A study on mixed Hodge structures on the relative log de Rham cohomology groups of log smooth degenerations

在2006年，我研究了紧致Kahler复流形的半稳定退化，作为对数光滑退化的一个重要例子。特别地，对构造在奇异纤维上的混合Hodge复形的权过滤进行了深入的研究. 2007年，我一直在研究这些权滤子，发现在对数光滑退化的情形下，权滤子的构造中，单值作用起着重要的作用，上同调群上的单值作用一般是拟幂幺的。然而，我希望我们可以构造混合的Hodge复形，对于log smooth退化的情况下，通过减少到单值作用是单幂的情况下，一旦我找到了适当的条件，确保单值的单幂性。为了找到这样的条件，我计算了相关的log de Rham复形的上同调层的log smooth退化。

In 2006, I studied the case of semi-stable degeneration of compact Kahler complex manifolds, as an important example of log smooth degenerations. In particular, weight filtrations on the mixed Hodge complexes constructed on the singular fibers were considered deeply. I found that the properties of these weight filtrations were related to the properties of monodromy logarithmsIn 2007, I have been studying these weight filtrations and found that the monodromy logarithms played important roles to construct the weight filtrations for the case of log smooth degenerations.The monodromy action on the cohomology groups is quasi-unipotent in general. However I expect that we can construct mixed Hodge complexes, for the case of log smooth degeneration, by the reduction to the case where the monodromy actions are unipotent, once I find out the appropriate conditions which assures the unipotency of the monodromy.In order to find out such conditions, I calculated the cohomology sheaves of the relative log de Rham complexes for log smooth degenerations.