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The moduli space of Enriques surfaces and the fake monster Lie superalgebra. (English) Zbl 0886.14015
It 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.

##### MSC:
 14J28 $$K3$$ surfaces and Enriques surfaces 17B70 Graded Lie (super)algebras 14J10 Families, moduli, classification: algebraic theory 11F22 Relationship to Lie algebras and finite simple groups
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