I shall report my recent joint work with Ke Chen, Xin Lu and Sheng-Li Tan. Motived by Coleman conjecture on finiteness of CM jacobians of smooth curves of genus g>7 and Ekedahl-Serre conjecture on finiteness of completely decomposable jacobians of smooth curves of higher genus.we introduce the notion of curves with given Hodge type by  Mumford Tate group. We try to classify the possible  Hodge types of smooth projective curves. As the first step: by applying Xiao Gang's inequality we show that there exists at most finitely many curves with some Hodge types.  Equivalently,  any special sub variety with those type Mumford-Tate group  intersects Torelli locus of curves in at most
finitely many points. Note that this statement is stronger than Oort conjecture for those type special sub varieties.

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