Modulo Automorphisms Research Articles

Moduli spaces of vector bundles with fixed determinant over a real curve

Let $$(\Sigma ,\tau )$$ denote a Riemann surface of genus $$g \ge 2$$ equipped with an anti-holomorphic involution $$\tau $$. In this paper we study the topology of the moduli space $$M(r,\xi )^\tau $$ of stable Real vector bundles over $$(\Sigma ,\tau )$$ of rank r and fixed determinant $$\xi $$ of degree coprime to r. We prove that $$M(r,\xi )^{\tau }$$ is an orientable and monotone Lagrangian submanifold of the complex moduli space $$M(r,\xi )$$ so it determines an object in the appropriate Fukaya category. We derive recursive formulas for the $${\mathbb {Z}}_2$$-Betti numbers of $$M(r,\xi )^\tau $$ and compute $${\mathbb {Z}}_p$$-Betti numbers for odd p through a range of degrees. We deduce that if r is even and $$ g>>0$$, then $$M(r,\xi )^{\tau }$$ and $$M(r,\xi ')^{\tau }$$ have non-isomorphic cohomology groups unless $$\xi $$ and $$\xi '$$ have equivalent Stieffel–Whitney classes modulo automorphisms of $$(\Sigma ,\tau )$$. If r is even, and $$g>>0$$ is even, we prove that the Betti numbers of $$M(r,\xi )^{\tau }$$ distinguish topological types of $$(\Sigma , \tau ; \xi )$$. If $$r=2$$ and g is odd, we compute all $${\mathbb {Z}}_p$$-Betti numbers of $$M(2,\xi )^\tau $$.

Conjugacy search problem and the Andrews–Curtis conjecture

AbstractWe develop new computational methods for studying potential counterexamples to the Andrews–Curtis conjecture, in particular, Akbulut–Kurby examples{\operatorname{AK}(n)}. We devise a number of algorithms in an attempt to disprove the most interesting counterexample{\operatorname{AK}(3)}. That includes an efficient implementation of the folding procedure for pseudo-conjugacy graphs, based on the original modification of a classic disjoint-set data structure. To improve metric properties of the search space (the set of balanced presentations of the trivial group), we introduce a new transformation, called an ACM-move, that generalizes the original Andrews–Curtis transformations and discuss details of a practical implementation. To reduce growth of the search space, we introduce a strong equivalence relation on balanced presentations and study the space modulo automorphisms of the underlying free group. We prove that automorphism moves can be applied to Akbulut–Kurby presentations. The improved technique allows us to enumerate balanced presentations AC-equivalent to{\operatorname{AK}(3)}with relations of lengths up to 20 (previous record was 17).

Geodesics, Retracts, and the Norm-Preserving Extension Property in the Symmetrized Bidisc

A set in a domain in has the norm-preserving extension property if every bounded holomorphic function on has a holomorphic extension to with the same supremum norm. We prove that an algebraic subset of the symmetrized bidischas the norm-preserving extension property if and only if it is either a singleton, itself, a complex geodesic of , or the union of the set and a complex geodesic of degree in . We also prove that the complex geodesics in coincide with the nontrivial holomorphic retracts in . Thus, in contrast to the case of the ball or the bidisc, there are sets in which have the norm-preserving extension property but are not holomorphic retracts of . In the course of the proof we obtain a detailed classification of the complex geodesics in modulo automorphisms of . We give applications to von Neumann-type inequalities for -contractions (that is, commuting pairs of operators for which the closure of is a spectral set) and for symmetric functions of commuting pairs of contractive operators. We find three other domains that contain sets with the norm-preserving extension property which are not retracts: they are the spectral ball of matrices, the tetrablock and the pentablock. We also identify the subsets of the bidisc which have the norm-preserving extension property for symmetric functions.

On the strict convexity of the K-energy

Let (X, L) be a polarized projective complex manifold. We show, by a simple toric one-dimensional example, that Mabuchi's K-energy functional on the geodesically complete space of bounded positive (1, 1)-forms in c(1)(L), endowed with the Mabuchi-Donaldson-Semmes metric, is not strictly convex modulo automorphisms. However, under some further assumptions the strict convexity in question does hold in the toric case. This leads to a uniqueness result saying that a finite energy minimizer of the K-energy (which exists on any toric polarized manifold (X, L) which is uniformly K-stable) is uniquely determined modulo automorphisms under the assumption that there exists some minimizer with strictly positive curvature current.

