INFINITESIMAL MIXED TORELLI PROBLEM FOR ALGEBRAIC SURFACES WITH ORDINARY SINGULARITIES, I

In this paper, we formulate the infinitesimal mixed Torelli problem for an algebraic surface S with ordinary singularities. We use 2-cubic hyperresolution a_●: X_● →S (● ∈ □_2) of S in the sense of F. Guillen, V. Navarro Aznar et al. not only to describe the mixed Hodge structure on the cohomology of S, but also to describe the infinitesimal locally trivial deformation space H^1 (S, Θ_S) of S, where Θ_S: =Homo_S(Ω^1_S, O_S). For an analytic family π: G→ (M, o) of locally trivial deformations of S, parametrized by a pointed complex space (M, o), we define the Kodaira-Spencer map σ_o: T_oM →H^1 (S, Θ_S). We show that if each fiber of the family π: G→ (M, o) is projective, then the variation of mixed Hodge structures, arising from this family, can be described by taking 2-cubic hyper-resolution of its each fiber simultaneously. We give a formula which describes the relation between the Kodaira-Spencer map (σ_o: T_oM → H^1 (S, Θ_S) and the Jacobian map dφ_o of the so-called period map φ: M → M_<mix> (H^1(S)z))×M_<mix> (H^2(S)z)) at o ∈ M, where M_<mix> (H^e(S)z)), l=1, 2, denotes the modular variety of mixed Hodge structures on H^l (S) z: = H^l (S, Z) modulo torsion (Theorem 3.17)