We classify three fold isolated quotient Gorenstein singularity C3=G. These singularities are rigid, i.e. there is no non-trivial deformation, and we conjecture that they dene 4d N = 2 SCFTs which do not have a Coulomb branch.

We introduce a new concept of strong vanishing property and investigate this property for minimally elliptic singularities. Acomparison theorem about strong vanishing property is also obtained. We use Zariski decomposition to give a formula of plurigenera for Gorenstein singularities and use it to test whether a divisor is f-nef.

One of the fundamental questions in CR geometry is: Given two strongly pseudoconvex CR manifolds X1 and X2 of dimension 2n􀀀 1, is there a non-constant CR morphism between them? In this paper, we use Kohn{Rossi cohomology to show the non-existence of non-constant CR morphism between such two CR manifolds. Specically, if dimHp;q KR(X1) < dimHp;q KR(X2) for any (p; q) with 1  q  n 􀀀 2, then there is no non-constant CR morphism from X1 to X2.

Let R = C[x1, x2, . . . , xn]/(f1, . . . , fm) be a positively graded Artinian algebra. A long-standing conjecture in algebraic geometry, differential geometry, and rational homotopy theory is the non-existence of negative weight derivations on R. Alexsandrov conjectured that there are no negative weight derivations when R is a complete intersection algebra, and Yau conjectured there are no negative weight derivations on R when R is the moduli algebra of a weighted homogeneous hypersurface singularity. This problem is also important in rational homotopy theory and differential geometry. In this paper we prove the non-existence of negative weight derivations on R when the degrees of f1, . . . ,fm are bounded below by a constant C depending only on the weights of x1, . . . , xn. Moreover this bound C is improved in several special cases.

We study cluster categories arising from marked surfaces (with punctures and nonempty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting dimensions of Ext1 as intersection numbers of tagged curves and Auslander-Reiten translation as tagged rotation. An important consequence is that the cluster(-tilting) exchange graphs of such cluster categories are connected.