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Semyon Alesker

Publications and source records attributed to Semyon Alesker.

At least 19 recordsLinked to original sources
arXiv2026

Towards Hodge-Riemann relations for non-Archimedean analogs of valuations on convex sets

In [8], a non-Archimedean analogue of the space of translation-invariant even valuations on convex sets was introduced. In [7], motivated by a further analogy with the classical theory, this space was equipped with two multiplicative structures, the product and the convolution. Both structures satisfy Poincare duality and the (non-mixed) hard Lefschetz theorem. In this paper, we formulate a conjecture concerning a more general mixed versions of the hard Lefschetz theorem and the Hodge-Riemann relations. We prove the non-mixed Hodge-Riemann relations in degree 1 for the product and, equivalently, in codegree 1 for the convolution.

math.MG↗
arXiv2024

Octonionic Calabi-Yau theorem

A new class of Riemannian metrics, called octonionic Kähler, is introduced and studied on a certain class of 16-dimensional manifolds. It is an octonionic analogue of Kähler metrics on complex manifolds and of HKT-metrics of hypercomplex manifolds. Then for this class of metrics an octonionic version of the Monge-Ampère equation is introduced and solved under appropriate assumptions. The latter result is an octonionic version of the Calabi-Yau theorem from Kähler geometry.

math.DG↗
arXiv2024

Non-Archimedean analogue of the space of valuations on convex sets

In the last two decades a number of structures on the classical space of translation invariant valuations on convex bodies were discovered, e.g. product, convolution, a Fourier type transform. In this paper a non-Archimedean analogue of the space of such (even) valuations with similar structures is constructed. It is shown that, like in the classical case, the new space equipped with either product or convolution satisfies Poincaré duality and hard Lefschetz theorem.

math.DG↗
arXiv2022

New invariants of Gromov-Hausdorff limits of Riemannian surfaces with curvature bounded below

Let $\{X_i\}$ be a sequence of compact $n$-dimensional Alexandrov spaces (e.g. Riemannian manifolds) with curvature uniformly bounded below which converges in the Gromov-Hausdorff sense to a compact Alexandrov space $X$. In an earlier paper by the first author there was described (without a proof) a construction of an integer valued function on $X$; this function carries additional geometric information on the sequence such as the limit of intrinsic volumes of $X_i$'s. In this paper we consider sequences of closed 2-surfaces and (1) prove the existence of such a function in this situation; and (2) classify the functions which may arise from the construction.

math.DG↗
arXiv2021

On convergence of intrinsic volumes of Riemannian manifolds

In 1939 H. Weyl has introduced the so called intrinsic volumes $V_i(M^n), i=0,\dots,n$, (known also as Lipschitz-Killing curvatures) for any closed smooth Riemannian manifold $M^n$. Given a Riemmanian submersion of compact smooth Riemannian manifolds $M\to B$, $B$ is connected. For $\varepsilon >0$ let us define a new Riemannian metric on $M$ by multiplying the original one by $\varepsilon$ along the vertical directions and keeping it the same along the (orthogonal) horizontal directions. Denote the corresponding Riemannian manifold by $M_\varepsilon$. The main result says that $\lim_{\varepsilon\to +0} V_i(M_\varepsilon)=χ(Z) V_i(B)$, where $χ(Z)$ is the Euler characteristic of a fiber of the submersion. This result is consistent with more general open conjectures on convergence of intrinsic volumes formulated previously by the author.

math.DG↗
arXiv2021

The multiplicative structure on polynomial continuous valuations

We introduce a canonical structure of a commutative associative filtered algebra with the unit on polynomial smooth valuations, and study its properties. The induced structure on the subalgebra of translation invariant smooth valuations has especially nice properties (it is the structure of the Frobenius algebra). We also present some applications.

math.MG↗
arXiv2020

Kotrbaty's theorem on valuations and geometric inequalities for convex bodies

Very recently J. Kotrbaty has proven general inequalities for translation invariant smooth valuations formally analogous to the Hodge- Riemann bilinear relations in the Kahler geometry. The goal of this note is to apply Kotrbaty's theorem to obtain a few apparently new inequalities for mixed volumes of convex bodies.

math.MG↗
arXiv2017

Some conjectures on intrinsic volumes of Riemannian manifolds and Alexandrov spaces

