On the modified Futaki invariant of complete intersections in projective spaces (Q2828017)

The author proposes a method for computing the modified Futaki invariant for Fano complete intersections in projective spaces.NEWLINENEWLINELet \(M\) be a compact complex Fano manifold, that is, \(c_1(M)\) is represented by a Kähler form. A \textit{Kähler-Ricci soliton} is a Kähler metric \(\omega\in c_1(M)\) satisfying \(\mathrm{Ric}(\omega)-\omega=L_V \omega\) for some holomorphic vector field \(V\). Tian and Zhu proved that the vanishing of the \textit{modified Futaki invariant} is a necessary condition for the existence of Kähler-Ricci solitons. Berman and Nyström extended this obstruction to Fano varieties, that is, projective normal varieties with log terminal singularities and with \(-K_M\) ample \(\mathbb Q\)-line bundle.NEWLINENEWLINEThe author provides a method to compute the modified Futaki invariant for Fano complete intersections in projective spaces, in terms of the degrees of the defining polynomials, the weights of the actions induced by the vector field, and some integrals. With respect to the localization formula for orbifolds by \textit{W. Ding} and \textit{G. Tian} [Invent. Math. 110, No. 2, 315--335 (1992; Zbl 0779.53044)], the author does not need to assume orbifold singularities, or the explicit geometric knowledge of \(M\), \(V\), and \(\omega\). The formula in the paper generalizes the result in [\textit{Z. Lu}, Port. Math. (N.S.) 60, No. 3, 263--268 (2003; Zbl 1058.53043)] for the modified Futaki invariant for smooth hypersurfaces in projective spaces.