Some enumerative global properties of variations of Hodge structures

The global enumerative invariants of a variation of polarized Hodge structures over a smooth quasi-projective curve reflect the global twisting of the family and numerical measures of the complexity of the limiting mixed Hodge structures that arise when the family degenerates. We study several of these global enumerative invariants and give applications to questions such as: Give conditions under which a non-isotrivial family of Calabi–Yau threefolds must have singular fibres? Determine the correction terms arising from the limiting mixed Hodge structures that are required to turn the classical Arakelov inequalities into exact equalities.