arXiv subjects
Publications and source records attributed to Vasily Bolbachan.
Let $\mathbb K$ be a field of characteristic zero. We prove that its motivic cohomology in degree $m-1$ and weight $m$ is rationally isomorphic to the cohomology of the polylogarithmic complex. This gives a partial extension of A. Suslin theorem describing the indecomposable $K_3$ of a field.
Chow polylogarithms are some special functions arising in explicit description of the Beilinson regulator map. The most interesting functional equation for this function reflects its vanishing on the boundary in the Bloch's cycle complex. We show that this functional equation formally follows from more simple ones, namely skew-symmetry, functoriality and multiplicativity. To prove this, we study some analogue of Bloch's cycle complex and establish for this complex an analogue Beilinson-Soule vanishing conjecture. A. Goncharov defined a group of functional equations for classical polylogarithms. We show that any such functional equation formally follows from functional equations for Chow polylogarithms stated above.
We construct some analog of cubical Bloch's higher Chow groups. Instead of considering cycles in $X\times\mathbb A^n$ we consider varieties $Y$ over $X$ together with a distinguished element in the $n$-th exterior power of the multiplicative group of the field of fraction on $Y$. This definition allows us to make sense of a cycle in $X\times\mathbb A^n$ intersecting faces improperly as an element in this complex. We prove that this complex is well-defined and study its basic properties: flat pullback, the localization sequence etc. As an application we prove that the cohomology of this complex in degrees $m-1, m$ and weight $m$ isomorphic to the cohomology of polylogarithmic complex.
We prove a conjecture of A. Goncharov concerning strong Suslin reciprocity law. The main idea of the proof is the construction of the norm map on so-called lifted reciprocity maps. This construction is similar to the construction of the norm map on Milnor $K$-theory. As an application, we express Chow dilogarithm in terms of Bloch-Wigner dilogarithm. Also, we obtain a new reciprocity law for four rational functions on an arbitrary algebraic surface with values in the pre-Bloch group.
Let $E$ be an elliptic curve over an algebraically closed field of characteristic 0. We prove that the pre-Bloch group of the function field of $E$ can be generated by the functions of degree not higher than 3. We apply this result to the elliptic dilogarithm function defined by S. Bloch. He has shown that any element of the pre-Bloch group gives a (so-called elliptic Bloch) relation between the values of the so-called Elliptic dilogarithm. We conclude that any elliptic Bloch relation can be reduced to the antisymmetry relation and the elliptic Bloch relations for the functions of degree 3.
To any cubic surface, one can associate a cubic threefold given by a triple cover of $\mathbb P^3$ branched in this cubic surface. D. Allcock, J. Carlson, and D. Toledo used this construction to define the period map for cubic surfaces. It is interesting to calculate this map for some specific cubic surfaces. In this paper, we have calculated it in the case when the cubic surface is given by a triple covering of $\mathbb P^2$ branched in a smooth elliptic curve. In this case, the periods can be expressed through periods of the corresponding elliptic curve.
D. Rudenko proved the homotopy invariance of the truncated polylogarithmic complexes. It follows that on these complexes there is the norm map with good proprieties. We apply his result and get the explicit formula for the norm map in the case of quadratic extensions.
We obtain two sequences of rational numbers which converge to the Euler-Gompertz constant. Denote by the integral of f(x)e^{-x} from 0 to infinity. Recall that the Euler-Gompertz constant δis . Main idea. Let P_n(x) be a polynomial with integer coefficients. It is easy to prove that =a_n+ b_n$ for some integers a_n, b_n. Hence if /b_n converges to zero, a_n/b_n converges to - δ. Main Theorem. Let u be positive real. There exists polynomials P_n(x)(they are explicitly given in the paper) such that tends to u as n tends to infinity. Proof of Main Theorem is elementary.