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.

Based on large N Chern-Simons/topological string duality, in a series of papers, Labastida, Mario, Ooguri, and Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invariants. In this paper, we provide a proof of this conjecture. Moreover, we also show these new integer invariants vanish at large genera.

By using Yau's Schwarz lemma and the Quillen determinant line bundles, several results about fibered algebraic surfaces and the moduli spaces of curves are improved and reproved.

The mirror theorem is generalized to any smooth projective variety X. That is, a fundamental relation between the Gromov-Witten invariants of X and Gromov-Witten invariants of complete intersections Y in X is established.

No abstract uploaded!