Embeddings of Demi-Normal Varieties

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 $S_2$, $G_1$, and seminormality. Equivalently, $X$ has Serre's $S_2$ 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 Koll\'ar in \cite{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 \cite{Berq14}. The original form of this vanishing result appears in \cite{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 \cite{Berq14}. The analogous result for quasi-projective varieties is settled here. Semi-rational surface singularities have also been studied in \cite{vS87}.