Shoetsu Ogata

Hirzebruch's conjecture on cusp singularities

Hirzebruch conjectured in [H1] that the signature defect of a Hilbert modular cusp should coincide with the special value of the Shimizu L-function corresponding to the cusp. Atiyah et al. lADS1, ADS2] and Miiller [Mu] proved the conjecture. In this paper we prove a relation between some geometric invariants associated with a cusp, which together with a recent result of Ishida [ I ] gives an algebro-geometric proof of the Hirzebruch conjecture. We treat more general cusps, called Tsuchihashi cusps, including Hilbert modular cusps. For this cusp singularity (V, p) we define two invariants, namely, the contribution of the cusp X~ (P) to the arithmetic genus and the signature defect a(p) of the cusp. The signature defect was defined by Hirzebruch [H1], which coincides with ours (see Sect. 1) in the case of Hilbert modular cusps. We can also find other generalizations by Morita [Mo] and Looijenga [L]. The invariant Z~(P) of a Hilbert modular cusp is called the q~-invariant by Ehlers [E]. Satake fSa] defined Z~(P) in a more general context, showing that it coincides with the contribution of the cusp to the dimension formula of the space of cusp forms as calculated by means of the Riemann-Roch Theorem.

Normality of 3-dimensional lattice polytopes

Abstract We construct examples of ample lattice polytope of dimension n ⩾ 3 such that the n − 2 times multiple of them are very ample but not normal. We also obtain two classes of normal 3-dimensional nonsingular lattice polytopes. One is a class of polytopes with relatively small internal polytopes. The other is that of polytopes which are the Minkowski sums with line segments.