Narrow Results

In this paper, we fix the complex structure and explore the moduli space of the heterotic system by considering two different yet “dual” deformation paths starting from a Kähler solution. They correspond to deformation along the Bott–Chern cohomology class and the Aeppli cohomology class, respectively. Using the implicit function theorem, we prove the stability of the existence of heterotic solutions under these two deformations and hence establish an initial step in constructing local moduli coordinates around a Kähler solution.

A framework of connections, gauge transformations, and more, is developed over cell complexes. The relationship between this and the continuum theory is explained. A 2-form over the space of flat connections over certain types of cell complexes is defined and its significant properties proved; this 2-form is induced by the Atiyah–Bott symplectic structure on the space of connections over closed, oriented surfaces.

This paper is a review of current developments in the study of moduli spaces of G₂ manifolds. G₂ manifolds are seven-dimensional manifolds with the exceptional holonomy group G₂. Although they are odd-dimensional, in many ways they can be considered as an analogue of Calabi–Yau manifolds in seven dimensions. They play an important role in physics as natural candidates for supersymmetric vacuum solutions of M-theory compactifications. Despite the physical motivation, many of the results are of purely mathematical interest. Here we cover the basics of G₂ manifolds, local deformation theory of G₂ structures and the local geometry of the moduli spaces of G₂ structures.

We construct the moduli space of Spin(7)-instantons on a hermitian complex vector bundle over a closed 8-dimensional manifold endowed with a (possibly non-integrable) Spin(7)-structure. We find suitable perturbations that achieve regularity of the moduli space, so that it is smooth and of the expected dimension over the irreducible locus.

Allcock and Freitag recently showed that the moduli space of marked cubic surfaces is a subvariety of a nine dimensional projective space which is defined by cubic equations. They used the theory of automorphic forms on ball quotients to obtain these results. Here we describe the same embedding using Naruki's toric model of the moduli space. We also give an explicit parametrization of the tritangent divisors, we discuss another way to find equations for the image and we show that the moduli space maps, with degree at least ten, onto the unique quintic hypersurface in a five dimensional projective space which is invariant under the action of the Weyl group of the root system E₆.

Let be a moduli space of stable parabolic Higgs bundles of rank two over a Riemann surface X. It is a smooth variety defined over equipped with a holomorphic symplectic form. Fix a projective structure on X. Using , we construct a quantization of a certain Zariski open dense subset of the symplectic variety .

In this paper we consider coherent systems (E,V) on an elliptic curve which are α-stable with respect to some value of a parameter α. We show that the corresponding moduli spaces, if non-empty, are smooth and irreducible of the expected dimension. Moreover we give precise conditions for non-emptiness of the moduli spaces. Finally we study the variation of the moduli spaces with α.

In this paper, we continue the investigation of coherent systems of type (n,d,k) on the projective line which are stable with respect to some value of a parameter α. We work mainly with k < n and obtain existence results for arbitrary k in certain cases, together with complete results for k = 3. Our methods involve the use of the "flips" which occur at critical values of the parameter.

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic pair on X is a couple (E, ϕ), where E is a holomorphic bundle over X of rank n and degree d, and ϕ ∈ H⁰(E) is a holomorphic section. In this paper, we determine the Hodge polynomials of the moduli spaces of rank 2 pairs, using the theory of mixed Hodge structures. We also deal with the case in which E has fixed determinant.

Let X be a general smooth projective algebraic curve of genus g ≥ 2 over ℂ. We prove that the moduli space G(α:n,d,k) of α-stable coherent systems of type (n,d,k) over X is empty if k > n and the Brill–Noether number β := β(n,d,n + 1) = β(1,d,n + 1) = g - (n + 1)(n - d + g) < 0. Moreover, if 0 ≤ β < g or β = g, n ∤g and for some α > 0, G(α : n,d,k) ≠ ∅ then G(α : n,d,k) ≠ ∅ for all α > 0 and G(α : n,d,k) = G(α′ : n,d,k) for all α,α′ > 0 and the generic element is generated. In particular, G(α : n,d,n + 1) ≠ ∅ if 0 ≤ β ≤ g and α > 0. Moreover, if β > 0 G(α : n,d,n + 1) is smooth and irreducible of dimension β(1,d,n + 1). We define a dual span of a generically generated coherent system. We assume d < g + n₁ ≤ g + n₂ and prove that for all α > 0, G(α : n₁,d, n₁ + n₂) ≠ ∅ if and only if G(α : n₂,d, n₁ + n₂) ≠ ∅. For g = 2, we describe G(α : 2,d,k) for k > n.

