Some Siegel Threefolds with a Calabi-Yau Model II

Abstract

2011It also contains Z andH := H/Z is a group of order 12 288 = 2 12 · 3 which acts faithfully on X and which leaves the holomorphic 3-form invariant. In [FS2] has been proved:Theorem. Let G be any subgroup ofH. Then there exists a desingularization (in the category of complex spaces) of X /G that is a weak Calabi-Yau threefold.

Calabi‐Yau Threefolds with Small Hodge Numbers

“…34 The two semi-direct products Z 8 Z 4 (1) and Z 8 Z 4 (2) correspond to the presentations a, b | a 8 = b 4 = e, bab −1 = a 3 and a, b | a 8 = b 4 = e, bab −1 = a 5 . Continued on the following page (0, 4) (2, 2) P 7 [2 2 2 2] /G , |G| divides 64 [14,20,44] (4, 2) (2, 0) P 7 [2 2 2 2] /G, |G| = 16 [29]…”

Siegel modular forms of genus 2 with the simplest divisor

“…We note that Δ 1 ∈ S 1 (Γ 3 , χ 6 ) and the Siegel modular variety ker χ 3 6 \ H 2 is the moduli space of (1, 3)-polarized abelian surfaces with a level 2 structure. In the recent paper [14], Freitag and Salvati Manni found a finite series of Siegel modular varieties of the principle polarized abelian surfaces with different two level structures such that the cusp form ∇ 3 defines a canonical differential form without zeros on their smooth compact models.…”

On isomorphisms between Siegel modular threefolds

“…Roughly speaking, the map ϕ sends "B" to "2B". Set G := Γ 0 (2)/Γ 2 2 (2, 4) ∼ = Γ 0 0 (2)/Γ 2 (2,4). If S is a symmetric 2×2 matrix with integer coefficients, then from [4, section 2] we know that Γ 0 (2) is generated by matrices of the form t γ 2S , γ ′ = A 0 0 t A −1 and γ S .…”

“…If S is a symmetric 2×2 matrix with integer coefficients, then from [4, section 2] we know that Γ 0 (2) is generated by matrices of the form t γ 2S , γ ′ = A 0 0 t A −1 and γ S . The classes of these matrices are then generators for the group Γ 0 (2)/Γ 2 2 (2, 4) and their images under ϕ are generators for the group Γ 0 0 (2)/Γ 2 (2,4). Omitting the weight and the multiplier in the notation of (10), by an easy computation it follows that f a…”

“…A first consequence of these relations is that κ(γ) 2 = κ(γ) 2 . Hence the square of v Θ is a non-trivial multiplier system on Γ g (2,4). Moreover, it also follows that the vector space of modular forms spanned by {Θ[a] 2 } a∈F g 2 coincides with the one spanned by The space of such functions will be denoted by [Γ, ρ].…”