature and holomorphic sectional curvature introduced by Chu–Lee–Tam [Trans. Amer. Math. Soc. 375 (2022), no. 11, 7925-7944]. We prove that if a compact complex manifold $�M$ admits a Kähler metric with quasi-positive mixed curvature and $��3�a�+�2�b�⩾�0�$, then it is projective. If $��a�,�b�⩾�0�$, then $�M$ is rationally connected. As a corollary, the same result holds for $�k$-Ricci curvature. We also show that any compact Kähler manifold with quasi-positive 2-scalar curvature is projective. Lastly, we generalize the result to the Hermitian case. In particular, any compact Hermitian threefold with quasi-positive real bisectional curvature has vanishing Hodge number $�h^{�2�,�0�}$. Furthermore, if it is Kählerian, then it is projective.

For a compact relative Kähler fibration over a compact Kähler manifold with negative holomorphic sectional curvature, if the relative Kähler form on each fiber also exhibits the negative holomorphic sectional curvature, we can construct Kähler metrics with the negative holomorphic sectional curvature on the total space. Additionally, if this form induces a Griffiths negative Hermitian metric on the relative tangent bundle, and the base admits a Kähler metric with the negative holomorphic bisectional curvature, we can also construct Kähler metrics with the negative holomorphic bisectional curvature on the total space. As an application, for a non-trivial fibration where both the fibers and base have Kähler metrics with negative holomorphic bisectional curvature, and the fibers are one-dimensional, we can explicitly construct Kähler metrics of the negative holomorphic bisectional curvature on the total space, thus resolving a question posed by To and Yeung for the case where the fibers have dimension one.

This paper establishes the noncommutative maximal inequality for discrete spherical averages over a lacunary sequence. Our main result extends the work of Hughes [J. Anal. Math. 138 (2019), no. 1, 1-21] to the noncommutative setting and, meanwhile, strengthens the recent study of Chen and Hong [arXiv: 2410.06035 (2024)] to the case of $�Z^4$.

Let $�p$ be an odd prime, and suppose that $�f_1$ and $�f_2$ are weight two newforms sharing the same irreducible Galois representation modulo $�p$. We establish a transition formula relating the $�\lambda$-invariants $��\lambda �\left(�f_1�\right)�$ and $��\lambda �\left(�f_2�\right)�$ in the case where their underlying modular varieties have the same dimension. For abelian extensions $��F�/�Q�$, this essentially removes the $��\mu �\left(�f_i�\right)�=�0�$ condition present in the earlier work of Greenberg–Vatsal and Emerton–Pollack–Weston.