Volume 2014, Issue 697

The first goal of this paper is to study the question of finiteness of integral points on a cofinite non-compact complex two-dimensional ball quotient defined over a number field. Along the process we will also consider some negatively curved compact surfaces. Using some fundamental results of Faltings, the question is to reduce to a conjecture of Borel about existence of virtual holomorphic one-forms on cofinite non-cocompact complex ball quotients. The second goal of this paper is to study the conjecture on such non-compact surfaces.

We establish a correspondence between the rank of Mordell–Weil group of the complex elliptic threefold associated with a plane curve 𝒞 ⊂ ℙ 2 (𝔻) with equation F = 0, certain roots of the Alexander polynomial associated with the fundamental group π 1 (ℙ 2 (𝔻)∖𝒞) and the polynomial solutions for the functional equation of type h 1 p F 1 + h 2 q F 2 + h 3 r F 3 = 0 where F = F 1 F 2 F 3 . This correspondence is obtained for curves in a certain class which includes the curves having introduced here δ-essential singularities and in particular for all curves with ADE singularities. As a consequence we find a linear bound for the degree of the Alexander polynomial in terms of the degree of 𝒞 for curves with δ-essential singularities and in particular arbitrary ADE singularities.

Based on a recent extension theorem for reflexive differential forms, that is, regular differential forms defined on the smooth locus of a possibly singular variety, we study the geometry and cohomology of sheaves of reflexive differentials. First, we generalise the extension theorem to holomorphic forms on locally algebraic complex spaces. We investigate the (non-)existence of reflexive pluri-differentials on singular rationally connected varieties, using a semistability analysis with respect to movable curve classes. The necessary foundational material concerning this stability notion is developed in an appendix to the paper. Moreover, we prove that Kodaira–Akizuki–Nakano vanishing for sheaves of reflexive differentials holds in certain extreme cases, and that it fails in general. Finally, topological and Hodge-theoretic properties of reflexive differentials are explored.

We introduce moduli spaces of abelian varieties which are arithmetic models of Shimura varieties attached to unitary groups of signature (n-1,1)$(n-1, 1)$. We define arithmetic cycles on these models and study their intersection behavior. In particular, in the non-degenerate case, we prove a relation between their intersection numbers and Fourier coefficients of the derivative at s = 0 of a certain incoherent Eisenstein series for the group U(n,n)$\textup {U}(n, n)$. This is done by relating the arithmetic cycles to their formal counterpart from part I [Invent. Math. 184 (2011), 629–682] via non-archimedean uniformization, and by relating the Fourier coefficients to the derivatives of representation densities of hermitian forms. The result then follows from the main theorem of part I and a counting argument.

This article studies the algebraic structure of homology theories defined by a left Hopf algebroid U over a possibly noncommutative base algebra A , such as for example Hochschild, Lie algebroid (in particular Lie algebra and Poisson), or group and étale groupoid (co)homology. Explicit formulae for the canonical Gerstenhaber algebra structure on ExtU(A,A)$ \textup {Ext}_U(A,A)$ are given. The main technical result constructs a Lie derivative satisfying a generalised Cartan–Rinehart homotopy formula whose essence is that TorU(M,A)$\textup {Tor}^U(M,A)$ becomes for suitable right U -modules M a Batalin–Vilkovisky module over ExtU(A,A)$ \textup {Ext}_U(A,A)$, or in the words of Nest, Tamarkin, Tsygan and others, that ExtU(A,A)$ \textup {Ext}_U(A,A)$ and TorU(M,A)$\textup {Tor}^U(M,A)$ form a differential calculus. As an illustration, we show how the well-known operators from differential geometry in the classical Cartan homotopy formula can be obtained. Another application consists in generalising Ginzburg's result that the cohomology ring of a Calabi–Yau algebra is a Batalin–Vilkovisky algebra to twisted Calabi–Yau algebras.

We characterize 𝒞1$\mathcal {C}^1$ embedded hypersurfaces of 𝐑n$\mathbf {R}^n$ as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most m<3/2$m&lt;3/2$. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is 𝒞1$\mathcal {C}^1$. In the real analytic case the same conclusion holds under the weakened hypothesis that each tangent cone be a hypersurface. In particular, any convex real analytic hypersurface X⊂𝐑n$X\subset \mathbf {R}^n$ is 𝒞1$\mathcal {C}^1$. Furthermore, if X is real algebraic, strictly convex, and unbounded, then its projective closure is a 𝒞1$\mathcal {C}^1$ hypersurface as well, which shows that X is the graph of a function defined over an entire hyperplane. Finally we show that the last property is a special feature of real algebraic sets, in the sense that it does not hold in the real analytic category.