The motivation for this paper is the study of arithmetic properties of Shimura varieties, in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure. This is closely related to the structure of RapoportZink spaces and of affine DeligneLusztig varieties. We prove a HodgeNewton decomposition for affine DeligneLusztig varieties and for the special fibers of RapoportZink spaces, relating these spaces to analogous ones defined in terms of Levi subgroups, under a certain condition (HodgeNewton decomposability) which can be phrased in combinatorial terms. Second, we study the Shimura varieties in which every non-basic\sigma-isogeny class is HodgeNewton decomposable. We show that (assuming the axioms of He and Rapoport in Manuscr. Math. 152 (34): 317343, 2017) this condition is equivalent to nice conditions on either the basic locus or on

The main results of this paper involve general algebraic differentials ω on a general pencil of algebraic curves. We show how to determine whether ω is integrable in elementary terms for infinitely many members of the pencil. In particular, this corrects an assertion of James Davenport from 1981 and provides the first proof, even in rather strengthened form. We also indicate analogies with work of Andre and Hrushovski and with the Grothendieck–Katz Conjecture. To reach this goal, we first provide proofs of independent results which extend conclusions of relative Manin–Mumford type allied to the Zilber–Pink conjectures: we characterise torsion points lying on a general curve in a general abelian scheme of arbitrary relative dimension at least 2. In turn, we present yet another application of the latter results to a rather general pencil of Pell equations A^2−DB^2=1 over a polynomial ring. We determine whether the Pell equation (with squarefree D) is solvable for infinitely many members of the pencil.

We present a new method to solve certain ar {\partial} -equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a ar {\partial} -lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne's degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at ar {\partial} -level, as well as certain injectivity theorem on compact Kahler manifolds.

Let $(V, 0)$ be an isolated hypersurface singularity. We introduce a series of new derivation Lie algebras $L_{k}(V)$ associated to $(V,0)$. Its dimension is denoted as $\lambda_{k}(V)$. The $L_{k}(V)$ is a generalization of the Yau algebra $L(V)$ and $L_{0}(V)=L(V)$. These numbers $\lambda_{k}(V)$ are new numerical analytic invariants of an isolated hypersurface singularity. In this article we compute $L_1(V)$ for fewnomial isolated singularities (Binomial, Trinomial) and obtain the formulas of $L_{1}(V)$. We also formulate a sharp upper estimate conjecture for the $L_k(V)$ of weighted homogeneous isolated hypersurface singularities and we prove this conjecture for large class of singularities. Furthermore, we formulate another conjecture: $\cdots >\lambda^{(i+1)}(V) > \lambda^i(V) \cdots >\lambda^0(V) \geq \lambda_l (V),\; l=1,2,\cdots,n$. We partially prove it for binomial and trinomial singularities.

We use the mirror theorem for toric Deligne-Mumford stacks, proved recently by the authors and by Cheong-Ciocan-Fontanine-Kim, to compute genus-zero Gromov-Witten invariants of a number of toric orbifolds and gerbes. We prove a mirror theorem for a class of complete intersections in toric Deligne-Mumford stacks, and use this to compute genus-zero Gromov-Witten invariants of an orbifold hypersurface.