The conifold point, conjecture O, and related problems

Conjecture O describes the geometry in the complex line of the eigenvalues u_i
of the operator of quantum multiplication by the first Chern class acting on the
cohomology of a Fano manifold. In particular, it says that eigenvalues with maximal
absolute value have multiplicity one and one of them is real and positive number T.

Fano manifolds tend to have mirror dual Ginzburg-Landau potentials f, which
tend to have a distinguished non-degenerate critical point which we name the
conifold point. Explicitly the conifold point is the unique critical point P on real
positive locus, and the respective critical value T_{con} = f(P) is the global minimum
on the real positive locus. In this case it is conjectured that T_{con} conicides with T,
that is any eigenvalue u_i has absolute value at most T_{con}, and that the conifold
point is the unique critical point with value T_{con}.

In most cases the existence of the conifold point is clear and the conjectures
can be checked to be true, however we do not know how to prove them even
for abstract toric Fano manifolds, or complete intersections therein. These conjectures
are basic for formulation of Gamma conjectures about the appearance of Gamma
unction in the symplectic topology of Fano manifolds. Two references are arXiv:1404.7388
and my joint work with Golyshev and Iritani arXiv:1404:6407.