We prove that the signature of the Milnor fiber of smoothings of a $2$-dimensional isolated complete intersection singularity does not exceed the negative number determined by the geometric genus, the embedding dimension, and the number of irreducible components of the exceptional set of the minimal resolution, which implies Durfee’s weak conjecture and a partial answer to Kerner–Némethi’s conjecture.

A rigid meromorphic cocycle is a class in the first cohomology of the discrete group $\Gamma :={\mathrm{SL}}_2\left(\mathbb{Z}\right[1/p\left]\right)$ with values in the multiplicative group of nonzero rigid meromorphic functions on the $p$-adic upper half-plane ${\mathcal{H}}_p:={\mathbb{P}}_1\left({\mathbb{C}}_p\right)-{\mathbb{P}}_1\left({\mathbb{Q}}_p\right)$. Such a class can be evaluated at the real quadratic irrationalities in ${\mathcal{H}}_p$, which are referred to as “RM points.” Rigid meromorphic cocycles can be envisaged as the real quadratic counterparts of Borcherds’ singular theta lifts: their zeroes and poles are contained in a finite union of $\Gamma$-orbits of RM points, and their RM values are conjectured to lie in ring class fields of real quadratic fields. These RM values enjoy striking parallels with the values of modular functions on ${\mathrm{SL}}_2\left(\mathbb{Z}\right)\backslash \mathcal{H}$ at complex multiplication (CM) points: in particular, they seem to factor just like the differences of classical singular moduli, as described by Gross and Zagier. A fast algorithm for computing rigid meromorphic cocycles to high $p$-adic accuracy leads to convincing numerical evidence for the algebraicity and factorization of the resulting singular moduli for real quadratic fields.

In this article, we describe the structure of codimension $1$ foliations with canonical singularities and numerically trivial canonical class on varieties with terminal singularities, extending a result of Loray, Pereira, and Touzet to this context.