On Kodaira energy of polarized log varieties (Q1915546)

The author considers couples \((V,B)\) consisting of a normal variety \(V\) with a \(\mathbb{Q}\)-Weil divisor \(B= \sum b_iB_i\), \(0\leq b_i\leq 1\), having only log-terminal singularities. In this situation, the log-canonical \(\mathbb{Q}\)-bundle \(\kappa(V,B)= K_V+B\) is well defined, and for any big \(\mathbb{Q}\)-bundle \(L\) on \(V\), \[ \kappa\varepsilon(V,b,L):= -\inf\{t\in\mathbb{Q}\mid \kappa(K(V,B)+ tL)\geq 0\} \] is said to be the log-Kodaira energy of \((V,B,L)\). Using the log minimal model theory, the existence of a Fano fibration structure for \(\dim(V)=3\) is established. In the second part of the paper, the possible values of \(\kappa\varepsilon(V,B,L)\) for fixed \(n= \dim(V)\) are studied (``spectrum set''). For \(B=0\) and \(V\) with at most terminal singularities, the ``spectrum conjecture'' asserts that there are no negative limit points. This is shown for the case \(n\leq 3\) and \(V\) \(\mathbb{Q}\)-factorial. Counterexamples are provided for the cases \(B\neq 0\) and \(V\) with other than log-terminal singularities, respectively.