Let \(A\) be a Noetherian commutative ring of finite cohomological dimension, and let \(U\) be a smooth complex analytic variety with fundamental group \(G\). Let \(\mathcal L_U\) be the universal local system on \(U\), defined by \(\mathcal L_U:=p_! A_{\tilde U}\), where \(p:\tilde U\to U\) is the universal cover of \(U\), and \(A_{\tilde U}\) is the constant sheaf with stalk \(A\) on \(\tilde U\). Note that the stalks of \(\mathcal L_U\) can be identified with the group ring \(A[G]\). Since \(G\) acts on \(\tilde U\) on the right by deck transformations, \(\mathcal L_U\) is a local system of rank one free right \(A[G]\)-modules. This local system is called universal because any other local system of \(A\)-modules with stalk \(V\) can be obtained from \(\mathcal L_U\) by tensoring it with \(V\) over \(A[G]\).
Let \(q:U\to \mathrm{pt}\) be the projection to a point, let \(D^b_c(U,A)\) be the bounded derived category of constructible sheaves of \(A\)-modules on \(U\), and let \(D^b(A[G])\) be the bounded derived category of \(A[G]\)-modules. The authors define the Mellin transformation by \[ \begin{array}{rcl} \mathfrak{M}_*^U: D_c^b(U,A)&\longrightarrow & D^b(A[G])\\ \mathcal F&\longmapsto& Rq_*(\mathcal F\otimes_A \mathcal L_U) \end{array} \] This is a non-abelian counterpart to Gabber and Loesers’s Mellin transformation [O. Gabber and F. Loeser, Duke Math. J. 83, No. 3, 501–606 (1996; Zbl 0896.14009)] for complex affine tori and field coefficients.
The main result of the paper is a generalization to certain Stein manifolds \(U\) of Gabber-Loeser’s result on the t-exactness of \(\mathfrak{M}_*^{\mathbb{C}^n}\) (see Theorem 1.1 and Remark 1.2 in the paper for details), which says that the Mellin transform of a perverse sheaf is concentrated in degree zero. For this result to hold, the authors impose some hypotheses on the geometry of \(U\), namely that it has a smooth compactification \(X\) whose boundary divisor \(E\) is a simple normal crossings divisor, that the usual stratification on \(E\) induced by its irreducible components is made up of Stein manifolds, and that the local fundamental groups of \(U\) at any point in \(E\) inject into the fundamental group of \(U\). These extra hypothesis hold for complex affine tori (Gabber-Loeser’s setting), but also hold for other important classes of varieties, such as complements of essential hyperplane arrangements, toric arrangements ane elliptic arrangements (in their respective wonderful compactifications), and complements of at least \(n+1\) general hyperplane sections in a projective manifold of dimension \(n\). In particular, this paper generalizes Gabber-Loeser’s t-exactness result from field coefficients to more general coefficients \(A\), a fact that does not directly follow from Gabber and Loeser’s proof.
In order to prove the main result, the authors reduce it to a certain local vanishing result for the multivariable Sabbah specialization functor, which they are able to obtain after first proving the t-exactness of the (global) multivariable Sabbah specialization functor (see Corollary 3.4). The (univariable) Sabbah specialization functor with field coefficients is non-canonically isomorphic to Deligne’s perverse nearby cycle functor, so the authors generalize the \(t\)-exactness of the nearby cycle functor to more general coefficients.
Lastly, the authors apply their main result to construct new families of duality spaces (in the sense of [R. Bieri and B. Eckmann, Invent. Math. 20, 103–124 (1973; Zbl 0274.20066)]). These are spaces \(U\) that are homotopy equivalent to a connected finite-type CW complex such that \(H^k(U,\mathcal L_U)\) (for \(A=\mathbb Z\)) is non-zero only in one degree, in which it is torsion-free. Smooth connected algebraic varieties that satisfy the extra geometric hypothesis imposed in the main result of this paper are duality spaces by work of Denham-Suciu [G. Denham and A. I. Suciu, Forum Math. Sigma 6, Paper No. e6, 20 p. (2018; Zbl 1400.55002)]. The authors generalize these examples providing non-affine and singular examples (see Proposition 6.4 and Corollary 6.6).