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The goal of this article is to provide an explicit algorithmic construction of formal $F$-manifold structures, formal Frobenius manifold structures, and higher residue pairings on the primitive middle-dimensional cohomology $\mathbb{H}$ of a smooth projective Calabi-Yau complete intersection variety $X$ defined by homogeneous polynomials $G_1(\underline x), \dots, G_k(\underline x)$. Our main method is to analyze a certain dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra $\mathcal{A}$ obtained from the twisted de Rham complex which computes $\mathbb{H}$. More explicitly, we introduce a notion of \textit{a weak primitive form} associated to a solution of the Maurer-Cartan equation of $\mathcal{A}$ and the Gauss-Manin connection, which is a weakened version of Saito's primitive form (\cite{Saito}). In addition, we provide an explicit algorithm for a weak primitive form based on the Gr\"obner basis in order to achieve our goal. Our approach through the weak primitive form can be viewed as a unifying link (based on Witten's gauged linear sigma model, \cite{W93}) between the Barannikov-Kontsevich's approach to Frobenius manifolds via dGBV algebras (non-linear topological sigma model, \cite{BK}) and Saito's approach to Frobenius manifolds via primitive forms and higher residue pairings (Landau-Ginzburg model, \cite{ST})., Comment: 39 pages

We provide an explicit algorithm to compute a Massey triple product relative to a defining system for a smooth projective plane curve $X$ defined by a homogeneous polynomial $G(\underline x)$ over a field. The main idea is to use the description (due to Carlson and Griffiths) of the cup product for $H^1(X,\mathbb{C})$ in terms of the multiplications inside the Jacobian ring of $G(\underline x)$ and the Cech-deRham complex of $X$. Our algorithm gives a criterion whether a Massey triple product vanishes or not in $H^2(X)$ under a particular non-trivial defining system of the Massey triple product and thus can be viewed as a generalization of the vanishing criterion of the cup product in $H^2(X)$ of Carlson and Griffiths. Based on our algorithm, we provide explicit numerical examples by running the computer program., Comment: 13 pages, 1 figure

We study the crossover behavoir on restricted surface depostions., Comment: Latex