The first author’s previous work established Solomon’s WDVV-type relations for Welschinger’s invariant curve counts in real symplectic fourfolds by lifting geometric relations over possibly unorientable morphisms. We apply her framework to obtain WDVV-style relations for the disk invariants of real symplectic sixfolds with some symmetry, in particular confirming Alcolado’s prediction for \({\mathbb {P}}^3\) and extending it to other spaces. These relations reduce the computation of Welschinger’s invariants of many real symplectic sixfolds to invariants in small degrees and provide lower bounds for counts of real rational curves with positive-dimensional insertions in some cases. In the case of \({\mathbb {P}}^3\), our lower bounds fit perfectly with Kollár’s vanishing results.

第一作者的先前工作通过将几何关系提升到可能不可定向的射影上，从而为威尔辛格的不变辛曲线数建立了所罗门的WDVV型关系。我们应用她的框架来获得具有对称性的实辛六重圆盘不变量的WDVV风格关系，尤其是确认Alcolado对\（{\ mathbb {P}} ^ 3 \）的预测 并将其扩展到其他空间。这些关系将许多实辛六倍的Welschinger不变量的计算量减小为小程度的不变量，并在某些情况下为带正维插入的实有理曲线的计数提供了下界。在 \（{\ mathbb {P}} ^ 3 \）的情况下，我们的下限与Kollár消失的结果完全吻合。