Counter-examples to Gamma conjecture I

We investigate Gamma conjecture I and its underlying Conjecture 𝒪 for the ℙ^1-bundles X_n=ℙ_ℙ^n(𝒪⊕𝒪(n)) with n≥ 3. We show that Conjecture 𝒪 does not hold if n is odd, and that Gamma conjecture I does not hold if n is even. Led by this example, we propose modifications for Gamma conjecture I, discuss Gamma conjecture I over the Kahler moduli space, and identify the corresponding principal asymptotic class.