Weak Landau-Ginzburg models for smooth Fano threefolds (Q2854097)

Let \(X\) be a smooth Fano variety of dimension \(n\) and Picard rank 1, and \(QH^*(X)\) be the quantum cohomology of \(X\). For \(H=-K_X\) define \(QH_H^*(X)\subset QH^*(X)\) the minimal subring containing \(H\). Then \(X\) is called quantum minimal if the dimension of \(QH_H^*\) over the Novikov ring is equal to \(n+1\). The fundamental term of the regularized \(I\)-series of \(X\) denoted by \(\tilde{I}_{H^0}^X\in \mathbb{C}[[t]]\) is defined in terms of the generating function of genus 0 Gromov-Witten invariants of the quantum minimal variety \(X\), having the descendants of \(H^0\) (dual of the fundamental class of \(X\)) as insertions. For the regular function \(f\in \mathbb{C}[x_1^{\pm1},\dots,x_n^{\pm1}]\) defined on the torus \(\mathbb{G}^n_m\), we denote by \(\Phi_f \in \mathbb{C}[[t]]\) the constant-term series of \(f\) (with coefficient of \(t^i\) equal to constant term of \(f^i\)). Then \(f\) is called a weak Landau-Ginzburg model for \(X\) if \(\Phi_f=\tilde{I}_{H^0}^X\), and furthermore if there is a fiberwise smooth (open) Calabi-Yau compactification of the family \(f:\mathbb{C}^{*n}\to \mathbb{C}\) model for \(X\) if \(\Phi_f=\tilde{I}_{H^0}^X\). Finding a weak Landau-Ginzburg model for \(X\) provides a coincidence of the invariants of categories involved in the Homological Mirror Symmetry.NEWLINENEWLINEThere are 17 families of smooth Fano threefolds of Picard rank 1. The paper under review provides a weak Landau-Ginzburg model as a Laurent polynomial in three variables as discussed above for all 17 families, and surveys all the known methods for finding such Laurent polynomials. These Laurent polynomials are shown to have Calabi-Yau compactifications to families of \(K3\)-surfaces. The Calabi-Yau compactifications of any of these families differ by flops. Following the ideas of Katzarkov, the paper under review proves that the numbers of irreducible components of central fibers of these Calabi-Yau compactifications are always 1 less than the Hodge number \(h^{12}(X)\), and in particular they are independent of the choice of the compactification. This result enables one to reconstruct the Hodge numbers of these Fano threefolds.