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

Abstract For a compact Kähler manifold $X$ of complex dimension $n$ and a positive line bundle $L$, we study the binomial Dolbeault zeta moments $Z_k(p)=\sum_q(-1)^q\binom qk\zeta_{p,q}(0)$, $1\le k\le n$. For large $p$ they are exact polynomials of degree $n$ in $p$. We give the coefficients of $p^n$ and $p^{n-1}$ explicitly, then the metric multiplier of the coefficient of $p^{n-2}$, namely $\gamma_{n,k}=\binom{n-1}{k-1}\frac{(k-1)(6-k)}{8k(k+1)(k+2)(k+3)}$, which vanishes exactly at $k=1$ and $k=6$. At the exceptional weight six, metric dependence reappears at order $p^{n-3}$, with coefficients $-1/3780$ and $\eta=1/1008$. In complex dimension three and for $k=2$, the linear coefficient is expressed through $s^2$, $|\mathrm{Ric}|^2$ and $|\mathrm{Rm}|^2$, and confirmed on $\mathbb{CP}^3$. Weighted heat-kernel remainder estimates justify the logarithmic expansion of $\tfrac12Z_k(p)\log p$. 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 $(Z_k(p)){1\le k\le n}$, a triangular transform of the values $\zeta{p,q}(0)$: since $\binom qk=0$ for $q<k$, $Z_k$ uses only degrees $q\ge k$, and $Z_n$ 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 $p^n$ and $p^{n-1}$ (sections 1 and 2). Second, the multiplier $\gamma_{n,k}$ of the Calabi energy at order $p^{n-2}$ and its vanishing at weight six (section 3). Third, the explicit coefficient of $\int|\nabla K|^2dA$ at the next order (section 4) and the complete computation in dimension three (section 5), checked on $\mathbb{CP}^3$ (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 $Z_6$ 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 $n$, $(L,h^L)$ positive, $\omega=2\pi c_1(L,h^L)$, $\ell=c_1(L)$, $dV=\omega^n/n!$. Let $\square_{p,q}$ be the Dolbeault Laplacian on $(0,q)$-forms with values in $L^p$, and $\zeta_{p,q}(z)=\sum_{\lambda\neq0}\lambda^{-z}$. The binomial moments are $$Z_k(p)=\sum_{q=0}^{n}(-1)^q\binom{q}{k},\zeta_{p,q}(0)=[t^0],\mathrm{Tr}\big(W_k e^{-t\square_p}\big),\qquad W_k=(-1)^N\binom{N}{k},\quad 1\le k\le n .$$ For large $p$, $Z_k(p)=\sum_{j=0}^{n}A_{k,j},p^{n-j}$ is an exact polynomial of degree exactly $n$. Leading coefficients On a product of flat elliptic curves, with $D=\prod\delta_i$: $Z_k(p)=\binom nk a_k D,p^n$, where $$a_k=(-1)^kx^0^{-k}=\frac1{k!}\int_0^1u(u+1)\cdots(u+k-1),du .$$ Values: $a_1=\tfrac12,\ a_2=\tfrac5{12},\ a_3=\tfrac38,\ a_4=\tfrac{251}{720},\ a_5=\tfrac{95}{288},\ a_6=\tfrac{19087}{60480}$. In general: $$[p^n]Z_k=\binom nk a_k\int_X\frac{\ell^n}{n!},\qquad [p^{n-1}]Z_k=b_{n,k}\int_X\frac{c_1(TX),\ell^{n-1}}{(n-1)!},$$ $$b_{n,k}=\binom{n-1}{k-1}\Big(\tfrac32a_k-a_{k+1}\Big)+\tfrac12\binom{n-1}{k}a_k,\qquad b_{n,1}=\frac{3n+1}{12},\quad b_{3,2}=\frac{17}{24}.$$ Order $p^{n-2}$ and the exceptional weight $k=6$ With $E(\omega)=(2\pi)^{-n}\int_X s^2,dV$ (Calabi energy, $s$ the Riemannian scalar curvature), $$[p^{n-2}]Z_k=u_{n,k}\int_X\frac{c_1(TX)^2\ell^{n-2}}{(n-2)!}+v_{n,k}\int_X\frac{c_2(TX)\ell^{n-2}}{(n-2)!}+\gamma_{n,k},E(\omega),$$ $$\gamma_{n,k}=\binom{n-1}{k-1}\frac{(k-1)(6-k)}{8,k(k+1)(k+2)(k+3)},\qquad \gamma_{n,k}=0\ \text{ for } k=1,\ k=6 .$$ The multiplier comes from the residue $d_k=(-1)^kx^0$ with $A=(e^x-1)^{-1}$: $$d_k=-\frac12\Big(\frac1k-\frac7{k+1}+\frac{12}{k+2}-\frac6{k+3}\Big)=\frac{(k-1)(6-k)}{2k(k+1)(k+2)(k+3)},\qquad \gamma_{n,k}=\tfrac14\tbinom{n-1}{k-1}d_k .