Geometric interpretation for exact triangles consisting of projectively flat bundles on higher dimensional complex tori

zbMATH Open1498.14040arXiv1705.04007MaRDI QIDQ2119660

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Publication date: 30 March 2022

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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 $E(L,mathcalL)ightarrow{X}^n$ is constructed from each affine Lagrangian submanifold $L$ in $check{X}^n$ with a unitary local system $mathcalLightarrowL$. In this paper, we first interpret these simple projectively flat bundles $E(L,mathcalL)$ in the language of factors of automorphy. Furthermore, we give a geometric interpretation for exact triangles consisting of three simple projectively flat bundles $E(L,mathcalL)$ 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 $X^n$ ($ngeq2$) is obtained as the pullback of an exact triangle on $X^1$ by a suitable holomorphic projection $X^{n}ightarrow{X}^1$.

Full work available at URL: https://arxiv.org/abs/1705.04007

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