Let $X$ be a smooth projective variety, and $Z\subset X$ a smooth closed subvariety of $X$. The first order deformations of $Z$ in $X$ are parameterized by $H^0(Z, N_{Z/X})$, while the first order deformations of $Z$ as a variety are parameterized by $H^1(Z,T_Z)$. Thus "forgetting $X$" gives a map $H^0(Z, N_{Z/X}) \rightarrow H^1(Z,T_Z)$. I am looking for a reference for the fact (certainly well-known) that this map is the coboundary map in the cohomology exact sequence associated to the exact sequence $$0\rightarrow T_Z\rightarrow T_{X|Z}\rightarrow N_{Z/X}\rightarrow 0\,.$$
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$\begingroup$ Is a (precise) reference in Illusie okay? I can track it down soon . . . $\endgroup$Jason Starr– Jason Starr2025-11-15 13:08:51 +00:00Commented Nov 15 at 13:08
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2$\begingroup$ In Illusie, it is Proposition III.2.2.4 on p. 198 (after unravelling all the definitions). Illusie cites 11.1.5 in Grothendieck's LNM on the cotangent complex for the original statement (but possibly it goes back to Kodaira -- Spencer, etc.). $\endgroup$Jason Starr– Jason Starr2025-11-15 14:09:41 +00:00Commented Nov 15 at 14:09
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1$\begingroup$ Great, @Jason, thanks! It needs indeed some unravelling, but the statement is definitely there. $\endgroup$abx– abx2025-11-15 15:59:46 +00:00Commented Nov 15 at 15:59
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