We compute the mod $2$ cohomology groups of real Lagrangians in Calabi-Yau threefolds using well-behaved torus fibrations constructed by Gross. To do this we study a long exact sequence introduced by Castano-Bernard and Matessi, which relates the cohomology of the Lagrangians to the cohomology of the Calabi-Yau. We show that the connecting homomorphism in this sequence is given by the map squaring divisor classes in the mirror Calabi-Yau. Applying this result, we compute the mod $2$ cohomology groups of real Lagrangians in the quintic threefold and in its mirror.

中文翻译：

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卡拉比丘的真正拉格朗日教徒有三倍

我们使用格罗斯（Gross）构造的行为良好的圆环纤维，计算了卡拉比丘（Calabi-Yau）真实拉格朗日人的mod $ 2 $同调群。为此，我们研究了Castano-Bernard和Matessi引入的长序列，该序列将拉格朗日派的同调与卡拉比丘的同调联系起来。我们表明，此序列中的连接同态由镜子Calabi-Yau中的地图平方除数类给出。应用此结果，我们可以计算出三次拉格朗日函数及其镜像中的实际拉格朗日人的mod $ 2 $同调群。