ABSTRACT. The present paper is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of singularities. In fact, a much more general theorem for log canonical pairs is established. The proof relies on vanishing theorems for log canonical varieties and on methods of the minimal model program. In addition, a theory of differential forms on dlt pairs is developed. It is shown that many of the fundamental theorems and techniques known for sheaves of logarithmic differentials on smooth varieties also hold in the dlt setting.Immediate applications include the existence of a pull-back map for reflexive differentials, generalisations of Bogomolov-Sommese type vanishing results, and a positive answer to the Lipman-Zariski conjecture for klt spaces.
Part I. Preparations 8 3. Notation, conventions, and facts used in the proof 8 4. Chern classes on singular varieties 11 5. Bertini-type theorems for sheaves and their moduli 14 Part II. Quasi-étale covers of klt spaces 16 6. Proof of Theorems 2.1 and 1.1 16 7. Direct applications 18 8. Flat sheaves on klt base spaces 21 9. Varieties with vanishing Chern classes 23 10. Varieties admitting polarised endomorphisms 25 11. Examples, counterexamples, and sharpness of results 26 Appendices 29 Appendix A. Zariski's Main Theorem in the equivariant setting 29 Appendix B. Galois closure 31 References 31
We resolve pathological wall-crossing phenomena for moduli spaces of sheaves on higher-dimensional complex projective manifolds. This is achieved by considering slopesemistability with respect to movable curves rather than divisors. Moreover, given a projective n-fold and a curve C that arises as the complete intersection of n − 1 very ample divisors, we construct a modular compactification of the moduli space of vector bundles that are slope-stable with respect to C. Our construction generalises the algebro-geometric construction of the Donaldson-Uhlenbeck compactification by Le Potier and Li. Furthermore, we describe the geometry of the newly constructed moduli spaces by relating them to moduli spaces of simple sheaves and to Gieseker-Maruyama moduli spaces.
We introduce a notion of stability for sheaves with respect to several polarisations that generalises the usual notion of Gieseker-stability. We prove, under a boundedness assumption, which we show to hold on threefolds or for rank two sheaves on base manifolds of arbitrary dimension, that semistable sheaves have a projective coarse moduli space that depends on a natural stability parameter. We then give two applications of this machinery. First, we show that given a real ample class $\omega \in N^1(X)_\mathbb{R}$ on a smooth projective threefold $X$ there exists a projective moduli space of sheaves that are Gieseker-semistable with respect to $\omega$. Second, we prove that given any two ample line bundles on $X$ the corresponding Gieseker moduli spaces are related by Thaddeus-flips.Comment: 51 pages; v2: added discussion concerning characteristic of the base field, fixed some typos and inconsistencies; removed "I" from title as second part has a new title; to appear in Geometry & Topolog
The classical Beauville-Bogomolov Decomposition Theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover that decomposes into a product of a torus, and irreducible, simply-connected Calabi-Yau-and holomorphic-symplectic manifolds. The decomposition of the simply-connected part corresponds to a decomposition of the tangent bundle into a direct sum whose summands are integrable and stable with respect to any polarisation.Building on recent extension theorems for differential forms on singular spaces, we prove an analogous decomposition theorem for the tangent sheaf of projective varieties with canonical singularities and numerically trivial canonical class.In view of recent progress in minimal model theory, this result can be seen as a first step towards a structure theory of manifolds with Kodaira dimension zero. Based on our main result, we argue that the natural building blocks for any structure theory are two classes of canonical varieties, which generalise the notions of irreducible Calabi-Yau-and irreducible holomorphic-symplectic manifolds, respectively.
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