On Topological Properties of Some Coverings. An Addendum to a Paper of Lanteri and Struppa

1992 ◽  
Vol 44 (1) ◽  
pp. 206-214
Author(s):  
Jarosław A. Wiśniewski
Keyword(s):  
Line Bundle ◽  
Projective Space ◽  
Betti Numbers ◽  
Ample Line Bundle ◽  
Double Cover ◽  
Topological Properties ◽  
Surjective Morphism ◽  
Projective Manifolds ◽  
Dimensional Projective Space ◽  
Complex Projective

AbstractLet π: X′ → X be a finite surjective morphism of complex projective manifolds which can be factored by an embedding of X′ into the total space of an ample line bundle 𝓛 over X. A theorem of Lazarsfeld asserts that Betti numbers of X and X′ are equal except, possibly, the middle ones. In the present paper it is proved that the middle numbers are actually non-equal if either 𝓛 is spanned and deg π ≥ dim X, or if X is either a hyperquadric or a projective space and π is not a double cover of an odd-dimensional projective space by a hyperquadric.

scholarly journals On the set of divisors with zero geometric defect

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Dinh Tuan Huynh ◽  
Duc-Viet Vu
Keyword(s):  
Line Bundle ◽  
Zero Divisor ◽  
Ample Line Bundle ◽  
Holomorphic Section ◽  
Projective Manifold ◽  
Holomorphic Curve ◽  
Very Ample Line Bundle ◽  
Complex Projective ◽  
Geometric Defect

AbstractLet {f:\mathbb{C}\to X} be a transcendental holomorphic curve into a complex projective manifold X. Let L be a very ample line bundle on {X.} Let s be a very generic holomorphic section of L and D the zero divisor given by {s.} We prove that the geometric defect of D (defect of truncation 1) with respect to f is zero. We also prove that f almost misses general enough analytic subsets on X of codimension 2.

Double Covers and Metastable Immersions of Spheres

1974 ◽  
Vol 26 (1) ◽  
pp. 145-176 ◽  
Author(s):  
Robert Wells
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Projective Space ◽  
Unit Sphere ◽  
Real Line ◽  
Double Cover ◽  
Sphere Bundle ◽  
Canonical Line Bundle ◽  
Projective Space Bundle ◽  
Double Covers

The real line will be R, Euclidean n-space will be Rn, the unit ball in Rn will be En, the unit sphere in Rn+1 will be Sn, and real projective n-space will be Pn. The canonical line bundle associated with the double cover Sn → Pn will be ηn. If γ is a vector bundle, E(γ) will be its associated cell bundle, S(γ) its associated sphere bundle, P(γ) its associated projective space bundle (P(γ) = S(γ) / (-1)) and T(γ) = E(γ)/S(γ) its Thorn space.

scholarly journals ADJUNCTION FOR VARIETIES WITH A ℂ* ACTION

2020 ◽  
Author(s):  
ELEONORA A. ROMANO ◽  
JAROSŁAW A. WIŚNIEWSKI
Keyword(s):  
Fixed Point ◽  
Line Bundle ◽  
Normal Bundle ◽  
Ample Line Bundle ◽  
Projective Manifold ◽  
Fano Manifolds ◽  
Adjunction Theory ◽  
Rank 2 ◽  
General Orbit ◽  
Complex Projective

Abstract Let X be a complex projective manifold, L an ample line bundle on X, and assume that we have a ℂ* action on (X;L). We classify such triples (X; L;ℂ*) for which the closure of a general orbit of the ℂ* action is of degree ≤ 3 with respect to L and, in addition, the source and the sink of the action are isolated fixed points, and the ℂ* action on the normal bundle of every fixed point component has weights ±1. We treat this situation by relating it to the classical adjunction theory. As an application, we prove that contact Fano manifolds of dimension 11 and 13 are homogeneous if their group of automorphisms is reductive of rank ≥ 2.

