Hirzebruch's 1954 Problem List — Problem 5

Definitions and mathematical statement

Let $M$ be a smooth manifold with an almost-complex structure $J\colon TM\to TM$, satisfying $J^2=-\mathrm{id}_{TM}$. A compatible Hermitian structure is represented by a Riemannian metric $g$ satisfying \[ g(JX,JY)=g(X,Y) \] for all tangent vectors $X,Y$. Write $\dim_{{\mathbb{R}}}M=2m$; the source writes $M^n$ without specifying its dimension convention.

The source refers to numbers $h^{p,q}$ "defined as above", but the supplied record does not contain that preceding definition. Denote these unspecified numbers provisionally by $h^{p,q}_g(M,J)$, for the bidegrees to which the missing definition applies.

The first question is whether, for every almost-complex manifold $(M,J)$ in the intended class and every pair of compatible Hermitian metrics $g_0,g_1$, one has \[ h^{p,q}_{g_0}(M,J)=h^{p,q}_{g_1}(M,J) \] for every relevant $(p,q)$.

If this independence fails, the source asks for another definition of numbers $\widetilde h^{p,q}(M,J)$ determined solely by the almost-complex structure and generalizing the $h^{p,q}$ of complex manifolds. Under the usual Dolbeault interpretation of complex Hodge numbers, the required agreement for integrable $J$ would be \[ \widetilde h^{p,q}(M,J)=\dim_{{\mathbb{C}}}H^{p,q}_{\bar\partial}(M,J), \] where \[ H^{p,q}_{\bar\partial}(M,J)= \frac{\ker\left(\bar\partial\colon\Omega^{p,q}(M)\to\Omega^{p,q+1}(M)\right)} {\operatorname{im}\left(\bar\partial\colon\Omega^{p,q-1}(M)\to\Omega^{p,q}(M)\right)} \] and $\Omega^{p,q}(M)$ denotes the space of smooth complex differential forms of type $(p,q)$. This interpretation of the extension requirement is conditional: the omitted preceding source text must be checked.

The supplied record does not specify compactness, boundary conditions, or the construction of $h^{p,q}_g$. Consequently, it does not determine a complete precise formulation of the first question or establish that the displayed dimensions are finite.

Short English statement

For a fixed almost-complex structure, ask whether the source's numbers $h^{p,q}$ are independent of the compatible Hermitian metric. If they are not, seek replacements determined by the almost-complex structure that recover the complex-manifold numbers. The defining passage and geometric hypotheses are missing, so this is a conditional statement-only draft requiring review.

Sources

• Local UnsolvedMath registry, supplied complete record with permanent ID $800056$, exact source problem code catalogue entry, title "Hirzebruch's 1954 Problem List --- Problem 5". The registry attributes the entry to Kotschick (2013) and labels it open; that label is supplied metadata, not an independently verified mathematical assessment.
• Kotschick, Updates on Hirzebruch's 1954 Problem List (2013), Problem 5. Supplied source URL: https://arxiv.org/abs/1305.4623. The background records extraction by arxiv_latex and an access date of 2026-10-16. These are supplied provenance details; no source visit or independent verification is asserted here.