  The article proves the Infinitesimal Torelli theorem for surfaces subject to
the following conditions:
  1) the canonical bundle of a surface is ample and generated by its global
sections,
  2)the geometric genus $p_g \geq 4$,
  3) the irregularity $q=0$ .
  The main novelty is a realization of the Kodaira-Spencer classes lying in the
kernel of the cohomology cup-product controlling the derivative of the period
map of weight 2 in the category of the coherent sheaves of a surface.
