Spherical fibrations in algebraic geometry

Lefschetz Fibrations in Symplectic Geometry

One of the cornerstones of complex geometry is the link between positivity of curvature and ampleness. Let X be a compact complex manifold and L ! X be a holomorphic line bundle over X. Suppose that L has a unitary connection whose curvature form is ?2i! where ! is a positive (1; 1)-form on X. Then for large k the line bundle L k has many holomorphic sections. More precisely, the holo-morphic s...

Special Lagrangian Fibrations II: Geometry

This paper is a progress report on work surrounding the Strominger-Yau-Zaslow mirror symmetry conjecture [25]. Roughly put, this conjecture suggests the following program for attacking an appropriate form of the mirror conjecture. Let X be a Calabi-Yau n-fold, with a large complex structure limit point p in a compactification of the complex moduli space of X . One expects mirrors of X to be ass...

The sectional category of spherical fibrations

We give homological conditions which determine sectional category, secat, for rational spherical fibrations. In the odd dimensional case the secat is the least power of the Euler class which is trivial. In the even dimensional case secat is one when a certain homology class in twice the dimension of the sphere is −1 times a square. Otherwise secat is two. We apply our results to construct a fib...

Special Lagrangian fibrations. I. Topology,” Integrable systems and algebraic geometry (Kobe/Kyoto

Yau and Zaslow made a surprising conjecture about pairs of mirror manifolds, which, if true, should at last provide a true geometric understanding of mirror symmetry. Simply put, string theory suggests that if X andˇX are mirror pairs of n-dimensional Calabi-Yau manifolds, then on X there should exist a special Lagrangian n-torus fibration f : X → B, (with some singular fibres) such thatˇX is o...

Introductory Geometry: Algebraic Geometry