Partition functions in algebraic geometry

Abstract

In geometry, a common theme is to construct canonical objects or invariants related to a given geometric object. For complex projective varieties — defined by polynomial equations over the complex numbers — this is exemplified by the hunt for canonical Kähler metrics. Past and recent progress has connected the existence of various canonical metrics with intricate, but algebraic, stability conditions of the underlying variety. For an arithmetic variety — a scheme defined by polynomial equations with integer coefficients — one can define its height, roughly measuring its arithmetic complexity, in terms of Arakelov geometry and with respect to a metric over the corresponding complex projective variety. When the metric is taken to be canonical, one obtains an invariant, its canonical height. These notions are studied in this thesis from the perspective of a canonical statistical mechanical system — an N-point process in the form of a Gibbs measure for a certain interaction energy — defined on either a Fano or a canonically polarized variety. In the canonically polarized case, the canonical Kähler–Einstein metric with constant negative Ricci curvature has been shown to emerge in the large N-limit. Given a statistical mechanical system, a lot can be learned from studying its \emph{partition function}. In the large-N or thermodynamic limit, the same role is played by the systems free energy. The thesis consists of five papers exploring this direction. In the first two, a sharp height bound is conjectured for arithmetic Fano varieties whose complexifications are K-semistable, in analogy with K. Fujita's bound over C. In Paper I this conjecture is established for the canonical integral model of a toric Fano manifold of dimension at most 6. The general case is shown to hinge on a conjectural gap hypothesis for the degree, recently established by Li–Miao. Paper II establishes the conjecture for diagonal hypersurfaces, and orbifold curves. The conjecture makes contact with Kähler–Einstein metrics, K-stability, and the aforementioned canonical height, coinciding with a version of the free energy. In Paper III, canonical heights are studied for Fano and canonically polarized arithmetic log pairs. Here the full statistical mechanical approach is leveraged to give a limit formula in terms of periods for the canonical height. It is valid in the canonically polarized case, and in the Fano but K-stable case in dimension one. Several applications are drawn from this result: an explicit formula in the case of log pairs consisting of the projective line with three points; based on an explicit period-formula originating in conformal field theory. Explicit formulas for the height of some Shimura curves; which combined with a recently established related height formula by X. Yuan yields information on the canonical integral model of some Shimura curves. Finally, an explicit formula for the canonical height of a Fermat curve. Paper IV takes a slightly different perspective. A general Coulomb gas on the sphere is studied, which in a special case coincides with the aforementioned system in case of the projective line. Instead of using a statistical mechanical framework to study problems in algebraic, Arakelov and Kähler geometry, methods of algebraic geometry are used to study problems in mathematical physics regarding this system. Among the results are a formula for the critical temperature at which the partition function diverges and asymptotics for the Gibbs measure and partition function as the critical temperature is approached. Some applications to well-studied versions of the systems, such as the two-component plasma, are given. In Paper V, the statistical mechanical approach is extended to the case of Fano manifolds with non-discrete automorphism group. While the large-N limit is still conjectural, just as in the higher-dimensional Fano case in general, several applications of the introduced framework are obtained. In particular, a notion of Gibbs polystability is defined in terms of log canonical thresholds of certain canonically defined divisors, refining the notion of Gibbs stability, and ensuring the existence of the statistical mechanical system for N large. Conjecturally, Gibbs polystability should by equivalent to the existence of a Kähler–Einstein metric, and the large-N limit of the thresholds for Gibbs polystability should coincide with an analytic coercivity threshold, yet to have been shown to equal an algebraically defined invariant. We compute all these invariants and verify the latter conjectures for log Fano curves. This in turn yields applications to a sharp and improved logarithmic Hardy–Littlewood–Sobolev inequality on the 2-sphere for probability measures with vanishing barycenter. Lastly, the limit period formula for the canonical height is extended to the case of K-polystable log Fano curves with non-discrete automorphism group.