12 research outputs found
On compactness of K\"ahler metrics with bounded entropy and bounded scalar curvature
In their seminal work (\cite{CC}, \cite{CC2}), Chen and Cheng proved apriori
estimates for the constant scalar curvature metrics on compact K\"ahler
manifolds. They also proved estimate for the potential of the
\ka metrics under boundedness assumption on the scalar curvature and the
entropy. The goal of this short note is to slightly relax the boundedness
condition on the scalar curvature
Balanced Metrics and Chow Stability of Projective Bundles over K\"ahler Manifolds
In 1980, I. Morrison proved that slope stability of a vector bundle of rank 2
over a compact Riemann surface implies Chow stability of the projectivization
of the bundle with respect to certain polarizations. Using the notion of
balanced metrics and recent work of Donaldson, Wang, and Phong-Sturm, we show
that the statement holds for higher rank vector bundles over compact algebraic
manifolds of arbitrary dimension that admit constant scalar curvature metric
and have discrete automorphism group
Quantization of the Laplacian operator on vector bundles I
Let be a holomorphic Hermitian vector bundle over a polarized
manifold. We provide a canonical quantization of the Laplacian operator acting
on sections of the bundle of Hermitian endomorphisms of . If is simple
we obtain an approximation of the eigenvalues and eigenspaces of the Laplacian
Quantization of Donaldson's heat flow over projective manifolds
33 pagesInternational audienceConsider a holomorphic vector bundle over a projective manifold polarized by an ample line bundle . Fix large enough, the holomorphic sections provide embeddings of in a Grassmanian space. We define the \textit{balancing flow for bundles} as a flow on the space of projectively equivalent embeddings of . This flow can be seen as a flow of algebraic type hermitian metrics on . At the quantum limit , we prove the convergence of the balancing flow towards the Donaldson heat flow, up to a conformal change. As a by-product, we obtain a numerical scheme to approximate the Yang-Mills flow in that context