arXiv subjects

Steven S. Y. Lu

Publications and source records attributed to Steven S. Y. Lu.

7 recordsLinked to original sources

arXiv2017

Reduction of manifolds with semi-negative holomorphic sectional curvature

In this note, we continue the investigation of a projective Kähler manifold $M$ of semi-negative holomorphic sectional curvature $H$. We introduce a new differential geometric numerical rank invariant which measures the number of linearly independent {\it truly flat} directions of $H$ in the tangent spaces. We prove that this invariant is bounded above by the nef dimension and bounded below by the numerical Kodaira dimension of $M$. We also prove a splitting theorem for $M$ in terms of the nef dimension and, under some additional hypotheses, in terms of the new rank invariant.

math.AG↗

arXiv2015

Kähler manifolds of semi-negative holomorphic sectional curvature

In an earlier work, we investigated some consequences of the existence of a Kähler metric of negative holomorphic sectional curvature on a projective manifold. In the present work, we extend our results to the case of semi-negative (i.e., non-positive) holomorphic sectional curvature. In doing so, we define a new invariant that records the largest codimension of maximal subspaces in the tangent spaces on which the holomorphic sectional curvature vanishes. Using this invariant, we establish lower bounds for the nef dimension and, under certain additional assumptions, for the Kodaira dimension of the manifold. In dimension two, a precise structure theorem is obtained.

math.AG↗

arXiv2015

Positivity criteria for log canonical divisors and hyperbolicity

Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria guaranteeing various positivity conditions for the log-canonical divisor K_X+D. By adjunction and running the log minimal model program, natural to our setting, we obtain a geometric criterion for K_X+D to be numerically effective as well as a geometric version of the cone theorem, generalizing to the context of log pairs these results of Mori. A criterion for K_X+D to be pseudo-effective with mild hypothesis on D follows. We also obtain, assuming the abundance conjecture and the existence of rational curves on Calabi-Yau manifolds, an optimal geometric sharpening of the Nakai-Moishezon criterion for the ampleness of a divisor of the form K_X+D, a criterion verified under a canonical hyperbolicity assumption on (X,D). Without these conjectures, we verify this ampleness criterion with assumptions on the number of ample and non ample components of D.

math.AG↗

arXiv2010

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 by 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.

hep-ph↗

arXiv2009

On the canonical line bundle and negative holomorphic sectional curvature

We prove that a smooth complex projective threefold with a Kähler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, ampleness is again obtained. The methods used come from both complex differential geometry and complex algebraic geometry.

math.AG↗

arXiv2008

On surfaces of general type with maximal Albanese dimension

Given a minimal surface equipped with a generically finite map to an Abelian variety, we give an optimal bound on the canonical degree of a rational or an elliptic curve. As a corollary, we obtain the finiteness of rational and elliptic curves on any surface of general type with two linearly independent regular one forms.

math.AG↗

arXiv2002

A refined Kodaira dimension and its canonical fibration

Given a (meromorphic) fibration $f:X\to Y$ where $X$ and $Y$ are compact complex manifolds of dimensions $n$ and $m$, we define $L_f$ to be the invertible subsheaf of the sheaf of holomorphic $m$-forms of $X$ given by the saturation of $f^*K_Y$, where $K_Y$ is the canonical sheaf of $Y$. We define the Kodaira dimension of the orbifold base $(Y,f)$ by that of $L_f$ and call the fibration to be of general type if this dimension is the maximal $\dime Y$. We call $X$ special if it does not have a general type fibration with positive dimensional base and call $f$ special if its general fibers are. We note that Iitaka and rationally connected fibrations are special. Our main theorem is that any compact complex $X$ has a special fibration of general type which dominates any fibration of general type and factors through any special fibration of general type from $X$ in the birational category, thus unique in this category. Also, the fibration has positive dimensional fibers if $X$ is not of general type thus resolving a problem in Mori's program for varieties with negative Kodaira dimension. Our main theorem is stated in the full orbifold context of log pairs as in Mori's program via which we give an application to the Albanese map of projective manifolds with nef anticanonical bundle a part of which was derived using positive charateristic techniques by Qi Zhang.

math.AG↗