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MR690367 (84f:53039) 53C25 (53C50 57R20 58G10)
Matsushita, Yasuo Thorpe-Hitchin inequality for compact Einstein $4$-manifolds of metric signature $(++--)$ and the generalized Hirzebruch index formula. J. Math. Phys. 24 (1983), no. 1, 36–40.
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