Is there a Siegel analogue of $E_2$?

The almost holomorphic elliptic modular form $E_2$ undoubtedly is the easiest non-holomorphic modular forms. Its image under the lowering operator the constant function 1. We show that for Siegel modular forms there is no analogue of $E_2$, and in particular, classify all almost holomorphic Siegel modular forms. In degree greater than 1, they arise without exception from derivatives of holomorphic ones.

We introduce almost meromorphic Siegel modular forms. In degree 2 we describe an almost meromorphic preimage under the lowering operator of the constant Siegel modular form 1. The Ramanujan differential equation for $E_2$ generalizes to this setting. In topological string theory, it can be interpreted as a propagator identity for genus 2 curves.