Tian Invariants on G2,4(C) Blown Up Along P1(C) ∼= S2

In this paper, we prove the existence of an extremal functionthat minorizes all admissible functions with zero supremum ona non-toric Fano manifold, namely the complex GrassmannianG2,4(C) blown up along P1(C). The functions considered areinvariant under a suitably chosen automorphism group. Thislower bound allows us to calculate the Tian invariant for this classof functions on these manifolds. More precisely, we show thatthe Tian invariant αG(X) for the blow-up X equals 1/2, and weexplicitly construct the extremal function achieving the optimallower bound. We provide all detailed calculations, including theanalysis of the Pl¨ucker embedding, the metric potentials, the Jacobiansof coordinate changes, the reduction to diagonal matricesvia polar decomposition, the maximization of the extremal function,the maximum principle arguments for the lemmas, and theconvergence analysis of the integral defining the Tian invariant.