Fundamental groups of algebraic varieties

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We generalized the classical Zariski-van Kampen theorem on the fundamental group of an open algebraic variety. As an application, we obtained a hyperplane-section theorem of Zariski type for Grassmannian varieties, and revealed a subtle relation between fundamental groups and Chow forms. As another application, we calculated the fundamental group of the complement to the discriminant variety of a sufficiently ample line bundle on a compact Riemann surface.Hoping to verify the generalized Hodge conjecture for certain varieties, we investigated the cylinder homomorphism associated with a family of algebraic cycles, and gave a sufficient condition for the image of the cylinder homomorphism contains the module of vanishing cycles. Using Grobner bases, we proved the generalized Hodge conjecture for certain Fano complete intersections.We made the complete list of maximal configurations of rational double points on supersingular K3 surfaces by means of lattice theory and heavy use of computers. The complete list of extremal (quasi-) elliptic fibrations on supersingular K3 surfaces was also obtained. As a corollary, it was shown that every supersingular K3 surface in characteristic 2 is birational to a purely inseparable double cover of a projective plane, which has 21 ordinary nodes. We described the configuration of the 21 nodes in terms of certain binary codes of length 21. By the same method, we also showed that every supersingular K3 surface in odd characteristic is also birational to a double cover of a projective plane.