Binomial Moments of Dolbeault Zeta Functions for High Powers of a Positive Line Bundle

For a compact Kähler manifold X of complex dimension n and a positive line bundle L, we study the binomial Dolbeault zeta moments Zₖ(p)=∑q(-1)q qkζp,q(0), 1≤ k≤ n. For large p they are exact polynomials of degree n in p. We give the coefficients of pⁿ and pⁿ⁻¹ explicitly, then the metric multiplier of the coefficient of pⁿ⁻², namely γn,k=n-1k-1(k-1)(6-k)/8k(k+1)(k+2)(k+3), which vanishes exactly at $k=1$ and $k=6$. At the exceptional weight six, metric dependence reappears at order pⁿ⁻³, with coefficients $-1/3780$ and η=1/1008. In complex dimension three and for $k=2$, the linear coefficient is expressed through s², |Ric|² and |Rm|², and confirmed on CP³. Weighted heat-kernel remainder estimates justify the logarithmic expansion of 12Zₖ(p)log p. Introduction Large-p asymptotics of spectral invariants attached to positive line bundles form a central chapter of analytic torsion theory. Bismut and Vasserot established the leading behavior of the Ray–Singer torsion, and Finski obtained a full expansion whose first coefficients he computes in the Kähler setting. The local tools are the heat kernel calculus (Gilkey, Vassilevich), the Bergman kernel expansion (Dai–Liu–Ma) and the determinant bundle framework (Bismut–Gillet–Soulé). Here we consider the family (Zₖ(p))1≤ k≤ n, a triangular transform of the values ζp,q(0): since qk=0 for $q<k$, Zₖ uses only degrees q≥ k, and Zₙ is, up to sign, the top-degree value alone. These moments are combinations of constant heat coefficients, and their polynomial coefficients are computed explicitly here. The central question is which coefficients depend effectively on the Kähler metric, and up to which order. The contributions are as follows. First, exact polynomiality and the coefficients of pⁿ and pⁿ⁻¹ (sections 1 and 2). Second, the multiplier γn,k of the Calabi energy at order pⁿ⁻² and its vanishing at weight six (section 3). Third, the explicit coefficient of ∫|∇ K|²dA at the next order (section 4) and the complete computation in dimension three (section 5), checked on CP³ (section 6). Finally, weighted remainder estimates and the logarithmic expansion (section 7). Scope. Metric dependence is established on product models (one curved curve times flat tori), not for an arbitrary Kähler manifold. It concerns a specific coefficient of Z₆ and should not be identified with Finski's universal coefficients for the analytic torsion itself without an explicit comparison formula. The extension to binomial weights is obtained by finite linear combination of Finski's degreewise estimates. Definition Let X be compact Kähler of complex dimension n, (L,hL) positive, ω=2π c₁(L,hL), =c₁(L), dV=ωⁿ/n!. Let p,q be the Dolbeault Laplacian on $(0,q)$-forms with values in Lᵖ, and ζp,q(z)=∑λ≠0λ⁻ᶻ. The binomial moments are Zₖ(p)=∑q=0ⁿ(-1)qqk,ζp,q(0)=[t⁰],Tr(Wₖ e-tₚ), Wₖ=(-1)NNk, 1≤ k≤ n . For large p, Zₖ(p)=∑ⱼ₌₀ⁿAk,j,pⁿ⁻ʲ is an exact polynomial of degree exactly n. Leading coefficients On a product of flat elliptic curves, with D=∏δᵢ: Zₖ(p)= nk aₖ D,pⁿ, where aₖ=(-1)ᵏx⁰⁻ᵏ=1k!