The alpha invariant of twisted Milnor hypersurfaces

In this talk, we give a computable formula about alpha invariant of twisted Milnor hypersurfaces with spin structure and dimension no less than 5. Using the result, we can also give the formula about $\hat A$-genus, and we do not restrict conditions on the spin structure and the dimension in this formula. As an important application, we show that Milnor hypersurfaces always admit PSC and give the sufficient and necessary conditions for twisted Milnor hypersurfaces with dimension restrictions non-existing PSC. During the calculation, we mentioned two methods, and one is concerned with important binomial numbers such as Bernoulli numbers, and the other can be applied in more general cases.