scholarly journals Rank 3 rigid representations of projective fundamental groups

2018 ◽  
Vol 154 (7) ◽  
pp. 1534-1570 ◽  
Author(s):  
Adrian Langer ◽  
Carlos Simpson
Keyword(s):  
Irreducible Representation ◽  
Projective Variety ◽  
Fundamental Groups ◽  
Projective Varieties ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Geometric Origin ◽  
Rank 2 ◽  
Complex Projective ◽  
Smooth Projective Varieties

Let$X$be a smooth complex projective variety with basepoint$x$. We prove that every rigid integral irreducible representation$\unicode[STIX]{x1D70B}_{1}(X\!,x)\rightarrow \operatorname{SL}(3,\mathbb{C})$is of geometric origin, i.e., it comes from some family of smooth projective varieties. This partially generalizes an earlier result by Corlette and the second author in the rank 2 case and answers one of their questions.

scholarly journals Demailly’s notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1645-1672 ◽  
Author(s):  
Ariyan Javanpeykar ◽  
Ljudmila Kamenova
Keyword(s):  
Moduli Spaces ◽  
Projective Variety ◽  
Closed Field ◽  
Projective Varieties ◽  
Complex Projective Variety ◽  
Hyperbolic Complex ◽  
Closed Subvariety ◽  
Algebraically Closed Fields ◽  
Complex Projective ◽  
Smooth Projective Varieties

Abstract Demailly’s conjecture, which is a consequence of the Green–Griffiths–Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly’s conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly’s definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety Y to a projective algebraically hyperbolic variety X that map a fixed closed subvariety of Y onto a fixed closed subvariety of X is finite. As an application, we obtain that $${{\,\mathrm{Aut}\,}}(X)$$ Aut ( X ) is finite and that every surjective endomorphism of X is an automorphism. Finally, we explore “weaker” notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green–Griffiths–Lang conjecture on hyperbolic projective varieties.

Vector Fields and the Cohomology Ring of Toric Varieties

2005 ◽  
Vol 48 (3) ◽  
pp. 414-427 ◽  
Author(s):  
Kiumars Kaveh
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Projective Variety ◽  
Cohomology Ring ◽  
Holomorphic Vector Field ◽  
Zero Set ◽  
Graded Algebras ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Complex Projective

AbstractLetXbe a smooth complex projective variety with a holomorphic vector field with isolated zero setZ. From the results of Carrell and Lieberman there exists a filtrationF0⊂F1⊂ · · · ofA(Z), the ring of ℂ-valued functions onZ, such thatas graded algebras. In this note, for a smooth projective toric variety and a vector field generated by the action of a 1-parameter subgroup of the torus, we work out this filtration. Our main result is an explicit connection between this filtration and the polytope algebra ofX.

scholarly journals Hyperbolicity related problems for complete intersection varieties

2013 ◽  
Vol 150 (3) ◽  
pp. 369-395 ◽  
Author(s):  
Damian Brotbek
Keyword(s):  
Projective Space ◽  
Existence Theorem ◽  
Complete Intersection ◽  
Cotangent Bundle ◽  
Projective Variety ◽  
Complete Intersections ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Complex Projective ◽  
Intersection Surface

AbstractIn this paper we examine different problems regarding complete intersection varieties of high multidegree in a smooth complex projective variety. First we prove an existence theorem for jet differential equations that generalizes a theorem of Diverio. Then we show how one can deduce hyperbolicity for generic complete intersections of high multidegree and high codimension from the known results on hypersurfaces. Finally, motivated by a conjecture of Debarre, we focus on the positivity of the cotangent bundle of complete intersections, and prove some results towards this conjecture; among other things, we prove that a generic complete intersection surface of high multidegree in a projective space of dimension at least four has an ample cotangent bundle.

Ample Vector Bundles of Curve Genus One

1999 ◽  
Vol 42 (2) ◽  
pp. 209-213 ◽  
Author(s):  
Antonio Lanteri ◽  
Hidetoshi Maeda
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Projective Variety ◽  
Ample Vector Bundle ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Curve Genus ◽  
Genus One ◽  
Complex Projective ◽  
Zero Locus

AbstractWe investigate the pairs (X, ε) consisting of a smooth complex projective variety X of dimension n and an ample vector bundle ε of rank n − 1 on X such that ε has a section whose zero locus is a smooth elliptic curve.

scholarly journals MINIMAL MODEL PROGRAM WITH SCALING AND ADJUNCTION THEORY

2013 ◽  
Vol 24 (02) ◽  
pp. 1350007 ◽  
Author(s):  
MARCO ANDREATTA
Keyword(s):  
Minimal Model ◽  
Projective Variety ◽  
Model Program ◽  
Minimal Model Program ◽  
Projective Varieties ◽  
Complex Projective Variety ◽  
Adjunction Theory ◽  
Birational Equivalence ◽  
Big Line Bundle ◽  
Complex Projective

Let (X, L) be a quasi-polarized pair, i.e. X is a normal complex projective variety and L is a nef and big line bundle on it. We study, up to birational equivalence, the positivity (nefness) of the adjoint bundles KX + rL for high rational numbers r. For this we run a Minimal Model Program with scaling relative to the divisor KX + rL. We give then some applications, namely the classification up to birational equivalence of quasi-polarized pairs with sectional genus 0, 1 and of embedded projective varieties X ⊂ ℙN with degree smaller than 2 codim ℙN(X) + 2.