∫₀¹u(u+1)⋯(u+k-1),du . Values: a₁=12,\ a₂=512,\ a₃=38,\ a₄=251/720,\ a₅=95/288,\ a₆=19087/60480. In general: [pⁿ]Zₖ= nk aₖ∫Xⁿ/n!, [pⁿ⁻¹]Zₖ=bn,k∫Xc₁(TX),ⁿ⁻¹(n-1)!, bn,k=n-1k-1(32aₖ-aₖ₊₁)+12n-1kaₖ, bn,1=3n+1/12, b3,2=17/24. Order pⁿ⁻² and the exceptional weight $k=6$ With E(ω)=(2π)⁻ⁿ∫X s²,dV (Calabi energy, s the Riemannian scalar curvature), [pⁿ⁻²]Zₖ=un,k∫Xc₁(TX)²ⁿ⁻²(n-2)!+vn,k∫Xc₂(TX)ⁿ⁻²(n-2)!+γn,k,E(ω), γn,k=n-1k-1(k-1)(6-k)/8,k(k+1)(k+2)(k+3), γn,k=0\ for k=1,\ k=6 . The multiplier comes from the residue dₖ=(-1)ᵏx⁰ with A=(eˣ-1)⁻¹: dₖ=-12(1k-7k+1+12/k+2-6k+3)=(k-1)(6-k)/2k(k+1)(k+2)(k+3), γn,k=14n-1k-1dₖ . For k=2,,8: dₖ=160,1120,1280,1840,0,-11680,-77920. Chern–Weil identities used: $$∫_X{c_1^2ⁿ⁻²}{(n-2)!}=1{(2π)^n}∫_X({s^2}4-{|Ric|^2}2)dV, ∫_X{c_2ⁿ⁻²}{(n-2)!}=1{(2π)^n}∫_X({s^2}8-{|Ric|^2}2+{|Rm|^2}8)dV .$$ For $2≤ k≤ n$, $k≠6$, the metric dependence is effective (a curve deformed at fixed area times flat curves). Weight six at order $pⁿ⁻³$ For the sphere of curvature $K$: $F_1(x/p)~δ(pA+KB+{K^2}pC+K^3/p^2J)$, with $$J(x)=x^2/630+x^3/1260-x^4/2520-x^5/3780+O(x^6), [x^0](A^5J)=-1{3780}.$$ On $C× E_2×⋯× E_n$ (a curve times flat tori), for $n≥6$, with $D_f=∏i≥2δ_i$: $$[pⁿ⁻³]Z_6={n-1}{5}D_f/2π(-1{3780}∫_C K^3dA+η∫_C|∇ K|^2dA), η=1{1008}.$$ The local oscillator computation gives $G(x)=-x^2/2240-x^3/4480+x^4/40320+x^5/32256+O(x^6)$, hence $$[x^0],(e^x-1)⁻⁵G(x)=1{4032}, η=4·1{4032}=1{1008}.$$ For an area-preserving conformal deformation of the round sphere, the second variation is $(-K_0/1260+ηλ)∫ K^2dA$; with $λ=6K_0$ and $λ=12K_0$ it equals $13K_0/2520$ and $K_0/90$, both nonzero. Dimension three, weight two $$Z_2(p)=5{24}p^3!∫_X!^3+17/48p^2!∫_X!c_1(TX)^2+A(ω),p+B(ω),$$ $$A(ω)=1{360(4π)^3}∫_X(130,s^2-266,|Ric|^2+27,|Rm|^2)dV=79/720!∫ c_1^2+3{40}!∫ c_2+1{960π^3}!∫ s^2dV,$$ so $γ3,2=1/120$. On $C_1× C_2× C_3$ (curvatures $K_i$, $D=δ_1δ_2δ_3$): $$Z_2(p)=D[54p^3+17/24p^2∑ K_i+p(1{30}∑ K_i^2+13/36∑i<jK_iK_j)+1{120}∑i≠ jK_i^2K_j+16K_1K_2K_3].$$ Test on $CP^3$ (Fubini–Study) With $L= O(1)$: $s=24$, $|Ric|^2=|Rm|^2=96$, so $|Rm|^2-2|Ric|^2=-96≠0$. The exact spectrum follows from $2p,q=C_G(λ)-C_K(σ_q)+p(2q-3)+4q$ on $U(4)/(U(3)× U(1))$. Result, computed in rational arithmetic: $$Z_2(p)=5{24}p^3+17/12p^2+541/180p+39/20.$$ The first three coefficients agree with the formulas of section 5; the value $39/20$ is a target for $B(ω)$. Logarithmic asymptotics and determinants With $S_k(p)=-12,Ξ_k'(0)$ and $Ξ_k(z)=∑_q(-1)^q qkζp,q(z)$, for every order $M$: $$S_k(p)=12Z_k(p)log p+∑ⱼ₌₀Mβk,j,pⁿ⁻ʲ+O(pn-M-1),$$ with no $p⁻ʲlog p$ terms. Interpretation: $e2S_k(p)=∏q=0ⁿ({}'p,q)(-1)^q qk$, and $S_k(c,_p)-S_k(_p)=12Z_k(p)log c$. Status of the proofs Exact: polynomiality, $a_k$, $bn,k$, $γn,k$, $-1/3780$ (finite extraction). Numerically corroborated: $η=1/1008$, through a direct computation accurate to $10⁻¹³$. Still open: an independent analytic recomputation of $g_4$ and a specialist review of the local-to-global transfer. Metric dependence is established on product models, not on every Kähler manifold.