Searcharxiv⌕ Search

arXiv subjects

Dmitri Panov

Publications and source records attributed to Dmitri Panov.

At least 19 recordsLinked to original sources
arXiv2026

Euclidean $\vee$-systems and real PK arrangements

We establish a correspondence between two structures arising in the geometry of hyperplane arrangements: Euclidean $\vee$-systems and real polyhedral K\"ahler (PK) arrangements. We prove that every irreducible Euclidean $\vee$-system determines a real PK arrangement, and conversely that every real PK arrangement arises this way. As a consequence, we show that, up to equivalence, there are exactly three irreducible rank-three Euclidean $\vee$-systems whose vectors have equal length; their arrangements are the mirrors of the reflection groups of the regular tetrahedron, cube, and icosahedron. The correspondence also yields a description of the moduli space of Euclidean $\vee$-systems in a fixed projective class: it is homeomorphic to the relative interior of a polytope. We also give a direct proof that the hyperplane arrangement associated with a Euclidean $\vee$-system is simplicial. Among the currently known simplicial line arrangements, we identify precisely those that arise from $\vee$-systems. As a consequence, we prove that the Schreiber--Veselov catalog is complete for irreducible rank-three Euclidean $\vee$-systems with at most $27$ vectors.

math.DG↗
arXiv2025

Polyhedral K\"ahler metrics on $\mathbb{CP}^n$

We give necessary and sufficient conditions for the existence of polyhedral K\"ahler metrics on $\mathbb{CP}^n$ whose singular set is a hyperplane arrangement and whose cone angles are in $(0, 2\pi)$. These conditions take the form of linear and quadratic constraints on the cone angles and are entirely determined by the intersection poset of the arrangement. Our proof of existence relies on a parabolic version of the Kobayashi-Hitchin correspondence, due to T. Mochizuki.

math.DG↗
arXiv2024

A Miyaoka-Yau inequality for hyperplane arrangements in $\mathbb{CP}^n$

Let $\mathcal{H}$ be a hyperplane arrangement in $\mathbb{CP}^n$. We define a quadratic form $Q$ on $\mathbb{R}^{\mathcal{H}}$ that is entirely determined by the intersection poset of $\mathcal{H}$. Using the Bogomolov-Gieseker inequality for parabolic bundles, we show that if $\mathbf{a} \in \mathbb{R}^{\mathcal{H}}$ is such that the weighted arrangement $(\mathcal{H}, \mathbf{a})$ is stable, then $Q(\mathbf{a}) \leq 0$. As an application, we consider the symmetric case where all the weights are equal. The inequality $Q(a, \ldots, a) \leq 0$ gives a lower bound for the total sum of multiplicities of codimension $2$ intersection subspaces of $\mathcal{H}$. The lower bound is attained when every $H \in \mathcal{H}$ intersects all the other members of $\mathcal{H} \setminus \{H\}$ along $(1-2/(n+1))|\mathcal{H}| + 1$ codimension $2$ subspaces; extending from $n=2$ to higher dimensions a condition found by Hirzebruch for line arrangements in the complex projective plane.

math.AG↗
arXiv2024

Simplicial arrangements with few double points

In their solution to the orchard-planting problem, Green and Tao established a structure theorem which proves that in a line arrangement in the real projective plane with few double points, most lines are tangent to the dual curve of a cubic curve. We provide geometric arguments to prove that in the case of a simplicial arrangement, the aforementioned cubic curve cannot be irreducible. It follows that Gr\"{u}nbaum's conjectural asymptotic classification of simplicial arrangements holds under the additional hypothesis of a linear bound on the number of double points.

