Geodesic
Analytic torsion for log-Enriques surfaces and Borcherds product
Forum of Mathematics, Sigma, Tome 10 (2022)

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We introduce a holomorphic torsion invariant of log-Enriques surfaces of index two with cyclic quotient singularities of type $\frac {1}{4}(1,1)$. The moduli space of such log-Enriques surfaces with k singular points is a modular variety of orthogonal type associated with a unimodular lattice of signature $(2,10-k)$. We prove that the invariant, viewed as a function of the modular variety, is given by the Petersson norm of an explicit Borcherds product. We note that this torsion invariant is essentially the BCOV invariant in the complex dimension $2$. As a consequence, the BCOV invariant in this case is not a birational invariant, unlike the Calabi-Yau case. 
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     author = {Xianzhe Dai and Ken-Ichi Yoshikawa},
     title = {Analytic torsion for {log-Enriques} surfaces and {Borcherds} product},
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Xianzhe Dai; Ken-Ichi Yoshikawa. Analytic torsion for log-Enriques surfaces and Borcherds product. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.66

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