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Motivicity of the mixed Hodge structure of some degenerations of curves. (English) Zbl 1202.32025

The general fibre of a 1-parameter degeneration of compact Kähler manifolds \(f: X\to D\) carries a limit mixed Hodge structure. Its weight filtration \(W_{\bullet}\) together with the Hodge filtration \(F^{\bullet}\) of the original manifold \(X_t\) does not always give a mixed Hodge structure (MHS). In this paper \((W_{\bullet},F^{\bullet})\) is expcitly computed for a degeneration of genus two curves (two Weierstraßpoints coming together). It is shown that in this case they define a MHS, which is motivic, in the sense that it can be realised as (relative) \(H^1\) of some singular non-compact curve (here an elliptic curve minus two points, relative to two other points). In fact, one has a variation of MHS, which after a base change of order two can be realised by a family of such elliptic curves.

MSC:

32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
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