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Stable Ulrich bundles on cubic fourfolds

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Abstract

In this paper, we examine the presence of Ulrich bundles on cubic fourfolds. We establish necessary and sufficient conditions for the existence of Ulrich bundles of a specific rank r. As a consequence, we show the existence of a family of non-decomposable Ulrich bundles of rank r on certain cubic fourfolds, which are dependent on approximately r parameters and have wild representation type. Our study also encompasses examples of arithmetically Cohen–Macaulay smooth surfaces that are not intersections of cubic fourfolds and codimension two subvarieties.

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Acknowledgements

We would like to thank Frank-Olaf Schreyer as well as Yeongrak Kim for their valuable advice and helpful discussions. Part of this work was done while H.L.T. was at Universität des Saarlandes. We would like to thank the referees for their several suggestions on how to improve the paper. H.L.T was partially supported by Prof. Thomas Hales during the Covid-19 pandemic. H.L.T was partially supported by Tosio Kato Fellowship and the Vietnam Academy of Science and Technology (VAST) under Grant Numbers NCVCC01.04/22-23, and CTTH00.03/24-25. The authors was partially supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.04-2023.02 and the Vietnam Academy of Science and Technology (VAST) under Grant Number CTTH00.03/24-25.

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  1. Institute of Mathematics, VAST, 18 Hoang Quoc Viet Road, Hanoi, 10307, Vietnam

    Truong Le Hoang

  2. Department of Mathematics, College of Humanities and Sciences, Nihon University, 3-25-40 Sakurajosui, Setagaya-Ku, Tokyo, 156-8550, Japan

    Truong Le Hoang

  3. The Department of Mathematics, Thai Nguyen University of Education, 20 Luong Ngoc Quyen Street, Thai Nguyen City, Thai Nguyen Province, Vietnam

    Yen Ngoc Hoang

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  1. Truong Le Hoang
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Hoang, T.L., Hoang, Y.N. Stable Ulrich bundles on cubic fourfolds. manuscripta math. 174, 243–267 (2024). https://doi.org/10.1007/s00229-023-01499-y

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