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Assume that we have a finite covering $$f \colon X \longrightarrow Y,$$ where $X$ and $Y$ are smooth, complex projective varieties of dimension $n$. Therefore we obtain a splitting $$f_* \mathscr{O}_X = \mathscr{O}_Y \oplus \mathscr{E},$$ where $\mathscr{E}$ is a vector bundle on $Y$ of rank $n-1$. Moreover, we have a short exact sequence $$0 \longrightarrow T_X \stackrel{df}{\longrightarrow} f^*T_Y \longrightarrow N_f \longrightarrow 0,$$ where $N_f$ is a coherent sheaf supported on the ramification locus of $f$ and called the normal sheaf of $f$.

Passing to cohomology, we obtain a linear map $$\psi \colon H^1(X, T_X) \longrightarrow H^1(X, \, f^*T_Y).$$ By standard deformation theory, this is a map between the space of first order deformations of $X$ to the obstruction space to deforming the map $f$ (keeping both $X$ and $Y$ fixed). I admit that the geometrical meaning of $\psi$ is still not clear to me: in fact, given a first order deformation of $X$, I do not see any natural way to associate to it an obstruction to deforming $f$.

Furthermore, using projection formula, the map $\psi$ becomes a map $$\eta \colon H^1(X, \, T_X) \longrightarrow H^1(Y, \, T_Y) \oplus H^1(Y, \, T_Y \otimes \mathscr{E}).$$

Question. Is there any natural interpretation of the maps $\psi$ and $\eta$ in terms of deformations of $X$, $Y$ and deformations of the map $f$?

For instance, what are the first order deformations of $X$ mapping into $H^1(Y, \, T_Y)$ via $\eta$?

Any reference to the existing literature will be particularly appreciated.

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    $\begingroup$ A good reference is Section 6 of K. Behrend, B. Fantechi, "The Intrinsic Normal Cone", Invent. Math. 128 (1997), 45-88. The adjoint of $df$ gives a perfect complex concentrated in amplitude $[-1,0]$ that is quasi-isomorphic to the cotangent complex $L_f$ of $f$. The hyperext group $RHom^1_{\mathcal{O}_X}(L_f,\mathcal{O}_X[0])$ is the space of first-order deformations of the pair $(X,f:X\to Y)$ with $Y$ held fixed. In your setting, this is the same as $H^0(X,N_f)$. Thus the image of an element under $\psi$ equals $0$ if and only if there is a corresponding first-order deformation of $f$. $\endgroup$
    – Jason Starr
    Commented Sep 2, 2015 at 18:28
  • $\begingroup$ I guess the reference is Prop. III.2.1.2.3, p. 186, Luc Illusie, "Complexe cotangent et deformations, I", Lect. Notes in Math. 239, Springer-Verlag, Berlin - New York, 1971. $\endgroup$
    – Jason Starr
    Commented Sep 2, 2015 at 19:12
  • $\begingroup$ @JasonStarr: This is interesting, thank you for the references. Do you know anything about $\eta$? For instance, can one characterize the first order deformations of $X$ mapping onto the first summand $H^1(Y, \, T_Y)$? $\endgroup$
    – Francesco Polizzi
    Commented Sep 3, 2015 at 10:10
  • $\begingroup$ I've made an edit of the question about this specific point. $\endgroup$
    – Francesco Polizzi
    Commented Sep 3, 2015 at 10:12
  • $\begingroup$ It should be in Illusie -- I will check. I believe the point is that, for a given flat family of Y's over a base, the obstruction to deforming $(X,f:X\to Y)$ to match that $Y$ lives in the same group. So probably that has something to do with the map from $H^1(Y,T_Y)$ to the obstruction group. $\endgroup$
    – Jason Starr
    Commented Sep 3, 2015 at 10:14

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