ADELIC TORSION WITH CURRENT INSERTIONS AND SIU-SATURATION: SINGULAR BOTT–CHERN LOCALIZATION ON COMPACT SHIMURA VARIETIES

We introduce a current-inserted adelic analytic torsion functional for compact Kähler Shimura varieties. The construction extends the philosophy of projected adelic analytic torsion from smooth automorphic data to singular geometric data represented by positive closed currents. Let X be a compact Kähler Shimura variety of complex dimension m, and let T be a positive closed (p,p)-current. For p = 1, such a current can often be realized as the curvature current of a singular Hermitian line bundle. For p > 1, however, a (p,p)-current is not, in general, the curvature of a line bundle. We therefore define the torsion by a heat-trace insertion formalism: the current is paired with the (m − p,m −p)-component of the local Dolbeault heat supertrace, and the resulting inserted trace is regularized by a finite-part Mellin transform. For currents admitting admissible regularizations, we prove a localization formula governed by the Siu decomposition T = λi[Zi] + R. i The current-inserted torsion decomposes into three pieces: the intrinsic adelic torsions of the analytic strata Zi, a singular Bott–Chern anomaly measuring the normal and metric discrepancy of the embeddings Zi → X, and a residual spectral term attached to the diffuse component R. We then isolate the precise analytic condition under which the residual term disappears. A positive closed current is called Siu-saturated if its analytic Siu part carries the full cohomology class. We prove that Siu-saturation is equivalent to the vanishing of the residual current: [T] = [Talg] ⇐⇒ R=0. This implication is unconditional and follows from positivity: a positive closed current with zero cohomology class has zero mass and hence vanishes. Finally, in the Shimura setting, we separate the analytic mechanism from the arithmetic input. Vector generation of rational Hodge classes by special cycles implies the Hodge property in the corresponding degree, while effective cone generation produces positive Siu-saturated representatives. The current-inserted torsion does not replace these arithmetic generation theorems; it provides a spectral-analytic certificate that, once Siu-saturation is known, no diffuse residual component remains.