In this note we identify two complex structures (one is given by algebraic geometry, the other by gauge theory) on the set of isomorphism classes of holomorphic bundles with section on a given compact complex manifold. In the case of line bundles, these complex spaces are shown to be isomorphic to a space of effective divisors on the manifold.
In this paper we study the problem of extension of holomorphic sections of line bundles/vector bundles from reduced unions of strata of divisors. An extension theorem of Ohsawa--Takegoshi type is proved. As consequences we deduce several qualitative results on extension from snc divisors and generic global generation of vector bundles.
Finite-dimensional construction of self-duality and related moduli spaces over a closed Riemann surface as stratified holomorphic symplectic spaces
Universit\'e de Lille - Sciences et Technologies D\'epartement de Math\'ematiques CNRS-UMR 8524, Labex CEMPI (ANR-11-LABX-0007-01) 59655 Villeneuve d'Ascq Cedex, France
Let X be a compact complex manifold whose anti-canonical line bundle is big. We show that X admits no non-trivial holomorphic vector fields if it is Gibbs stable (at any level). The proof is based on a vanishing result for measure preserving holomorphic vector fields on X of independent interest. As an application it shown that, in general, if the anti-canonical line bundle is big, there are no holomorphic vector fields on X that are tangent to a non-singular irreducible anti-canonical divisor S on X. More generally, the result holds for varieties with log terminal singularities and log pairs.…
We establish normal form theorems for a large class of singular flat connections on complex manifolds, including connections with logarithmic poles along weighted homogeneous Saito free divisors. As a result, we show that the moduli spaces of such connections admit the structure of algebraic quotient stacks. In order to prove these results, we introduce homogeneous Lie groupoids and study their representation theory. In this direction, we prove two main results: a Jordan-Chevalley decomposition theorem, and a linearization theorem. We give explicit normal forms for several examples of free div…
We study birational projective models of ${\mathcal M}_{2,2}$ obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of ${\mathcal Z}$-stable curves $\overline{\mathcal M}_{2,2}({\mathcal Z})$ defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space $\overline{M}_{2,2}({\mathcal Z})$.
We prove that the moduli b-divisor of an lc-trivial fibration from a log canonical pair is log abundant. The result follows from a theorem on the restriction of the moduli b-divisor, based on a theory of lc-trivial morphisms, which allows us to treat $\mathbb{R}$-divisors and proper morphisms possibly with disconnected fibres. We also prove a theorem on extending a finite cover over a closed subvariety to that over a variety in arbitrary characteristic.
In this note, we extend work of Farkas and Rimányi on applying quadric rank loci to finding divisors of small slope on the moduli space of curves by instead considering all divisorial conditions on the hypersurfaces of a fixed degree containing a projective curve. This gives rise to a large family of virtual divisors on $\overline{\mathcal{M}_g}$. We determine explicitly which of these divisors are candidate counterexamples to the Slope Conjecture. The potential counterexamples exist on $\overline{\mathcal{M}_g}$, where the set of possible values of $g\in \{1,\ldots,N\}$ has density $Ω(\log(N)…
Let $X$ be a smooth projective complex curve of genus $g \geq 2$ and let $\M_X(2,Λ)$ be the moduli space of semi-stable rank-$2$ vector bundles over $X$ with fixed determinant $Λ$. We show that the wobbly locus, i.e., the locus of semi-stable vector bundles admitting a non-zero nilpotent Higgs field is a union of divisors $\Ww_k \subset \M_X(2,Λ)$. We show that on one wobbly divisor the set of maximal subbundles is degenerate. We also compute the class of the divisors $\Ww_k$ in the Picard group of $\M_X(2,Λ)$.
Let $f \colon (X,Δ) \to Y$ be a fibration such that $K_X + Δ$ is torsion along the fibres of $f$. Assume that $Y$ has dimension 2, or that $Y$ has dimension 3 and the fibres have dimension at most 3. Then the restriction of the moduli part to its augmented base locus is semiample.
