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2018•Proceedings of the London Mathematical SocietyOpen access

Towards the Green–Griffiths–Lang conjecture via equivariant localisation

Gergely Bérczi

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Abstract

Green, Griffiths and Lang conjectured that for every complex projective algebraic variety X of general type there exists a proper algebraic subvariety of X containing all nonconstant entire holomorphic curves f : C → X . Using equivariant localisation we develop an iterated residue formula for cohomological pairings on the Demailly–Semple jet bundle. We apply this formula and a strategy of Demailly to give affirmative answer to the Green–Griffiths–Lang conjecture for generic projective hypersurfaces X ⊂ P n + 1 of degree deg ( X ) ⩾ n 9 n .

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Green, Griffiths and Lang conjectured that for every complex projective algebraic variety X of general type there exists a proper algebraic subvariety of X containing all nonconstant entire holomorphic curves f : C → X . Using equivariant localisation we develop an iterated residue formula for cohomological pairings on the Demailly–Semple jet bundle. We apply this formula and a strategy of Demailly to give affirmative answer to the Green–Griffiths–Lang conjecture for generic projective hypersurfaces X ⊂ P n + 1 of degree deg ( X ) ⩾ n 9 n .

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Available abstract

Green, Griffiths and Lang conjectured that for every complex projective algebraic variety X of general type there exists a proper algebraic subvariety of X containing all nonconstant entire holomorphic curves f : C → X . Using equivariant localisation we develop an iterated residue formula for cohomological pairings on the Demailly–Semple jet bundle. We apply this formula and a strategy of Demailly to give affirmative answer to the Green–Griffiths–Lang conjecture for generic projective hypersurfaces X ⊂ P n + 1 of degree deg ( X ) ⩾ n 9 n .

Key concepts: Equivariant map, Mathematics, Conjecture, Combinatorics, Pure mathematics

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