scholarly journals Algebraic varieties with quasi-projective universal cover

2013 ◽  
Vol 2013 (679) ◽  
pp. 207-221 ◽  
Author(s):  
Benoît Claudon ◽  
Andreas Höring ◽  
János Kollár
Keyword(s):  
Abelian Variety ◽  
Fiber Bundle ◽  
Projective Variety ◽  
Universal Cover ◽  
Algebraic Varieties ◽  
Simply Connected ◽  
Connected Fiber

Abstract We prove that the universal cover of a normal projective variety X is quasi-projective iff a finite étale cover of X is a fiber bundle over an Abelian variety with simply connected fiber.

The Hodge Theory of Maps

2017 ◽  
Author(s):  
Mark Andrea de Cataldo ◽  
Luca Migliorini Lectures 1–3 ◽  
Luca Migliorini
Keyword(s):  
Projective Variety ◽  
Cohomology Ring ◽  
Betti Numbers ◽  
Hodge Theory ◽  
Topological Invariants ◽  
Algebraic Varieties ◽  
Topological Constraints ◽  
Algebraic Maps ◽  
Simply Connected ◽  
Compact Kähler Manifold

This chapter summarizes the classical results of Hodge theory concerning algebraic maps. Hodge theory gives nontrivial restrictions on the topology of a nonsingular projective variety, or, more generally, of a compact Kähler manifold: the odd Betti numbers are even, the hard Lefschetz theorem, the formality theorem, stating that the real homotopy type of such a variety is, if simply connected, determined by the cohomology ring. Similarly, Hodge theory gives nontrivial topological constraints on algebraic maps. This chapter focuses on the latter, as it considers how the existence of an algebraic map f : X → Y of complex algebraic varieties is reflected in the topological invariants of X.

scholarly journals Continuous CM-regularity of semihomogeneous vector bundles

2020 ◽  
Vol 20 (3) ◽  
pp. 401-412
Author(s):  
Alex Küronya ◽  
Yusuf Mustopa
Keyword(s):  
Vector Bundle ◽  
Abelian Variety ◽  
Upper Bound ◽  
Vector Bundles ◽  
Projective Variety ◽  
Constant Function ◽  
Piecewise Constant Function ◽  
Piecewise Constant ◽  
Generic Vanishing

AbstractWe ask when the CM (Castelnuovo–Mumford) regularity of a vector bundle on a projective variety X is numerical, and address the case when X is an abelian variety. We show that the continuous CM-regularity of a semihomogeneous vector bundle on an abelian variety X is a piecewise-constant function of Chern data, and we also use generic vanishing theory to obtain a sharp upper bound for the continuous CM-regularity of any vector bundle on X. From these results we conclude that the continuous CM-regularity of many semihomogeneous bundles — including many Verlinde bundles when X is a Jacobian — is both numerical and extremal.

scholarly journals Abelian varieties attached to cycles of intermediate dimension

1979 ◽  
Vol 75 ◽  
pp. 95-119 ◽  
Author(s):  
Hiroshi Saito
Keyword(s):  
Abelian Variety ◽  
Projective Variety ◽  
Classical Case ◽  
Degree Zero ◽  
Algebraic Construction ◽  
Jacobian Varieties ◽  
Codimension One ◽  
Picard Variety ◽  
Complex Tori ◽  
Intermediate Jacobian

The group of cycles of codimension one algebraically equivalent to zero of a nonsingular projective variety modulo rational equivalence forms an abelian variety, i.e., the Picard variety. To the group of cycles of dimension zero and of degree zero, there corresponds an abelian variety, the Albanese variety. Similarly, Weil, Lieberman and Griffiths have attached complex tori to the cycles of intermediate dimension in the classical case. The aim of this article is to give a purely algebraic construction of such “intermediate Jacobian varieties.”

scholarly journals DISTINGUISHED MODELS OF INTERMEDIATE JACOBIANS

2018 ◽  
Vol 19 (3) ◽  
pp. 891-918 ◽  
Author(s):  
Jeffrey D. Achter ◽  
Sebastian Casalaina-Martin ◽  
Charles Vial
Keyword(s):  
Abelian Variety ◽  
Hilbert Scheme ◽  
Projective Variety ◽  
Galois Representation ◽  
Complete Intersections ◽  
Base Field ◽  
Field Of Definition ◽  
Level 1 ◽  
Albanese Variety ◽  
Coniveau Filtration

We show that the image of the Abel–Jacobi map admits functorially a model over the field of definition, with the property that the Abel–Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also recover a result of Deligne on complete intersections of Hodge level 1.

scholarly journals Geometric properties of projective manifolds of small degree

2015 ◽  
Vol 160 (2) ◽  
pp. 257-277 ◽  
Author(s):  
SIJONG KWAK ◽  
JINHYUNG PARK
Keyword(s):  
Projective Variety ◽  
Smooth Projective Variety ◽  
Small Degree ◽  
Point Of View ◽  
Geometric Properties ◽  
Projective Varieties ◽  
Cox Rings ◽  
Projective Manifolds ◽  
Simply Connected ◽  
Smooth Projective Varieties