math.CO↗
arXiv2023

On decorated representation spaces associated to spherical surfaces

We analyse local features of the spaces of representations of the fundamental group of a punctured surface in $\mathrm{SU}_2$ equipped with a decoration, namely a choice of a logarithm of the representation at peripheral loops. Such decorated representations naturally arise as monodromies of spherical surfaces with conical points. Among other things, in this paper we determine the smooth locus of such absolute and relative decorated representation spaces: in particular, in the relative case (with few special exceptions) such smooth locus is dense, connected, and exactly consists of non-coaxial representations. The present study sheds some light on the local structure of the moduli space of spherical surfaces with conical points, which is locally modelled on the above-mentioned decorated representation spaces.

math.DG↗
arXiv2022

Frieze patterns

We discuss here the geometry of frieze patterns, and add a few words about Greek vases, molecular symmetry, and 2D crystallography. The work is written primarily for school students.

math.HO↗
arXiv2022

Dunkl connections on $\mathbb{C}^2$ and spherical metrics

We show that general Dunkl connections on $\mathbb{C}^2$ do not preserve non-zero Hermitian forms. Our proof relies on recent understanding of the non-trivial topology of the moduli space of spherical tori with one conical point.

math.DG↗
arXiv2021

Parabolic bundles and spherical metrics

We use the Kobayashi-Hitchin correspondence for parabolic bundles to reprove the results of Troyanov and Luo-Tian regarding existence and uniqueness of conformal spherical metrics on the Riemann sphere with prescribed cone angles in the interval $(0, 2π)$ at a given configuration of three or more points.

math.DG↗
arXiv2021

Symplectic and Kähler structures on $\mathbb CP^1$-bundles over $\mathbb CP^2$

We show that there exist symplectic structures on a $\mathbb CP^1$-bundle over $\mathbb CP^2$ that do not admit a compatible Kähler structure. These symplectic structures were originally constructed by Tolman and they have a Hamiltonian $\mathbb T^2$-symmetry. Tolman's manifold was shown to be diffeomorphic to a $\mathbb CP^1$-bundle over $\mathbb CP^{2}$ by Goertsches, Konstantis, and Zoller. The proof of our result relies on Mori theory, and on classical facts about holomorphic vector bundles over $\mathbb CP^{2}$.

math.SG↗
arXiv2021

Moduli spaces for Lamé functions and Abelian integrals of the second kind

The space of Lamé functions of order m is isomorphic to the space of pairs (elliptic curve, Abelian differential) where the differential has a single zero of order 2m at the origin and m double poles with vanishing residues. We describe the topology of this space: it is a Riemann surface of finite type; we find the number of components and the genus and Euler characteristic of each component. As an application we find the degrees of Cohn's polynomials confirming a conjecture by Robert Maier. As another application we partially describe the degeneration locus of the space of spherical metrics on tori with one conic singularity where the conic angle is an odd multiple of 2$π$.

math.CV↗
arXiv2021

Polyhedral Kähler cone metrics on $\mathbb{C}^n$ singular at hyperplane arrangements

Let $X$ be a complex manifold and let $g$ be a polyhedral metric on it inducing its topology. We say that $g$ is a polyhedral Kähler (PK) metric on $X$ if it is Kähler outside its singular set. The local geometry of PK metrics is modelled on PK cones, and in this article we focus on an interesting class of examples of these. Following work of Couwenberg-Heckman-Looijenga, we consider a special kind of flat torsion free meromorphic connections on $\mathbb{C}^n$ with simple poles at the hyperplanes of a linear arrangement. In the case of unitary holonomy we show that, under suitable numerical conditions, the metric completion is a PK cone metric on $\mathbb{C}^n$. We apply our results to the essential braid arrangement, extending to higher dimensions the classical story of spherical metrics on $\mathbb{CP}^1$ with three cone points.

