A remark on elementary contractions

1995 ◽  
Vol 118 (1) ◽  
pp. 183-188
Author(s):  
Qi Zhang
Keyword(s):  
Projective Variety ◽  
Intersection Number ◽  
Smooth Projective Variety ◽  
Complex Numbers ◽  
Canonical Bundle ◽  
Small Contraction ◽  
Dimension One ◽  
Extremal Ray

Let X be a smooth projective variety of dimension n over the field of complex numbers. We denote by Kx the canonical bundle of X. By Mori's theory, if Kx is not numerically effective (i.e. if there exists a curve on X which has negative intersection number with Kx), then there exists an extremal ray ℝ+[C] on X and an elementary contraction fR: X → Y associated with ℝ+[C].fR is called a small contraction if it is bi-rational and an isomorphism in co-dimension one.

scholarly journals Dynamical construction of Kähler-Einstein metrics

2010 ◽  
Vol 199 ◽  
pp. 107-122
Author(s):  
Hajime Tsuji
Keyword(s):  
Einstein Metrics ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Einstein Metric ◽  
Canonical Bundle ◽  
General Fiber ◽  
New Construction ◽  
Kähler Einstein Metrics ◽  
Singular Hermitian Metric ◽  
Projective Morphism

AbstractIn this article, we give a new construction of a Kähler-Einstein metric on a smooth projective variety with ample canonical bundle. As a consequence, for a dominant projective morphismf:X→Swith connected fibers such that a general fiber has an ample canonical bundle, and for a positive integerm, we construct a canonical singular Hermitian metrichE,monwith semipositive curvature in the sense of Nakano.

scholarly journals Formality of -objects

2019 ◽  
Vol 155 (5) ◽  
pp. 973-994
Author(s):  
Andreas Hochenegger ◽  
Andreas Krug
Keyword(s):  
Projective Variety ◽  
Smooth Projective Variety ◽  
Complex Numbers ◽  
Cohomology Algebra ◽  
Triangulated Category ◽  
Endomorphism Algebra ◽  
Structure Sheaf

We show that a$\mathbb{P}$-object and simple configurations of$\mathbb{P}$-objects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient triangulated category. We also observe that the category generated by the structure sheaf of a smooth projective variety over the complex numbers only depends on its graded cohomology algebra.

scholarly journals A note on cubic equivalences

1986 ◽  
Vol 101 ◽  
pp. 1-26
Author(s):  
Hiroshi Saito
Keyword(s):  
Projective Variety ◽  
Present Note ◽  
Smooth Projective Variety ◽  
Complex Numbers

The present note is intended to be a supplement to [9], in which the following is proven: Let V be a smooth projective variety over the field of complex numbers C, T a smooth quasi-projective variety, Z a cycle in T × V of codimension p. If Z(t) is l-cube equivalent to zero for general t e T, then, setting r = dim V − p,vanishes for l′ < l, where {tZ} is the correspondence defined by Z.

scholarly journals On a theorem of Campana and P\u{a}un

2017 ◽  
Vol Volume 1 ◽  
Author(s):  
Christian Schnell
Keyword(s):  
General Type ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Complex Numbers ◽  
Tensor Power

Let $X$ be a smooth projective variety over the complex numbers, and $\Delta \subseteq X$ a reduced divisor with normal crossings. We present a slightly simplified proof for the following theorem of Campana and P\u{a}un: If some tensor power of the bundle $\Omega_X^1(\log \Delta)$ contains a subsheaf with big determinant, then $(X, \Delta)$ is of log general type. This result is a key step in the recent proof of Viehweg's hyperbolicity conjecture. Comment: 9 pages. Formatted using epigamath.sty

Relations of Normal Bundles and Flips of Small Contractions on Projective Manifolds

2010 ◽  
Vol 17 (01) ◽  
pp. 11-16
Author(s):  
Jihong Su ◽  
Yicai Zhao
Keyword(s):  
Irreducible Component ◽  
Complex Number ◽  
Normal Bundle ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Normal Bundles ◽  
Exceptional Locus ◽  
Small Contraction ◽  
Projective Manifolds ◽  
Complex Number Field

Let X be a smooth projective variety over the complex number field. Let f : X → Y be a small contraction, and suppose that each irreducible component Ei of the exceptional locus E of f is a smooth subvariety. Assume that dim E ≤ ½ ( dim X + 1), and the normal bundle [Formula: see text]. Then each Ei ≅ P dim Ei or Q dim Ei. Moreover, the flip f+ : X+ → Y of f exists.

scholarly journals Dynamical construction of Kähler-Einstein metrics

2010 ◽  
Vol 199 ◽  
pp. 107-122 ◽  
Author(s):  
Hajime Tsuji
Keyword(s):  
Einstein Metrics ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Einstein Metric ◽  
Canonical Bundle ◽  
General Fiber ◽  
New Construction ◽  
Kähler Einstein Metrics ◽  
Singular Hermitian Metric ◽  
Projective Morphism

AbstractIn this article, we give a new construction of a Kähler-Einstein metric on a smooth projective variety with ample canonical bundle. As a consequence, for a dominant projective morphism f: X → S with connected fibers such that a general fiber has an ample canonical bundle, and for a positive integer m, we construct a canonical singular Hermitian metric hE,m on with semipositive curvature in the sense of Nakano.

scholarly journals Homotopy abelianity of the DG-Lie algebra controlling deformations of pairs (variety with trivial canonical bundle, line bundle)

2021 ◽  
pp. 2150086
Author(s):  
Donatella Iacono ◽  
Marco Manetti
Keyword(s):  
Line Bundle ◽  
Lie Algebra ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Canonical Bundle ◽  
Algebraically Closed Field

We investigate the deformations of pairs [Formula: see text], where [Formula: see text] is a line bundle on a smooth projective variety [Formula: see text], defined over an algebraically closed field [Formula: see text] of characteristic 0. In particular, we prove that the DG-Lie algebra controlling the deformations of the pair [Formula: see text] is homotopy abelian whenever [Formula: see text] has trivial canonical bundle, and so these deformations are unobstructed.

scholarly journals Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type

2021 ◽  
Author(s):  
Lie Fu ◽  
Robert Laterveer ◽  
Charles Vial
Keyword(s):  
Projective Variety ◽  
Arbitrary Dimension ◽  
Smooth Projective Variety ◽  
The Other ◽  
Fano Variety ◽  
Fano Varieties ◽  
Intersection Product ◽  
Canonical Class ◽  
Main Input

AbstractGiven a smooth projective variety, a Chow–Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow–Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow–Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Küchle fourfolds of type c7. The main input in the cubic hypersurface case is the Franchetta property for the square of the Fano variety of lines; this was established in our earlier work in the fourfold case and is generalized here to arbitrary dimension. On the other end of the spectrum, we also give evidence that varieties with ample canonical class and with cohomology of K3 type might admit a multiplicative Chow–Künneth decomposition, by establishing this for two families of Todorov surfaces.

scholarly journals Vector bundles trivialized by proper morphisms and the fundamental group scheme

2010 ◽  
Vol 10 (2) ◽  
pp. 225-234 ◽  
Author(s):  
Indranil Biswas ◽  
João Pedro P. Dos Santos
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Characteristic Zero ◽  
Vector Bundles ◽  
Projective Variety ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Surjective Morphism ◽  
Finite Vector

AbstractLet X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro-finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.

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