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

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 $E(L,\mathcal{L})\to X^n$ is constructed from each affine Lagrangian submanifold $L$ in ${\check{X}}^n$ with a unitary local system $\mathcal{L}\to L$. 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 $X^n$ ($n\ge 2$) is obtained as the pullback of an exact triangle on $X^1$ by a suitable holomorphic projection $X^n\to X^1$.

To appear in Osaka Journal of Mathematics, 44 pages

References in corpus (1)

• $A_{\infty }$-structures on an elliptic curve