scholarly journals The Bogomolov-Beauville-Yau decomposition for klt projective varieties with trivial first Chern class - without tears

2021 ◽  
Vol 149 (1) ◽  
pp. 1-13
Author(s):  
Frédéric Campana
Keyword(s):  
Chern Class ◽  
Characteristic Zero ◽  
Vector Fields ◽  
Positive Characteristic ◽  
Decomposition Theorem ◽  
Canonical Bundle ◽  
Projective Varieties ◽  
Holomorphic Vector Fields ◽  
First Chern Class ◽  
Complex Projective

We give a simplified proof (in characteristic zero) of the decomposition theorem for connected complex projective varieties with klt singularities and a numerically trivial canonical bundle. The proof mainly consists in reorganizing some of the partial results obtained by many authors and used in the previous proof but avoids those in positive characteristic by S. Druel. The single, to some extent new, contribution is an algebraicity and bimeromorphic splitting result for generically locally trivial fibrations with fibers without holomorphic vector fields. We first give the proof in the easier smooth case, following the same steps as in the general case, treated next. The last two words of the title are plagiarized from [4].

scholarly journals The Kodaira Problem for Kähler Spaces with Vanishing First Chern Class

2021 ◽  
Vol 9 ◽  
Author(s):  
Patrick Graf ◽  
Martin Schwald
Keyword(s):  
Chern Class ◽  
Cohomology Group ◽  
Canonical Bundle ◽  
Deformation Space ◽  
Projective Varieties ◽  
Quotient Singularities ◽  
Finite Quotient ◽  
First Chern Class ◽  
Small Deformations

Abstract Let X be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that X admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if X has toroidal singularities, if X has finite quotient singularities and if the cohomology group ${\mathrm {H}^{2} \!\left ( X, {\mathscr {T}_{X}} \right )}$ vanishes.

Nef vector bundles on a quadric surface with the first Chern class (2, 1)

2020 ◽  
Vol 20 (1) ◽  
pp. 109-116
Author(s):  
Masahiro Ohno
Keyword(s):  
Chern Class ◽  
Characteristic Zero ◽  
Vector Bundles ◽  
Closed Field ◽  
Quadric Surface ◽  
Algebraically Closed Field ◽  
First Chern Class ◽  
Smooth Quadric Surface ◽  
Nef Vector Bundles

AbstractWe classify nef vector bundles on a smooth quadric surface with the first Chern class (2, 1) over an algebraically closed field of characteristic zero; we see in particular that such nef bundles are globally generated.

PRINCIPAL BUNDLES ON RATIONALLY CONNECTED FIBRATIONS OVER ABELIAN VARIETIES

2009 ◽  
Vol 20 (02) ◽  
pp. 167-188
Author(s):  
INDRANIL BISWAS
Keyword(s):  
Algebraic Group ◽  
Chern Class ◽  
Borel Subgroup ◽  
Linear Algebraic Group ◽  
Line Bundles ◽  
Projective Manifold ◽  
Galois Cover ◽  
Rationally Connected ◽  
First Chern Class ◽  
Complex Projective

Let f : M → A be a smooth surjective algebraic morphism, where M is a connected complex projective manifold and A a complex abelian variety, such that all the fibers of f are rationally connected. We show that an algebraic principal G-bundle EG over M admits a flat holomorphic connection if EG admits a holomorphic connection; here G is any connected reductive linear algebraic group defined over ℂ. We also show that EG admits a holomorphic connection if and only if any of the following three statements holds. (1) The principal G-bundle EG is semistable, c2( ad (EG)) = 0, and all the line bundles associated to EG for the characters of G have vanishing rational first Chern class. (2) There is an algebraic principal G-bundle E'G on A such that f*E'G = EG, and all the translations of E'G by elements of A are isomorphic to E'G itself. (3) There is a finite étale Galois cover [Formula: see text] and a reduction of structure group [Formula: see text] to a Borel subgroup B ⊂ G such that all the line bundles associated to ÊB for the characters of B have vanishing rational first Chern class. In particular, the above three statements are equivalent.

scholarly journals On a theorem of Lichnerowicz

1978 ◽  
Vol 72 ◽  
pp. 65-69
Author(s):  
Jun-Ichi Hano
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Vector Field ◽  
Vector Fields ◽  
Elliptic Partial Differential Equation ◽  
Holomorphic Vector Field ◽  
Holomorphic Vector Fields ◽  
Kähler Form ◽  
First Chern Class ◽  
Compact Kähler Manifold

In his study on the structure of the complex Lie algebra of holomorphic vector fields on a compact Kähler manifold, Lichnerowicz ([3] Theorem 2, see also [1] and [4]) shows that if the first Chern class of the manifold is positive semi-definite, then to each harmonic (O.l)-form (i.e. anti-holomorphic 1-form) η, there exists a holomorphic vector field X such that the (O.1)-form i(X)k is d″-cohomologous to η, where k is the Kähler form. The purpose of this note is to indicate that this result is a consequence of an existence theorem for solutions of a certain self-adjoint elliptic partial differential equation.

A Module-theoretic Characterization of Algebraic Hypersurfaces

2018 ◽  
Vol 61 (1) ◽  
pp. 166-173
Author(s):  
Cleto B. Miranda-Neto
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Vector Fields ◽  
Exterior Power ◽  
Algebraic Hypersurfaces ◽  
Theoretic Characterization

AbstractIn this note we prove the following surprising characterization: if X ⊂ is an (embedded, non-empty, proper) algebraic variety deûned over a field k of characteristic zero, then X is a hypersurface if and only if the module of logarithmic vector fields of X is a reflexive -module. As a consequence of this result, we derive that if is a free -module, which is shown to be equivalent to the freeness of the t-th exterior power of for some (in fact, any) t ≤ n, then necessarily X is a Saito free divisor.

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