Rational curves on Calabi-Yau varieties and Abelian fibrations -- the structure of complex projective manifolds without rational curves
15:45:00 - 17:00:00
103 , Mathematics Research Center Building (ori. New Math. Bldg.)
An important problem on the geometry of Calabi-Yau manifold, and to the structure of algebraic varieties in general, is the problem of the existence of rational curves on compact simply connected kahler manifold with trivial canonical bundle. Even if we generalize the problem to the holomorphic world to the existence of entire curves or to the complete vanishing of any holomorphically invariant pseudo-metric, the problem is unknown beyond dimension two already in the projective case. I will first describe an attempt to resolve the latter via differential geometry leading to a structure theory for projective Kahler manifolds with seminegative holomorphic sectional curvature, joint work with Gordon Heier and Bun Wong. The key tool used here is just a version of the classical maximum principle (due to Chern, Yau and others). I will then describe the analogous problem for projective manifold with few, or no, rational curves leading to the result that such a manifold always admits unramified finite covers by abelian schemes over a canonical polarized manifolds, assuming standard conjectures in Mori classification program of algebraic varieties, joint work with Ying Zong. The is done, apart from break and bend, via local techniques by a detailed study of torsors in the Grothendieck topology and uses only classical results of Grothendieck, Neron, Raynaud and others.
(1) with Heier and Wong: Preprint http://front.math.ucdavis.edu/1403.4210" target="_blank"> arXiv:1403.4210 and also http://front.math.ucdavis.edu/0909.0106" target="_blank"> arXiv:0909.0106 published in Math. Res. Lett 17 (2010), no. 6, 1101-1110
(2) with Ying Zong http://front.math.ucdavis.edu/1410.0063" target="_blank"> arXiv:1410.0063
(3) Also of possible relevance but not directly related is the joint work with L. Kamenova and M. Verbitsky: "Kobayashi pseudometric on hyperkähler manifolds," J. London Math. Soc. (2014) 90 (2): 436-450 .