Narrow Results

We investigate the relative lie algebroid connections on a holomorphic vector bundle over a family of compact complex manifolds (or smooth projective varieties over $ℂ$). We provide a sufficient condition for the existence of a relative Lie algebroid connection on a holomorphic vector bundle over a complex analytic family of compact complex manifolds. We show that the relative Lie algebroid Chern classes of a holomorphic vector bundle admitting relative Lie algebroid connection vanish, if each of the fibers of the complex analytic family is compact and Kähler. Moreover, we consider the moduli space of relative Lie algebroid connections and we show that there exists a natural relative compactification of this moduli space.

We generalize the descriptions of vortex moduli spaces in [4] to more than one section with adiabatic constant $s$. The moduli space is topologically independent of $s$ but is not compact with respect to $C^{\infty }$ topology. Following [17], we construct a Gromov limit for vortices of fixed energy, and attempt to compactify the moduli space via bubble trees with possibly conical bubbles (or raindrops).

This paper aims to investigate the geometry of the moduli spaces of harmonic maps of compact Riemann surfaces into compact Lie groups or compact symmetric spaces. The approach here is to study the gauge theoretic equations for such harmonic maps and the moduli space of their solutions. We discuss the S¹-action, the hyper-presymplectic structure, the energy function, the Hitchin map, the flag transforms on the moduli space, several kinds of subspaces in the moduli space, and the relationship among them, especially the structure of the critical point subset for the energy function on the moduli space. As results, we show that every uniton solution is a critical point of the energy function on the moduli space, and moreover we give a characterization of the fixed point subset fixed by S¹-action in terms of a flag transform.

The DoCarmo–Wallach theory studies isometric minimal immersions f : M → Sⁿ of a compact Riemannian homogeneous space M = G/K into Euclidean n-spheres. The parameter space of such immersions is a compact convex body in a representation space for the Lie group G. In this article we give a very general definition of the moduli space and study its geometric properties such as the distortion (as a convex set). In addition, we introduce a general notion of operators, derive various criteria under which they map the moduli into one another, and finally, we show that, under general conditions, the operators are distortion decreasing.

Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve C; we study the moduli problem of parametrizing certain pairs consisting of a sheaf ℰ on S and a map of ℰ to a fixed reference sheaf on S. We prove that the full moduli stack for this problem is representable by a scheme in some cases. Moreover, the moduli stack admits an action by the group C*, and we determine its fixed-point set, which leads to explicit formulas for the rational homology of the moduli space.

We prove that the space of mathematical instantons with second Chern class 5 over ℙ₃ is smooth and irreducible. Unified and simple proofs for the same statements in case of second Chern class ≤ 4 are contained.

In this paper we extend the result on base point freeness of the powers of the determinant bundle on the moduli space of vector bundles on a curve. We describe the parabolic analogues of parabolic theta functions, then we determine a uniform bound depending only on the rank of the parabolic bundles. In order to get this bound, we construct a parabolic analogue of Grothendieck's scheme of quotients, which parametrizes quotient bundles of a parabolic bundle, of fixed parabolic Hilbert polynomial. We prove an estimate for its dimension, which extends the result of Popa and Roth on the dimension of the Quot scheme. As an application of the theorem on base point freeness, we characterize parabolic semistability on the algebraic stack of quasi-parabolic bundles as the base locus of the linear system of the parabolic determinant bundle.

Generalized Calabi–Yau structures, a notion recently introduced by Hitchin, are studied in the case of K3 surfaces. We show how they are related to the classical theory of K3 surfaces and to moduli spaces of certain SCFT as studied by Aspinwall and Morrison. It turns out that K3 surfaces and symplectic structures are both special cases of this general notion. The moduli space of generalized Calabi–Yau structures admits a canonical symplectic form with respect to which the moduli space of symplectic structures is Lagrangian. The standard theory of K3 surfaces implies surjectivity of the period map and a weak form of the Global Torelli theorem.

We generalize Bertram's work on rank two vector bundles to an irreducible projective nodal curve C. We use the natural rational map defined by to study a compactification of the moduli space of semi-stable vector bundles of rank 2 and determinant L on C. In particular, we resolve the indeterminancy of ϕ_L in the case deg L = 3,4 via a sequence of three blow-ups with smooth centers.

We construct a Hecke correspondence for a moduli space of symplectic vector bundles over a curve. As an application we prove the following. Let X be a complex smooth projective curve of genus g(X) > 2 and L a line bundle over X. Let be the moduli space parametrizing stable pairs of the form (E,φ), where E is a vector bundle of rank 2n over X and φ : E ⊗ E → L a skew-symmetric nondegenerate bilinear form on the fibers of E. If deg(E) ≥ 4n(g(X)-1), then there is a projectivized Picard bundle on , which is a projective bundle whose fiber over any point is ℙ(H⁰(X,E)). We prove that this projective bundle is stable.

We define and study a family of cubic surfaces in the projectivized tangent bundle over a four-dimensional projective space associated to the root system D₅. The 27 lines are rational over the base and we determine the classifying map to the moduli space of marked cubic surfaces. This map has degree two and we use it to get short proofs for some results on the Chow group of the moduli space of marked cubic surfaces.

