Fano流形的镜像对称物的研究引起推广经典Hodge理论的热潮。调和丛缺乏对Hodge过滤的刻画，更好的推广- - -纯的极化TERP结构被C. Hertling提出，并由C. Sabbah，C.Hertling，和H.Iritani等人得到很好的发展。 tt*几何刻画这种结构的微分几何性质，它已被证明以某种相容方式存在于某类Frobeniu 流形，Hertling公理化该结构并将其定义为CDV-结构。根据Hertling判别法得到的CDV-结构被称为典范的。本项目主要研究以下问题：1.把CDV结构中的厄米特度量求出来，而不仅仅是证明其存在性；2. 找出CDV-结构中的典范结构，参考Hertling的判别法和已被完成的Tate 类型CDV-结构的典范结构的充要条件，继续深入探讨非Tate类型的情况。3. 探索CDV-结构所成模空间的合理定义。4.构造纯的极化TERP-结构的例子。

Mirror side of Fano manifolds interests us to do the research on generalization of VHS. Harmonic bundles lose the information on Hodge filtrations of Hodge structures. Pure polarized TEPR-structure, introduced by C. Hertlng,becomes a much better generalization, which was developed by Hertling,Sabbah,Iritani and so on. Tt*-geometry, which was firstly introduced by Cecotti and Vafa, was seen as the differential geometry properties of pure polarized TERP-structures. It was proved they exist on many interesting Frobenius manifolds in a compatible way. That was axiomaticated by Hertling to be CDV-structures. Hertling give a criterion for a Frobenius manifold together with a tt*-geometry to be a CDV-structure, which was called a canonical CDV-structure. In my project, I will work on the following problems: 1. try to write explicitly the Hermitian metrics in CDV-structures; 2.For any non-Tate CDV-structures, try to find the sufficient and neccesaey conditions such that they are canonical; 3. Explore the natural definition of the moduli space of all CDV-structures under some condtions; 4. Try to construct the examples of pure polarized TERP-structures.