Nonexistence of global solutions for a class of complex vector fields on two-torus

The goal of this paper is study the global solvability of a class of complex vector fields of the special form L = ∂ / ∂ t + ( a + i b ) ( x ) ∂ / ∂ x , a , b ∈ C ∞ ( S 1 ; R ) , defined on two-torus T 2 ≅ R 2 / 2 π Z 2 . The kernel of transpose operator L t is described and the solvability near the characteristic set is also studied.