Let $X$ be a compact complex manifold with structure sheaf $\mathscr{O}_X$ (the sheaf of holomorphic functions on $X$).

What is the geometric meaning (if any) of $H^2(X, \mathscr{O}_X)$?

In the comments, Piotr Achinger mentions that $H^2(X, \mathscr{O}_X)$ is the tangent space of the (formal) Brauer group. Let me ask a follow up question:

What is the geometric meaning (if any) of $H^2(X, \mathscr{O}_X)$ if the Brauer group, and the cohomological Brauer group are both trivial?

Let $X$ be a compact complex manifold with structure sheaf $\mathscr{O}_X$ (the sheaf of holomorphic functions on $X$).

What is the geometric meaning (if any) of $H^2(X, \mathscr{O}_X)$?

In the comments, Piotr Achinger mentions that $H^2(X, \mathscr{O}_X)$ is the tangent space of the (formal) Brauer group. Let me ask a follow up question:

What is the geometric meaning (if any) of $H^2(X, \mathscr{O}_X)$ if the Brauer group, and the cohomological Brauer group are both trivial?

Let $X$ be a compact complex manifold with structure sheaf $\mathscr{O}_X$ (the sheaf of holomorphic functions on $X$).

What is the geometric meaning (if any) of $H^2(X, \mathscr{O}_X)$?

Let $X$ be a compact complex manifold with structure sheaf $\mathscr{O}_X$ (the sheaf of holomorphic functions on $X$).

What is the geometric meaning (if any) of $H^2(X, \mathscr{O}_X)$?

In the comments, Piotr Achinger mentions that $H^2(X, \mathscr{O}_X)$ is the tangent space of the (formal) Brauer group. Let me ask a follow up question:

What is the geometric meaning (if any) of $H^2(X, \mathscr{O}_X)$ if the Brauer group, and the cohomological Brauer group are both trivial?