Diophantine approximation for subvarieties


Julie Tzu-Yueh Wang ( Academia Sinica )
2016-11-11  13:30 - 14:30
Room 103, Mathematics Research Center Building (ori. New Math. Bldg.)

In their recent Invent. Math. paper (see [5]), McKinnon and Roth introduced the approximation constant αx(L) to an algebraic point x on an algebraic variety V with an ample line bundle bundle L. The invariant αx(L) measures how well x can be approximated by rational points on X with respect to the height function associate to L. They showed that αx(L) is closely related to the Seshadri constant x(L) measuring the local positivity of L at x. They also showed that the invariant αx(L) can be computed through another invariant βx(L) in the height inequality they established (see Theorem 5.1 and Theorem 6.1) in [5]. By computing the Seshadri constant x(L) for the case of V = P 1 , their result recovers the Roth’s theorem, so the height inequality they established can be viewed as the generalization of the Roth’s theorem to arbitrary projective varieties. In my recent joint work with Min Ru, we give such results a short and simpler proof. Furthermore, we extend the results from the points of a projective variety to subschemes. The generalized result in terms of subschemes connects, as well as gives a clearer explanation, the above mentioned result of McKinnon and Roth [5] with the recent Diophantine approximation results in term of the divisors obtained by Corvarja-Zannier [1], Evertse-Ferretti [3] , Ru [8], and Ru-Vojta [9]. References [1] P. Corvaja and U. Zannier, On a general Thue’s equation, Amer. J. Math., 126(5) (2004),1033–1055. [2] J.-H. Evertse and R. G. Ferretti, Diophantine inequalities on projective varieties, Int. Math. Res. Notices 25 (2002), 1295–1330. [3] J.-H. Evertse and R. Ferretti, A generalization of the Subspace Theorem with polynomials of higher degree, In Diophantine approximation, volume 16 of Dev. Math., pages 175–198. SpringerWienNewYork, Vienna, 2008. [4] G. Faltings and G. Wustholz, ¨ Diophantine approximations on projective varieties, Invent. Math. 116 (1994), 109–138. [5] D. McKinnon and M. Roth, Seshadri constants, diophantine approximation and Roth’s theorem for arbitrary varieties, Invent. Math. 200 (2015), 513–583. [6] M. Ru, A defect relation for holomorphic curves intersecting hypersurfaces, Amer. J. Math. 126 (2004), 215–226. [7] M. Ru, Holomorphic curves into algebraic varieties, Annals of Mathematics, 169 (2009), 255–267. [8] M. Ru, A general Diophantine inequality, Functiones et Approximatio, to appear. [9] M. Ru and P. Vojta, Birational Nevanlinna constant and its consequences, Preprint, arXiv:1608.05382 [math.NT].

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