The moduli space of Enriques surfaces and the fake monster Lie superalgebra.

*(English)*Zbl 0886.14015It is known that the moduli space \(D^0\) of an Enriques surface is the quotient \(D\) of a 10-dimensional hermitian symmetric space \(\Omega\) by a discrete group, with a divisor \(H_d\) removed. Hence \(D^0\) is quasi-projective.

The author proves that \(D^0\) is quasi-affine, more precisely it is an affine variety with an affine line removed. The author considers a function \(\Phi\) defined previously by him [see R. E. Borcherds, Invent. Math. 109, No. 2, 405-444 (1992; Zbl 0799.17014)] and related to the fake monster Lie superalgebra. He proves that \(\Phi\) is an automorphic form of degree \(4\) over \(\Omega\) and has a simple zero along \(H_d\) and has no other zeros.

The proof uses the explicit description of \(\Phi\) as either an infinite product or an infinite sum. It follows that \(\Phi\) gives a trivialization over \(D^0\) of the ample line bundle which gives the projective embedding of \(D^0\), so that \(D^0\) is quasi-affine. The precise structure of \(D^0\) is studied by using the Bailey-Borel compactification of \(D\) obtained by H. Sterk [Math. Z. 207, No. 1, 1-36 (1991; Zbl 0736.14017)].

The paper contains some interesting speculations about possible generalizations of the form \(\Phi\) over other symmetric bounded domains, with the hope to get period spaces of some families of Calabi-Yau n-dimensional manifolds.

The author proves that \(D^0\) is quasi-affine, more precisely it is an affine variety with an affine line removed. The author considers a function \(\Phi\) defined previously by him [see R. E. Borcherds, Invent. Math. 109, No. 2, 405-444 (1992; Zbl 0799.17014)] and related to the fake monster Lie superalgebra. He proves that \(\Phi\) is an automorphic form of degree \(4\) over \(\Omega\) and has a simple zero along \(H_d\) and has no other zeros.

The proof uses the explicit description of \(\Phi\) as either an infinite product or an infinite sum. It follows that \(\Phi\) gives a trivialization over \(D^0\) of the ample line bundle which gives the projective embedding of \(D^0\), so that \(D^0\) is quasi-affine. The precise structure of \(D^0\) is studied by using the Bailey-Borel compactification of \(D\) obtained by H. Sterk [Math. Z. 207, No. 1, 1-36 (1991; Zbl 0736.14017)].

The paper contains some interesting speculations about possible generalizations of the form \(\Phi\) over other symmetric bounded domains, with the hope to get period spaces of some families of Calabi-Yau n-dimensional manifolds.

Reviewer: G.Ottaviani (Firenze)