On the Briançon-Skoda theorem for analytic local rings with singularities

Grauert-Riemenschneider multiplier ideal sheaves and the (optimal) Briançon-Skoda number

“…Moreover, we have the following: Proposition 1.1. ( [22], Proposition 3.3). Let ϕ ∈ Psh(X) be a plurisubharmonic function on X such that ϕ −∞ on every irreducible component of X.…”

“…In [22], the author introduced a notion of Grauert-Riemenschneider multiplier ideal sheaves on any (reduced) complex space of pure dimension (not necessarily normal or Q-Gorenstein), motivated by the definition of Grauert-Riemenschneider canonical sheaf. Relying on the new multiplier ideals, the author established the original Briançon-Skoda theorem for analytic local rings with normal weakly rational singularities.…”

“…In the present note, we are going to investigate the Grauert-Riemenschneider multiplier ideals further and prove some related properties (e.g., the weak holomorphicity and Skoda-type division theorem, etc.). As applications, following the idea adopted in [22], we also obtain the Briançon-Skoda number for Noetherian rings of weakly holomorphic functions (i.e., germs of locally bounded meromorphic functions on singular complex spaces) with weakly rational singularities, and give some characterizations of validity of the original Briançon-Skoda theorem.…”

“…[8], Definition 5.4), the Grauert-Riemenschneider multiplier ideal sheaf is in general a fractional ideal sheaf, which is an ideal sheaf when X is a normal complex space and agrees with the usual multiplier ideal if X is smooth (cf. [22]). So it is interesting for us to understand how "singular" the multiplier ideals might be.…”

“…Combining with a relative version of Grauert-Riemenschneider vanishing theorem for the higher direct images, we formulate a Skoda-type division theorem for Grauert-Riemenschneider multiplier ideal sheaves as follows (see also [22]).…”