On a Lemma of Varchenko and Higher Bilinear Forms Induced by Grothendieck Duality on the Milnor Algebra of an Isolated Hypersurface Singularity

Abstract

For an isolated hypersurface singularity $f:(\mathbb {C}^{n+1},0)\rightarrow (\mathbb {C},0)$ with Milnor number $\mu$ and good representative $f:(X,0)\rightarrow (\Delta ,0)$ canonical $\mu$-dimensional $\mathbb {C}$-bilinear vector spaces are associated: the Jacobian module, $\Omega ^{f}$, which is isomorphic to the Milnor algebra $A_f$ up to a choice of coordinates; and the cohomology of the canonical Milnor fiber, H. Indeed, one has defined on $\Omega ^f$, and hence in $A_f$, the non-degenerate Grothendieck pairing $res_{f,0}$ which is a symmetric $\mathbb {C}$-bilinear form, and on the vanishing cohomology H it is defined a non-degenerate $\mathbb {C}$-bilinear form $\mathbb {S}$, induced by Poincaré duality, which is $(-1)^{n+1}$-symmetric on the generalized monodromy eigenspace $H_{1}$ and $(-1)^{n}$-symmetric on the direct sum of generalized monodromy eigenspaces $H_{\ne 1}:=\oplus _{\lambda \ne 1}H_{\lambda }$. On the other hand, there are two nilpotent $\mathbb {C}$-linear maps defined on $\Omega ^f$ and H, respectively; the first one is the map $\{\mathbf {f}\}$ given by multiplication with f, which is $res_{f,0}$-symmetric, and the other one is the $\mathbb {S}$-antisymmetric endomorphism N given by the logarithm of the unipotent part of the monodromy transformation. New bilinear forms can be constructed by composing on the left (or equivalently on the right) with powers of such nilpotent maps: $res_{f,0}(\{\mathbf {f}\}^{\ell }\bullet ,\bullet )$ and $\mathbb {S}(N^{\ell }\bullet ,\bullet )$ for each integer $\ell \ge 1$. These new bilinear forms are called higher bilinear forms on $\Omega ^f$ resp. on H. In this paper, we show a formula which relates the powers $\{\mathbf {f}\}^{\ell }$, $\ell \ge 1$, to the powers $N^{j}$, $j\ge 1$. Our proof, which is inspired by a result of Varchenko obtained in 1981, uses the Laurent series (asymptotic) expansions of elements in the Jacobian module with respect to the Malgrange–Kashiwara’s $\mathcal {V}$-filtration. Finally, when the relation between Saito pairing and Grothendieck pairing is considered such a formula provides us with a result that gives an additive expansion for each higher bilinear form on $\Omega ^f$ expressed in terms of the higher bilinear forms on H and depending on the asymptotic expansions for the top forms on $\Omega ^f$ where these bilinear forms act.