Very ampleness and the infinitesimal Torelli problem

According to P. Griffiths, an important problem in higher-dimensional geometry is to find classes of complex varieties of general type X for which the infinitesimal Torelli map is injective. His deep ideas are at the origin of a large literature. If dim X = 1, very ampleness of the canonical sheaf is a sufficient condition. In this paper we prove that for any natural number N ≥ 5 there exists a generically smooth irreducible (N + 9)-dimensional component \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\mathcal S_{N}]$$\end{document} of the moduli space of algebraic surfaces such that for a general element [X] of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ [\mathcal S_{N}]$$\end{document} , the canonical sheaf is very ample and the 2-infinitesimal Torelli map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\Phi_2 : H^1(X, T_X)\longrightarrow\,{\rm Hom}\,\left(H^{2,0}(X), H^{1,1}(X)\right)$$\end{document}has kernel of dimension at least 1. This shows that contrary to the curve case, the task of finding conditions which imply the injectivity of the infinitesimal Torelli map is very open.