Rigidity Subvarieties And Special Subvarieties In Moduli Spaces

In this thesis, two properties of moduli spaces are studied.The first property is about the rigidity of subvarieties of moduli space of degree d hypersurfaces in Pn. We prove the following theorem定理0.0.4Let f:X�'> U be a family of smooth hypersurfaces in P" of degree d> n+1over a smooth curve U. Assume that θnf≠0. Then f is rigid.We also give a generalization of this result.定理0.0.5Let f:X�'U be a family of smooth hypersurfaces in Pn of degree d> n+1over a smooth curve U. If l(θf)=l(θ), where l(θ) and l(θ) are the length of iterated Kodaira-Spencer maps of f and universal family respectively, then f is rigid.The method relies on the construction of the new variation of Hodge structure and poly-stability of Higgs bundles.Second property is about the special subvarieties and Torelli locus T8o:=j(M8) in A8. If B is a special curve, and it has a Zariski open set in T8o, then we obtain a fibred surface S�'B. The invariants of some surfaces of this kind are calculated. We get the following theorem定理0.0.6Let P�'P1be the algebraic surface induced by y5=x(x-1)(x-A) whose base curve is a special in A4. After a base change of5times ramified at{0,1,∞}, we get a fibred surface f:S�'C whose invariants are KS2＝45, χtop(S)＝15,χ(DS)＝5, so it is a ball quotient.At the same time, the algebraic surfaces induced by base changes of y7=x(x-1)(x-λ) and y9=x(x-1)(x-A) are not ball quotient, though the base curves are special in A6, and A7. We get these results by applying the properties of fiberwise invariants under base change.