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Moduli of log mixed Hodge structures. (English) Zbl 1209.14008

This short paper announces the construction of toroidal partial compactifications for moduli spaces of graded-polarized mixed Hodge structures. As in the pure case [Classifying spaces of degenerating polarized Hodge structures. Ann. Math. Stud. 169. Princeton, NJ: Princeton University Press (2009; Zbl 1172.14002)], the idea is to add points corresponding to certain mixed nilpotent orbits, and to describe the resulting space using log geometry.
The main result is that for a neat subgroup \(\Gamma \subseteq G_{\mathbb{Z}}\), the space \(\Gamma \backslash D_{\Sigma}\) is a log manifold and Hausdorff. Moreover, it is a fine moduli space for log mixed Hodge structures with polarized graded quotients. One interesting application is the following analyticity result: given a collection of period maps \(f_j : S^{\ast} \to \Gamma \backslash D\) on the complement \(S^{\ast}\) of a smooth divisor in a complex manifold \(S\), the set of points where \(f_1(s) = \dotsb = f_n(s)\) has an analytic closure in \(S\). In the special case of admissible normal functions, this gives a new proof for a theorem by P. Brosnan and G. Pearlstein [Duke Math. J. 150, No. 1, 77–100 (2009; Zbl 1187.14015)].

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14D07 Variation of Hodge structures (algebro-geometric aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
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