Abstract
Abstract We define and study a notion of minimal exponent for a local complete intersection subscheme đ of a smooth complex algebraic variety đ, extending the invariant defined by Saito in the case of hypersurfaces. Our definition is in terms of the KashiwaraâMalgrange đ-filtration associated to đ. We show that the minimal exponent describes how far the Hodge filtration and order filtration agree on the local cohomology H Z r âą ( O X ) \mathcal{H}^{r}_{Z}(\mathcal{O}_{X}) , where đ is the codimension of đ in đ. We also study its relation to the BernsteinâSato polynomial of đ. Our main result describes the minimal exponent of a higher codimension subscheme in terms of the invariant associated to a suitable hypersurface; this allows proving the main properties of this invariant by reduction to the codimension 1 case. A key ingredient for our main result is a description of the KashiwaraâMalgrange đ-filtration associated to any ideal ( f 1 , ⊠, f r ) (f_{1},\ldots,f_{r}) in terms of the microlocal đ-filtration associated to the hypersurface defined by â i = 1 r f i âą y i \sum_{i=1}^{r}f_{i}y_{i} .
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