Geodesic
The local Torelli theorem for varieties with divisible canonical class
Izvestiya. Mathematics , Tome 12 (1978) no. 1, pp. 53-67

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In this paper the local Torelli theorem is proved for a number of varieties with divisible canonical class: surfaces with a family of elliptic curves and base $P^1$, fiberings by varieties with $K=0$ whose base is a curve, complete intersections, cyclic coverings, and surfaces with $V$, $K>0$, $\operatorname{Pic}V=Z$ and $h^{1,0}=0$, whose invariants satisfy certain conditions. Bibliography: 7 titles. 
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     author = {K. I. Kii},
     title = {The local {Torelli} theorem for varieties with divisible canonical class},
     journal = {Izvestiya. Mathematics },
     pages = {53--67},
     publisher = {mathdoc},
     volume = {12},
     number = {1},
     year = {1978},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1978_12_1_a2/}
}
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K. I. Kii. The local Torelli theorem for varieties with divisible canonical class. Izvestiya. Mathematics , Tome 12 (1978) no. 1, pp. 53-67. http://geodesic.mathdoc.fr/item/IM2_1978_12_1_a2/