Varieties with \(q(X)=\dim(X)\) and \(P_2(X)=2\) (Q2915214)

The classification problem is one of the major problems in algebraic geometry. In classification theory, the irregularity \(q(X)=h^1(X, {\mathcal{O}}_X)\) and the plurigenera \(P_m(X)=h^0(X, \omega_X^m)\) of a smooth projective variety \(X\) are commonly used. Kawamata proved that \(X\) is birational to an abelian variety if and only if \(q(X)=\dim (X)\) and the Kodaira dimension \(\kappa(X)=0\).NEWLINENEWLINEDue to \textit{J. A. Chen, C. D. Hacon} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 3, No. 2, 399--425 (2004; Zbl 1104.14015)] and the author [Commun. Contemp. Math. 13, No. 3, 509--532 (2011; Zbl 1311.14015)], Kawamata's theorem has been improved: \(X\) is birational to an abelian variety if and only if \(q(X)=\dim (X)\) and \(P_2(X)=1\) or \(0<P_m(X)\leq m-2\) for some \(m\geq 3\). If \(q(X)=\dim (X)\) and the numerical invariants of \(X\) are a little bit higher than these bounds, a complete birational description of \(X\) can be obtained [\textit{J. A. Chen, C. D. Hacon}, loc. cit; Zbl 1073.14507; Zbl 1104.14015; \textit{C. D. Hacon} and \textit{R. Pardini}, J. Reine Angew. Math. 546, 177--199 (2002; Zbl 0993.14005)].NEWLINENEWLINEIn this paper, by the method of Chen and Hacon and the result of the author, the following theorem is proved:NEWLINENEWLINE Let \(X\) be a smooth projective variety with \(q(X)=\dim (X)\). (1) If \(P_2(X)=2\), then \(\kappa(X)=1\) and \(X\) is birational to a quotient \((K\times C)/G\), where \(K\) is an abelian variety, \(C\) is a curve, \(G\) is a finite group which acts diagonally and freely on \(K\times C\), and \(C\rightarrow C/G\) is branched at \(2\) points. (2) If \(0<P_m(X)\leq 2m-2\), for some \(m\geq 4\), then \(\kappa(X)\leq 1\).