On characteristic forms of holomorphic foliations (Q1304928)

Let \(M\) be a complex manifold of dimension \(n\geq 2\) and let \({\mathcal F}\) be a holomorphic foliation of codimension \(q\geq 1\) on \(M\). The normal bundle of \({\mathcal F}\) is denoted by \(\nu({\mathcal F})\). There always exists a \(C^{\infty}\) affine connection \(a=\{a_{\alpha}\}\) of \(\nu({\mathcal F})\), by which the authors define the Chern forms \(\{c_k(a)\}_{k=1}^q\) of \(\nu({\mathcal F})\), and similarly, there always exist a \(C^{\infty}\) (normal reduced) projective connection \(\pi=\{p_{\alpha}\}\) of \(\nu({\mathcal F})\), by which they define a kind of characteristic forms \(\{P_k(\pi)\}_{k=1}^q\) of \(\nu{\mathcal F})\) and which they call the projective Weyl forms. The authors prove that for any \(C^{\infty}\) normal reduced projective connection \(\pi=\{p_{\alpha}\}\) of \(\nu({\mathcal F})\), the associated projective Weyl forms are \(d\)-closed, and that there exists a \(C^{\infty}\) affine connection \(a=\{a_{\alpha}\}\) of \(\nu({\mathcal F})\) satisfying the following formulas : \[ \sum_{k=0}^q c_k(a)t^k={(1+\alpha t)^{q+1}\over 1+(\alpha-\beta)t} \sum_{k=0}^q P_k(\pi)\biggl({t\over 1+\alpha t}\biggr)^k \] and \[ \sum_{k=0}^q P_k(\pi)t^k=(1-\alpha t)^q (1-\beta t) \sum_{k=0}^q c_k(a)\biggl({t\over 1-\alpha t}\biggr)^k, \] where \(c_k(a)\) is the \(k\)-th Chern form defined by the affine connection \(a\), and both \(\alpha\) and \(\beta\) are \(d\)-closed 2-forms which represent the de Rham cohomology class \([{1\over q+1} c_1(a)]\).