The curvature of a Hessian metric

In this paper, inspired by P.M.H. Wilson’s paper on sectional curvatures of Kähler moduli [31], we concentrate on the case where f is a homogeneous polynomial (also called a “form”) of degree d at least 2. Following Okonek and van de Ven [23], Wilson considers the “index cone,” the open subset where the Hessian matrix of f is Lorentzian (that is, of signature (1, ∗)) and f is positive. He restricts the indefinite metric −1/d(d − 1)∂f/∂xi∂xj to the hypersurface M := {f = 1} in the index cone, where it is a Riemannian metric, which he calls the Hodge metric. (In affine differential geometry, this metric is known as the “centroaffine metric” of the hypersurface M , up to a constant factor.) Wilson considers two main questions about the Riemannian manifold M . First, when does M have nonpositive sectional curvature? (It does have nonpositive sectional curvature in many examples.) Next, when does M have constant negative curvature? On the first question, Wilson gave examples of cubic forms f on R to show that the surface M need not have nonpositive curvature everywhere. But he showed that for every cubic form on R such that M is nonempty (that is, the index cone is nonempty), M has nonpositive curvature somewhere ([31], Prop. 5.2). One result of this paper is to confirm Wilson’s suggestion that this statement should fail for forms of higher degree or on a higher-dimensional space. Namely, we give examples of a quartic form on R and a cubic form on R such that M is nonempty and M has positive sectional curvature on some 2-plane at every point (Lemmas 4.1 and 5.1). If Wilson’s conjecture that the Kähler moduli space of a Kähler manifold has nonpositive sectional curvature is correct, then these forms cannot occur as the intersection form on H1,1(X,R) for a Kähler 4-fold with h1,1 = 3, or a Kähler 3-fold with h1,1 = 4 (respectively), although they would be allowed by the Hodge index theorem. Wilson showed that the Riemannian manifold M has constant negative sectional curvature when f is a Fermat form x1−x d 2−· · ·−x d n ([31], Introduction, Example 2). More generally, we show that M has constant negative curvature −d2/4 when f is a sum of forms of degree d in at most two variables, f = α1(x1, x2)+α2(x3, x4)+ · · · . The problem of finding forms f such that the surface M has constant curvature is a special case of the WDVV equations of string theory, as explained in section 2. In fact, section 2 lists a whole series of natural problems of differential geometry that are essentially equivalent to the WDVV equations. The problem of finding all forms f on R such that the surface M has constant curvature −d2/4 also has a close relation to classical invariant theory, in particular