Steven S. Y. Lu

Patterns in the fermion mixing matrix, a bottom-up approach

We first obtain the most general and compact parametrization of the unitary transformation diagonalizing any 3×3 Hermitian matrix H, as a function of its elements and eigenvalues. We then study a special class of fermion mass matrices, defined by the requirement that all of the diagonalizing unitary matrices (in the up, down, charged lepton, and neutrino sectors) contain at least one mixing angle much smaller than the other two. Our new parametrization allows us to quickly extract information on the patterns and predictions emerging from this scheme. In particular we find that the phase difference between two elements of the two mass matrices (of the sector in question) controls the generic size of one of the observable fermion mixing angles: i.e. just fixing that particular phase difference will predict the generic value of one of the mixing angles, irrespective of the value of anything else.

Kobayashi pseudometric on hyperkähler manifolds

The Kobayashi pseudometric on a complex manifold M is the maximal pseudometric such that any holomorphic map from the Poincare disk to M is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this result for any hyperk¨ahler manifold if it admits a deformation with a Lagrangian fibration, and its Picard rank is not maximal. The SYZ conjecture claims that any parabolic nef line bundle on a deformation of a given hyperkahler manifold is semi-ample. We prove that the Kobayashi pseudometric vanishes for all hyperkahler manifolds satisfying the SYZ property. This proves the Kobayashi conjecture for K3 surfaces and their Hilbert schemes.

On surfaces of general type with maximal Albanese dimension

Abstract Given a minimal surface S equipped with a generically finite map to an Abelian variety and C ⊂ S a rational or an elliptic curve, we show that the canonical degree of C is bounded by four times the self-intersection of the canonical divisor of S. As a corollary, we obtain the finiteness of rational and elliptic curves with an optimal uniform bound on their canonical degrees on any surface of general type with two linearly independent regular one forms.

Quasiprojective varieties admitting Zariski dense entire holomorphic curves

Abstract. Let X

Reduction of manifolds with semi-negative holomorphic sectional curvature

X

On the Kobayashi Pseudometric, Complex Automorphisms and Hyperkähler Manifolds

be a complex quasiprojective variety. A result of Noguchi–Winkelmann–Yamanoi shows that if X

On the canonical line bundle and negative holomorphic sectional curvature

X

Kähler manifolds of semi-negative holomorphic sectional curvature

admits a Zariski dense entire curve, then its quasi-Albanese map is a fiber space. We show that the orbifold structure induced by a proper birationally equivalent map on the base is special in this case. As a consequence, if X

On semistability of Albanese maps

X

Positivity criteria for log canonical divisors and hyperbolicity

is of log-general type with q (X)dimX