Vol 302, No 1 (2018)

We apply the general construction developed in our previous papers to the first nontrivial case of Gr^TP(2, 4). In particular, we construct finite-gap real quasi-periodic solutions of the KP-II equation in the form of a soliton lattice corresponding to a smooth M-curve of genus 4 which is a desingularization of a reducible rational M-curve for soliton data in Gr^TP(2, 4).

The Hirzebruch functional equation is \(\sum\nolimits_{i = 1}^n {\prod\nolimits_{j \ne i} {(1/f({z_j} - {z_i})) = c} } \) with constant c and initial conditions f(0) = 0 and f'(0) = 1. In this paper we find all solutions of the Hirzebruch functional equation for n ≤ 6 in the class of meromorphic functions and in the class of series. Previously, such results have been known only for n ≤ 4. The Todd function is the function determining the two-parameter Todd genus (i.e., the χ_a,b-genus). It gives a solution to the Hirzebruch functional equation for any n. The elliptic function of level N is the function determining the elliptic genus of level N. It gives a solution to the Hirzebruch functional equation for n divisible by N. A series corresponding to a meromorphic function f with parameters in U ⊂ ℂ^k is a series with parameters in the Zariski closure of U in ℂ^k, such that for the parameters in U it coincides with the series expansion at zero of f. The main results are as follows: (1) Any series solution of the Hirzebruch functional equation for n = 5 corresponds either to the Todd function or to the elliptic function of level 5. (2) Any series solution of the Hirzebruch functional equation for n = 6 corresponds either to the Todd function or to the elliptic function of level 2, 3, or 6. This gives a complete classification of complex genera that are fiber multiplicative with respect to ℂPⁿ⁻¹ for n ≤ 6. A topological application of this study is an effective calculation of the coefficients of elliptic genera of level N for N = 2,..., 6 in terms of solutions of a differential equation with parameters in an irreducible algebraic variety in ℂ⁴.

We extend the category of (super)manifolds and their smooth mappings by introducing a notion of microformal, or “thick,” morphisms. They are formal canonical relations of a special form, constructed with the help of formal power expansions in cotangent directions. The result is a formal category so that its composition law is also specified by a formal power series. A microformal morphism acts on functions by an operation of pullback, which is in general a nonlinear transformation. More precisely, it is a formal mapping of formal manifolds of even functions (bosonic fields), which has the property that its derivative for every function is a ring homomorphism. This suggests an abstract notion of a “nonlinear algebra homomorphism” and the corresponding extension of the classical “algebraic–functional” duality. There is a parallel fermionic version. The obtained formalism provides a general construction of L_∞-morphisms for functions on homotopy Poisson (P_∞) or homotopy Schouten (S_∞) manifolds as pullbacks by Poisson microformal morphisms. We also show that the notion of the adjoint can be generalized to nonlinear operators as a microformal morphism. By applying this to L_∞-algebroids, we show that an L_∞-morphism of L_∞-algebroids induces an L_∞-morphism of the “homotopy Lie–Poisson” brackets for functions on the dual vector bundles. We apply this construction to higher Koszul brackets on differential forms and to triangular L_∞-bialgebroids. We also develop a quantum version (for the bosonic case), whose relation to the classical version is like that of the Schrödinger equation to the Hamilton–Jacobi equation. We show that the nonlinear pullbacks by microformal morphisms are the limits as ħ → 0 of certain “quantum pullbacks,” which are defined as special form Fourier integral operators.

Earlier (2000) the authors introduced the notion of the integral with respect to the Euler characteristic over the space of germs of functions on a variety and over its projectivization. This notion allowed the authors to rewrite known definitions and statements in new terms and also turned out to be an effective tool for computing the Poincar´e series of multi-index filtrations in some situations. However, the “classical” (initial) notion can be applied only to multi-index filtrations defined by so-called finitely determined valuations (or order functions). Here we introduce a modified version of the notion of the integral with respect to the Euler characteristic over the projectivization of the space of function germs. This version can be applied in a number of settings where the “classical approach” does not work. We give examples of the application of this concept to definitions and computations of the Poincar´e series (including equivariant ones) of collections of plane valuations which contain valuations not centred at the origin.

It is shown that, for the discretization of complex analysis introduced earlier by S. P. Novikov and the present author, there exists a rich family of bounded discrete holomorphic functions on the hyperbolic (Lobachevsky) plane endowed with a triangulation by regular triangles whose vertices have even valence. Namely, it is shown that every discrete holomorphic function defined in a bounded convex domain can be extended to a bounded discrete holomorphic function on the whole hyperbolic plane so that the Dirichlet energy be finite.

A theory and corresponding algorithms are developed for fast and exact calculation of the L_∞-locality (i.e., the greatest cube-to-linear ratio in the maximum metric) for polyfractal three-dimensional Peano curves.

V. V. Batyrev constructed a family of Calabi–Yau hypersurfaces dual to the first Chern class in toric Fano varieties. Using this construction, we introduce a family of Calabi–Yau manifolds whose SU-bordism classes generate the special unitary bordism ring \({\Omega ^{SU}}[\frac{1}{2}] \cong Z[\frac{1}{2}][{y_i}:i \geqslant 2]\). We also describe explicit Calabi–Yau representatives for multiplicative generators of the SU-bordism ring in low dimensions.

The concept of polynomial Lie algebra of finite rank was introduced by V. M. Buchstaber in his studies of new relationships between hyperelliptic functions and the theory of integrable systems. In this paper we prove the following theorem: the Lie subalgebra generated by the frame of a polynomial Lie algebra of finite rank has at most polynomial growth. In addition, important examples of polynomial Lie algebras of countable rank are considered in the paper. Such Lie algebras arise in the study of certain hyperbolic partial differential equations, as well as in the construction of self-similar infinite-dimensional Lie algebras (such as the Fibonacci algebra).

We consider non-diagonalizable hydrodynamic-type systems integrable by the extended hodograph method. We restrict the analysis to non-diagonalizable hydrodynamic reductions of the three-dimensionalMikhalev equation. We show that families of these hydrodynamictype systems are reducible to the heat hierarchy. Then we construct new particular explicit solutions for the Mikhalev equation.

A smooth stably complex manifold is called a totally tangentially/normally split manifold (TTS/TNS manifold for short) if the respective complex tangential/normal vector bundle is stably isomorphic to a Whitney sum of complex line bundles, respectively. In this paper we construct manifolds M such that any complex vector bundle over M is stably equivalent to a Whitney sum of complex line bundles. A quasitoric manifold shares this property if and only if it is a TNS manifold. We establish a new criterion for a quasitoric manifold M to be TNS via non-semidefiniteness of certain higher degree forms in the respective cohomology ring of M. In the family of quasitoric manifolds, this generalizes the theorem of J. Lannes about the signature of a simply connected stably complex TNS 4-manifold. We apply our criterion to show the flag property of the moment polytope for a nonsingular toric projective TNS manifold of complex dimension 3.