Abstract
Consider E a holomorphic vector bundle over a projective manifold X polarized by an ample line bundle L. Fix k large enough, the holomorphic sections \(H^0(E\otimes L^k)\) provide embeddings of X in a Grassmanian space. We define the balancing flow for bundles as a flow on the space of projectively equivalent embeddings of X. This flow can be seen as a flow of algebraic type hermitian metrics on E. At the quantum limit \(k\rightarrow \infty \), 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.
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Acknowledgments
The first author is very grateful to Simon Donaldson, Joel Fine and Julius Ross for illuminating conversations on the subject of balanced embeddings throughout the years. He would also like to thank Xiaonan Ma and Xiaowei Wang. This work has been carried out in the framework of the Labex Archimède (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government programme managed by the French National Research Agency (ANR). The first author work was also partially supported by supported by the ANR Project MNGNK, decision No. ANR-10-BLAN-0118 and the ANR Project EMARKS, decision No. ANR-14-CE25-0010.
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Keller, J., Seyyedali, R. Quantization of Donaldson’s heat flow over projective manifolds. Math. Z. 282, 839–866 (2016). https://doi.org/10.1007/s00209-015-1567-8
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DOI: https://doi.org/10.1007/s00209-015-1567-8
Keywords
- Yang–Mills flow
- Donaldson heat flow
- Balanced metric
- Balancing flow
- Quantization
- Projective
- Holomorphic vector bundle