We present a new method to solve certain ar {\partial} -equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a ar {\partial} -lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne's degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at ar {\partial} -level, as well as certain injectivity theorem on compact Kahler manifolds.

Let $(V, 0)$ be an isolated hypersurface singularity. We introduce a series of new derivation Lie algebras $L_{k}(V)$ associated to $(V,0)$. Its dimension is denoted as $\lambda_{k}(V)$. The $L_{k}(V)$ is a generalization of the Yau algebra $L(V)$ and $L_{0}(V)=L(V)$. These numbers $\lambda_{k}(V)$ are new numerical analytic invariants of an isolated hypersurface singularity. In this article we compute $L_1(V)$ for fewnomial isolated singularities (Binomial, Trinomial) and obtain the formulas of $L_{1}(V)$. We also formulate a sharp upper estimate conjecture for the $L_k(V)$ of weighted homogeneous isolated hypersurface singularities and we prove this conjecture for large class of singularities. Furthermore, we formulate another conjecture: $\cdots >\lambda^{(i+1)}(V) > \lambda^i(V) \cdots >\lambda^0(V) \geq \lambda_l (V),\; l=1,2,\cdots,n$. We partially prove it for binomial and trinomial singularities.

We prove that the image of period map is algebraic, as conjectured by Griffiths.

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We use the mirror theorem for toric Deligne-Mumford stacks, proved recently by the authors and by Cheong-Ciocan-Fontanine-Kim, to compute genus-zero Gromov-Witten invariants of a number of toric orbifolds and gerbes. We prove a mirror theorem for a class of complete intersections in toric Deligne-Mumford stacks, and use this to compute genus-zero Gromov-Witten invariants of an orbifold hypersurface.