Moduli spaces and algebraic cycles in real algebraic geometry

TL;DR

This thesis advances real algebraic geometry by studying algebraic cycles, moduli spaces, and proving new cases of the Hodge conjecture for real and complex abelian varieties.

Contribution

It constructs integral Fourier transforms on Chow rings, proves the integral Hodge conjecture for certain abelian varieties, and analyzes the topology and structure of real moduli spaces.

Findings

Proved the integral Hodge conjecture for one-cycles on complex Jacobians.

Established the real integral Hodge conjecture modulo torsion for real abelian threefolds.

Described the moduli space of stable real binary quintics as a non-arithmetic ball quotient.

Abstract

This thesis intends to make a contribution to the theories of algebraic cycles and moduli spaces over the real numbers. In the study of the subvarieties of a projective algebraic variety, smooth over the field of real numbers, the cycle class map between the Chow ring and the equivariant cohomology ring plays an important role. The image of the cycle class map remains difficult to describe in general; we study this group in detail in the case of real abelian varieties. To do so, we construct integral Fourier transforms on Chow rings of abelian varieties over any field. They allow us to prove the integral Hodge conjecture for one-cycles on complex Jacobian varieties, and the real integral Hodge conjecture modulo torsion for real abelian threefolds. For the theory of real algebraic cycles, and for several other purposes in real algebraic geometry, it is useful to have moduli spaces of real varieties to our disposal. Insight in the topology of a real moduli space provides insight in the geometry of a real variety that defines a point in it, and the other way around. In the moduli space of real abelian varieties, as well as in the Torelli locus contained in it, we prove density of the set of moduli points attached to abelian varieties containing an abelian subvariety of fixed dimension. Moreover, we provide the moduli space of stable real binary quintics with a hyperbolic orbifold structure, compatible with the period map on the locus of smooth quintics. This identifies the moduli space of stable real binary quintics with a non-arithmetic ball quotient.