Viacheslav V. Nikulin

Igusa modular forms and 'the simplest' Lorentzian Kac-Moody algebras

Automorphic corrections for the?Lorentzian Kac-Moody algebras with the?simplest generalized Cartan matrices of rank?3, ? ?????and????are found. For this correction, which is a?generalized Kac-Moody Lie super algebra, is delivered by , the Igusa -modular form of weight 35, while for it is given by some Siegel modular form of weight?30 with respect to a 2-congruence subgroup of . Expansions of and in infinite products are obtained and the?multiplicities of all the?roots of the corresponding generalized Lorentzian Kac-Moody superalgebras are calculated. These multiplicities are determined by the Fourier coefficients of certain Jacobi forms of weight?0 and index?1. The?method adopted for constructing and leads in a?natural way to an?explicit construction (as infinite products or sums) of Siegel modular forms whose divisors are Humbert surfaces with fixed discriminants. A geometric construction of these forms was proposed by van der Geer in 1982. To show the?prospects for further studies, the?list of all hyperbolic symmetric generalized Cartan matrices with the?following properties is presented: is a?matrix of rank?3 and of elliptic or parabolic type, has a?lattice Weyl vector, and contains a?parabolic submatrix .

AUTOMORPHIC FORMS AND LORENTZIAN KAC–MOODY ALGEBRAS PART I

Using the general method which was applied to prove finiteness of the set of hyperbolic generalized Cartan matrices of elliptic and parabolic type, we classify all symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices of elliptic type and of rank 3 with a lattice Weyl vector. We develop the general theory of reflective lattices T with 2 negative squares and reflective automorphic forms on homogeneous domains of type IV defined by T. We consider this theory as mirror symmetric to the theory of elliptic and parabolic hyperbolic reflection groups and corresponding hyperbolic root systems. We formulate Arithmetic Mirror Symmetry Conjecture relating both these theories and prove some statements to support this Conjecture. This subject is connected with automorphic correction of Lorentzian Kac--Moody algebras. We define Lie reflective automorphic forms which are the most beautiful automorphic forms defining automorphic Lorentzian Kac--Moody algebras, and we formulate finiteness Conjecture for these forms. Detailed study of automorphic correction and Lie reflective automorphic forms for generalized Cartan matrices mentioned above will be given in Part II.

The Arithmetic Mirror Symmetry and¶Calabi–Yau Manifolds

Abstract:We extend our variant of mirror symmetry for K3 surfaces [GN3] and clarify its relation with mirror symmetry for Calabi–Yau manifolds. We introduce two classes (for the models A and B) of Calabi–Yau manifolds fibrated by K3 surfaces with some special Picard lattices. These two classes are related with automorphic forms on IV type domains which we studied in our papers [GN1]–[GN6]. Conjecturally these automorphic forms take part in the quantum intersection pairing for model A, Yukawa coupling for model B and mirror symmetry between these two classes of Calabi–Yau manifolds. Recently there were several papers by physicists where it was shown on some examples. We propose a problem of classification of introduced Calabi–Yau manifolds. Our papers [GN1]–[GN6] and [N3]–[N14] give hope that this is possible. They describe possible Picard or transcendental lattices of general K3 fibers of the Calabi–Yau manifolds.

Reflection groups in Lobachevskii spaces and the denominator identity for Lorentzian Kac-Moody algebras

We develop a theory of Lorentzian Kac-Moody algebras based on the theory of reflection groups in Lobachevskii spaces and on recent results of R. Borcherds.

On classification of Lorentzian Kac-Moody algebras

The general theory of Lorentzian Kac-Moody algebras is considered. This theory must serve as a hyperbolic analogue of the classical theories of finite-dimensional semisimple Lie algebras and affine Kac-Moody algebras. The first examples of Lorentzian Kac-Moody algebras were found by Borcherds. Here general finiteness results for the set of Lorentzian Kac-Moody algebras of rank ≥3 are considered along with the classification problem for these algebras. As an example, a classification is given for Lorentzian Kac-Moody algebras of rank 3 with hyperbolic root lattice , symmetry lattice , and symmetry group , , where and are given by and is trivial on , is the extended paramodular group. This is perhaps the first example in which a large class of Lorentzian Kac-Moody algebras has been classified.

Finiteness of the number of arithmetic groups generated by reflections in Lobachevsky spaces

After results of the author (1980, 1981) and Vinberg (1981), the finiteness of the number of maximal arithmetic groups generated by reflections in Lobachevsky spaces remained unknown in dimensions only. It was proved recently (2005) in dimension 2 by Long, Maclachlan and Reid and in dimension 3 by Agol. Here we use the results in dimensions 2 and 3 to prove the finiteness in all remaining dimensions . The methods of the author (1980, 1981) are more than sufficient for this using a very short and very simple argument.

Weil Linear Systems on Singular K3 Surfaces

We recall that K3 surface is a smooth projective algebraic surface X over an algebraically closed field k with K X =0 and H 1(X, 0 X )=0.

On ground fields of arithmetic hyperbolic reflection groups. III

The paper continues from the work of Nikulin. Using our methods of 1980 and 1981, we define some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimensions at least 3, and we give good upper bounds for their degrees (over Q). This extends the earlier results of Nikulin for dimensions at least 4. This finally delivers a possibility, in principle, of effective finite classification of maximal arithmetic hyperbolic reflection groups (more generally, of reflective hyperbolic lattices) in all dimensions. Our results also give another proof of finiteness in dimension 3. In fact, using our methods, we show that finiteness in dimension 3 follows from finiteness in dimension 2.

Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups. II

Using the results of [1]-[3] on Kahlerian K3 surfaces and Niemeier lattices, we classify degenerations of Kahlerian K3 surfaces with finite symplectic automorphism groups with emphasis on degenerations of codimension 1.

Self-correspondences of K3 Surfaces via Moduli of Sheaves and Arithmetic Hyperbolic Reflection Groups

In a series of our papers with Carlo Madonna (2002–2008), we described self-correspondences of a K3 surface over ℂ via moduli of sheaves with primitive isotropic Mukai vectors for the Picard number 1 or 2 of the K3 surfaces. Here we give a natural and functorial answer to the same problem for an arbitrary Picard number. As an application, we characterize, in terms of self-correspondences via moduli of sheaves, K3 surfaces with reflective Picard lattice, that is, when the automorphism group of the lattice is generated by reflections up to finite index. It is known since 1981 that the number of reflective hyperbolic lattices is finite. We also formulate some natural unsolved related problems.