On the existence of logarithmic and orbifold jet differentials

Campana, Frédéric; Darondeau, Lionel; Demailly, Jean-Pierre; Rousseau, Erwan

Annales Henri Lebesgue, Tome 7 (2024), pp. 1-67

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 $V\langle D\rangle$ 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.

Nous introduisons le concept d’orbifoldes dirigées, à savoir les triplets $(X,V,D)$ formés par une variété dirigée algébrique ou analytique $(X,V)$, et un diviseur de ramification $D$, où $V$ est un sous-faisceau cohérent du fibré tangent $T_X$. Dans ce contexte, nous introduisons une algèbre de différentielles de jets orbifoldes et leurs sections. Ces sections peuvent être vues comme des opérateurs différentiels algébriques agissant sur les germes de courbes, à coefficients méromorphes, dont les pôles sont supportés par $D$ et les multiplicités sont bornées par les indices de ramification des composantes de $D$. Nous estimons avec précision le tenseur de courbure de la structure dirigée correspondante $V\langle D\rangle$ dans le cas orbifolde général – avec une attention particulière pour le cas compact $D=0$ et le cas logarithmique où les indices de ramifications sont infinis. En utilisant les inégalités de Morse holomorphes sur le fibré en droites tautologique du fibré projectivisé orbifolde de Green–Griffiths, nous obtenons finalement des conditions suffisantes pour l’existence de différentielles de jets orbifoldes globales.

Classification : 32Q45, 32H30, 14F06
Keywords: Projective variety, directed variety, orbifold, ramification divisor, entire curve, jet differential, Green–Griffiths conjecture, algebraic differential operator, holomorphic Morse inequalities, Chern curvature, Chern form

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