Reflective modular forms in algebraic geometry

Abstract

We prove that the existence of a strongly reflective modular form of a large weight implies that the Kodaira dimension of the corresponding modular variety is negative or, in some special case, it is equal to zero. Using the Jacobi lifting we construct three towers of strongly reflective modular forms with the simplest possible divisor. In particular we obtain a Jacobi lifting construction of the Borcherds-Enriques modular form Phi_4 and Jacobi liftings of automorphic discriminants of the Kähler moduli of Del Pezzo surfaces constructed recently by Yoshikawa. We obtain also three modular varieties of dimension 4, 6 and 7 of Kodaira dimension 0.

28 pages

References in corpus (4)

• The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras
• Exact Gravitational Threshold Correction in the FHSV Model
• The Siegel modular forms of genus 2 with the simplest divisor
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Cited by in corpus (14)

• Lorentzian Kac-Moody algebras with Weyl groups of 2-reflections
• Birational geometry of the moduli space of quartic K3 surfaces
• 24 faces of the Borcherds modular form Phi_{12}
• Quasi-pullback of Borcherds products
• The singularities and Kodaira dimension of unitary Shimura varieties
• Root systems for Lorentzian Kac-Moody algebras in rank 3 (arXiv version)
• Uniruledness of orthogonal modular varieties
• Finiteness of stable orthogonal modular varieties of non-general type
• Finiteness of 2-reflective lattices of signature (2,n)
• Reflective modular forms: A Jacobi forms approach
• Modular forms for the $A_1$-tower
• Borcherds Products Everywhere
• Jacobi forms of theta type
• Examples of lattice-polarized K3 surfaces with automorphic discriminant, and Lorentzian Kac--Moody algebras