scholarly journals ARITHMETIC-GEOMETRIC MEANS FOR HYPERELLIPTIC CURVES AND CALABI–YAU VARIETIES

2010 ◽  
Vol 21 (07) ◽  
pp. 939-949 ◽  
Author(s):  
KEIJI MATSUMOTO ◽  
TOMOHIDE TERASOMA
Keyword(s):  
Projective Space ◽  
Hyperelliptic Curve ◽  
Geometric Mean ◽  
Hyperelliptic Curves ◽  
Double Cover ◽  
The Other ◽  
Geometric Means ◽  
Dimensional Projective Space ◽  
Period Integral ◽  
Generalized Arithmetic

In this paper, we define a generalized arithmetic-geometric mean μg among 2g terms motivated by 2τ-formulas of theta constants. By using Thomae's formula, we give two expressions of μg when initial terms satisfy some conditions. One is given in terms of period integrals of a hyperelliptic curve C of genus g. The other is by a period integral of a certain Calabi–Yau g-fold given as a double cover of the g-dimensional projective space Pg.

scholarly journals EXPECTED TOPOLOGY OF RANDOM REAL ALGEBRAIC SUBMANIFOLDS

2014 ◽  
Vol 14 (4) ◽  
pp. 673-702 ◽  
Author(s):  
Damien Gayet ◽  
Jean-Yves Welschinger
Keyword(s):  
Holomorphic Vector Bundle ◽  
Betti Numbers ◽  
Ample Line Bundle ◽  
Connected Components ◽  
Projective Manifold ◽  
Positive Probability ◽  
Random Real ◽  
Smooth Complex ◽  
Real Locus ◽  
Complex Projective

Let$X$be a smooth complex projective manifold of dimension$n$equipped with an ample line bundle$L$and a rank$k$holomorphic vector bundle$E$. We assume that$1\leqslant k\leqslant n$, that$X$,$E$and$L$are defined over the reals and denote by$\mathbb{R}X$the real locus of$X$. Then, we estimate from above and below the expected Betti numbers of the vanishing loci in$\mathbb{R}X$of holomorphic real sections of$E\otimes L^{d}$, where$d$is a large enough integer. Moreover, given any closed connected codimension$k$submanifold${\it\Sigma}$of$\mathbb{R}^{n}$with trivial normal bundle, we prove that a real section of$E\otimes L^{d}$has a positive probability, independent of$d$, of containing around$\sqrt{d}^{n}$connected components diffeomorphic to${\it\Sigma}$in its vanishing locus.

Projective manifolds of sectional genus three as zero loci of sections of ample vector bundles

2008 ◽  
Vol 144 (1) ◽  
pp. 109-118 ◽  
Author(s):  
ANTONIO LANTERI ◽  
HIDETOSHI MAEDA
Keyword(s):  
Vector Bundles ◽  
Projective Variety ◽  
Ample Line Bundle ◽  
Global Section ◽  
Sectional Genus ◽  
Ample Vector Bundle ◽  
Complex Projective Variety ◽  
Projective Manifolds ◽  
Complex Projective ◽  
Zero Locus

AbstractLet ϵ be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n such that there exists a global section of ϵ whose zero locus Z is a smooth subvariety of dimension n-r ≥ 2 of X. Let H be an ample line bundle on X such that the restriction HZ of H to Z is very ample. Triplets (X, ϵ, H) with g(Z, HZ) = 3 are classified, where g(Z, HZ) is the sectional genus of (Z, HZ).

