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Complex Non-Kähler Geometry pp 57–120Cite as

Calabi–Yau Manifolds with Torsion and Geometric Flows

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Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2246))

Abstract

The main theme of these lectures is the study of Hermitian metrics in non-Kähler complex geometry. We will specialize to a certain class of Hermitian metrics which generalize Kähler Ricci-flat metrics to the non-Kähler setting. These non-Kähler Calabi–Yau manifolds have their origins in theoretical physics, where they were introduced in the works of C. Hull and A. Strominger. We will introduce tools from geometric analysis, namely geometric flows, to study this non-Kähler Calabi–Yau geometry. More specifically, we will discuss the Anomaly flow, which is a version of the Ricci flow customized to this particular geometric setting. This flow was introduced in joint works with Duong Phong and Xiangwen Zhang. Section 2.1 contains a review of Hermitian metrics, connections, and curvature. Section 2.2 is dedicated to the geometry of Calabi–Yau manifolds equipped with a conformally balanced metric. Section 2.3 introduces the Anomaly flow in the simplest case of zero slope, where the flow can be understood as a deformation path connecting non-Kähler to Kähler geometry. Section 2.4 concerns the Anomaly flow with α′ corrections, which is motivated from theoretical physics and canonical metrics in non-Kähler geometry.

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Acknowledgements

I would like to first thank D.H. Phong, my former Ph.D. advisor, for guiding me through this material over the course of many years, and whose style shaped the presentation of this course. I thank Xiangwen Zhang and Teng Fei, whose joint work is discussed here, for countless inspiring discussions on the content of these notes. I also thank Daniele Angella, Giovanni Bazzoni, Slawomir Dinew, Kevin Smith, Freid Tong, and Yuri Ustinovskiy for valuable comments and corrections. These lecture notes were prepared for a course given at the CIME Summer School on complex non-Kähler geometry in 2018, and I would like to thank D. Angella, L. Arosio and E. Di Nezza for the invitation and for organizing a wonderful conference.

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  1. Department of Mathematics, Harvard University, Cambridge, MA, USA

    Sébastien Picard

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  1. Sébastien Picard
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  1. Dipartimento di Matematica e Informatica “Ulisse Dini”, Università di Firenze, Firenze, Italy

    Daniele Angella

  2. Dipartimento di Matematica, Università di Roma Tor Vergata, Roma, Italy

    Leandro Arosio

  3. Sorbonne Université, Paris, France

    Eleonora Di Nezza

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Picard, S. (2019). Calabi–Yau Manifolds with Torsion and Geometric Flows. In: Angella, D., Arosio, L., Di Nezza, E. (eds) Complex Non-Kähler Geometry. Lecture Notes in Mathematics(), vol 2246. Springer, Cham. https://doi.org/10.1007/978-3-030-25883-2_2

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