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Stable maps in higher dimensions

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Abstract

We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich’s definition when the domain is a curve and Tian-Donaldson’s definition of K-stability when the target is a point. We give some examples, such as Kodaira embeddings and fibrations. We prove the existence of a projective moduli space of canonically polarised stable maps, generalising the Kontsevich-Alexeev moduli space of stable maps in dimensions one and two. We also state an analogue of the Yau–Tian-Donaldson conjecture in this setting, relating stability of maps to the existence of certain canonical Kähler metrics.

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Acknowledgements

The authors would like to thank Giulio Codogni, Kento Fujita and Jacopo Stoppa and Gabor Székelyhidi for helpful discussions. The first author especially thanks Roberto Svaldi and Chenyang Xu for birational advice. We also thank the referee for their comments.

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Authors and Affiliations

  1. DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK

    Ruadhaí Dervan

  2. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL, 60607, USA

    Julius Ross

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  1. Ruadhaí Dervan
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Correspondence to Julius Ross.

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Communicated by Ngaiming Mok.

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Dervan, R., Ross, J. Stable maps in higher dimensions. Math. Ann. 374, 1033–1073 (2019). https://doi.org/10.1007/s00208-018-1706-8

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