The subject of counting positive lattice points in n-dimensional simplexes has interested mathematicians for decades due to its applications in singularity the- ory and number theory. Enumerating the lattice points in a right-angled simplex is equivalent to determining the geometric genus of an isolated singularity of a weighted homogeneous complex polynomial. It is also a method to shed insight into large gaps in the sequence of prime numbers. Seeking to contribute to these applications, in this paper, we prove the Yau Geometric Conjecture in six dimensions, a sharp upper bound for the number of positive lattice points in a six-dimensional tetrahedron. The main method of proof is summing existing sharp upper bounds for the number of points in 5-dimensional simplexes over the cross sections of the six-dimensional simplex. Our new results pave the way for the proof of a fully general sharp upper bound for the number of lattice points in a simplex. It also sheds new light on proving the Yau Geometric and Yau Number-Theoretic Conjectures in full generality.

It is well known that the geometric genus andmultiplicity are two important invariants for isolated singularities. In this paper we give a sharp lower estimate of the geometric genus in terms of the multiplicity for isolated hypersurface singularities. In 1971, Zariski asked whether the multiplicity of an isolated hypersurface singularity depends only on its embedded topological type. This problem remains unsettled. In this paper we answer positively Zariski’s multiplicity question for isolated hypersurface singularity if Milnor number or geometric genus is small.

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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.