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We prove an obstruction at the level of rational cohomology in small degreesto the existence of positively curved metrics with large symmetry rank. Thesymmetry rank bound is logarithmic in the dimension of the manifold. As anapplication, we provide evidence for a generalized conjecture of Hopf that saysthat no symmetric space of rank at least two admits a metric with positivecurvature.

Positively curved metrics on symmetric spaces with large symmetry rank