AI 中文总结
本文证明在满足同调镜像对称的K3曲面上，Bridgeland稳定性蕴含特殊Lagrangian球面存在，验证Thomas-Yau-Joyce猜想，并推广至乘积Calabi-Yau三维流形，为紧流形上首例。
AI 中文摘要
我们考虑由特定Lagrangian球面定义的一类K3曲面的Fukaya范畴中的对象。假设K3曲面的同调镜像对称以足够强的（预期的）性质成立，我们证明在此情形下，相对于合适的Bridgeland稳定性条件的稳定性蕴含同构的特殊Lagrangian球面的存在性，正如一般Thomas-Yau-Joyce猜想所预测的。特别地，这无条件地适用于$\mathbb{P}^3$中合适的四次曲面或$\mathbb{P}(3,1,1,1)$中的六次曲面及其镜像。一个变体在由我们的K3曲面与椭圆曲线乘积得到的Calabi-Yau三维流形上也成立。这些似乎是紧流形上此类结果的首例。
英文摘要
We consider objects in the Fukaya category of a class of K3 surfaces defined by certain Lagrangian spheres. Assuming that homological mirror symmetry for K3 surfaces holds with sufficiently strong (expected) properties, we prove that, in this case, stability with respect to a suitable Bridgeland stability condition implies the existence of an isomorphic special Lagrangian sphere, as predicted by the general Thomas-Yau-Joyce conjectures. In particular this holds unconditionally for suitable quartic surfaces in $\mathbb{P}^3$ or sextics in $\mathbb{P}(3,1,1,1)$ and their mirrors. A variant holds on the Calabi-Yau threefolds obtained by taking the product of our K3 surfaces with an elliptic curve. These seem to be the first results of this type on compact manifolds.
