Embeddings of Demi-Normal Varieties

الأصل · EN

Our primary result is that a demi-normal quasi-projective variety can be embedded in a demi-normal projective variety. Recall that a demi-normal variety X is a variety with properties S₂, G₁, and seminormality. Equivalently, X has Serre's S₂ property and there is an open subvariety U with complement of codimension at least 2 in X, such that the only singularities of U are (analytically) double normal crossings. The term demi-normal was coined by Kollár in Kol13. As a consequence of this embedding theorem, we prove a semi-smooth Grauert-Riemenschneider vanishing theorem for quasi-projective varieties, the projective case having been settled in Berq14. The original form of this vanishing result appears in GR70. We prove an analogous result for semi-rational singularities. The definition of semi-rationality requires that the choice of a semi-resolution is immaterial. This also has been established in the projective case in Berq14. The analogous result for quasi-projective varieties is settled here. Semi-rational surface singularities have also been studied in vS87.