Limits of Hodge structures (after Steenbrink) 1

Speaker: Feng Hao (KU Leuven)

Abstract: For a family projective varieties degenerating to a singular fiber over a disc, a limit of pure Hodge structures of general fibers exists as a mixed Hodge when general fibers approach to the singular fiber. The existence of the limit is first given by Schmid in his celebrated paper “Variation of Hodge Structure: The Singularities of the Period Mapping”. There are many applications of the existence of limit mixed Hodge structures in the study of singular fibers of degenerations, compactification of moduli spaces, milnor fibers associated to isolated singular points, cycle theory, etc. In this learning seminar, I will intoduce the algebraic construction of limit mixed Hodge structures given by Steenbrink. The weight filtration and Hodge filtration are defined over a double complex, which resolves the cohomology of nearby fiber. Also, I will cover some basic properties of the limit mixed Hodge structure, and the integral structure via log structures.

References: 1. Steenbrink, Joseph. "Limits of Hodge structures." Inventiones mathematicae 31.3 (1976): 229-257.

2. Steenbrink, Joseph. "Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures." Mathematische Annalen 301.1 (1995): 105-118