Hermitian matrices and cohomology of Kähler varieties (Q2433974)

Let \(X\) be an \(n\)-dimensional compact Kähler variety of strict Albanese type. This means that the Albanese map \(\alpha:X\rightarrow A(X)\) is generically finite and \(\dim X<\dim A(X)=:q\). The authors assume that there is no rational fibration \(f:X\rightarrow Y\) onto any Kähler variety \(Y\) of strict Albanese type with \(\dim Y<\dim X\). This condition is equivalent to injectivity properties of the Hodge structure maps \(\bigwedge^kH^{1,0}X\rightarrow H^{k,0}(X)\) [see \textit{F. Catanese}, Invent. Math. 104, No. 2, 263--289; Appendix 289 (1991; Zbl 0743.32025)] and yields in particular upper bounds for the kernel of the cup product \(\phi:\bigwedge^2 H^1(X;\mathbb C)\rightarrow H^2(X,\mathbb C)\). If \(q\leq 2n-1\), then \(\phi\) is injective, if \(q=2n=2^c(b+1)\), \(b,c\in\mathbb N\), then \(\ker\phi\leq 2c+3\). If \(q=2n+1=5\), then \(\dim\ker\phi\leq 14\). The authors get these estimates by analyzing the connection of real \((1,1)\)-forms on \(X\) with Hermitian matrices and applying advanced results in matrix theory by \textit{J. Adams, P. Lax} and \textit{R. Phillips} [Proc. Am. Math. Soc. 16, 318--322 (1965); Correction. Ibid. 17, 945--947 (1966; Zbl 0168.02404)] and \textit{S. Friedland} and \textit{A. Libgober} [Isr. J. Math. 136, 353--371 (2003; Zbl 1064.14070)].