Numerically effectiveness and principal bundles on Kähler manifolds
Let \((X,\omega)\) be a compact connected Kähler manifold, \(G\) a connected simple linear algebraic group over \(\mathbb C\) and fix a proper parabolic subgroup \(P\subset G\). Moreover, consider a principal \(G\)-bundle \(E_G\) over \(X\) and any strictly antidominant character \(\chi\) of \(P\) (recall that a character \(\chi\) of \(P\) is said to be strictly antidominant if the associated line bundle over \(G/P\) is ample). Finally, a holomorphic vector bundle \(F\to X\) is said to be pseudostable, if there exists a filtration of holomorphic subbundles of \(F\) \[ 0=F_0\subset F_1\subset\cdots\subset F_{\ell-1}\subset F_\ell=F \] such that for each \(i=1,\dots,\ell\), the quotient \(F_i/F_{i-1}\) is a polystable vector bundle (i.e. a direct sum of stable vector bundles) with \(\mu(F_i/F_{i-1})=\mu(F)\), where \(\mu(\bullet)\) is the slope. In this paper the authors investigate the nefness of the line bundle \(E_G(\chi)\) over the quotient space \(E_G/P\), which is obtained by taking the quotient of \(E_G\times\mathbb C\) by \(P\) acting on the right as follows: the action of \(g\in P\) sends any \((z,c)\in E_G\times\mathbb C\) to \((zg,\chi(g^{-1})c)\). More precisely, they prove the equivalence of the following five facts. \(\bullet\quad\) The line bundle \(E_G(\chi)\to E_G/P\) is nef. \(\bullet\quad\) The \(G\)-bundle \(E_G\) is pseudostable and the second Chern class \(c_2(\operatorname{ad}(E_G))\in H^4(X,\mathbb Z)\) vanishes, where \(\operatorname{ad}(E_G)\) is the holomorphic vector bundle over \(X\) associated to the principal \(G\)-bundle \(E_G\) for the adjoint action of \(G\) on its own Lie algebra. \(\bullet\quad\) The vector bundle \(\operatorname{ad}(E_G)\) is numerically flat. \(\bullet\quad\) Take any triple \((Y,\omega_Y,\beta)\), where \((Y,\omega_Y)\) is a reduced irreducible compact Kähler analytic space of dimension \(m\) and \(\beta\colon Y\to X\) a holomorphic map, and let \(E_P\subset\beta^*E_G\) be any holomorphic reduction of the structure group of \(\beta^*E_G\) to the subgroup \(P\). Denote by \(E_P(\chi)\) the line bundle over \(Y\) associated to the principal \(P\)-bundle \(E_P\) for the character \(\chi\). Then, the inequality \[ \int_Y c_1(E_P(\chi))\wedge\omega_Y^{m-1}\geq 0 \] holds. \(\bullet\quad\) Fix a Kähler form \(\omega_0\) on \(E_G/P\) (this quotient space always admits one). For every triple \((Y,\beta,E_P\subset\beta^*E_G)\) as above, the inequality \[ \int_Y c_1(E_P(\chi))^j\wedge\widehat\beta^*\omega_0^{m-k}\geq 0 \] holds for each \(k=1,\dots,m\), where \(\widehat\beta\colon Y\to E_G/P\) is the tautological lift of \(\beta\) defined by the reduction of the structure group \(E_P\) of \(\beta^*E_G\). The main ingredients in the proof are the numerical criteria for a line bundle on a compact Kähler manifold to be nef contained in \textit{J.-P. Demailly} and \textit{M. Păun} [Ann. Math. (2) 159, No. 3, 1247--1274 (2004; Zbl 1064.32019)] and a result of \textit{C. Mourougane} [Publ. Res. Inst. Math. Sci. 33, No. 6, 893--916 (1997; Zbl 0926.14004)] about the positivity of direct images of adjoint line bundles. To finish with, in the last section the authors prove an analogous result for principal bundles whose structure groups are just assumed to be reductive.
scientific article; zbMATH DE number 5324299