We consider the moduli space of log smooth pairs formed by a cubic surface and an anticanonical divisor. We describe all compactifications of this moduli space which are constructed using Geometric Invariant Theory and the anticanonical polarization. The construction depends on a weight on the divisor. For smaller weights the stable pairs consist of mildly singular surfaces and very singular divisors. Conversely, a larger weight allows more singular surfaces, but it restricts the singularities on the divisor. The one-dimensional space of stability conditions decomposes in a wall-chamber struct…
We establish a new moduli theory for divisors, that interpolates between the Hilbert scheme and the Cayley-Chow variety. This completes the last step in the construction of a good moduli theory for stable pairs $(X,Δ)$.
A holomorphic chain on a compact Riemann surface is a tuple of vector bundles together with homomorphisms between them. We show that the moduli space of holomorphic chains of rank one is identified with a fiber product of projective space bundles. We compute the Euler characteristic of the moduli space. The stability of chains involves real vector parameters. We also show that the variation of parameters corresponds to the characters of Gm.
Let $k$ be a field with algebraic closure $\bar{k}$ and $D \subset \mathbb{P}^{1}_{\bar{k}}$ a reduced, effective divisor of degree $n \ge 3$, write $k(D)$ for the field of moduli of $D$. A. Marinatto proved that when $n$ is odd, or $n = 4$, $D$ descends to a divisor on $\mathbb{P}^{1}_{k(D)}$. We analyze completely the problem of when $D$ descends to a divisor on a smooth, projective curve of genus $0$ on $k(D)$, possibly with no rational points. In particular, we study the remaining cases $n \ge 6$ even, and we obtain conceptual proofs of Marinatto's results and of a theorem by B. Huggins ab…
Let M_g be the moduli space of smooth curves of genus g >= 3, and \bar{M}_g the Deligne-Mumford compactification in terms of stable curves. Let \bar{M}_g^{[1]} be an open set of \bar{M}_g consisting of stable curves of genus g with one node at most. In this paper, we determine the necessary and sufficient condition to guarantee that a Q-divisor D on \bar{M}_g is nef over \bar{M}_g^{[1]}, that is, (D . C) >= 0 for all irreducible curves C on \bar{M}_g with C \cap \bar{M}_g^{[1]} \not= \emptyset.
We provide methods to construct explicit examples of $K3$ surfaces. This leads to unirational constructions of Noether--Lefschetz divisors inside the moduli space of $K3$ surfaces of genus $g$. We implement Mukai's unirationality construction of the moduli spaces of $K3$ surfaces of genus $g\in\{6,\dots, 10, 12\}$, and we also present a new constructive proof of the unirationality of the moduli space of $K3$ surfaces of genus $11$. Furthermore, we show the existence of three unirational hypersurfaces in any moduli space of $K3$ surfaces of genus $g$.
There is an isomorphism between the moduli spaces of $σ$-stable holomorphic triples and some of the critical submanifolds of the moduli space of $k$-Higgs bundles of rank three, whose elements $(E,φ^k)$ correspond to variations of Hodge structure, VHS. There are special embeddings on the moduli spaces of $k$-Higgs bundles of rank three. The main objective here is to study the cohomology of the critical submanifolds of such moduli spaces, extending those embeddings to moduli spaces of holomorphic triples.
Let $X$ be a compact Riemann surface of genus $g \geq 3$. We consider the moduli space of holomorphic connections over $X$ and the moduli space of logarithmic connections singular over a finite subset of $X$ with fixed residues. We determine the Chow group of these moduli spaces. We compute the global sections of the sheaves of differential operators on ample line bundles and their symmetric powers over these moduli spaces, and show that they are constant under certain condition. We show the Torelli type theorem for the moduli space of logarithmic connections. We also describe the rational con…
We prove that actions of complex reductive Lie groups on a holomorphic vector bundle over a complex compact manifold are locally extendable to its local moduli space.
