Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics. (English) Zbl 1243.14038

Let us consider a projective complex manifold \(X\) and an ample line bundle \(L\) on \(X\). The Mabuchi energy \(\nu\) is a crucial object to detect the existence of a constant scalar curvature Kähler metric (cscK for short) in \(c_1(L)\). It is an energy functional defined on the Kähler potentials that enjoys some natural geometric properties when there exists a cscK metric in \(c_1(L)\). Given the polarization \(L\), one can also consider the space of holomorphic sections \(H^0(X,L ^k)\) and the associated symmetric space of Bergman metrics \(H_k=\mathrm{GL}(N_k,\mathbb{C})/U(N_k)\) where \(N_k=\dim H^0(X,L^k)\). It is well known from the work of G. Tian than the space of Bergman metrics \(H_k\) is dense in the set of Kähler metrics in the class \(c_1(L)\). Therefore it is natural to understand the behavior of the Mabuchi energy over the space of Bergman metrics and to relate this behavior to the geometry of \((X,L)\).
This paper provides important results in that direction by giving a simple formula for the restriction of the Mabuchi energy \(\nu\) to the space of Bergman metrics. This formula depends on the log norms of the Chow form \(R_X\) and the hyperdiscriminant \(\Delta_X\) of \((X,L^k)\). Let us recall that the Chow form of \(X\) is given by the equation of the divisor \([R_X]=\{ L \in \mathrm{Gr}(N-1-n,N)| L \cap X\neq \emptyset\}\) where \(n=\dim X\). The hyperdiscriminant \(\Delta_X\) of \(X\) is the equation of the dual variety of \(X \times \mathbb{P}^n\) in the corresponding Segre embedding. Namely, if \(\phi_{\sigma}\) is the potential of the Bergman metric induced by \(\sigma\in \mathrm{SL}(N,\mathbb{C})\), then the author proves that \[ \nu(\phi_{\sigma})=\deg(R_X)\log \frac{\| \sigma \cdot \Delta_X \|^2}{\| \Delta_X\|^2}- \deg(\Delta_X)\log \frac{\|\sigma \cdot R_X\|^2}{\| R_X\|^2}. \] There are several nice consequences of this result. For instance, one can see the asymptotic behavior of the Mabuchi energy along any algebraic one parameter subgroup of a maximal algebraic torus of \(\mathrm{SL}(N,\mathbb{C})\). It is completely determined by the Chow polytope and the hyperdiscriminant polytope associated to \(R_X\) and \(\Delta_X\). Furthermore, the boundedness (or properness) of the Mabuchi energy along degenerations in \(\mathrm{SL}(N,\mathbb{C})\) can be rewritten in a geometric way in terms of inclusions of those polytopes (but unfortunately one needs to check the inclusion for all maximal algebraic tori). This leads the author to give a new definition of the notion of K-stability (which is a priori different from the one introduced by G. Tian and S.K. Donaldson) in terms of inclusion of polytopes. Contrarily to the classical notion of K-stability, the deformation of \(X\) does not play the role and so does not use the delicate notion of test-configuration, but requires to check a property on all maximal tori of \(\mathrm{SL}(N,\mathbb{C})\). Then the stability of \((X,L^k)\) as introduced by the author is equivalent to a condition on the positions of the relative polytopes of \(R_X\) and \(\Delta_X\) and hence involves only classical projective complex geometry.


14L24 Geometric invariant theory
32Q20 Kähler-Einstein manifolds
51M20 Polyhedra and polytopes; regular figures, division of spaces
Full Text: DOI arXiv


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