Complete geodesic metrics in big classes
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- by Prakhar Gupta;
- Trans. Amer. Math. Soc. 378 (2025), 4129-4172
- DOI: https://doi.org/10.1090/tran/9360
- Published electronically: March 19, 2025
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Abstract:
Let $(X,\omega )$ be a compact Kähler manifold and $\theta$ be a smooth closed real $(1,1)$-form that represents a big cohomology class. In this paper, we show that for $p\geq 1$, the high energy space $\mathcal {E}^{p}(X,\theta )$ can be endowed with a metric $d_{p}$ that makes $(\mathcal {E}^{p}(X,\theta ),d_{p})$ a complete geodesic metric space. The weak geodesics in $\mathcal {E}^{p}(X,\theta )$ are the metric geodesic for $(\mathcal {E}^{p}(X,\theta ), d_{p})$. Moreover, for $p > 1$, the geodesic metric space $(\mathcal {E}^{p}(X,\theta ), d_{p})$ is uniformly convex.References
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Bibliographic Information
- Prakhar Gupta
- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland
- ORCID: 0009-0002-1619-0446
- Email: pgupta8@umd.edu
- Received by editor(s): January 11, 2024
- Received by editor(s) in revised form: October 1, 2024, and October 14, 2024
- Published electronically: March 19, 2025
- Additional Notes: This research was partially supported by NSF CAREER grant DMS-1846942.
- © Copyright 2025 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 378 (2025), 4129-4172
- MSC (2020): Primary 32U05; Secondary 32Q15, 53C55
- DOI: https://doi.org/10.1090/tran/9360