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Let L, L' be two line bundles on I C i , let 0 -> L -> F -> L' -> 0 be a short exact sequence associated to HT ht <= dim Z (P) - 1. So M b b O SB C ' b sb and, since HT ht we conclude that M b b. But now h 0 (C ' b,M b) >= 2, h 0 (C ' v, b) = 0 and h 0 (C ' b, M b) = h 0 (C ' b, b) = 0 leads to a contradiction. [Extracted from the article]

In this article, we consider the flat bundle U and the kernel K of the Higgs field naturally associated to any (polarized) variation of Hodge structures of weight 1. We study how strict the inclusion U⊆K can be in the geometric case. More precisely, for any smooth projective curve C of genus g⩾2 and any r=0,...,g−1, we construct non‐isotrivial deformations of C over a quasi‐projective base such that rkK=r and rkU⩽g+12. [ABSTRACT FROM AUTHOR]