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Pursuing McQuillan's philosophy in proving the Green-Griffiths conjecture forcertain surfaces of general type, we deal with the algebraic degeneracy ofentire curves tangent to holomorphic foliations by curves. Inspired by therecent work of Paun and Sibony, we study the intersection of Ahlfors currentwith the tangent bundle of the foliation, and derive some consequences. Inparticular, we introduce the definition of weakly reduced singularities forfoliations by curves, which requires less work than the exact classificationfor foliations. Finally we discuss the strategy to prove the Green-Griffithsconjecture for complex surfaces.

Degeneracy of entire curves into higher dimensional complex manifolds