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Abstract We develop a Hodge theoretic invariant for families of projective manifolds that measures the potential failure of an Arakelov-type inequality in higher dimensions, one that naturally generalizes the classical Arakelov inequality over regular quasi-projective curves.We show that, for families of manifolds with ample canonical bundle, this invariant is uniformly bounded.As a consequence, we establish that such families over a base of arbitrary dimension satisfy the aforementioned Arakelov inequality, answering a question of Viehweg. 

Abstract KSB stability holds at codimension$$1$$points trivially, and it is quite well understood at codimension$$2$$points because we have a complete classification of$$2$$-dimensional slc singularities. We show that it is automatic in codimension$$3$$. 

The analytic aspects of multiplier ideals, log canonical thresholds and log canonical centers played an important role in several papers of Demailly, including [DEL00, Dem01, DK01, Dem12, DP14, Dem16, CDM17, Dem18]. Log canonical centers are seminormal by [Amb03, Fuj17], even Du Bois by [KK10, KK20]. This has important applications to birational geometry and moduli theory; see [KK10, KK20] or [Kol23, Sec.2.5]. We recall the concept of Du Bois singularities in Definition-Theorem 4. An unusual aspect is that this notion makes sense for complex spaces that have irreducible components of different dimension. This is crucial even for the statement of our theorem. In this note we generalize the results of [KK10, KK20] by showing that if a closed subset is 'close enough' to being a union of log canonical centers, then it is Du Bois. The minimal discrepancy is a nonnegative rational number, that measures the deviation from being a union of log canonical centers. The log canonical gap gives the precise notion of 'closeness.' 

Abstract We show that, for any fixed weight, there is a natural system of Hodge sheaves, whose Higgs field has no poles, arising from a flat projective family of varieties parametrized by a regular complex base scheme, extending the analogous classical result for smooth projective families due to Griffiths. As an application, based on positivity of direct image sheaves, we establish a criterion for base spaces of rational Gorenstein families to be of general type. A key component of our arguments is centered around the construction of derived categorical objects generalizing relative logarithmic forms for smooth maps and their functorial properties.