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January 2022 Geometric Interpretation for Exact Triangles Consisting of Projectively Flat Bundles on Higher Dimensional Complex Tori
Kazushi Kobayashi
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Osaka J. Math. 59(1): 75-112 (January 2022).

Abstract

Let $(X^n, \check{X}^n)$ be a mirror pair of an $n$-dimensional complex torus $X^n$ and its mirror partner $\check{X}^n$. Then, a simple projectively flat bundle $E(L,\mathcal{L})\rightarrow X^n$ is constructed from each affine Lagrangian submanifold $L$ in $\check{X}^n$ with a unitary local system $\mathcal{L}\rightarrow L$. In this paper, we first interpret these simple projectively flat bundles $E(L,\mathcal{L})$ in the language of factors of automorphy. Furthermore, we give a geometric interpretation for exact triangles consisting of three simple projectively flat bundles $E(L,\mathcal{L})$ and their shifts by focusing on the dimension of intersections of the corresponding affine Lagrangian submanifolds $L$. Finally, as an application of this geometric interpretation, we discuss whether such an exact triangle on $X^n$ ($n\geq 2$) is obtained as the pullback of an exact triangle on $X^1$ by a suitable holomorphic projection $X^n\rightarrow X^1$.

Acknowledgments

I would like to thank Hiroshige Kajiura for various advices in writing this paper. I am also grateful to Satoshi Sugiyama for helpful comments. Finally, I would like to thank the referee for reading this paper carefully. This work was supported by Grant-in-Aid for JSPS Research Fellow 18J10909 and JSPS KAKENHI Grant Number JP16H06337.

Citation

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Kazushi Kobayashi. "Geometric Interpretation for Exact Triangles Consisting of Projectively Flat Bundles on Higher Dimensional Complex Tori." Osaka J. Math. 59 (1) 75 - 112, January 2022.

Information

Received: 9 October 2020; Published: January 2022
First available in Project Euclid: 31 January 2022

Subjects:
Primary: 14F08
Secondary: 14J33

Rights: Copyright © 2022 Osaka University and Osaka City University, Departments of Mathematics

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Vol.59 • No. 1 • January 2022
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