Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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

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