$$ For $k=2,\dots,8$: $d_k=\tfrac1{60},\tfrac1{120},\tfrac1{280},\tfrac1{840},0,-\tfrac1{1680},-\tfrac7{7920}$. Chern–Weil identities used: $$\int_X\frac{c_1^2\ell^{n-2}}{(n-2)!}=\frac1{(2\pi)^n}\int_X\Big(\frac{s^2}4-\frac{|\mathrm{Ric}|^2}2\Big)dV,\quad \int_X\frac{c_2\ell^{n-2}}{(n-2)!}=\frac1{(2\pi)^n}\int_X\Big(\frac{s^2}8-\frac{|\mathrm{Ric}|^2}2+\frac{|\mathrm{Rm}|^2}8\Big)dV .$$ For $2\le k\le n$, $k\neq6$, the metric dependence is effective (a curve deformed at fixed area times flat curves). Weight six at order $p^{n-3}$ For the sphere of curvature $K$: $F_1(x/p)\sim\delta\big(pA+KB+\tfrac{K^2}pC+\tfrac{K^3}{p^2}J\big)$, with $$J(x)=\frac{x^2}{630}+\frac{x^3}{1260}-\frac{x^4}{2520}-\frac{x^5}{3780}+O(x^6),\qquad [x^0]\big(A^5J\big)=-\frac1{3780}.$$ On $C\times E_2\times\cdots\times E_n$ (a curve times flat tori), for $n\ge6$, with $D_f=\prod_{i\ge2}\delta_i$: $$[p^{n-3}]Z_6=\binom{n-1}{5}\frac{D_f}{2\pi}\Big(-\frac1{3780}\int_C K^3dA+\eta\int_C|\nabla K|^2dA\Big),\qquad \eta=\frac1{1008}.$$ The local oscillator computation gives $G(x)=-\frac{x^2}{2240}-\frac{x^3}{4480}+\frac{x^4}{40320}+\frac{x^5}{32256}+O(x^6)$, hence $$[x^0],(e^x-1)^{-5}G(x)=\frac1{4032},\qquad \eta=4\cdot\frac1{4032}=\frac1{1008}.$$ For an area-preserving conformal deformation of the round sphere, the second variation is $\big(-\tfrac{K_0}{1260}+\eta\lambda\big)\int\dot K^2dA$; with $\lambda=6K_0$ and $\lambda=12K_0$ it equals $\tfrac{13K_0}{2520}$ and $\tfrac{K_0}{90}$, both nonzero. Dimension three, weight two $$Z_2(p)=\tfrac5{24}p^3!\int_X!\ell^3+\tfrac{17}{48}p^2!\int_X!c_1(TX)\ell^2+A(\omega),p+B(\omega),$$ $$A(\omega)=\frac1{360(4\pi)^3}\int_X\big(130,s^2-266,|\mathrm{Ric}|^2+27,|\mathrm{Rm}|^2\big)dV=\tfrac{79}{720}!\int c_1^2\ell+\tfrac3{40}!\int c_2\ell+\frac1{960\pi^3}!\int s^2dV,$$ so $\gamma_{3,2}=1/120$. On $C_1\times C_2\times C_3$ (curvatures $K_i$, $D=\delta_1\delta_2\delta_3$): $$Z_2(p)=D\Big[\tfrac54p^3+\tfrac{17}{24}p^2\sum K_i+p\Big(\tfrac1{30}\sum K_i^2+\tfrac{13}{36}\sum_{i<j}K_iK_j\Big)+\tfrac1{120}\sum_{i\ne j}K_i^2K_j+\tfrac16K_1K_2K_3\Big].$$ Test on $\mathbb{CP}^3$ (Fubini–Study) With $L=\mathcal O(1)$: $s=24$, $|\mathrm{Ric}|^2=|\mathrm{Rm}|^2=96$, so $|\mathrm{Rm}|^2-2|\mathrm{Ric}|^2=-96\ne0$. The exact spectrum follows from $2\square_{p,q}=C_G(\lambda)-C_K(\sigma_q)+p(2q-3)+4q$ on $U(4)/(U(3)\times U(1))$. Result, computed in rational arithmetic: $$Z_2(p)=\frac5{24}p^3+\frac{17}{12}p^2+\frac{541}{180}p+\frac{39}{20}.$$ The first three coefficients agree with the formulas of section 5; the value $39/20$ is a target for $B(\omega)$. Logarithmic asymptotics and determinants With $S_k(p)=-\tfrac12,\Xi_k'(0)$ and $\Xi_k(z)=\sum_q(-1)^q\binom qk\zeta_{p,q}(z)$, for every order $M$: $$S_k(p)=\frac12Z_k(p)\log p+\sum_{j=0}^{M}\beta_{k,j},p^{n-j}+O(p^{n-M-1}),$$ with no $p^{-j}\log p$ terms. Interpretation: $e^{2S_k(p)}=\prod_{q=0}^{n}\big(\det{}'\square_{p,q}\big)^{(-1)^q\binom qk}$, and $S_k(c,\square_p)-S_k(\square_p)=\tfrac12Z_k(p)\log c$. Status of the proofs Exact: polynomiality, $a_k$, $b_{n,k}$, $\gamma_{n,k}$, $-1/3780$ (finite extraction). Numerically corroborated: $\eta=1/1008$, 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.