Search for: All records

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 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$$3$$. 

Abstract We prove a generalization of the Fulton–Hansen connectedness theorem, where$${\mathbb {P}}^n$$ $P^n$is replaced by a normal variety on which an algebraic group acts with a dense orbit. 

Abstract We prove Grauert–Riemenschneider–type vanishing theorems for excellent divisiorally log terminal threefolds pairs whose closed points have perfect residue fields of positive characteristic $p>5$. Then we discuss applications to dlt singularities and to Mori fiber spaces of three-folds.