We prove that the count of Maslov index $2$ $J$-holomorphic discs passing through a generic point of a real Lagrangian submanifold with minimal Maslov number at least two in a closed spherically monotone symplectic manifold must be even. As a corollary, we exhibit a genuine real symplectic phenomenon in terms of involutions, namely that the Chekanov torus $T_{\operatorname{Chek}}$ in $S^2 \times S^2$, which is a monotone Lagrangian torus not Hamiltonian isotopic to the Clifford torus $T_{\operatorname{Clif}}$, can be seen as the fixed point set of a smooth involution, but not of an antisymplectic involution.

中文翻译：

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$ S ^ 2 \ times S ^ 2 $中的Chekanov圆环不是真实的

我们证明，在封闭的球面单调辛流形中，通过最小马斯洛夫数的实拉格朗日子流形的通有至少2个Maslov指数$ 2 $ $ J $-全同态圆盘的数量必须是偶数。作为推论，我们展示出一个真正的辛格现象，涉及对合，即$ S ^ 2 \ times S ^ 2 $中的Chekanov圆环$ T _ {\ operatorname {Chek}} $，这是一个单调的拉格朗日圆环，而不是Clifford圆环$ T _ {\ operatorname {Clif}} $的哈密顿同位素，可以看作是平滑对合的定点集，而不是反对称对合的不动点集。