Moduli spaces and algebraic cycles in real algebraic geometry

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.

arXiv comment234 pages, PhD thesis

References in corpus (15)

• The Torelli locus and special subvarieties
• On Albanese torsors and the elementary obstruction
• On the integral Hodge conjecture for real varieties, I
• The Schwartz space of a smooth semi-algebraic stack
• The integral Hodge conjecture for 3-folds of Kodaira dimension zero
• On the Geometry of the Moduli Space of Real Binary Octics
• Sums of three squares and Noether-Lefschetz loci
• Stacks of cyclic covers of projective spaces
• Hyperbolic Geometry and Moduli of Real Curves of Genus Three
• Arithmetic occult period maps
• Some elementary theorems about divisibility of 0-cycles on abelian varieties defined over finite fields
• Compact moduli of K3 surfaces
• Moduli Spaces of Unordered $n\ge 5$ Points on the Riemann Sphere and Their Singularities
• Pointwise surjective presentations of stacks
• On the integral Hodge conjecture for real abelian threefolds

Cited by in corpus (1)

• On the integral Hodge conjecture for real abelian threefolds