MathSciDoc: An Archive for Mathematician ∫

We classify three dimensional isolated weighted homogeneous rational complete intersection singularities, which define many new four dimensional N = 2 superconformal field theories. We also determine the mini-versal deformation of these singularities, and therefore solve the Coulomb branch spectrum and Seiberg-Witten solution.

The estimate of integral points in an n-dimensional polyhedron has many applications in singularity theory, number theory and toric geometry. The third named author formulated a conjecture (i.e., Yau Number Theoretic Conjecture) which gives a sharp polynomial upper estimate on the number of positive integral points in n-dimensional (n  3) real right-angled simplices. The previous results on the conjecture in low dimension cases (n  6) have been proved by using the sharp GLY conjecture. However, it is only valid in low dimension. The Yau Number Theoretic Conjecture for n = 7 has been shown with a completely new method in [22]. In this paper, on the one hand, the similar method has been applied to prove the conjecture for n = 8, but with more meticulous analyses. The main method of proof is summing existing sharp upper bounds for the number of points in seven-dimensional simplex over the cross sections of eight-dimensional simplex. This is a signicant progress since it sheds light on proving the Yau Number Theoretic Conjecture in full generality. On the other hand, we give a new sharper estimate of the Dickman-De Bruijn function (x; y) for 5  y < 23, compared with the result obtained by Ennola.

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.

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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.

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.

No abstract uploaded!

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.

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.

In this paper, we give an almost complete classication of toric surface codes of dimension less than or equal to 7, according to monomially equivalence. This is a natural extension of our previous work [YZ], [LYZZ]. More pairs of monomially equivalent toric codes constructed from non-equivalent lattice polytopes are discovered. A new phenomenon appears, that is, the monomially non-equivalence of two toric codes C P(10)7and CP(19)7 can be discerned on Fq, for all q  8, except q = 29. This sudden break seems to be strange and interesting. Moreover, the parameters, such as the numbers of codewords with dierent weights, depends on q heavily. More meticulous analyses have been made to have the possible distinct families of reducible polynomials.