Quantisation and the Hessian of Mabuchi energy

Details Quantisation and the Hessian of Mabuchi energy Quantisation and the Hessian of Mabuchi energy

2010-09-23

arxiv.org/abs/1009.4543v1

Mathematics

Differential Geometry

40 pages

Scientific paper

Let L be an ample line bundle over a compact complex manifold X. Fix a Hermitian metric in L whose curvature defines a Kaehler metric on X. The Hessian of Mabuchi energy is a fourth-order elliptic operator D on functions which arises in the study of scalar curvature. We quantise D by an operator E(k) defined on the space of Hermitian endomorphisms of H^0(X,L^k) endowed with the L^2-inner-product. E(k) is the Hessian of a function appearing in the study of balanced embeddings. We consider the behaviour of E(k) as k tends to infinity. We first prove that the leading order term in the asymptotic expansion of E(k) is D. We next show that if either the first m eigenvalues of D are simple, or if there is uniform control over the first m spectral gaps of E(k), then the first m+1 eigenvalues and eigenvectors of E(k) converge to those of D. It is a consequence of our results that these two alternative hypotheses are in fact equivalent.

Affiliated with

Also associated with

No associations

Rating

Quantisation and the Hessian of Mabuchi energy does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Quantisation and the Hessian of Mabuchi energy, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Quantisation and the Hessian of Mabuchi energy will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-338386