7-2【Acuna Ricardo Jaime】Hodge adjacency conditions for singularities

We introduce a theory of weak polarized relations governing which other strata of a discriminant locus can be in the closure of a given stratum. This theory applies to both families of varieties (e.g., over MMP-type compactifications of a moduli space) and deformations of singularities (initially just isolated ones). These relations can be encoded in directed graphs with admissible Hodge diamonds (or diamonds modulo the k+1 Hodge flag) as vertices. We study the combinatorial properties of the weak graphs, and give sufficient conditions for a family with Hodge numbers h to have a saturated weak graph (meaning all the admissible edges in the graph appear).

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