In the paper

• Friedrich Hirzebruch, Some Problems on Differentiable and Complex Manifolds, Annals of Mathematics Second Series, Vol. 60, No. 2 (1954) pp. 213-236, doi:10.2307/1969629

Hirzebruch collected problems and questions on smooth and complex manifolds presented at a conference the year prior.

For an almost complex manifold $M$ equipped with a hermitian metric, one can form the Laplacian $\Delta_{\bar{\partial}}$. Even though $\bar{\partial}$ need not square to zero on an almost complex manifold, this Laplacian is an elliptic operator and so the kernel of the Laplacian is finite dimensional if $M$ is compact. Denote the dimension of this kernel restricted to $(p,q)$ forms by $h^{p,q}$.

Problem 20 in Hirzebruch's list, attributed to Kodaira and Spencer, asks the following about compact almost complex manifolds:

Let $M^n$ be an almost-complex manifold. Choose an Hermitian structure and consider the numbers $h^{p,q}$ defined as above. Is $h^{p,q}$ independent of the choice of the Hermitian structure? If not, give some other definition of the $h^{p,q}$ of $M^n$ which depends only on the almost-complex structure and which generalizes the $h^{p,q}$ of a complex manifold.

(Note that in the case of an integrable complex structure, the numbers $h^{p,q}$ are metric independent as they are the dimension of the Dolbeault cohomology group $H_{\bar{\partial}}^{p,q}$.)

In 2020, Holt and Zhang posted a preprint

• Tom Holt, Weiyi Zhang, Harmonic Forms on the Kodaira–Thurston Manifold, arXiv:2001.10962,

showing that the numbers $h^{p,q}$ are in general metric-dependent. The underlying manifold they work with is the four-dimensional Kodaira–Thurston nilmanifold.

In the paper

• Friedrich Hirzebruch, Some Problems on Differentiable and Complex Manifolds, Annals of Mathematics Second Series, Vol. 60, No. 2 (1954) pp. 213-236, doi:10.2307/1969629

Hirzebruch collected problems and questions on smooth and complex manifolds presented at a conference the year prior.

For an almost complex manifold $M$ equipped with a hermitian metric, one can form the Laplacian $\Delta_{\bar{\partial}}$. Even though $\bar{\partial}$ need not square to zero on an almost complex manifold, this Laplacian is an elliptic operator and so the kernel of the Laplacian is finite dimensional if $M$ is compact. Denote the dimension of this kernel restricted to $(p,q)$ forms by $h^{p,q}$.

Problem 20 in Hirzebruch's list, attributed to Kodaira and Spencer, asks the following about compact almost complex manifolds:

Let $M^n$ be an almost-complex manifold. Choose an Hermitian structure and consider the numbers $h^{p,q}$ defined as above. Is $h^{p,q}$ independent of the choice of the Hermitian structure? If not, give some other definition of the $h^{p,q}$ of $M^n$ which depends only on the almost-complex structure and which generalizes the $h^{p,q}$ of a complex manifold.

(Note that in the case of an integrable complex structure, the numbers $h^{p,q}$ are metric independent as they are the dimension of the Dolbeault cohomology group $H_{\bar{\partial}}^{p,q}$.)

In 2020, Holt and Zhang posted a preprint (update: the paper is now published in Advances in Mathematics)

• Tom Holt, Weiyi Zhang, Harmonic Forms on the Kodaira–Thurston Manifold, arXiv:2001.10962,

showing that the numbers $h^{p,q}$ are in general metric-dependent. The underlying manifold they work with is the four-dimensional Kodaira–Thurston nilmanifold.

In a 1954 paper https://www.jstor.org/stable/pdf/1969629.pdf, Hirzebruch collected problems and questions on smooth and complex manifolds presented at a conference the year prior.

For an almost complex manifold $M$ equipped with a hermitian metric, one can form the Laplacian $\Delta_{\bar{\partial}}$. Even though $\bar{\partial}$ need not square to zero on an almost complex manifold, this Laplacian is an elliptic operator and so the kernel of the Laplacian is finite dimensional if $M$ is compact. Denote the dimension of this kernel restricted to $(p,q)$ forms by $h^{p,q}$.

Problem 20 in Hirzebruch's list, attributed to Kodaira and Spencer, asks the following about compact almost complex manifolds:

Let $M^n$ be an almost-complex manifold. Choose an Hermitian structure and consider the numbers $h^{p,q}$ defined as above. Is $h^{p,q}$ independent of the choice of the Hermitian structure? If not, give some other definition of the $h^{p,q}$ of $M^n$ which depends only on the almost-complex structure and which generalizes the $h^{p,q}$ of a complex manifold.

(Note that in the case of an integrable complex structure, the numbers $h^{p,q}$ are metric independent as they are the dimension of the Dolbeault cohomology group $H_{\bar{\partial}}^{p,q}$.)

In 2020, Holt and Zhang posted a preprint showing that the numbers $h^{p,q}$ are in general metric dependent https://arxiv.org/pdf/2001.10962.pdf. The underlying manifold they work with is the four-dimensional Kodaira-Thurston nilmanifold.

In the paper

• Friedrich Hirzebruch, Some Problems on Differentiable and Complex Manifolds, Annals of Mathematics Second Series, Vol. 60, No. 2 (1954) pp. 213-236, doi:10.2307/1969629

Hirzebruch collected problems and questions on smooth and complex manifolds presented at a conference the year prior.

For an almost complex manifold $M$ equipped with a hermitian metric, one can form the Laplacian $\Delta_{\bar{\partial}}$. Even though $\bar{\partial}$ need not square to zero on an almost complex manifold, this Laplacian is an elliptic operator and so the kernel of the Laplacian is finite dimensional if $M$ is compact. Denote the dimension of this kernel restricted to $(p,q)$ forms by $h^{p,q}$.

Problem 20 in Hirzebruch's list, attributed to Kodaira and Spencer, asks the following about compact almost complex manifolds:

Let $M^n$ be an almost-complex manifold. Choose an Hermitian structure and consider the numbers $h^{p,q}$ defined as above. Is $h^{p,q}$ independent of the choice of the Hermitian structure? If not, give some other definition of the $h^{p,q}$ of $M^n$ which depends only on the almost-complex structure and which generalizes the $h^{p,q}$ of a complex manifold.

(Note that in the case of an integrable complex structure, the numbers $h^{p,q}$ are metric independent as they are the dimension of the Dolbeault cohomology group $H_{\bar{\partial}}^{p,q}$.)

In 2020, Holt and Zhang posted a preprint

• Tom Holt, Weiyi Zhang, Harmonic Forms on the Kodaira–Thurston Manifold, arXiv:2001.10962,

showing that the numbers $h^{p,q}$ are in general metric-dependent. The underlying manifold they work with is the four-dimensional Kodaira–Thurston nilmanifold.