Borcherds products on \(O(2,l)\) and Chern classes of Heegner divisors
Let \((V,q)\) be a quadratic space of signature \((2,\ell)\), and let \(L \subset V\) be an even unimodular lattics, which implies that \(\ell\) is even. In a pioneering paper, \textit{R. Borcherds} [Invent. Math. 120, 161-213 (1995; Zbl 0932.11028)] discovered a lifting from nearly holomorphic modular forms of weight \(1-\ell/2\) for \(SL(2,\mathbb Z)\) to meromorphic modular forms for the orthogonal group of \(L\), which have infinite product expansions analogous to the Dedekind eta-function. This book discusses geometric aspects of the Borcherds lifting. Let \(O^+ (V)\) be the connected component of the identity of the orthogonal group \(O(V)\) of \(V\), and let \(H\) be a maximal compact subgroup of \(O^+ (V)\). Then the quotient space \(O^+ (V) /H\) can be realized as a tube domain \(\mathbb H_\ell\) in \(\mathbb C^\ell\). The orthogonal group \(O(L)\) of the lattice \(L\) is an arithmetic subgroup of \(O(V)\), and the subgroup \(\Gamma (L) = O(L) \cap O^+ (V)\) acts on \(\mathbb H_\ell\). The associated quotient space \(\mathcal X_L = \mathbb H_L \backslash \Gamma (L)\) has the structure of a quasi-projective variety over \(\mathbb C\). To each nonnegative integer \(m\), one can associate a \(\Gamma (L)\)-invariant divisor \(H(m)\), known as the Heegner divisor of discriminant \(m\). The Borcherds lifting determines explicit relations between Heegner divisors in the divisor class group \(\text{Cl} (\mathcal X_L)\) of \(\mathcal X_L\), which can be used for the study of the geometry of \(\mathcal X_L\). In fact, it is more convenient to work with a slightly modified divisor group \(\widetilde{\text{Cl}} (\mathcal X_L)\), which is defined as the quotient of \(\text{Cl} (\mathcal X_L)\) by the subgroup of divisors coming from meromorphic modular forms for \(\Gamma (L)\) of rational weight with some multiplier system. Let \(\kappa = 1+\ell/2\), and denote by \(S_\kappa\) the space of cusp forms of weight \(\kappa\) for \(SL(2,\mathbb Z)\). Let \(\mathcal A_\kappa (\mathbb Z)\) be the \(\mathbb Z\)-submodule of the dual space \(S_\kappa^\ast\) of \(S_\kappa\) generated by the functional \(a_r: h \mapsto b(r)\) for \(h = \sum_{n \geq 1} b(n) q^n \in S_\kappa\). Then the assignment \(a_r \mapsto H(-r)\) defines a homomorphism \(\eta: \mathcal A_\kappa (\mathbb Z) \to \widetilde{\text{Cl}} (\mathcal X_L)\). Let \(f\) be a meromorphic modular form for \(\Gamma (L)\) with some multiplier system, whose divisor is a linear combination of Heegner divisors \(H(m)\). This book studies the condition for \(f\) to be a Borcherds product by using the fact that it is equivalent to the condition that the map \(\eta\) is injective. In Chapters 1 to 3, the regularized theta lifting to Maass wave forms with singularities at the cusps is extended to find for each Heegner divisor some interesting analytic object. In Chapter 4, the map \(\eta\) is composed with the Chern class map to the second cohomology of the space \(\mathcal X_L\). An automorphic description of the resulting map from \(S_\kappa\) to the cohomology in terms of square integrable harmonic differential forms is discussed in Chapter 5, which is used to obtain a criterion for the injectivity of \(\eta\).
scientific article; zbMATH DE number 1749034