arXiv subjects
Publications and source records attributed to Carlos T. Simpson.
The moduli space $M(c_2)$, of stable rank two vector bundles of degree one on a very general quintic surface $X\subset {\mathbb P}^3$, is irreducible for all $c_2\geq 4$ and empty otherwise.
In this paper we continue our study of the moduli space of stable bundles of rank two and degree 1 on a very general quintic surface. The goal in this paper is to understand the irreducible components of the moduli space in the first case in the "good" range, which is $c_2=10$. We show that there is a single irreducible component of bundles which have seminatural cohomology, and conjecture that this is the only component for all stable bundles.
We look at natural foliations on the Painlevé VI moduli space of regular connections of rank 2 on $\pp ^1 -{t_1,t_2,t_3,t_4}$. These foliations are fibrations, and are interpreted in terms of the nonabelian Hodge filtration, giving a proof of the nonabelian Hodge foliation conjecture in this case. Two basic kinds of fibrations arise: from apparent singularities, and from quasiparabolic bundles. We show that these are transverse. Okamoto's additional symmetry, which may be seen as Katz's middle convolution, exchanges the quasiparabolic and apparent-singularity foliations.
We consider bounded 2-metric spaces satisfying an additional axiom, and show that a contractive mapping has either a fixed point or a fixed line.
We use coverings by smooth projective varieties then apply nonabelian Hodge techniques to study the topology of proper Deligne-Mumford stacks as well as more general simplicial varieties.
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 saying when the moduli space is good.
This is the first draft of a book about higher categories approached by iterating Segal's method, as in Tamsamani's definition of $n$-nerve and Pelissier's thesis. If $M$ is a tractable left proper cartesian model category, we construct a tractable left proper cartesian model structure on the category of $M$-precategories. The procedure can then be iterated, leading to model categories of $(\infty, n)$-categories.
We construct a Mixed Hodge Structure on the local complete ring of the representation scheme at the holonomy of a VHS on a compact Kähler manifold and prove that the corresponding tautological representation is the holonomy of a VMHS. In order to carry out this construction we revisit the well-known work of Goldman and Millson on the deformation theory of representations of Kähler groups.
Given a vector bundle with integrable connection $(V,\nabla)$ on a curve, if $V$ is not itself semistable as a vector bundle then we can iterate a construction involving modification by the destabilizing subobject to obtain a Hodge-like filtration $F^p$ which satisfies Griffiths transversality. The associated graded Higgs bundle is the limit of $(V,t\nabla)$ under the de Rham to Dolbeault degeneration. We get a stratification of the moduli space of connections, with as minimal stratum the space of opers. The strata have fibrations whose fibers are Lagrangian subspaces of the moduli space.
We construct a locally geometric $\infty$-stack $M_{Hod}(X,Perf)$ of perfect complexes with $λ$-connection structure on a smooth projective variety $X$. This maps to $A ^1 / G_m$, so it can be considered as the Hodge filtration of its fiber over 1 which is $M_{DR}(X,Perf)$, parametrizing complexes of $D_X$-modules which are $O_X$-perfect. We apply the result of Toen-Vaquie that $Perf(X)$ is locally geometric. The proof of geometricity of the map $M_{Hod}(X,Perf) \to Perf(X)$ uses a Hochschild-like notion of weak complexes of modules over a sheaf of rings of differential operators. We prove a strictification result for these weak complexes, and also a strictification result for complexes of sheaves of $O$-modules over the big crystalline site.
The twistor space of representations on an open variety maps to a weight two space of local monodromy transformations around a divisor component at infinty. The space of $σ$-invariant sections of this slope-two bundle over the twistor line is a real 3 dimensional space whose parameters correspond to the complex residue of the Higgs field, and the real parabolic weight of a harmonic bundle.
In this paper, we prove a generalization of Reznikov's theorem which says that the Chern-Simons classes and in particular the Deligne Chern classes (in degrees $>1$) are torsion, of a flat bundle on a smooth complex projective variety. We consider the case of a smooth quasi--projective variety with an irreducible smooth divisor at infinity. We define the Chern-Simons classes of Deligne's canonical extension of a flat vector bundle with unipotent monodromy at infinity, which lift the Deligne Chern classes and prove that these classes are torsion.
Suppose $X$ is a smooth quasiprojective variety over $\cc$ and $ρ: π_1(X,x) \to SL(2,\cc)$ is a Zariski-dense representation with quasiunipotent monodromy at infinity. Then $ρ$ factors through a map $X\to Y$ with $Y$ either a DM-curve or a Shimura modular stack.
In this paper, we obtain an explicit formula for the Chern character of a locally abelian parabolic bundle in terms of its constituent bundles. Several features and variants of parabolic structures are discussed. Parabolic bundles arising from logarithmic connections form an important class of examples. As an application, we consider the situation when the local monodromies are semi-simple and are of finite order at infinity. In this case the parabolic Chern classes of the associated locally abelian parabolic bundle are deduced to be zero in the rational Deligne cohomology in degrees $\geq 2$.
This is an expository account of Katz's middle convolution operation on local systems over ${\bf P}^1-\{q\_1,..., q\_n\}$. We describe the Betti and de Rham versions, and point out that they give isomorphisms between different moduli spaces of local systems, following Völklein, Dettweiler-Reiter, Haraoka-Yokoyama. Kostov's program for applying the Katz algorithm is to say that in the range where middle convolution no longer reduces the rank, one should give a direct construction of local systems. This has been done by Kostov and Crawley-Boevey. We describe here an alternative construction using the notion of cyclotomic harmonic bundles: these are like variations of Hodge structure except that the Hodge decomposition can go around in a circle.
In this paper, we consider the weight $i$ de Rham--Gauss--Manin bundles on a smooth variety arising from a smooth projective morphism $f:X\_U\lrar U$ for $i\geq 0$. We associate to each weight $i$ de Rham bundle, a certain parabolic bundle on $S$ and consider their parabolic Chern characters in the rational Chow groups, for a good compactification $S$ of $U$. We show the triviality of the alternating sum of these parabolic bundles in the (positive degree) rational Chow groups. This removes the hypothesis of semistable reduction in the original result of this kind due to Esnault and Viehweg.
This preprint contains the Coq proof files for Gabriel-Zisman localization, bundled with the source. The text of this preprint consists of the definitions and lemma statements of the main files, with proofs removed. See the other preprint ``Explaining GZ localization to the computer'' for explanation and discussion.
This explains a computer formulation of Gabriel-Zisman localization of categories in the proof assistant Coq. It includes both the general localization construction with the proof of GZ's Lemma 1.2, as well as the construction using calculus of fractions. The proof files are bundled with the other preprint "Files for GZ localization" posted simultaneously.