The Modular Form of the Barth-Nieto Quintic

Barth and Nieto have found a remarkable quintic threefold which parametrizes Heisenberg invariant Kummer surfaces which belong to abelian surfaces with a (1,3)-polarization and a lecel 2 structure. A double cover of this quintic, which is also a Calabi-Yau variety, is birationally equivalent to the moduli space {\cal A}_3(2) of abelian surfaces with a (1,3)-polarization and a level 2 structure. As a consequence the corresponding paramodular group \Gamma_3(2) has a unique cusp form of weight 3. In this paper we find this cusp form which is \Delta_1^3. The form \Delta_1 is a remarkable weight 1 cusp form with a character with respect to the paramodular group \Gamma_3. It has several interesting properties. One is that it admits an infinite product representation, the other is that it vanishes of order 1 along the diagonal in Siegel space. In fact \Delta_1 is an element of a short series of modular forms with this last property. Using the fact that \Delta_1 is a weight 3 cusp form with respect to the group \Gamma_3(2) we give an independent construction of a smooth projective Calabi-Yau model of the moduli space {\cal A}_3(2).Comment: 20 pages, Latex2e RIMS Preprint 120

Moduli spaces of polarised symplectic O'Grady varieties and Borcherds products

We study moduli spaces of O'Grady's ten-dimensional irreducible symplectic manifolds. These moduli spaces are covers of modular varieties of dimension 21, namely quotients of hermitian symmetric domains by a suitable arithmetic group. The interesting and new aspect of this case is that the group in question is strictly bigger than the stable orthogonal group. This makes it different from both the K3 and the K3^[n] case, which are of dimension 19 and 20 respectively

Abelianisation of orthogonal groups and the fundamental group of modular varieties

We study the commutator subgroup of integral orthogonal groups belonging to indefinite quadratic forms. We show that the index of this commutator is 2 for many groups that occur in the construction of moduli spaces in algebraic geometry, in particular the moduli of K3 surfaces. We give applications to modular forms and to computing the fundamental groups of some moduli spaces

Moduli spaces of irreducible symplectic manifolds

We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of 2d polarised (split type) symplectic manifolds which are deformation equivalent to degree 2 Hilbert schemes of a K3 surface is of general type if d is at least 12.Comment: Exposition improved, Reference to work of Debarre and Voisin added, Corollary 1.6 remove

Experimental Studies of Cylinder Group State During Motoring

AbstractThe paper describes a method for diagnosing the cylinder-piston group of internal combustion engines (ICE). The method is based on analyzing the results of measuring the dynamic compression during crankshaft motoring. The position of the maximum pressure in the combustion chamber, its magnitude and phase parameters give a complete picture of the technical state of the engine cylinder group. Measured parameters of the pressure signal were compared to instrumental measurements of air leakage in the cylinder-piston group as well as to the compression values. The developed method allows to combine all the positive aspects of existing and proposed methods and to solve the long-standing problem of improving the accuracy of estimating the technical state of ICE units and mechanisms. cylinders, rings, pistons, valves, displacement of phases timing. Experimental results are presented as graphs showing the inter-dependency of parameters of the processes under consideration