In this paper, we continue the investigation of coherent systems of type (n, d, k) on the projective line which are stable with respect to some value of a parameter α. We consider the case k = 1 and study the variation of the moduli spaces with α. We determine inductively the first and last moduli spaces and the flip loci, and give an explicit description for ranks 2 and 3. We also determine the Hodge polynomials explicitly for ranks 2 and 3 and in certain cases for arbitrary rank.

We investigate topological properties of the moduli space of spin structures over genus two curves. In particular, we provide a combinatorial description of this space and give a presentation of the (rational) cohomology ring via generators and relations.

A Riemann surface X of genus g > 2 is elliptic-hyperelliptic if it admits a conformal involution h such that the orbit space X/〈h〉 has genus one. This elliptic-hyperelliptic involution h is unique for g > 5 [1]. In a previous article [3], we established the non-connectedness of the subspace of real elliptic-hyperelliptic algebraic curves in the moduli space of Riemann surfaces of genus g, when g is even and > 5. In this paper we improve this result and give a complete answer to the connectedness problem of the space of real elliptic-hyperelliptic surfaces of genus > 5: we show that is connected if g is odd and has exactly two connected components if g is even; in both cases the closure of in the compactified moduli space is connected.

We show that the log canonical bundle, κ, of is very ample, show the homogeneous coordinate ring is Koszul, and give a nice set of rank 4 quadratic generators for the homogeneous ideal: The embedding is equivariant for the symmetric group, and the image lies on many Segre embedded copies of ℙ¹ × ⋯ × ℙ^n-3, permuted by the symmetric group. The homogeneous ideal of is the sum of the homogeneous ideals of these Segre embeddings.

Let X be a compact connected Riemann surface of genus at least two. Let r be a prime number and ξ → X a holomorphic line bundle such that r is not a divisor of degree ξ. Let denote the moduli space of stable vector bundles over X of rank r and determinant ξ. By Γ we will denote the group of line bundles L over X such that L^⊗r is trivial. This group Γ acts on by the rule (E, L) ↦ E ⊗ L. We compute the Chen–Ruan cohomology of the corresponding orbifold.

We compare the Kontsevich moduli space of stable maps to projective space with the quasi-map space ℙ(Sym^d(ℂ²) ⊗ ℂⁿ)//SL(2). Consider the birational map

Let X be a smooth projective curve of genus g ≥ 2 over ℂ. Fix n ≥ 2, d ∈ ℤ. A pair (E, ϕ) over X consists of an algebraic vector bundle E of rank n and degree d over X and a section ϕ ∈ H⁰(E). There is a concept of stability for pairs which depends on a real parameter τ. Let be the moduli space of τ-semistable pairs of rank n and degree d over X. Here we prove that the cohomology groups of are Hodge structures isomorphic to direct summands of tensor products of the Hodge structure H¹(X). This implies a similar result for the moduli spaces of stable vector bundles over X.

In this note we consider the moduli space of stable bundles of rank two on a very general quintic surface. We study the potentially obstructed points of the moduli space via the spectral covering of a twisted endomorphism. This analysis leads to generically non-reduced components of the moduli space, and components which are generically smooth of more than the expected dimension. We obtain a sharp bound asked for by O'Grady on when the moduli space is good.

For a prime p, the gauge group of a principal U(p)-bundle over a compact, orientable Riemann surface is decomposed up to homotopy as a product of spaces, each of which is commonly known. This is used to deduce explicit computations of the homotopy groups of the moduli space of stable vector bundles through a range, answering a question of Daskalopoulos and Uhlenbeck.

Let X be an irreducible smooth complex projective curve of genus g ≥ 4. Fix a line bundle L on X. Let M_Sp(L) be the moduli space of semistable symplectic bundles (E, φ : E ⊗ E → L) on X, with the symplectic form taking values in L. We show that the automorphism group of M_Sp(L) is generated by the automorphisms of the form E ↦ E ⊗ M, where , together with the automorphisms induced by automorphisms of X.