Extremal Kähler metrics on projectivized vector bundles

Finsler Geometry of Projectivized Vector Bundles

Projectivized Rank Two Toric Vector Bundles Are Mori Dream Spaces

We prove that the Cox ring of the projectivization P(E) of a rank two toric vector bundle E , over a toric variety X, is a finitely generated k-algebra. As a consequence, P(E) is a Mori dream space if the toric variety X is projective and simplicial.

Cox Rings and Pseudoeffective Cones of Projectivized Toric Vector Bundles

We study projectivizations of a special class of toric vector bundles that includes cotangent bundles, whose associated Klyachko filtrations are particularly simple. For these projectivized bundles, we give generators for the cone of effective divisors and a presentation of the Cox ring as a polynomial algebra over the Cox ring of a blowup of a projective space along a sequence of linear subspa...

Canonical metrics on stable vector bundles

The problem of constructing moduli space of vector bundles over a projective manifold has attracted many mathematicians for decades. In mid 60’s Mumford first constructed the moduli space of vector bundles over algebraic curves via his celebrated GIT machinery. Later, in early 80’s Atiyah and Bott found an infinite dimensional symplectic quotient description of this moduli space. Since then, we...

Extremal Sasakian Metrics on S-bundles over S

In this note I prove the existence of extremal Sasakian metrics on S3-bundles over S2. These occur in a collection of open cones that I call a