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In Kollár's book, Rational curves on Algebraic Varieties, he states the following theorem [II Theorem 1.7].

For a reltative projective flat reduced curve $C$ over an irreducibles base $S$ and a smooth quasi-projective $S$-scheme $X$, given a subscheme $B$ of $C$ flat vover $S$, an $S$-morphism $g:B\rightarrow X$ and a morphism $f:C_s\rightarrow X_s$ that coincides with $g_s$ on $B_s$, there is a dimension estimate, $$\dim_{[f]}Mor_S(C,X;g)\geq \chi(C_s,f^*T_{X_s}\otimes I_{B_s})+\dim S.$$

This is the relative version of another theorem as folllowing which also holds for complex analytic spaces.

Let $C$ be a proper algebraic curve without embedded points and $f:C\rightarrow X$ a morphism to a compact complex manifold $X$ of $n$. Then $$\dim_{[f]} Mor (C, X)\geq \chi (C,f^* TX).$$

My question is that if we replace $X$ in the first theorem by a family of compact complex manifolds, does the conclusion still hold? By a theorem of Namba (see Families of Meromorphic Functions on Compact Riemann Surfaces, Proposition 3.3.1), if $H^1(C_s,f^*T_{X_s})=0$, then $[f]$ is a nonsingular point of $Hol_S(C,X)$ and $\dim_{[f]}Hol_S(C,X)=h^0(C_s,f^*T_{X_s})+\dim_s S$. But I am not sure if we can get a some dimension estimate in the singular case.

Any references and comments are welcome.

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    $\begingroup$ All of the same theorems hold: instead of the Hilbert scheme of $C\times_S X$, use the Douady space of $C\times_S X$. $\endgroup$ Jan 2, 2016 at 19:32
  • $\begingroup$ In general, for extensions to complex analytic spaces of deformation theory results, the main references are to the articles of Flenner and the articles of Bingener. $\endgroup$ Jan 2, 2016 at 20:33
  • $\begingroup$ @JasonStarr Thank you very much. I checked the articles of Flenner and Bingener, it seems that many of them are written in German. Unfortunately, I donot know any German at all. Do you know any survey or references in English or French? Thank you again. $\endgroup$
    – Chieh LIU
    Jan 2, 2016 at 22:04
  • $\begingroup$ I believe that Buchweitz and Flenner are writing (or perhaps have already completed) a textbook on deformation theory for complex analytic spaces that is written in English. $\endgroup$ Jan 3, 2016 at 0:49
  • $\begingroup$ @JasonStarr I find it. Thank you very much! $\endgroup$
    – Chieh LIU
    Jan 3, 2016 at 19:53

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