In this note we show that if a projective manifold admits a Khler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

Let K=Q(√(-q)), where q is any prime number congruent to 7 modulo 8, and let O be the ring of integers of K. The prime 2 splits in K, say 2O=pp∗, and there is a unique Z_2-extension K_∞ of K which is unramified outside p. Let H be the Hilbert class field of K, and write H_∞=HK_∞. Let M(H_∞) be the maximal abelian 2-extension of H_∞ which is unramified outside the primes above p, and put X(H_∞)=Gal(M(H_∞)/H_∞). We prove that X(H_∞) is always a finitely generated Z_2-module, by an elliptic analogue of Sinnott’s cyclotomic argument. We then use this result to prove for the first time the weak p-adic Leopoldt conjecture for the compositum J_∞ of K_∞ with arbitrary quadratic extensions J of H. We also prove some new cases of the finite generation of the Mordell–Weil group E(J_∞) modulo torsion of certain elliptic curves E with complex multiplication by O.

We relate previously defined quantum characteristic classes to Morse theoretic aspects of the Hofer length functional on Ham (M, ω). As an application we prove a theorem which can be interpreted as stating that this functional is “virtually” a perfect Morse-Bott functional. This can be applied to study the topology and Hofer geometry of Ham(M, ω). We also use this to give a prediction for the index of some geodesics for this functional, which was recently partially verified by Yael Karshon and Jennifer Slimowitz[5].

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