When Atiyah class and Chern class coincide?

Question

Let $X$ be a compact complex manifold, $L$ be a holomorphic line bundle on $X$, then the exponential exact sequence $0\to \mathbb Z\hookrightarrow \mathcal O\to \mathcal O^*\to 0$ induces the map $c:H^1(X,\mathcal O^*)\to H^2(X,\mathbb Z)$, it is well-known that the line bundle $L$ can be seen as an element in $H^1(X,\mathcal O^*)$, the image $c(L)\in H^2(X,\mathbb Z)$ is called the Chern class of $L$.

According to Atiyah's paper in 1957, p.196, the exact sequence $0\to \mathbb C\hookrightarrow \mathcal O\stackrel{d}\to \Omega^1\to 0$ together with the exponential exact sequence above induce the map $\mathcal O^*\to \Omega^1:f\mapsto \frac{1}{2\pi i}d\text{log}f$, which induces the map $\sigma:H^1(X,\mathcal O^*)\to H^1(X,\Omega)$, we call the image $\sigma(L)\in H^1(X,\Omega^1)$ the Atiyah class of the line bundle $L$.

In Atiyah's paper, the author assumed $X$ to be a compact Kähler manifold, then he concludes that the Atiyah class equals to Chern class, here my question is: can this condition be weakened a bit? For example, $X$ is a $\partial\bar\partial$-manifold or the Frölicher spectral sequence degenerates at $E_1$? Or equivalently, what's the sufficient and necessary condition of the Atiyah class coincides with the Chern class?

Answer 1

$\def\ZZ{\mathbb{Z}}\def\CC{\mathbb{C}}\def\cO{\mathcal{O}}$To spell out my comment a little more, let $Z^1$ be the sheaf of $\partial$-closed holomorphic $(1,0)$-forms. Since "holomorphic" means $\overline{\partial}$-closed, this can also be described as the space of closed $(1,0)$forms. Then we have a commutative diagram, with exact rows and columns:

$$\begin{matrix} && && 0 && 1 && \\ && && \downarrow && \downarrow && \\ 0 &\longrightarrow& \mathbb{Z} &\overset{2 \pi i}{\longrightarrow}& \mathbb{C} &\overset{\exp}{\longrightarrow}& \mathbb{C}^{\times} &\longrightarrow& 1 \\ & &=& &\downarrow& &\downarrow& \\ 0 &\longrightarrow& \mathbb{Z} &\overset{2 \pi i}{\longrightarrow}& \cO &\overset{\exp}{\longrightarrow}& \cO^{\times} &\longrightarrow& 1 \\ & && &\phantom{\partial} \downarrow \partial& &\phantom{\partial \log}\downarrow \partial \log& \\ && && Z^1 &=& Z^1 && \\ && && \downarrow && \downarrow && \\ && && 0 && 0 && \\ \end{matrix}$$ Using each of the $4$ short exact sequences, we have a diagram $$ \begin{matrix} H^1(\cO^{\times}) &\longrightarrow& H^2(\ZZ) \\ \downarrow && \downarrow \\ H^1(Z^1) &\longrightarrow& H^2(\CC) \\ \end{matrix}$$ and, with care, one can check that it commutes.

We also have a short exact sequence $0 \to Z^1 \to \Omega^1 \overset{\partial}{\longrightarrow} Z^2 \to 0$, where $Z^2$ is the closed $(2,0)$-forms. So we can extend this diagram to $$ \begin{matrix} H^1(\cO^{\times}) &\longrightarrow& H^2(\ZZ) \\ \downarrow && \downarrow \\ H^1(Z^1) &\longrightarrow& H^2(\CC) \\ \downarrow && \\ H^1(\Omega^1) && \\ \end{matrix}$$ The Atiyah class is the image of a class from $H^1(\cO^{\times})$ in $H^1(\Omega^1)$; the Chern class is the image in $H^2(\CC)$. This much is true without assuming anything about your complex manifold.

If you want to consider the Atiyah class and the Chern class to be "the same", then it seem like you want to ask is "when does it make sense to think of $H^1(\Omega^1)$ as a subspace of $H^2(\CC)$?" If your spectral sequence degenerates at $E^1$, then the map $H^1(Z^1) \to H^2(\CC)$ is an injection and $H^1(Z^1) \to H^1(\Omega^1)$ is a surjection, so $H^1(\Omega^1)$ is a subquotient of $H^2(\CC)$. But I don't know of anything which lets you identify $H^1(\Omega^1)$ with a subspace of $H^2(\CC)$ without Hodge theory. (With Hodge theory, on a compact Kahler manifold, $H^1(Z^1)$ is the piece $H^{20} \oplus H^{11}$ in the Hodge filtration, and the maps $H^1(Z^1) \to H^2(\CC)=H^{20} \oplus H^{11} \oplus H^{02}$ and $H^1(Z^1) \to H^1(\Omega^1)=H^{11}$ are the obvious ones.)

Answer 2

I guess this is always true, if you adjust the statement appropriately.

Consider the Bott–Chern cohomology $H^*_{BC}(M):=\dfrac{\ker d\cap \ker d^c}{\operatorname{im} dd^c}$. Since the curvature of a line bundle is a closed $(1,1)$-form, its Chern class can be considered as an element of the Bott–Chern cohomology. There are natural maps from $H^{1,1}_{BC}$ to the Dolbeault cohomology and to de Rham cohomology; the de Rham image of the curvature is $c_1(L)$, and the Dolbeault image is the cohomology class of $\bar\partial \partial\log\lvert f\rvert$, which is equal to the Dolbeault representative of the Atiyah class.

This is also seen from the commutative square that Atiyah writes

$$\require{AMScd}\begin{CD} H^1(O^*_M) @>>> H^2(M,{\Bbb Z}) \\ @VVV @VVV \\ H^1(\Omega^1_M) @>>> H^2(M,{\Bbb C}) \end{CD}$$

which is valid in general situation, non-compact or non-Kähler as well.