Homological equivalence relations on an algebraic curve

We study the collection of homological equivalence relations on a fixed curve. We construct a moduli space for pairs consisting of a curve of genus g and a homological equivalence relation of degree n on the curve, and a classifying set for homological equivalence relations of degree n on a fixed curve, modulo automorphisms of the curve. We identify a special type of homological equivalence relations, and we characterize the special homological equivalence relations in terms of the existence of elliptic curves in the Jacobian of the curve.

Convexity of the 𝐾-energy on the space of Kähler metrics and uniqueness of extremal metrics

We establish the convexity of Mabuchi’s K K -energy functional along weak geodesics in the space of Kähler potentials on a compact Kähler manifold, thus confirming a conjecture of Chen, and give some applications in Kähler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogeneous Monge-Ampère equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.

Uniqueness of constant scalar curvature Sasakian metrics

In this paper, we prove that the transverse Mabuchi K-energy functional is convex along the weak geodesic in the space of Sasakian metrics. As an application, we obtain the uniqueness of constant scalar curvature Sasakian metrics modulo automorphisms for the transverse holomorphic structure.

Asymptotic stability for Kähler–Ricci solitons

We show that the coercivity of the modified Ding functional leads to the existence of a certain kind of balanced metrics and their convergence to the K\"ahler-Ricci soliton modulo automorphisms. In our results, we do not assume that the vanishing of the higher order modified Futaki invariants introduced by Berman-Nystr\"om.

Characterization of 2D rational local conformal nets and its boundary conditions: the maximal case

Let \mathcal A be a completely rational local Möbius covariant net on S^1 , which describes a set of chiral observables. We show that local Möbius covariant nets \mathcal B_2 on 2D Minkowski space which contains \mathcal A as chiral left-right symmetry are in one-to-one correspondence with Morita equivalence classes of Q-systems in the unitary modular tensor category \mathrm{DHR}(\mathcal A) . The Möbius covariant boundary conditions with symmetry \mathcal A of such a net \mathcal B_2 are given by the Q-systems in the Morita equivalence class or by simple objects in the module category modulo automorphisms of the dual category. We generalize to reducible boundary conditions. To establish this result we define the notion of Morita equivalence for Q-systems (special symmetric \ast -Frobenius algebra objects) and non-degenerately braided subfactors. We prove a conjecture by Kong and Runkel, namely that Rehren's construction (generalized Longo–Rehren construction, \alpha -induction construction) coincides with the categorical full center. This gives a new view and new results for the study of braided subfactors.

Theorems of W. W. Stothers and the Jacobian Conjecture in two variables

Differential equations of the form rp(z)q'(z) ― sp'(z)q(z) = γp(z) or rp(z)q'(z) ― sp'(z)q(z) = ―, where ― is a nonzero complex number and r, s are positive integers, have arisen in attempts to solve the two-variable Jacobian Conjecture. Solutions of such equations, in which p(z) and q(z) are monic complex polynomials of positive degrees r and s, give rise to extra-special pairs of polynomials in the sense of W. W. Stothers. Stothers showed that, modulo automorphisms of ℂ[z], there are only finitely many extra-special pairs of a given degree n. This implies that, modulo automorphisms of ℂ[z], there are only finitely many solutions of the above differential equations in which p(z) and q(z) are monic polynomials of given degrees r and s.

Coble rational surfaces

A Coble surface is a smooth rational projective surface such that its anti-canonical linear system is empty while the anti-bicanonical linear system is nonempty. In this paper we shall classify Coble surfaces and consider the finiteness problem of the number of negative rational curves on it modulo automorphisms.

The Diameters Graph of the Root System E8 is Uniquely Geometrisable

The root system E8 has 120 diameters (i.e., lines joining pairs of opposite roots) any two of which make an angle of sixty or ninety degrees. The graph of the title has these diameters as its vertices, where two vertices are adjacent if and only if the corresponding diameters are at right angles. Known results in the literature imply that this graph is geometrisable. In this paper we prove that, modulo automorphisms of the graph, there is a unique way to construct this geometry out of the given graph. Along the way, we observe that this graph is locally the orthogonal graph O(7, 2). We also prove that, modulo automorphisms of the polar space, there is a unique spread of O(7,2).

Compact involutions of semisimple quantum groups

It is proved that a complex cosemisimple Hopf algebra has at most one compact involution modulo automorphisms.