Let $X$ be a compact complex manifold of complex dimension $n$. The Hodge-Frölicher spectral sequence starts with $$ E_1^{p,q}=H^{p,q}(X,\mathbb C) $$ and the limit term $E^{p,q}_\infty$ is the graded module associated to a filtration of the de Rham cohomology group $H^{p+q}_{\text{dR}}(X,\mathbb C)$. In particular $$ b_k=\sum_{p+q=k}\dim E_\infty^{p,q}\le\sum_{p+q=k}\dim E_1^{p,q}=\sum_{p+q=k}h^{p,q}. $$ The equality is equivalent to the degeneration of the spectral sequence at the $E_1^\bullet$ level.

On the other hand, an elementary lemma on bounded complexes of finite dimensional vector spaces applied to $E^\bullet_r$, tells you that you always have equality for the (topological) Euler characteristic: $$ \chi_{\text{top}}(X)=\sum_{k=0}^{2n}(-1)^kb_k=\sum_{p,q=0}^n(-1)^{p+q}h^{p,q}. $$

Let $X$ be a compact complex manifold of complex dimension $n$. The Hodge-Frölicher spectral sequence starts with $$ E_1^{p,q}=H^{p,q}(X,\mathbb C) $$ and the limit term $E^{p,q}_\infty$ is the graded module associated to a filtration of the de Rham cohomology group $H^{p+q}_{\text{dR}}(X,\mathbb C)$. In particular $$ b_k=\sum_{p+q=k}\dim E_\infty^{p,q}\le\sum_{p+q=k}\dim E_1^{p,q}=\sum_{p+q=k}h^{p,q}. $$ The equality is equivalent to the degeneration of the spectral sequence at the $E_1^\bullet$ level.

On the other hand, an elementary lemma on bounded complexes of finite dimensional vector spaces applied to $E^\bullet_r$, tells you that you always have equality for the (topological) Euler characteristic: $$ \chi_{\text{top}}(X)=\sum_{k=0}^{2n}(-1)^kb_k=\sum_{p,q=0}^n(-1)^{p+q}h^{p,q}. $$

Let $X$ be a compact complex manifold of complex dimension $n$. The Hodge-Frölicher spectral sequence starts with $$ E_1^{p,q}=H^{p,q}(X,\mathbb C) $$ and the limit term $E^{p,q}_\infty$ is the graded module associated to a filtration of the de Rham cohomology group $H^{p+q}_{\text{dR}}(X,\mathbb C)$. In particular $$ b_k=\sum_{p+q=k}\dim E_\infty^{p,q}\le\sum_{p+q=k}\dim E_1^{p,q}=\sum_{p+q=k}h^{p,q}. $$ The equality is equivalent to the degeneration of the spectral sequence at the $E_1^\bullet$ level.

On the other hand, an elementary lemma on bounded complexes of finite dimensional vector spaces applied to $E^\bullet_r$, tells you that you always have equality for the (topological) Euler characteristic: $$ \chi_{\text{top}}(X)=\sum_{k=0}^{2n}(-1)^kb_k=\sum_{p,q=0}^n(-\)^{p+q}h^{p,q}. $$

Let $X$ be a compact complex manifold of complex dimension $n$. The Hodge-Frölicher spectral sequence starts with $$ E_1^{p,q}=H^{p,q}(X,\mathbb C) $$ and the limit term $E^{p,q}_\infty$ is the graded module associated to a filtration of the de Rham cohomology group $H^{p+q}_{\text{dR}}(X,\mathbb C)$. In particular $$ b_k=\sum_{p+q=k}\dim E_\infty^{p,q}\le\sum_{p+q=k}\dim E_1^{p,q}=\sum_{p+q=k}h^{p,q}. $$ The equality is equivalent to the degeneration of the spectral sequence at the $E_1^\bullet$ level.

On the other hand, an elementary lemma on bounded complexes of finite dimensional vector spaces applied to $E^\bullet_r$, tells you that you always have equality for the (topological) Euler characteristic: $$ \chi_{\text{top}}(X)=\sum_{k=0}^{2n}(-1)^kb_k=\sum_{p,q=0}^n(-1)^{p+q}h^{p,q}. $$