Algebraic cycles, Hodge theory, and Arithmetic

The antique origins of Algebraic Geometry lie in the study of solution sets of polynomial equations, in which complex, symplectic, and arithmetic geometry are bound tightly together. Many of the most spectacular recent developments in the subject have occurred through the consideration of these aspects in tandem: for example, the duality between symplectic and complex geometry that is mirror symmetry, and the Beilinson conjectures [1] on the transcendental invariants of generalized algebraic cycles defined over number fields. Beilinson [2] (and independently Bloch) also made conjectures on the structure of algebraic cycle groups over C; here the arithmetic content only becomes apparent when one looks at recent efforts to construct the filtration they predict. Likewise, the arithmetic side to the Hodge Conjecture (cf. [3]) is revealed by its bifurcation into absoluteness of Hodge cycles (proved for abelian varieties by Deligne [4]) and validity of the (slightly weakened) conjecture for varieties over Q̄, emphasized in recent work of Voisin [5]. The invariants appearing in these longstanding and celebrated problems are formalized Hodge-theoretically — that is, in terms of integrals of algebraic differentials on the variety called periods. Roughly speaking, the conjectures can be thought of as predicting the existence of certain algebraic cycles explaining what algebraic structure these otherwise transcendental periods do have. But cycles are more ubiquitous (and useful!) than this formulation might suggest, manifesting themselves through generalized normal functions in diverse contexts, from number theory (modular forms [6] and higher Green’s functions [7]; Apéry numbers [8]) to mathematical physics (asymptotics of local instanton numbers [9]; prediction of open Gromov-Witten invariants [10]; Feynman integrals [11, 12]; topological string theory [13]) and differential equations. What is more, the discovery of a “motivating cycle” can be the key to proving properties of such functions and generating more examples. This is a first major theme of my research. The periods described above are typically packaged in a linear-algebraic object called a Hodge structure (or mixed Hodge structure), and this is used to do far more than predict the existence of cycles. For instance, they can be used to study degeneration of algebraic varieties in one or more variables via limiting mixed Hodge structures. A more representation-theoretic approach to the algebraic structure of periods is obtained by considering the (linear-algebraic) symmetry groups of Hodge structures. These Mumford-Tate groups and their classification [14] lead (particularly in higher weight) to strong constraints