Is very well known that the only algebraic curve which admits no deformations is the projective line.
Q. What are "rigid" smooth algebraic surfaces? Is there a sensible classification?
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There are several different notions of "rigidity" (local rigidity, global rigidity, infinitesimal rigidity, étale rigidity and strong rigidity) and it is possible to provide examples for each of them.
This topic is discussed in the paper by I. Bauer and F. Catanese On rigid compact complex surfaces and manifolds, you can have a look at it for more details and references to the literature.
answered Jun 28, 2017 at 8:22
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$\begingroup$ Thank you. I actually meant local rigidity (apparently the weakest). $\endgroup$ Commented Jun 28, 2017 at 9:39
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2$\begingroup$ It looks like this paper has an answer to my question, in a sense that a complete classification seems out of reach. $\endgroup$ Commented Jun 28, 2017 at 9:49