𝑉-filtrations and minimal exponents for local complete intersections

We define and study a notion of minimal exponent for a local complete intersection subscheme 𝑍 of a smooth complex algebraic variety 𝑋, extending the invariant defined by Saito in the case of hypersurfaces. Our definition is in terms of the Kashiwara–Malgrange 𝑉-filtration associated to 𝑍. We show that the minimal exponent describes how far the Hodge filtration and order filtration agree on the local cohomology <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi mathvariant="script">H</m:mi> <m:mi>Z</m:mi> <m:mi>r</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi mathvariant="script">O</m:mi> <m:mi>X</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> HʳZ(OX) , where 𝑟 is the codimension of 𝑍 in 𝑋. We also study its relation to the Bernstein–Sato polynomial of 𝑍. 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 Kashiwara–Malgrange 𝑉-filtration associated to any ideal <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>f</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>f</m:mi> <m:mi>r</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (f₁,…,fᵣ) in terms of the microlocal 𝑉-filtration associated to the hypersurface defined by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mo>∑</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>r</m:mi> </m:msubsup> <m:mrow> <m:msub> <m:mi>f</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:msub> <m:mi>y</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:mrow> </m:math> ∑ᵢ₌₁ʳfᵢyᵢ .