A Fourier--Dolbeault model for Kohn--Rossi geometry

Definitions and mathematical statement

Source scope. The complete Question 1.32 says only that Akahori discussed problems about the geometry of Kohn--Rossi cohomology. It supplies no manifold class, bidegree, conjecture, estimate or requested construction. It is therefore context only; the following regular-circle-bundle calculation is an auxiliary example from the aid, not a reconstructed missing question.

Let $B$ be a compact complex manifold of dimension $n\ge1$ and let $L\to B$ be a positive holomorphic line bundle with a positively curved Hermitian metric. Its dual unit circle bundle $X=S(L^*)$ has its standard regular strongly pseudoconvex CR structure and circle action $R_\theta(v)=e^{i\theta}v$. The scalar Kohn--Rossi groups are the cohomology of the tangential Cauchy--Riemann complex, \[ H_b^{0,q}(X)=\ker(\bar\partial_b:\Omega_b^{0,q}\to\Omega_b^{0,q+1}) /\operatorname{im}(\bar\partial_b:\Omega_b^{0,q-1}\to\Omega_b^{0,q}). \] Use weight $m\in{\mathbb{Z}}$ to mean $R_\theta^*u=e^{im\theta}u$. With this convention the aid asserts the natural Fourier--Dolbeault identification \[ H_b^{0,q}(X)_m\cong H^q(B,L^m),\qquad 0\le q\le n. \] In particular positive weight corresponds to $L^m$, not $L^{-m}$. Reversing the circle action reverses that sign.

For each interior degree $1\le q\le n-1$, the aid asserts that only finitely many weights contribute, giving \[ H_b^{0,q}(X)\cong\bigoplus_{m\in{\mathbb{Z}}}H^q(B,L^m), \] with a finite nonzero sum. The positive tail is controlled by Serre vanishing and the negative tail by Serre duality and vanishing for $K_B\otimes L^{-m}$; no effective uniform numerical cutoff is supplied. When $n=1$ the interior range is empty. The finite-window assertion does not extend to endpoint degrees; full smooth endpoint spaces require Fourier-series convergence/completion conventions rather than a naive algebraic direct sum of arbitrary modes.

For the standard sphere $S^{2n+1}=S(\mathcal O(-1))\to{\mathbb{P}}^n$, the aid's complete mode count is \[ \dim H_b^{0,q}(S^{2n+1})_m= \begin{cases} \binom{m+n}{n},&q=0,\ m\ge0,\\ \binom{-m-1}{n},&q=n,\ m\le-n-1,\\ 0,&\text{otherwise}. \end{cases} \] This fixes the weight sign and exhibits the infinite positive and negative endpoint tails. No conclusion about the unspecified Akahori problems is inferred from this special regular model.

Short English statement

The source records a topic discussed by Akahori but no recoverable mathematical question. The supplied auxiliary example identifies Fourier modes of Kohn--Rossi cohomology on a positive circle bundle with line-bundle Dolbeault cohomology. Interior degrees have only finitely many contributing modes, while sphere endpoint degrees have infinite tails.

Sources

• Local UnsolvedMath registry, permanent ID 20002090, source code catalogue entry; canonical aim-geometry-notes.json, notes[427], statement status reconstructed unverified.
• AIM Holomorphic curves in contact geometry, Question 1.32, discussion attributed to Akahori: https://aimath.org/WWN/contactgeom2/contactgeom2.pdf.
• No separate authoritative citation for a precise Akahori problem or for the Fourier--Dolbeault special case is supplied. The aid invokes standard line-bundle cohomology, Serre vanishing and duality without bibliographic entries.
• Registry research attempt 1, dated 2026-08-14: context only, status at run not a problem, rigor "proved", novelty confidence low. The machine-generated, non-peer-reviewed aid requires independent verification before citation.