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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?

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    $\begingroup$ It is the tangent space of the (formal) Brauer group. It contains obstruction classes to deforming line bundles on $X$ to infinitesimal deformations. $\endgroup$
    – Piotr Achinger
    Commented Dec 25, 2021 at 20:24
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    $\begingroup$ @PiotrAchinger Dear Piotr, thank you for this! I was not aware of Brauer groups at all and have begun looking into them. Do you have a reference suggestion for using Brauer groups to study the geometry of a complex manifold? I'm admittedly far from knowledgable about number theory and my understanding of commutative algebra is only at a working level (although, this is a great excuse to learn more!) :) $\endgroup$
    – AmorFati
    Commented Dec 25, 2021 at 21:42
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    $\begingroup$ It classifies gerbes with band $\mathbb{C}$ (also look at bundle gerbes). $\endgroup$
    – Josh Lackman
    Commented Dec 26, 2021 at 5:52
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    $\begingroup$ For the follow up question I encourage you to take a look at the cohomology sequence from the exponential sequence. $\endgroup$
    – Piotr Achinger
    Commented Dec 26, 2021 at 8:35
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    $\begingroup$ @Z.M Take the identity in $B^2\mathbb{G}_m$ as a $2$-group and look at the formal neighbourhood around it. That's the formal thing. Truncate it by killing the squares in the horizontal direction. That's the tangent space. Maybe it's a good idea to take a look at the usual definitions of a formal moduli problem. See, for instance, Lurie's ICM math.ias.edu/~lurie/papers/moduli.pdf . If you're unfamiliar with the basics of $\infty$/derived stuff, it's a good idea to start with the cotangent complex. See also ncatlab.org/nlab/show/Brauer+stack . $\endgroup$
    – user40276
    Commented Dec 26, 2021 at 19:07

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