Weekly Bulletin

In their celebrated work 30 years ago, Kudla and Millson constructed modular forms with values in the cohomology of locally symmetric spaces. For spaces that parametrize variations of Hodge structures of weight 2, a connection with work of Mathai and Quillen on superconnections was done by Garcia 10 years ago. A more careful examination of the work shows that the definition is in fact very simple. This simple formula allows us, in work with Garcia and Greer, to show modularity of Noether-Lefschetz cycles on $\mathcal A_g$. In the talk, I will review the main ideas behind the proof of modularity of special cycles using Mathai-Quillen forms with an emphasis on the case of K3 surfaces. This will serve as a preparation for the next talk, where the proof with Garcia and Greer will be discussed.

We study orbifold Calabi-Yau hypersurfaces in weighted projective stacks from the point of view of the gauged linear sigma model. We give a very brief introduction to the gauged linear sigma model which is a physics version of quasimap theory and review some general conjectures. We explain how these conjectures allow for determination of the orbifold Gromov-Witten invariants without using Gromov-Witten or quasimap theory.

The expectation, which was conjectured by Kudla and proven in several instances by several mathematicians, is that the Kudla-Millson forms extend to the toroidal compactifications of Shimura varieties and still preserve modularity in a certain way. In joint work with Garcia and Greer, we confirm this expectation on the compactification of $\mathcal A_1 \times \mathcal A_{g}$ obtained by adding the cusp and use it to show modularity of Noether-Lefschetz cycles. In the talk I will discuss approaches to Kudla's conjecture and give our proof of modularity. If time permits, I will talk about an extension to mixed Shimura varieties.