A sequence of Riemannian metrics \(\{g_i\}_{i \in \mathbb{N}}\) on a smooth manifold \(M\) is said to have almost non-negative Ricci curvature if the Ricci curvature of the \(i\)-th metric \(g_i\) is bounded below by \(-1\) while its diameter is bounded above by \(1/i\). Recall that the Todd genus of a complex manifold \(M\) of complex dimension \(n\) is defined by the alternating sum \(\sum_{p=0}^n (-1)^p\dim H^{0,p}(M,\mathbb{C})\) where \(H^{0,p}(M,\mathbb{C})\) denotes the Dolbeault cohomology group of \(M\).
The authors prove (A) that a compact complex manifold with an infinite fundamental group that admits a sequence of Kähler metrics with almost non-negative Ricci curvature has a vanishing Todd genus. The proof is given in the second section of the article.
Inspired by a result of J. Lott [J. Geom. Anal. 10, No. 3, 529–543 (2000; Zbl 1047.53024)] that a compact spin manifold that admits a sequence of metrics with almost nonnegative sectional curvature has vanishing \(\hat{A}\)-genus, the authors prove (B) that a compact spin manifold \(M^{4n}\) with an infinite fundamental group that admits a sequence of metrics with almost nonnegative Ricci curvature has vanishing \(\hat{A}\)-genus. The proof of (A) is given in the second section of the article. The details of the proof of (B) will be similar and are sketched there only in outline.
The fundamental group of \(M\) contains a finite index nilpotent subgroup, thus \(M\) has a finite cover \(M_1\) whose fundamental group is nilpotent. The authors argue that \(\pi_1(M_1)\) contains a nested sequence of finite index subgroups \(G_j\) whose index tends to infinity. Following an online forum post by S. Ivanov [“Diameter of \(m\)-fold cover”, https://mathoverflow.net/questions/7732/diameter-of-m-fold-cover/16939#16939], the authors show the diameter of the pullback of \(g_i\) to the \([\pi_1(M_1),G_j]\)-fold cover \(M_j\) of \(M_1\) is bounded by \([\pi_1(M_1),G_j]\) times the diameter of the pullback of \(g_i\) to \(M_1\).
Crucially, the size of the index of \(\partial+\partial^*\) for the lift of \(g_i\) is bounded on \(M_j\) by a bound \(C(n)\) that depends only on the dimension of \(M\). This follows from [P. Li, Ann. Sci. Éc. Norm. Supér. (4) 13, 451–468 (1980; Zbl 0466.53023)] and an argument using the Bochner technique. Since the size of the index of \(\partial+\partial^*\) on \(M_j\) is equal to the size of the index of \(\partial+\partial^*\) on \(M_1\) times \([\pi_1(M_1),G_j]\) it follows the size of the index of \(\partial+\partial^*\) on \(M_1\) is bounded above by \(C(n)/[\pi_1(M_1),G_j]\) which tends to zero as \(j\) tends to infinity. Hence the size of the index of \(\partial+\partial^*\) on \(M_1\) vanishes, and the Todd genus of \(M_1\) vanishes, and likewise for \(M\).