We study variants of the local models constructed by the second author and Zhu and consider corresponding integral models of Shimura varieties of abelian type. We determine all cases of good, resp. of semi-stable, reduction under tame ramification hypotheses.

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

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.

In this paper we study the quantum sheaf cohomology of Grassmannians with deformations of the tangent bundle. Quantum sheaf cohomology is a (0,2) deformation of the ordinary quantum cohomology ring, realized as the OPE ring in A/2-twisted theories. Quantum sheaf cohomology has previously been computed for abelian gauged linear sigma models (GLSMs); here, we study (0,2) deformations of nonabelian GLSMs, for which previous methods have been intractable. Combined with the classical result, the quantum ring structure is derived from the one-loop effective potential. We also utilize recent advances in supersymmetric localization to compute A/2 correlation functions and check the general result in examples. In this paper we focus on physics derivations and examples; in a companion paper, we will provide a mathematically rigorous derivation of the classical sheaf cohomology ring.

From the work of Lian, Liu, and Yau on" Mirror Principle", in the explicit computation of the Euler data Q=\{Q_0, Q_1,...\} for an equivariant concavex bundle Q=\{Q_0, Q_1,...\} over a toric manifold, there are two places the structure of the bundle comes into play:(1) the multiplicative characteric class Q=\{Q_0, Q_1,...\} of Q=\{Q_0, Q_1,...\} one starts with, and (2) the splitting type of Q=\{Q_0, Q_1,...\} . Equivariant bundles over a toric manifold has been classified by Kaneyama, using data related to the linearization of the toric action on the base toric manifold. In this article, we relate the splitting type of Q=\{Q_0, Q_1,...\} to the classifying data of Kaneyama. From these relations, we compute the splitting type of a couple of nonsplittable equivariant vector bundles over toric manifolds that may be of interest to string theory and mirror symmetry. A code in Mathematica that carries out the computation of some of these examples is attached.