Let R = C[x1; x2;


; xn]=(f) where f is a weighted homogeneous polynomial dening an isolated singularity at the origin. Then R and Der(R;R) are graded. It is well-known that Der(R;R) does not have a negatively graded component. Wahl conjectured that this is still true for R = C[x1; x2;


; xn]=(f1; f2;


; fm) which denes an isolated, normal and complete intersection singularity and f1; f2;


; fm weighted homogeneous polynomials with the same weight type (w1;w2;


;wn). Here we give a positive answer to the Wahl Conjecture and its generalization (without the condition of complete intersection singularity) for R when the degree of fi; 1  i  m are bounded below by a constant C depending only on the weights w1;w2;


;wn. Moreover this bound C is improved when any two of w1;w2;


;wn are coprime. Since there are counter-examples for the Wahl Conjecture and its generalization when fi are low degree, our theorem is more or less optimal in the sense that only the lower bound constant can be improved.

Let X1 and X2 be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann (CR) manifolds of dimensions 2m − 1 and 2n − 1 in Cm+1 and Cn+1, respectively. We introduce the Thom- Sebastiani sum X = X1⊕X2 which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension 2m+2n+1 in Cm+n+2. Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in Cn+1 for all n > 2 forms a semigroup. X is said to be an irreducible element in this semigroup if X cannot be written in the form X1 ⊕X2. It is a natural question to determine when X is an irreducible CR manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly, we show that if X = X1 ⊕ X2, then the Kohn-Rossi cohomology of the X is the product of those Kohn-Rossi cohomology coming from X1 and X2 provided that X2 admits a transversal holomorphic S1-action.

No abstract uploaded!

We discuss an approach to the problem of classifying zero-dimensional gradient quasihomogeneous singularities using simple properties of deformation theory. As an example, we enumerate all such singularities with modularity ℘ = 0 and with Milnor number not greater than 12. We also compute normal forms and monomial vector-bases of the first cotangent homology and cohomology modules, the corresponding Poincar´e polynomials, inner modality, inner modularity, primitive ideals, etc.

The Okounkov body is a construction which, to an effective divisor $D$ on an $n$-dimensional algebraic variety $X$, associates a convex body $\Delta(D)$ in the $n$-dimensional Euclidean space $\RR^n$. It may be seen as a generalization of the moment polytope of an ample divisor on a toric variety, and it encodes rich numerical information about the divisor $D$. When constructing the Okounkov body, an intermediate product is a lattice semigroup $\Gamma(D)\subset \NN^{n+1}$, which we will call the Okounkov semigroup. Recently it was discovered that finite generation of the Okounkov semigroup has interesting geometric implication for $X$ regarding toric degenerations and integrable systems, however the finite generation condition is difficult to establish except for some special varieties $X$. In this article, we show that smooth projective Fano varieties of coindex${}\le 2$ have finitely generated Okounkov semigroups, providing the first family of nontrivial higher dimensional examples that are not coming from representation theory. Our result also gives a partial answer to a question of Anderson, K\"uronya, and Lozovanu.