Bulletin of the London Mathematical Society最新文献

Let $�X$ be a normal algebraic variety over an algebraically closed field $�k$. We prove that the Kähler differential sheaf of $�X$ is torsion-free if and only if any regular section of the ideal sheaf of the first order deformation of $�X$ inside $��X�{\times }_k�X�$, defined outside the singular locus of $��X�{\times }_k�X�$, extends regularly to the singular locus.

We characterise Geometric Property (T) by the existence of a certain projection in the maximal uniform Roe algebra $��C_{�u�,�\max �}^{\ast }��\left(�X�\right)��$, extending the notion of Kazhdan projection for groups to the realm of metric spaces. We also describe this projection in terms of the decomposition of the metric space into coarsely connected components.

We construct a bipartite generalization of Alon and Szegedy's nearly orthogonal vectors, thereby obtaining strong bounds for several extremal problems involving the Lovász theta function, vector chromatic number, minimum semidefinite rank, nonnegative rank, and extension complexity of polytopes. In particular, we answer a question from our previous work together with Letzter and Sudakov, while also addressing a question of Hrubeš and of Kwan, Sauermann, and Zhao. Along the way, we derive a couple of general lower bounds for the vector chromatic number which may be of independent interest.

We extend the Fortuin–Kasteleyn–Ginibre (FKG) inequality to cover multiple events with equal pairwise intersections. We then apply this inequality to resolve Kahn's question on positive associated measures, as well as prove new inequalities concerning random graphs and probabilities of connection in Bernoulli percolation.

The equidistribution of roots of quadratic congruences with prime moduli depends crucially upon effective bounds for special Weyl linear forms. Duke, Friedlander and Iwaniec discovered strong estimates for these Weyl linear forms when the quadratic polynomial has negative discriminant. Tóth proved analogous but weaker bounds when the quadratic polynomial has positive discriminant. We establish strong estimates for these Weyl linear forms for quadratics of positive discriminants. As an application of our bounds, we derive a quantitative uniform distribution of modular square roots with integer moduli in an arithmetic progression.

This is a survey article, with essentially complete proofs, of a series of recent results concerning the geometry of the characteristic foliation on smooth divisors in compact hyperkähler manifolds, starting with work by Hwang–Viehweg, but also covering articles by Amerik–Campana and Abugaliev. The restriction of the holomorphic symplectic form on a hyperkähler manifold $�X$ to a smooth hypersurface $��D�\subset �X�$ leads to a regular foliation $��F�\subset �T_D�$ of rank 1, the characteristic foliation. The picture is complete in dimension 4 and shows that the behaviour of the leaves of $�F$ on $�D$ is determined by the Beauville–Bogomolov square $��q�\left(�D�\right)�$ of $�D$. In higher dimensions, some of the results depend on the abundance conjecture for $�D$.

We provide a complete classification of the algebraicity of (generalized) hypergeometric functions with no restriction on the set of their parameters. Our characterization relies on the interlacing criteria of Christol and Beukers–Heckman for globally bounded and algebraic hypergeometric functions, however, in a more general setting that allows arbitrary complex parameters with possibly integral differences. We also showcase the adapted criterion on a variety of different examples.

After one-point compactification, the collection of all unordered configuration spaces of a manifold admits a commutative multiplication by superposition of configurations. We explain a simple (derived) presentation for this commutative monoid object. Using this presentation, one can quickly deduce Knudsen's formula for the rational cohomology of configuration spaces, prove rational homological stability and understand how automorphisms of the manifold act on the cohomology of configuration spaces. Similar considerations reproduce the work of Farb–Wolfson–Wood on homological densities.

We prove that the number of quartic fields $�K$ with discriminant $���|��{\Delta }_K��|�⩽�X��$ whose Galois closure is $�D_4$ equals $��C�X�+�O�\left(�X^{�5�/�8�+�\epsilon �}�\right)�$, improving the error term in a well-known result of Cohen, Diaz y Diaz, and Olivier. We prove an analogous result for counting quartic dihedral extensions over an arbitrary base field.

We prove that any number of general fat points of any multiplicities impose the expected number of conditions on a linear system on a smooth projective surface, in several cases including primitive linear systems on very general K3 and abelian surfaces, “Du Val” linear systems on blowups of $�P^2$ at nine very general points, and certain linear systems on some ruled surfaces over elliptic curves. This is done by answering a question of the author about the case of only one fat point on a certain ruled surface, which follows from a circle of results due to Treibich–Verdier, Segal–Wilson, and others.