The geometry on smooth toroidal compactifications of Siegel varieties (Q2879437)

The paper under review deals with intersection numbers on a toroidal compactification \(\bar{A}_g\) of the Siegel moduli space \(A_g\). The complement \(\bar{A}_g-A_g\) is a simple normal crossing divisor \(D_\infty\) with irreducible components \(D_i\), and it admits the line-bundle \(\Omega^{\max}(D_\infty)\). The main result states that the intersection product of a power of \(\Omega^{\max}(D_\infty)\) with a monomial contains \(g\) different \(D_i\) and \(\Omega^{\max}(D_\infty)\) occurs at least once, or if there are \((g-1)\) different \(D_i\) and \(\Omega^{\max}(D_\infty)\) occurs with exponent at least two.NEWLINENEWLINEThe proof uses that \(\Omega^{\max}(D_\infty)\) admits a specific metric (with mild singularities which do not matter) such that in local coordinates \(z_j\) the norm of a generator is a power of \(\det(-\sum \log | z_j | \cdot b_j)\), where the \(b_j\) are positive semidefinite symmetric \(g \times g\) matrices (the volume of a degenerating abelian variety). The intersection numbers are obtained by integrating powers of \(\partial \bar{\partial}\) of the logarithm of this. Restriction to one \(D_i\) amounts to replacing this by the determinant of the restriction of \(-\sum \log | z_j | b_j\) to the kernel of \(b_i\). Repeating this \(g\) times gives the constant function 1 with trivial \(\partial \bar{\partial} \log\). Repeating only \(g-1\) times the \(\partial \bar{\partial} \log\) has square zero.NEWLINENEWLINEThis argument seems to suffice for strata parametrising totally degenerate abelian varieties. However, for mixed cases something seems to be missing. The first case is \(g=3\) and strata parametrising semiabelian varieties which are extensions of an elliptic curve by a two dimensional torus.