math.DG↗
arXiv2020

Moduli of spherical tori with one conical point

In this paper we determine the topology of the moduli space $\mathcal{MS}_{1,1}(\vartheta)$ of surfaces of genus one with a Riemannian metric of constant curvature $1$ and one conical point of angle $2π\vartheta$. In particular, for $\vartheta\in (2m-1,2m+1)$ non-odd, $\mathcal{MS}_{1,1}(\vartheta)$ is connected, has orbifold Euler characteristic $-m^2/12$, and its topology depends on the integer $m>0$ only. For $\vartheta=2m+1$ odd, $\mathcal{MS}_{1,1}(2m+1)$ has $\lceil{m(m+1)/6}\rceil$ connected components. For $\vartheta=2m$ even, $\mathcal{MS}_{1,1}(2m)$ has a natural complex structure and it is biholomorphic to $\mathbb{H}^2/G_m$ for a certain subgroup $G_m$ of $\mathrm{SL}(2,\mathbb{Z})$ of index $m^2$, which is non-normal for $m>1$.

math.DG↗
arXiv2019

Spherical surfaces with conical points: systole inequality and moduli spaces with many connected components

In this article we address a number of features of the moduli space of spherical metrics on connected, compact, orientable surfaces with conical singularities of assigned angles, such as its non-emptiness and connectedness. We also consider some features of the forgetful map from the above moduli space of spherical surfaces with conical points to the associated moduli space of pointed Riemann surfaces, such as its properness, which follows from an explicit systole inequality that relates metric invariants (spherical systole) and conformal invariant (extremal systole).

math.DG↗
arXiv2019

Symplectic domination

Let M be a compact oriented even-dimensional manifold. This note constructs a compact symplectic manifold S of the same dimension and a map f from S to M of strictly positive degree. The construction relies on two deep results: the first is a theorem of Ontaneda that gives a Riemannian manifold N of tightly pinched negative curvature which admits a map to M of degree equal to one; the second is a result of Donaldson on the existence of symplectic divisors. Given Ontaneda's negatively curved manifold N, the twistor space Z is symplectic. The manifold S is then a suitable multisection of the twistor space, found via Donaldson's theorem.

math.SG↗
arXiv2018

$S^{1}$-invariant symplectic hypersurfaces in dimension $6$ and the Fano condition

We prove that any symplectic Fano $6$-manifold $M$ with a Hamiltonian $S^1$-action is simply connected and satisfies $c_1 c_2(M)=24$. This is done by showing that the fixed submanifold $M_{\min}\subseteq M$ on which the Hamiltonian attains its minimum is diffeomorphic to either a del Pezzo surface, a $2$-sphere or a point. In the case when $\dim(M_{\min})=4$, we use the fact that symplectic Fano $4$-manifolds are symplectomorphic to del Pezzo surfaces. The case when $\dim(M_{\min})=2$ involves a study of $6$-dimensional Hamiltonian $S^1$-manifolds with $M_{\min}$ diffeomorphic to a surface of positive genus. By exploiting an analogy with the algebro-geometric situation we construct in each such $6$-manifold an $S^1$-invariant symplectic hypersurface ${\cal F}(M)$ playing the role of a smooth fibre of a hypothetical Mori fibration over $M_{\min}$. This relies upon applying Seiberg-Witten theory to the resolution of symplectic $4$-orbifolds occurring as the reduced spaces of $M$.

math.SG↗
arXiv2016

A gauge theoretic approach to Einstein 4-manifolds

This article investigates a new gauge theoretic approach to Einstein's equations in dimension 4. Whilst aspects of the formalism are already explained in various places in the mathematics and physics literature, our first goal is to give a single coherent account of the theory in purely mathematical language. We then explain why the new approach may have important mathematical applications: the possibility of using the calculus of variations to find Einstein 4-manifolds, as well as links to symplectic topology. We also carry out some of the technical groundwork to attack these problems.

math.DG↗
arXiv2016

Real line arrangements with Hirzebruch property

A line arrangement of $3n$ lines in $\mathbb CP^2$ satisfies Hirzebruch property if each line intersect others in $n+1$ points. Hirzebruch asked if all such arrangements are related to finite complex reflection groups. We give a positive answer to this question in the case when the line arrangement in $\mathbb CP^2$ is real, confirming that there exist exactly four such arrangements.

math.AG↗