Superconnections, theta series, and period domains

Abstract

We use superconnections to define and study some natural differential forms on period domains D that parametrize polarized Hodge structures of given type on a rational quadratic vector space V . These forms depend on a choice of vectors v 1 , . . . , v r ∈ V and have a Gaussian shape that peaks on the locus where v 1 , . . . , v r become Hodge classes. We show that they can be rescaled so that one can form theta series by summing over a lattice L r ⊂ V r . These series define differential forms on arithmetic quotients Γ\D. We compute their cohomology class explicitly in terms of the cohomology classes of Hodge loci in Γ\D. When the period domain is a hermitian symmetric domain of type IV, we show that the components of our forms of appropriate degree recover the forms introduced by Kudla and Millson. In particular, our results provide another way to establish the main properties of these forms. 1

Green forms and the arithmetic Siegel–Weil formula

“…The forms ϕ(v) and ν(v) were already defined and studied in the setting of general period domains in [11].…”

“…This paper is a contribution to the archimedean aspects of this theory. Building upon previous work of the first author [11], we use Quillen's formalism of superconnections [39] as developed by Bismut-Gillet-Soulé [2,3] to construct natural Green forms for special cycles, in all codimensions, on orthogonal and unitary Shimura varieties. We show that these forms have good functorial properties and are compatible with star products.…”

“…In recent work [11], the first author introduced a superconnection ∇ v on D and showed that the component of degree (r, r) of the corresponding Chern form agrees with ϕ o KM (v).…”

On the equidistribution of some Hodge loci

“…This result is reminiscent of the work of Kudla-Millson on modularity of special cycles, see [KM90], see also [Gar18] for a recent approach using superconnections. Indeed, in both these papers, it is proved that the formal generating series:…”

Transgressions of the Euler class and Eisenstein cohomology of GLN(Z)

“…Working GL N (R)-equivariantly the Gaussian Thom form can be represented by a cocycle in the (gl N (R), SO N )-complex of the representation of GL N (R) in the space of Schwartz functions on R N . This cocycle behaves in several ways like the cocycle constructed by Kudla and Millson in [38], and the construction of the transgression form produces a (N − 1)form that behaves like their form ψ; that both cocycles should be regarded as analogous follows from previous work of the third author [25], which contains an approach to Kudla and Millson's results using Mathai and Quillen's ideas. We provide explicit formulas for all these forms in Sections 6 and 7.…”