The Weighted Chern Character $ch_{\mathcal{N}}$: A Degree-Weighted Package for Alternating Sums, Bott–Chern Transgression, and Dolbeault Torsion Expressions

This paper introduces the weighted Chern character $ch_N$, an autonomous characteristic form that packages the degree-weighted alternating sums arising ubiquitously in holomorphic analytic torsion (Ray–Singer, Quillen) and the Bismut–Gillet–Soulé theory. The object $ch_N$ is defined as the ordinary Chern character of a rational virtual K-theory class $∑_q (−1)^q (q/2) [S^{(0,q)}]$ attached to a Z-graded holomorphic bundle $S = ⊕_q S^{(0,q)}$. This K-theoretic realisation yields clean functoriality, a canonical normalisation compatible with the number operator weight in torsion expressions, and a Bott–Chern transgression formula obtained purely by linearity from classical Bott–Chern theory. A key technical contribution is the establishment of precise conjugation sign rules (Lemma 6.1) for the passage between the holomorphic bundles $Ω^{q,0}$ and the smooth anti-holomorphic bundles $Ω^{0,q}$: the degree-$2p$ Chern–Weil form picks up the sign $(−1)^p$, while the corresponding Bott–Chern transgression form picks up the sign $(−1)^{p−1}$ (one fewer curvature factor). These distinct signs are verified by two independent arguments (dd^c consistency and explicit transgression integral) and are confirmed by sanity checks on $CP^1$ and for $p = 1$. The Dolbeault weighted character $ch_N^{Dol}$ is accordingly defined using the holomorphic bundles Ω^{q,0} with their Chern connections. The paper further develops a $λ_y-formalism$ viewpoint (Proposition 6.8) that makes the holomorphic/anti-holomorphic symmetry transparent at the generating-function level: the substitution $x_j → −x_j$ in the product $∏_j(1 + y e^{∓x_j})$ exchanges the two sides, and the degree weight is extracted by the operator $(1/2)(y d/dy)|_{y=−1}$. Universal low-degree formulas for $ch_N^{Dol}(X)$ are derived for complex surfaces and threefolds in terms of the Chern classes c_i(TX). Explicit computations are carried out on Hirzebruch surfaces $F_k$ (with twist by a line bundle), the flag manifold $Fl(3)$, and $P^3$. Detailed comparative examples demonstrate that the weighted package reduces lengthy degree-by-degree expansions to single substitutions, while also clarifying the sign pitfalls that arise when using literal $Ω^{0,q}$ in place of $Ω^{q,0}$, especially in the twisted case where no single global sign factor exists. The paper does not claim a new analytic torsion anomaly theorem; rather, it systematises the cohomological package underlying the degree-weighted sums produced by the number operator, providing universal reusable identities that reduce sign and combinatorial errors in torsion-related computations.