On the conjecture \(\mathcal{O}\) of GGI for \(G/P\) (Q728236)

Let \(G\) be a connected semisimple complex algebraic group. A subgroup \(P\) of \(G\) is said to be parabolic if the quotient \(X:=G/P\) is a (homogeneous) projective variety. The main result of the beautiful, important and deep paper under review is that all the complex homogeneous projective varieties, i.e. of the form \(G/P\), with \(G\) and \(P\) as above, satisfy the \({\mathcal O}\)-conjecture introduced by \textit{S. Galkin} et al. [Duke Math. J. 165, No. 11, 2005--2077 (2016; Zbl 1350.14041)]. The conjecture \({\mathcal O}\) concerns a property of the quantum cohomology of Fano manifolds. Recall that a Fano manifold is an algebraic variety whose anti-canonical series is ample. General results ensure that complex homogeneous varieties are Fano. In the case of the Grassmannians \(G(k,n)\), this can be checked by hands, as the anticanonical class is \(n\sigma_1\), where \(\sigma_1\) is the class of a hyperplane section in the Plücker embedding. The index \(r_X\) of a Fano manifold is the maximum integer \(n\) such that \(-K_X=nH\), for some divisor \(H\). The quantum cohomology \((H^\star(X,{\mathbb C}),\star)\) of a Fano manifold is a deformation of the ordinary cohomology \((H^*(X,{\mathbb C}),*)\) over a ring \(R:={\mathbb C}[q_1,\ldots,q_i]\) of formal power series, where the indeterminates \(q_i\) play the role of \textsl{quantum parameters}. If \(\sigma,\tau\in H^*(X,{\mathbb C})\) are cohomology classes, their quantum product \(\sigma\star\tau\) is of the form \(\sigma*\tau\) plus a formal power series in the \(q_i\), whose coefficients encode Gromov-Witten invariants, i.e. numbers of maps \({\mathbb P}^1\rightarrow X\) satisfying some prescribed incidence conditions. The ordinary product \(\sigma*\tau\) is recovered by setting to zero all the quantum parameters: this explains why one talks about \textsl{quantum deformation of the cup product}. The usual cohomology of \(X\) can be regarded as a commutative sub-ring of the algebra of the endomorphisms of \(H^\star(X,{\mathbb C})\), by attaching to \(\sigma\in H^*(X,{\mathbb C})\) the linear map \([\sigma]\) of \(H^\star(X,{}\mathbb C))\) to itself, given by \([\sigma]\tau=\sigma\star \tau\). Let \(c_1(X)\in H^*(X,{\mathbb C})\) be the first Chern class of the tangent bundle of \(X\) and let \(\delta_0\) be the positive real number defined to be the absolute value of an eigenvalue of \([c_1(X)]\) with maximum modulus. Then \(X\) satisfies the \({\mathcal O}\)-conjecture by GGI, if the real number \(\delta_0\) is itself a simple eigenvalue of \([c_1(X)]\) with the following property: if \(\delta\) is an eigenvalue of \([c_1(X)]\) for which \(|\delta|=\delta_0\), then \(\delta=\delta_0\xi\), where \(\xi\) is a \(r_X\)-th root of the unity. The paper is organized as follows. The introduction explains in a clear way the conjecture \({\mathcal O}\) and its relationship with the Gamma conjectures, having to do with the expression of a certain asymptotic quantum cohomology class in terms of the Euler Gamma function. The \({\mathcal O}\) conjecture underlies the Gamma conjectures, as the latter are based precisely on the existence of such a distinguished eigenvalue \(\delta_0\) of \([c_1(X)]\). Section 2 contains background material (quantum cohomology, basics on algebraic groups, representation theory and combinatorics) which turns the paper definitely self-contained. Section 3 is about the Perron-Frobenius theory of non-negative matrices that are employed in a clever way, together with the quantum Chevalley formula, to prove the main result; elements of graph theory useful for the proof are also explained in this section. Finally, Section 4 contains all the technical details of the proof of the main result, which is eventually stated in all of its beauty and brevity at the end of this last, but not the least, section. The paper ends itself with a rich reference list helping the reader to reconstruct all those pieces of background one needs to enjoy the beautiful story told in this beautiful paper.