On algebraic surfaces and threefolds whose cotangent bundles are generated by global sections (Q912941)

Let X be a smooth subvariety of general type of an abelian variety with \(\dim (X)=1\) or \(co\dim (X)=1\). Then it is a well-known fact that the tricanonical map is an embedding. The author generalizes the fact above to the case of \(\dim (X)=2\) or 3. Main results: Theorem A. Let S be a surface of general type whose cotangent sheaf \(\Omega^ 1_ S\) is generated by global sections. If \(h^ 0(\Omega^ 1_ S)\geq 4\), the bicanonical map of S is birational except the case where \(S=C_ 1\times C_ 2\), \(g(C_ 2)=2.\) Moreover if the Albanese torus of S is a simple abelian variety, the bicanonical map is an embedding. Theorem B. Let V be a projective threefold of general type whose cotangent sheaf \(\Omega^ 1_ V\) is generated by global sections. If \(h^ 0(\Omega^ 1_ V)\geq 5\), the tricanonical map of V is birational. Note that if S and V are smooth subvarieties of an abelian variety then \(\Omega^ 1_ S\) and \(\Omega^ 1_ V\) are generated by global sections. Related results was previously obtained by P. Francia and I. Reider in the 2-dimensional case and by J. Kollár in the 3-dimensional case.