\(c_1\)-cohomological rigidity for smooth toric Fano varieties of Picard number two (Q6665295)

The authors have previously formulated the following conjecture [\textit{Y. Cho} et al., J. Symplectic Geom. 21, No. 3, 439--462 (2023; Zbl 1530.53079)]\N:\N\NLet \(X\) and \(Y\) be smooth toric Fano varieties. If there exists a \(c_1\)-preserving graded ring isomorphism between their integral cohomology rings, then \(X\) and \(Y\) are isomorphic as varieties, where an isomorphism is said to be \(c_1\)-preserving if it preserves the first Chern classes of \(X\) and \(Y\). The conjecture is equivalent to saying that every smooth toric Fano variety is \(c_1\)-cohomologically rigid.\N\NThe \(c_1\)-cohomological rigidity is verified for Fano Bott manifolds and smooth toric Fano varieties of dimension up to 4 or of Picard number greater than or equal to \(2n-2\), where \(n\) is the complex dimension of the Fano variety. The authors prove the conjecture for smooth toric Fano varieties of Picard number 2.