# On the Bergman kernels of holomorphic vector bundles

@inproceedings{Lempert2021OnTB, title={On the Bergman kernels of holomorphic vector bundles}, author={L{\'a}szl{\'o} Lempert}, year={2021} }

Consider a very ample line bundle E → X over a compact complex manifold, endowed with a hermitian metric of curvature −iω, and the space O(E) of its holomorphic sections. The Fubini–Study map associates with positive definite inner products 〈 , 〉 on O(E) functions FS(〈 , 〉) ∈ Hω = {u ∈ C∞(X) : ω + i∂∂u > 0}. We prove that FS is an injective immersion, but its image in general is not closed in Hω. To obtain a closed range, FS has to be extended to certain degenerate inner products. This we do by…

## One Citation

### On the metric structure of section ring

- Mathematics
- 2022

. The main goal of this article is to study for a projective manifold and an ample line bundle over it the relation between metric and algebraic structures on the associated section ring. More…

## References

SHOWING 1-10 OF 11 REFERENCES

### Mapping properties of the Hilbert and Fubini–Study maps in Kähler geometry

- Mathematics
- 2017

Suppose that we have a compact Kahler manifold $X$ with a very ample line bundle $\mathcal{L}$. We prove that any positive definite hermitian form on the space $H^0 (X,\mathcal{L})$ of holomorphic…

### The Bergman Kernel and a Theorem of Tian

- Mathematics
- 1999

Given a domain Ω in ℂn, the Bergman kernel is the kernel of the projection operator from L 2 (Ω) to the Hardy space A 2 (Ω). When the boundary of Ω is strictly pseudoconvex and smooth, Fefferman [2]…

### Szego kernels and a theorem of Tian

- Mathematics
- 1998

We give a simple proof of Tian's theorem that the Kodaira embeddings associated to a positive line bundle over a compact complex manifold are asymptotically isometric. The proof is based on the…

### A Wess--Zumino--Witten type equation in the space of Kähler potentials in terms of Hermitian--Yang--Mills metrics

- Mathematics
- 2020

We prove that the solution of a Wess-Zumino-Witten type equation from a domain $D$ in $\mathbb{C}^m$ to the space of Kahler potentials can be approximated uniformly by Hermitian-Yang-Mills metrics on…

### On a set of polarized Kähler metrics on algebraic manifolds

- Mathematics
- 1990

A projective algebraic manifold M is a complex manifold in certain projective space CP, N > dim c M = n . The hyperplane line bundle of CP restricts to an ample line bundle L on M. This bundle L is a…

### Nonlinear analysis in geometry, L’Enseignement Math

- 1987

### extremality, interpolation of norms, and Kähler quantization, arXiv:1910.01782

- 1910