Let $mathcal{X}$ be a class of $R$-modules‎. ‎In this paper‎, ‎we investigate ;$mathcal{X}$-injective (projective) and DG-$mathcal{X}$-injective (projective) complexes which are generalizations of injective (projective) and DG-injecti‎‎ve (projective) complexes‎. ‎We prove that some known results can be extended to the class of ;$mathcal{X}$-injective (projective) and DG-$mathcal{X}$-injective ...

In this paper we explore the Isotropic Subspace Theorems of Catanese and Bauer, establishing relations between isotropic subspaces in the 1cohomology of a quasi-projective variety M and certain irrational pencils f : M → C, from the point of view of the Tangent Cone Theorem due to Papadima, Suciu and the author. In the proper case the picture is completely clear, and is described in section 3. ...

Given a projective variety X and a smooth projective curve C one may consider the moduli space of maps C → X. This space admits certain compactification whose points are called quasi-maps. In the last decade it has been discovered that in the case when X is a (partial) flag variety of a semi-simple algebraic group G (or, more generally, of any symmetrizable Kac-Moody Lie algebra) these compacti...

Let L be a local system on a smooth quasi projective variety over C . We see that L is semisimple if and only if there exists a tame pure imaginary pluri-harmonic metric on L. Although it is a rather minor refinement of a result of Jost and Zuo, it is significant for the study of harmonic bundles and pure twistor D-modules. As one of the application, we show that the semisimplicity of local sys...

Let X be a nonsingular complex projective variety and D a locally quasihomogeneous free divisor in X. In this paper we study a numerical relation between the Chern class of the sheaf of logarithmic derivations on X with respect to D, and the Chern-Schwartz-MacPherson class of the complement of D in X. Our result confirms a conjectural formula for these classes, at least after push-forward to pr...

We prove an analogue of Lowrey--Sch\"urg's algebraic Spivak's theorem when working over a base ring $A$ that is either field or nice enough discrete valuation ring, and after inverting the residual characteristic exponent $e$ in coefficients. By this result bordism groups quasi-projective derived $A$-schemes can be generated by classical cycles, leading to vanishing results for low degree $e$-i...

Let X be a smooth projective scheme of dimension d ≥ 1 over the field k, and let C be an indecomposable coherent sheaf on X . Then there is an Auslander-Reiten sequence in the category of quasi-coherent sheaves on X , 0 → (ΣC)⊗ ω −→ B −→ C → 0. Here ΣC is the (d−1)’st syzygy in a minimal injective resolution of C, and ω is the dualizing sheaf of X . 0. Introduction This note shows that Auslande...

In 1988 Simpson extended the Donaldson–Uhlenbeck–Yau theorem to context of Higgs bundles, and as an application he proved a uniformization which characterizes complex projective manifolds quasi-projective curves whose universal coverings are unit balls. this paper we give necessary sufficient condition for be uniformized by This generalizes Simpson. Several byproducts also obtained in paper.

Abstract We construct explicitly a finite cover of the moduli stack compact Riemann surfaces with given group symmetries by smooth quasi-projective variety.