Cohomology formula for obstructions to asymptotic Chow semistability (Q309676)

Let \(M\) be a compact complex \(n\)-dimensional manifold with an ample line bundle \(L\) and \(\omega\) a Kähler form in \(c_1(L)\). Denote also by \(\mathfrak h_0\) the Lie algebra of infinitesimal automorphisms of the polarized variety \((M, L)\) and, for each \(X \in \mathfrak h_0\), let \(u_X: M \to \mathbb C\) be the unique complex function such that \(\imath_X \omega = - \bar \partial u_X\) and satisfying the normalizing condition \(\int_M u_X \omega^n = 0\). We recall that, given \(1 \leq p \leq n\), the \textit{\(p\)-th order Futaki invariant of \((M, L)\)} is the map \(\mathcal F_p: \mathfrak h_0 \to \mathbb C\), given by NEWLINE\[NEWLINE \mathcal F_p(X) = (n- p +1) \int_M \text{Td}_p(\Theta) \wedge u_X \omega^{n-p} + \int_M \phi(L(X) + \Theta) \wedge \omega^{n-p+ 1}\;,NEWLINE\]NEWLINE introduced by \textit{A. Futaki} in [Int. J. Math. 15, No. 9, 967--979 (2004; Zbl 1074.53060)]. Here \(L(X)\) is the Nomizu operator of \(X\), i.e., \(L(X) = \nabla_X - \mathcal L_X\), \(\Theta\) is the curvature \(2\)-form of the connection \(\nabla\) and \(\text{Td}_p\) is the \(p\)-th Todd polynomial. The vanishing of all \(\mathcal F_p\) on a maximal reductive subalgebra \(\mathfrak h_r \subset \mathfrak h\) is an important necessary condition for asymptotically Chow stability.NEWLINENEWLINEThe main result of this paper is the proof of a formula which expresses the higher Futaki invariants \(\mathcal F_p(\mathcal X, \mathcal L)\) of a test configuration \((\mathcal X, \mathcal L)\) of \((M, L)\) in terms of the intersection numbers of \((M, L)\) and of the natural compactification \((\overline{\mathcal X}, \overline{\mathcal L})\) of \((\mathcal X, \mathcal L)\). It can be considered as a generalization of the Odaka and Wang intersection formula for the Donaldson-Futaki invariant.