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Geometric interpretation for exact triangles consisting of projectively flat bundles on higher dimensional complex tori. (English) Zbl 1498.14040

Summary: 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 \(X^n\rightarrow X^1\).

MSC:

14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
14J33 Mirror symmetry (algebro-geometric aspects)
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