On the existence of logarithmic and orbifold jet differentials

Frédéric Campana (1) , Lionel Darondeau (2) , Jean-Pierre Demailly (3) , Erwan Rousseau (4, 5)

1 IECL - Institut Élie Cartan de Lorraine
2 IMAG - Institut Montpelliérain Alexander Grothendieck
3 IF - Institut Fourier
4 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique
5 IUF - Institut universitaire de France

We introduce the concept of directed orbifold, namely triples (X, V, D) formed by a directed algebraic or analytic variety (X, V), and a ramification divisor D, where V is a coherent subsheaf of the tangent bundle T X. In this context, we introduce an algebra of orbifold jet differentials and their sections. These jet sections can be seen as algebraic differential operators acting on germs of curves, with meromorphic coefficients, whose poles are supported by D and multiplicities are bounded by the ramification indices of the components of D. We estimate precisely the curvature tensor of the corresponding directed structure V D in the general orbifold case-with a special attention to the compact case D = 0 and to the logarithmic situation where the ramification indices are infinite. Using holomorphic Morse inequalities on the tautological line bundle of the projectivized orbifold Green-Griffiths bundle, we finally obtain effective sufficient conditions for the existence of global orbifold jet differentials.

Mots clés: Projective variety; directed variety; orbifold; ramification divisor; entire curve; jet differential; Green-Griffiths conjecture; algebraic differential operator; holomorphic Morse inequalities; Chern curvature; Chern form

Domaines: Géométrie algébrique [math.AG]

Frédéric Campana, Lionel Darondeau, Jean-Pierre Demailly, Erwan Rousseau. On the existence of logarithmic and orbifold jet differentials. 2021. ⟨hal-03331607v2⟩