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

TL;DR

This paper provides a geometric interpretation of exact triangles formed by projectively flat bundles on higher-dimensional complex tori, linking them to affine Lagrangian submanifolds and their intersections, with applications to projections from higher to lower dimensions.

Contribution

It introduces a geometric perspective on exact triangles of projectively flat bundles on complex tori, relating them to affine Lagrangian intersections and their automorphy factors.

Findings

Interpretation of bundles via factors of automorphy

Geometric understanding of exact triangles and intersections

Application to projections from higher to lower dimensions

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 $X^n \rightarrow X^1$.