AbstractThe aim of this paper is to study geometric properties of non-degenerate smooth projective varieties of small degree from a birational point of view. First, using the positivity property of double point divisors and the adjunction mappings, we classify smooth projective varieties in $\mathbb{P}$r of degree d ⩽ r + 2, and consequently, we show that such varieties are simply connected and rationally connected except in a few cases. This is a generalisation of P. Ionescu's work. We also show the finite generation of Cox rings of smooth projective varieties in $\mathbb{P}$r of degree d ⩽ r with counterexamples for d = r + 1, r + 2. On the other hand, we prove that a non-uniruled smooth projective variety in $\mathbb{P}$r of dimension n and degree d ⩽ n(r − n) + 2 is Calabi–Yau, and give an example that shows this bound is also sharp.

scholarly journals Algebraic varieties with semialgebraic universal cover

2012 ◽  
Vol 5 (1) ◽  
pp. 199-212 ◽  
Author(s):  
János Kollár ◽  
John Pardon
Keyword(s):  
Universal Cover ◽  
Algebraic Varieties

scholarly journals THE UNIVERSAL COVER OF AN AFFINE THREE-MANIFOLD WITH HOLONOMY OF SHRINKABLE DIMENSION ≤ TWO

2000 ◽  
Vol 11 (03) ◽  
pp. 305-365 ◽  
Author(s):  
SUHYOUNG CHOI
Keyword(s):  
Affine Connection ◽  
Holonomy Group ◽  
Singular Locus ◽  
Universal Cover ◽  
Affine Structure ◽  
Affine Manifold ◽  
Cone Type ◽  
Parallel Volume ◽  
A Cell ◽  
Simply Connected

An affine manifold is a manifold with an affine structure, i.e. a torsion-free flat affine connection. We show that the universal cover of a closed affine 3-manifold M with holonomy group of shrinkable dimension (or discompacité in French) less than or equal to two is diffeomorphic to R3. Hence, M is irreducible. This follows from two results: (i) a simply connected affine 3-manifold which is 2-convex is diffeomorphic to R3, whose proof using the Morse theory takes most of this paper; and (ii) a closed affine manifold of holonomy of shrinkable dimension less or equal to d is d-convex. To prove (i), we show that 2-convexity is a geometric form of topological incompressibility of level sets. As a consequence, we show that the universal cover of a closed affine three-manifold with parallel volume form is diffeomorphic to R3, a part of the weak Markus conjecture. As applications, we show that the universal cover of a hyperbolic 3-manifold with cone-type singularity of arbitrarily assigned cone-angles along a link removed with the singular locus is diffeomorphic to R3. A fake cell has an affine structure as shown by Gromov. Such a cell must have a concave point at the boundary.

scholarly journals Wedderburn components, the index theorem and continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles

2021 ◽  
Vol 20 (1) ◽  
pp. 95-119
Author(s):  
Nathan Grieve
Keyword(s):  
New York ◽  
Abelian Variety ◽  
Vector Bundles ◽  
Index Theorem ◽  
Polynomial Functions ◽  
Algebraic Varieties ◽  
Wedderburn Decomposition ◽  
Generic Vanishing ◽  
Roch Theorem

Abstract We study the property of continuous Castelnuovo-Mumford regularity, for semihomogeneous vector bundles over a given Abelian variety, which was formulated in A. Küronya and Y. Mustopa [Adv. Geom. 20 (2020), no. 3, 401-412]. Our main result gives a novel description thereof. It is expressed in terms of certain normalized polynomial functions that are obtained via the Wedderburn decomposition of the Abelian variety’s endo-morphism algebra. This result builds on earlier work of Mumford and Kempf and applies the form of the Riemann-Roch Theorem that was established in N. Grieve [New York J. Math. 23 (2017), 1087-1110]. In a complementary direction, we explain how these topics pertain to the Index and Generic Vanishing Theory conditions for simple semihomogeneous vector bundles. In doing so, we refine results from M. Gulbrandsen [Matematiche (Catania) 63 (2008), no. 1, 123–137], N. Grieve [Internat. J. Math. 25 (2014), no. 4, 1450036, 31] and D. Mumford [Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, pp. 29-100].

scholarly journals Riemannian manifolds with local symmetry

2014 ◽  
Vol 06 (02) ◽  
pp. 211-236 ◽  
Author(s):  
Wouter van Limbeek
Keyword(s):  
Riemannian Manifolds ◽  
Local Symmetry ◽  
Universal Cover ◽  
Connected Components ◽  
Curved Manifolds ◽  
Locally Homogeneous ◽  
Simply Connected ◽  
Positively Curved ◽  
Aspherical Manifolds

We give a classification of many closed Riemannian manifolds M whose universal cover [Formula: see text] possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds M such that [Formula: see text] has noncompact connected components. We prove that in many cases, such a manifold is as a fiber bundle over a locally homogeneous space. This is inspired by work of Eberlein (for non-positively curved manifolds) and Farb-Weinberger (for aspherical manifolds), and generalizes work of Frankel (for a semisimple group action). As an application, we characterize simply-connected Riemannian manifolds with both compact and finite volume noncompact quotients.

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.

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