scholarly journals H1–semistability for projective groups

2016 ◽  
Vol 162 (1) ◽  
pp. 89-100 ◽  
Author(s):  
INDRANIL BISWAS ◽  
MAHAN MJ
Keyword(s):  
Fundamental Groups ◽  
Projective Varieties ◽  
Smooth Complex ◽  
Universal Covers ◽  
Complex Projective

AbstractWe initiate the study of the asymptotic topology of groups that can be realised as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers (these are called here as holomorphically convex groups). We prove the H1-semistability conjecture of Geoghegan for holomorphically convex groups. In view of a theorem of Eyssidieux, Katzarkov, Pantev and Ramachandran [EKPR], this implies that linear projective groups satisfy the H1-semistability conjecture.

scholarly journals Augmented base loci and restricted volumes on normal varieties, II: The case of real divisors

2015 ◽  
Vol 159 (3) ◽  
pp. 517-527
Author(s):  
ANGELO FELICE LOPEZ
Keyword(s):  
Open Subset ◽  
Projective Variety ◽  
Closed Field ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Smooth Complex ◽  
Normal Varieties ◽  
Restricted Volume ◽  
Complex Projective ◽  
Base Loci

AbstractLet X be a normal projective variety defined over an algebraically closed field and let Z be a subvariety. Let D be an ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on X. Given an expression (*) D$\sim_{\mathbb R}$t1H1 +. . .+ tsHs with ti ∈ ${\mathbb R}$ and Hi very ample, we define the (*)-restricted volume of D to Z and we show that it coincides with the usual restricted volume when Z$\not\subseteq$B+(D). Then, using some recent results of Birkar [Bir], we generalise to ${\mathbb R}$-divisors the two main results of [BCL]: The first, proved for smooth complex projective varieties by Ein, Lazarsfeld, Mustaţă, Nakamaye and Popa, is the characterisation of B+(D) as the union of subvarieties on which the (*)-restricted volume vanishes; the second is that X − B+(D) is the largest open subset on which the Kodaira map defined by large and divisible (*)-multiples of D is an isomorphism.

scholarly journals Higher-dimensional Clifford–Severi equalities

2019 ◽  
Vol 22 (08) ◽  
pp. 1950079 ◽  
Author(s):  
Miguel Ángel Barja ◽  
Rita Pardini ◽  
Lidia Stoppino
Keyword(s):  
Lower Bound ◽  
Line Bundle ◽  
Abelian Variety ◽  
Projective Variety ◽  
Line Bundles ◽  
Smooth Complex ◽  
Higher Dimensional ◽  
Complex Projective Variety ◽  
Irregular Varieties ◽  
Complex Projective

Let [Formula: see text] be a smooth complex projective variety, [Formula: see text] a morphism to an abelian variety such that [Formula: see text] injects into [Formula: see text] and let [Formula: see text] be a line bundle on [Formula: see text]; denote by [Formula: see text] the minimum of [Formula: see text] for [Formula: see text]. The so-called Clifford–Severi inequalities have been proven in [M. A. Barja, Generalized Clifford–Severi inequality and the volume of irregular varieties, Duke Math. J. 164(3) (2015) 541–568; M. A. Barja, R. Pardini and L. Stoppino, Linear systems on irregular varieties, J. Inst. Math. Jussieu (2019) 1–39; doi:10.1017/S1474748019000069]; in particular, for any [Formula: see text] there is a lower bound for the volume given by: [Formula: see text] and, if [Formula: see text] is pseudoeffective, [Formula: see text] In this paper, we characterize varieties and line bundles for which the above Clifford–Severi inequalities are equalities.

scholarly journals Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes

2015 ◽  
Vol 2015 (702) ◽  
pp. 1-40 ◽  
Author(s):  
Timo Schürg ◽  
Bertrand Toën ◽  
Gabriele Vezzosi
Keyword(s):  
Algebraic Geometry ◽  
Projective Variety ◽  
Obstruction Theory ◽  
Important Ingredient ◽  
Smooth Complex ◽  
Perfect Complexes ◽  
Derived Algebraic Geometry ◽  
Complex Projective Variety ◽  
Complex Projective

AbstractA quasi-smooth derived enhancement of a Deligne–Mumford stack 𝒳 naturally endows 𝒳 with a functorial perfect obstruction theory in the sense of Behrend–Fantechi. We apply this result to moduli of maps and perfect complexes on a smooth complex projective variety.ForWe give two further applications toAn important ingredient of our construction is a

Unramified Brauer group of the moduli spaces of PGLr(ℂ)-bundles over curves

2014 ◽  
Vol 12 (8) ◽  
Author(s):  
Indranil Biswas ◽  
Amit Hogadi ◽  
Yogish Holla
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Projective Variety ◽  
Brauer Group ◽  
Projective Curve ◽  
Connected Component ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Complex Projective ◽  
Complex Projective Curve

AbstractLet X be an irreducible smooth complex projective curve of genus g, with g ≥ 2. Let N be a connected component of the moduli space of semistable principal PGLr (ℂ)-bundles over X; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of N is trivial.

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