Maschke's Calabi-Yau three-fold is the double cover of projective three space branched along Maschke's octic surface. This surface is defined by the lowest degree invariant of a certain finite group acting on a four-dimensional (4D) vector space. Using this group, we show that the middle Betti cohomology group of the three-fold decomposes into the direct sum of 150 2D Hodge substructures. We exhibit 1D families of rational curves on the three-fold and verify that the associated Abel-Jacobi map is non-trivial. By counting the number of points over finite fields, we determine the rank of the Néron-Severi group of Maschke's surface and the Galois representation on the transcendental lattice of some of its quotients.We also formulate precise conjectures on the modularity of the Galois representations associated to Maschke's three-fold (these have now been proven by M. Schütt) and to a genus 33 curve, which parametrizes rational curves in the three-fold.The Hodge structure on the middle dimensional Betti cohomology group of a Calabi-Yau (CY) three-fold carries important information on the moduli and the one-dimensional (1D) algebraic cycles on the three-fold. However, if the three-fold is easy to define, say by one equation in a (weighted) projective space, the dimension h 3 of this vector space tends to be large. For example, a smooth quintic three-fold in P 4 has h 3 = 204 and a double octic, i.e., a double cover of P 3 branched along a smooth surface of degree 8, has h 3 = 300. Using automorphisms of the three-folds, one can decompose the cohomology into subrepresentations, which give rise to Hodge substructures. In this paper, we consider a double octic with a particularly large automorphism group G of order 16 ·
We consider Bekenstein-Hawking entropy and attractors in extremal BPS black holes of $$ \mathcal{N} $$ N = 2, D = 4 ungauged supergravity obtained as reduction of minimal, matter-coupled D = 5 supergravity. They are generally expressed in terms of solutions to an inhomogeneous system of coupled quadratic equations, named BPS system, depending on the cubic prepotential as well as on the electric-magnetic fluxes in the extremal black hole background. Focussing on homogeneous non-symmetric scalar manifolds (whose classification is known in terms of L(q, P, Ṗ) models), under certain assumptions on the Clifford matrices pertaining to the related cubic prepotential, we formulate and prove an invertibility condition for the gradient map of the corresponding cubic form (to have a birational inverse map which is given by homogeneous polynomials of degree four), and therefore for the solutions to the BPS system to be explicitly determined, in turn providing novel, explicit expressions for the BPS black hole entropy and the related attractors as solution of the BPS attractor equations. After a general treatment, we present a number of explicit examples with Ṗ = 0, such as L(q, P), 1 ⩽ q ⩽ 3 and P ⩾ 1, or L(q, 1), 4 ⩽ q ⩽ 9, and one model with Ṗ = 1, namely L(4, 1, 1). We also briefly comment on Kleinian signatures and split algebras. In particular, we provide, for the first time, the explicit form of the BPS black hole entropy and of the related BPS attractors for the infinite class of L(1, P) P ⩾ 2 non-symmetric models of $$ \mathcal{N} $$ N = 2, D = 4 supergravity.
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