When it emerged in the early 1990s, mirror symmetry was an aspect of theoretical physics, and specifically a duality between quantum field theories. Since then, many people have tried to place it on a mathematical foundation. Their labors have built up a fascinating but somewhat unruly subject. It describes some sort of relation between pairs of Calabi-Yau spaces, but there are several quite different formulations of this relation, with no strong links between them. Notable among these are the toric approach of Batyrev-Borisov, leading to a very large class of examples whose Hodge numbers behave as desired, and the symplectic approach of Strominger-Yau-Zaslow, hereinafter SYZ. The latter is inspired by the original physics, and holds out the remarkable promise of connecting mirror symmetry to the theory of integrable systems. But it is extremely difficult to find examples.
复射影直线上一个线丛的全空间 $\operatorname{Tot}\mathcal O_{\mathbb P^1}(-2)$ 大约与余切丛 $T^*S^2$ 对称. 两者的实维数都是 $4$.