@ARTICLE{Totaro_thecurvature, author = {Burt Totaro}, title = {The curvature of a Hessian metric}, journal = {Internat. J. Math}, year = {}, pages = {369--391} }

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Abstract

Given a smooth function f on an open subset of a real vector space, one can define the associated “Hessian metric ” using the second derivatives of f, gij: = ∂ 2 f/∂xi∂xj. 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)∂2f/∂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?