Let C be a curve of genus two. We denote by the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over C, and by J^d the variety of line bundles of degree d on C. In particular, J¹ has a canonical theta divisor Θ. The space is a double cover of ℙ⁸ = |3Θ| branched along a sextic hypersurface, the Coble sextic. In the dual , where J¹ is embedded, there is a unique cubic hypersurface singular along J¹, the Coble cubic. We prove that these two hypersurfaces are dual, inducing a non-abelian Torelli result. Moreover, by looking at some special linear sections of these hypersurfaces, we can observe and reinterpret some classical results of algebraic geometry in a context of vector bundles: the duality of the Segre–Igusa quartic with the Segre cubic, the symmetric configuration of 15 lines and 15 points, the Weddle quartic surface and the Kummer surface.

We construct stable sheaves over K3 fibrations using a relative Fourier-Mukai transform which describes the sheaves in terms of spectral data similar to the construction for elliptic fibrations. On K3 fibered Calabi–Yau threefolds we show that the Fourier-Mukai transform induces an embedding of the relative Jacobian of spectral line bundles on spectral covers into the moduli space of sheaves of given invariants. This makes the moduli space of spectral sheaves a generic torus fibration over the moduli space of curves of the given arithmetic genus on the Calabi–Yau manifold.

Let S^[n] be the Hilbert scheme of length n subschemes of a K3 surface S. H²(S^[n],ℤ) is endowed with the Beauville–Bogomolov bilinear form. Denote by Mon the subgroup of GL[H*(S^[n],ℤ)] generated by monodromy operators, and let Mon² be its image in OH²(S^[n],ℤ). We prove that Mon² is the subgroup generated by reflections with respect to +2 and -2 classes (Theorem 1.2). Thus Mon² does not surject onto OH²(S^[n],ℤ)/(±1), when n - 1 is not a prime power.

As a consequence, we get counterexamples to a version of the weight 2 Torelli question for hyperKähler varieties X deformation equivalent to S^[n]. The weight 2 Hodge structure on H²(X,ℤ) does not determine the bimeromorphic class of X, whenever n - 1 is not a prime power (the first case being n = 7). There are at least 2^{ρ(n - 1) - 1} distinct bimeromorphic classes of X with a given generic weight 2 Hodge structure, where ρ(n - 1) is the Euler number of n - 1.

The second main result states, that if a monodromy operator acts as the identity on H²(S^[n],ℤ), then it acts as the identity on H^k(S^[n],ℤ), 0 ≤ k ≤ n + 2 (Theorem 1.5). We conclude the injectivity of the restriction homomorphism Mon → Mon², if n ≡ 0 or n ≡ 1 modulo 4 (Corollary 1.6).

We provide a sketch of the GIT construction of the moduli spaces for the three classes of connections: the class of meromorphic connections with fixed divisor of poles D and its subclasses of integrable and integrable logarithmic connections. We use the Luna Slice Theorem to represent the germ of the moduli space as the quotient of the Kuranishi space by the automorphism group of the central fiber. This method is used to determine the singularities of the moduli space of connections in some examples.

When k < n, we study the coherent systems that come from a BGN extension in which the quotient bundle is strictly semistable. In this case we describe a stratification of the moduli space of coherent systems. We also describe the strata as complements of determinantal varieties and we prove that these are irreducible and smooth. These descriptions allow us to compute the Hodge polynomials of this moduli space in some cases. In particular, we give explicit computations for the cases in which (n, d, k) = (3, d, 1) and d is even, obtaining from them the usual Poincaré polynomials.

We construct monads for framed torsion-free sheaves on blow-ups of the complex projective plane at finitely many distinct points. Using these monads we prove that the moduli space of such sheaves is a smooth algebraic variety. Moreover, we construct monads for families of such sheaves parametrized by a noetherian scheme S of finite type. A universal monad on the moduli space is introduced and used to prove that the moduli space is fine.

Over the past 20 years, a great deal of work has been done on the moduli spaces of coherent systems on algebraic curves. Until recently, however, there has been very little work on the fixed determinant case, except for the special case of rank 2 and canonical determinant. This situation has changed due to two papers of Osserman, who has obtained lower bounds for the dimensions of the fixed determinant moduli spaces in some cases. Our object in this paper is to show that some of Osserman's bounds are sharp.

In order to unify the construction of the moduli space of vector bundles with different types of global decorations, such as Higgs bundles, framed vector bundles and conic bundles, A. H. W. Schmitt introduced the concept of a swamp. In this work, we consider vector bundles with both a global and a local decoration over a fixed point of the base. This generalizes the notion of parabolic vector bundles, vector bundles with a level structure and parabolic Higgs bundles. We introduce a notion of stability and construct the coarse moduli space for these objects as the GIT-quotient of a parameter space. In the case of parabolic vector bundles and vector bundles with a level structure our stability concept reproduces the known ones. Thus, our work unifies the construction of their moduli spaces.

A framed symplectic sheaf on a smooth projective surface $X$ is a torsion-free sheaf $E$ together with a trivialization on a divisor $D\subseteq X$ and a morphism ${\mathrm{\Lambda }}^{2}E\to {{𝒪}}_X$ satisfying some additional conditions. We construct a moduli space for framed symplectic sheaves on a surface, and present a detailed study for $X={\mathbb{P}}_{ℂ}^2$. In this case, the moduli space is irreducible and admits an ADHM-type description and a birational proper map onto the space of framed symplectic ideal instantons.