Journal of the London Mathematical Society-Second Series最新文献

We establish an intriguing relation of the smooth theta divisor $�{\Theta }^n$ with permutohedron $�{\Pi }^n$ and the corresponding toric variety $�X_{\Pi }^n$. In particular, we show that the generalised Todd genus of the theta divisor $�{\Theta }^n$ coincides with $�h$-polynomial of permutohedron $�{\Pi }^n$ and thus is different from the same genus of $�X_{\Pi }^n$ only by the sign $�{�(�-�1�)�}^n$. As an application we find all the Hodge numbers of the theta divisors in terms of the Eulerian numbers. We reveal also interesting numerical relations between theta divisors and Tomei manifolds from the theory of the integrable Toda lattice.

We study the asymptotic behavior of small data solutions to the screened Vlasov–Poisson equation on $��R^d�\times �R^d�$ near vacuum. We show that for dimensions $��d�⩾�2�$, under mild assumptions on localization (in terms of spatial moments) and regularity (in terms of at most three Sobolev derivatives) solutions scatter freely. In dimension $��d�=�1�$, we obtain a long-time existence result in analytic regularity.

We give the construction of the universal, natural up to homotopy Chern–Weil differential graded algebra homomorphism:

Moreno-Spiegelberg et al. [Proc. Natl. Acad. Sci., 122 (2025), no. 11] proposed a general model that incorporates positive feedback together with negative feedback mediated by an inhibitor, and successfully applied it to Posidonia oceanica meadows to explain the observed spatiotemporal phenomena. Their work via numerical simulations and theoretical analysis produced various spatiotemporal patterns, such as traveling pulses, wave trains, expanding rings, and spiral waves. Here, our work rigorously establishes the existence of traveling pulses, traveling fronts, wave trains, as well as pulled and pushed fronts for this model. Our approach, based on geometric singular perturbation theory, allows us to construct traveling waves far from equilibrium, and can reveal more organized underlying structures. This provides a rigorous mathematical foundation for the wave phenomena and contributes to a deeper understanding of the mechanisms generating complex spatiotemporal patterns in spatially extended ecological systems.

Homological mirror symmetry predicts that there is a relation between autoequivalence groups of derived categories of coherent sheaves on Calabi–Yau varieties and the symplectic mapping class groups of symplectic manifolds. In this paper, as an analogue of Dehn twists for closed oriented real surfaces, we study spherical twists for dg-enhanced triangulated categories. We introduce the intersection number and relate it to group-theoretic properties of spherical twists. Using an inequality analogous to a fundamental one in the theory of mapping class groups about the behavior of the intersection number via iterations of Dehn twists, we classify the subgroups generated by two spherical twists using the intersection number. As an application, we compute the center of autoequivalence groups of derived categories of K3 surfaces.

This paper studies rings of integral piecewise-exponential functions on rational fans. Motivated by lattice-point counting in polytopes, we introduce a special class of unimodular fans called Ehrhart fans, whose rings of integral piecewise-exponential functions admit a canonical linear functional that behaves like a lattice-point count. In particular, we verify that all complete unimodular fans are Ehrhart and that the Ehrhart functional agrees with lattice-point counting in corresponding polytopes, which can otherwise be interpreted as holomorphic Euler characteristics of vector bundles on smooth toric varieties. We also prove that all Bergman fans of matroids are Ehrhart and that the Ehrhart functional in this case agrees with the Euler characteristic of matroids, introduced recently by Larson, Li, Payne, and Proudfoot. A key property that we prove about the Ehrharticity of fans is that it only depends on the support of the fan, not on the fan structure, thus providing a uniform framework for studying $�K$-rings and Euler characteristics of complete fans and Bergman fans simultaneously.

We study the termination of minimal model programs for log canonical pairs in the complex analytic setting. By using the termination, we prove a relation between the minimal model theory for projective log canonical pairs and that for log canonical pairs in the complex analytic setting. The minimal model programs for algebraic stacks and analytic stacks are also discussed.

We compute the asymptotic number of cylinders, weighted by their area to any nonnegative power, on any cyclic branched cover of any generic translation surface in any stratum. Our formulae depend only on topological invariants of the cover and number-theoretic properties of the degree: in particular, the ratio of the related Siegel–Veech constants for the locus of covers and for the base stratum component is independent of the number of branch values. One surprising corollary is that this ratio for $��a�r�e�a^3�$ Siegel–Veech constants is always equal to the reciprocal of the degree of the cover. A key ingredient is a classification of the connected components of certain loci of cyclic branched covers.

We investigate the regularity of solutions to linear elliptic equations in both divergence and non-divergence forms, particularly when the principal coefficients have Dini mean oscillation. We show that if a solution $�u$ to a divergence-form equation satisfies $��D�u�\left(�x^o�\right)�=�0�$ at a point, then the second derivative $��D^2�u��\left(�x^o�\right)��$ exists and satisfies sharp continuity estimates. As a consequence, we obtain “$�C^{�2�,�\alpha �}$ regularity” at critical points when the coefficients of $�L$ are $�C^{\alpha }$. This result refines a theorem of Teixeira (Math. Ann. 358 (2014), no. 1–2, 241–256) in the linear setting, where both linear and nonlinear equations were considered. We also establish an analogous result for equations in the non-divergence form.