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Oct 31, 2017 at 16:21 history edited modnar CC BY-SA 3.0
Deleted part superseded by comment below.
Oct 31, 2017 at 16:14 history edited modnar CC BY-SA 3.0
Updated question taking comments into account and made more clear what my specific question was.
Oct 31, 2017 at 8:21 comment added Ariyan Javanpeykar Infinitesimal Torelli fails for curves of genus $>2$; see mathoverflow.net/questions/8938/is-the-torelli-map-an-immersion However, infinitesimal Torelli holds for all smooth projective varieties with trivial canonical bundle (e.g., abelian varieties). You can find this in Voisin's books; see the paragraph on infinitesimal Torelli for Calabi-Yau varieties.
Oct 31, 2017 at 7:16 history edited modnar CC BY-SA 3.0
added 2 characters in body
Oct 31, 2017 at 7:08 history edited modnar CC BY-SA 3.0
Found a partial answer and some additional sources of confusion.
Oct 31, 2017 at 4:52 history edited modnar CC BY-SA 3.0
Reading material explained
Oct 31, 2017 at 4:08 history edited modnar CC BY-SA 3.0
Clarified actual cases considered and what is known already vs. attempted computations
Oct 31, 2017 at 3:55 comment added user21574 See arxiv.org/abs/math/0206248
Oct 31, 2017 at 3:39 history asked modnar CC BY-SA 3.0