HOLOMORPHIC RODRIGUES–RAY–SINGER ENVELOPES FOR KÄHLER FOUR- AND SIX-MANIFOLDS

We develop a holomorphic analogue of the Rodrigues–Ray–Singer torsion envelope in real dimensions four and six. The de Rham complex of the odd-dimensional theory is replaced by the Dolbeault complex of a Hermitian holomorphic vector bundle over a compact Kähler manifold, and the Hodge spectral gap is replaced by a uniform Dolbeault spectral gap. Under bounded Kähler geometry of finite order and uniform bounds on the Chern curvature of the bundle, we prove a non-asymptotic linear-volume estimate for the holomorphic Ray–Singer torsion: |\log T_{hol}(X,E,\omega,h)| \le B_{hol}\text{Vol}(X,\omega). The proof follows the heat-kernel and Mellin architecture of the odd-dimensional Rodrigues–Ray–Singer envelope, but the even-dimensional setting introduces a critical local heat coefficient producing a logarithmic Mellin contribution. We isolate and control this term uniformly. The resulting estimate gives a bounded-geometry control of Quillen determinant norms in Kähler four- and six-manifolds, with applications to Calabi–Yau geometry, HSFG-type geometric flows, and BCOV-type determinant contributions in mathematical physics. More precisely, the novelty of the present paper is not the definition of holomorphic analytic torsion itself, which belongs to the Ray–Singer–Quillen–Bismut–Gillet–Soulé framework, but the construction of a uniform bounded-geometry envelope for it in real dimensions four and six. The estimate is non-asymptotic, depends only on finite-order Kähler bounded geometry, Chern-curvature control, and a Dolbeault spectral gap, and yields a linear-volume control of the holomorphic torsion and Quillen norm along geometric families and flows.