Additive Riemann–Hilbert Problem in Line Bundles Over ℂℙ1

2006 ◽  
Vol 49 (1) ◽  
pp. 72-81 ◽  
Author(s):  
Roman J. Dwilewicz
Keyword(s):  
Line Bundle ◽  
Projective Space ◽  
Complex Projective Space ◽  
Real Hypersurface ◽  
Hilbert Problem ◽  
Holomorphic Functions ◽  
Riemann Function ◽  
Line Bundles ◽  
Complex Projective ◽  
Riemann Hilbert Problem

AbstractIn this note we consider -problem in line bundles over complex projective space ℂℙ1 and prove that the equation can be solved for (0, 1) forms with compact support. As a consequence, any Cauchy-Riemann function on a compact real hypersurface in such line bundles is a jump of two holomorphic functions defined on the sides of the hypersurface. In particular, the results can be applied to ℂℙ2 since by removing a point from it we get a line bundle over ℂℙ1.

scholarly journals Index theorems for meromorphic self-maps of the projective space

2014 ◽  
Author(s):  
Marco Abate
Keyword(s):  
Line Bundle ◽  
Projective Space ◽  
Fixed Points ◽  
Index Theorem ◽  
Connected Components ◽  
Small Letter ◽  
Index Theorems ◽  
Dimensional Projective Space ◽  
Meromorphic Maps ◽  
Tangent To The Identity

This chapter uses techniques from the theory of local dynamics of holomorphic germs tangent to the identity to prove three index theorems for global meromorphic maps of projective space. More precisely, the chapter seeks to prove a particular index theorem: Let f : ℙⁿ ⇢ ℙⁿ be a meromorphic self-map of degree ν‎ + 1 ≥ 2 of the complex n-dimensional projective space. Let Σ‎(f) = Fix(f) ∪ I(f) be the union of the indeterminacy set I(f) of f and the fixed points set Fix(f) of f. Let Σ‎(f) = ⊔subscript Greek Small Letter AlphaΣ‎subscript Greek Small Letter Alpha be the decomposition of Σ‎ in connected components, and denote by N the tautological line bundle of ℙⁿ. After laying out the statements under this theorem, the chapter discusses the proofs.

scholarly journals Deligne pairing and Quillen metric

2014 ◽  
Vol 25 (14) ◽  
pp. 1450122 ◽  
Author(s):  
Indranil Biswas ◽  
Georg Schumacher
Keyword(s):  
Positive Integer ◽  
Line Bundle ◽  
Ample Line Bundle ◽  
Hermitian Structure ◽  
Line Bundles ◽  
Canonical Isomorphism ◽  
Surjective Morphism ◽  
Kähler Form ◽  
Hermitian Structures ◽  
Determinant Line Bundle

Let X → S be a smooth projective surjective morphism of relative dimension n, where X and S are integral schemes over ℂ. Let L → X be a relatively very ample line bundle. For every sufficiently large positive integer m, there is a canonical isomorphism of the Deligne pairing 〈L,…,L〉 → S with the determinant line bundle [Formula: see text] (see [D. H. Phong, J. Ross and J. Sturm, Deligne pairings and the knudsen–Mumford expansion, J. Differential Geom. 78 (2008) 475–496]). If we fix a hermitian structure on L and a relative Kähler form on X, then each of the line bundles [Formula: see text] and 〈L,…,L〉 carries a distinguished hermitian structure. We prove that the above mentioned isomorphism between 〈L,…,L〉 → S and [Formula: see text] is compatible with these hermitian structures. This holds also for the isomorphism in [Deligne pairing and determinant bundle, Electron. Res. Announc. Math. Sci. 18 (2011) 91–96] between a Deligne paring and a certain determinant line bundle.

scholarly journals Formality conjecture for K3 surfaces

2019 ◽  
Vol 155 (5) ◽  
pp. 902-911 ◽  
Author(s):  
Nero Budur ◽  
Ziyu Zhang
Keyword(s):  
Line Bundle ◽  
Stability Condition ◽  
Ample Line Bundle ◽  
Main Tool ◽  
K3 Surfaces ◽  
Derived Category ◽  
K3 Surface ◽  
Coherent Sheaves ◽  
Dg Algebra ◽  
Complex Projective

We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the differential graded (DG) algebra$\operatorname{RHom}^{\bullet }(F,F)$is formal for any sheaf$F$polystable with respect to an ample line bundle. Our main tool is the uniqueness of the DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.

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