For any closed smooth Riemannian manifold H. Weyl has defined a sequence of numbers called today intrinsic volumes. They include volume, Euler characteristic, and integral of the scalar curvature. We conjecture that absolute values of all intrinsic volumes are bounded by a constant depending only on the dimension of the manifold, upper bound on its diameter, and lower bound on the sectional curvature. Furthermore we conjecture that intrinsic volumes can be defined for some (so called smoothable) Alexandrov spaces with curvature bounded below and state few of the expected properties of them, particularly the behavior under the Gromov-Hausdorff limits. We suggest conjectural compactifications of the space of smooth closed connected Riemannian manifolds with given upper bounds on dimension and diameter and a lower bound on sectional curvature to which the intrinsic volumes extend by continuity. We discuss also known cases of some of these conjectures.

math.DG↗
arXiv2017

Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry

A hypercomplex manifold is a manifold equipped with a triple of complex structures $I, J, K$ satisfying the quaternionic relations. We define a quaternionic analogue of plurisubharmonic functions on hypercomplex manifolds, and interpret these functions geometrically as potentials of HKT (hyperkähler with torsion) metrics, and prove a quaternionic analogue of A.D. Aleksandrov and Chern-Levine-Nirenberg theorems.

math.CV↗
arXiv2017

Valuations on convex functions and convex sets and Monge-Ampere operators

The notion of a valuation on convex bodies is very classical. The notion of a valuation on a class of functions was recently introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translations invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite dimensional kernel. The proof uses the author's irreducibility theorem and few properties of the real Monge-Ampere operators due to A.D. Alexandrov and Z. Blocki. Fur- thermore we show how to use complex, quaternionic, and octonionic Monge-Ampere operators to construct more examples of continuous valuations on convex functions in an analogous way.

math.MG↗
arXiv2016

On a uniform estimate for the quaternionic Calabi problem

We establish a C^0 a priori bound on the solutions of the quaternionic Calabi-Yau equation (of Monge-Ampere type) on compact HKT manifolds with a locally flat hypercomplex structure. As an intermediate step, we prove a quaternionic version of the Gauduchon theorem.

math.CV↗
arXiv2016

Plurisubharmonic functions on the octonionic plane and Spin(9)-invariant valuations on convex sets

A new class of plurisubharmonic functions on the octonionic plane O^2= R^{16} is introduced. An octonionic version of theorems of A.D. Aleksandrov and Chern- Levine-Nirenberg, and Blocki are proved. These results are used to construct new examples of continuous translation invariant valuations on convex subsets of O^2=R^{16}. In particular a new example of Spin(9)-invariant valuation on R^{16} is given.

math.MG↗
arXiv2016

Solvability of the quaternionic Monge-Ampere equation on compact manifolds with a flat hyperKaehler metric

A quaternionic version of the Calabi problem was recently formulated by M. Verbitsky and the author. It conjectures a solvability of a quaternionic Monge-Ampere equation on a compact HKT manifold (HKT stays for HyperKaehler with Torsion). In this paper this problem is solved under an extra assumption that the manifold admits a flat hyperKaehler metric compactible with the underlying hypercomplex structure. The proof uses the continuity method and a priori estimates.

math.CV↗
arXiv2016

Quaternionic plurisubharmonic functions and their applications to convexity

The goal of this article is to present a survey of the recent theory of plurisubharmonic functions of quaternionic variables, and its applications to theory of valuations on convex sets and HKT-geometry (HyperKähler with Torsion). The exposition follows the articles math.CV/0104209, math.CV/0208005, math.MG/0401219 by the author and math.CV/0510140 by M. Verbitsky and the author.

math.MG↗
arXiv2016

Quaternionic Monge-Ampere equations

The main result of this paper is the existence and uniqueness of solution of the Dirichlet problem for quaternionic Monge-Ampere equations in quaternionic strictly pseudoconvex bounded domains in H^n. We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2].

math.CV↗
arXiv2016

Valuations on convex sets, non-commutative determinants, and pluripotential theory

A new method of constructing translation invariant continuous valuations on convex subsets of the quaternionic space $\HH^n$ is presented. In particular new examples of $Sp(n)Sp(1)$-invariant translation invariant continuous valuations are constructed. This method is based on the theory of plurisubharmonic functions of quaternionic variables.

math.MG↗