Let \(X\) denote a compact Kähler manifold of dimension \(n\). The cup product on \(H^2(X,{\mathbb Z})\) determines a degree \(n\) form on \(H^{1,1}(X,{\mathbb R})\). An index cone, denoted by \(W\), consists of elements \(D\) in the positive cone \(\{D\in H^{1,1}(X,{\mathbb R})\,:\,D^n>0\}\) for which the quadratic form on \(H^{1,1}(X,{\mathbb R})\) given by \(L\mapsto D^{n-2}\cup L^2\) has signature \((1,h^{1,1}-1)\). We denote by \(W_1\) the level set \(\{D\in W\,:\,D^n-1\}\) in the index cone. For a fixed complex structure, and volume form \(\omega_0^n/n!\) let \(\tilde\mathcal K\) be the curved Kähler forms \(\omega\) with \(\omega^n=c\omega_0^n\) for some \(c>0\), and let \(\mathcal K\subset H^1(X,\mathbb R)\) denote the cone of Kähler classes. The projection map from \(\tilde\mathcal K\) to \(\mathcal K\) is a bijection. We normalize the volume form so that \(\int_X\omega_0^n=1\), and set \(\tilde\mathcal K_1\) to consist of the Kähler forms \(\omega\) with \(\omega^n=\omega_0^n\). Thus, setting \(\mathcal K_1=\mathcal K\cap W_1\), the projection \(\tilde\mathcal K_1\longrightarrow\mathcal K_1\) is a bijection.
The main purpose of this paper is to study these spaces. In particular, the author studies their geodesics and sectional curvatures. He considers the question: for which compact Kähler manifolds \(X\) are the sectional curvatures of \(\tilde\mathcal K_1\) non-positive, and for which \(X\) are they bounded below by \(-n(n-1)/2\)? The author finds simple formulae for the sectional curvatures, and proves both the bounds hold for various classes of varieties, developing along the way a mirror to the Weil-Petersson theory of complex moduli. In the case of threefolds with \(h^{1,1}=3\) an explicit formula for this curvature in terms of the invariants of the cubic form is given. The bounds for a wide range of examples by using this formula and computer are checked. The implications of the non-positivity of these curvatures are also studied.