scholarly journals SNC Log Symplectic Structures on Fano Products

2020 ◽  
Vol 63 (4) ◽  
pp. 891-900
Author(s):  
Katsuhiko Okumura
Keyword(s):  
Projective Spaces ◽  
Fano Varieties ◽  
Poisson Structures ◽  
Picard Number ◽  
Symplectic Structures ◽  
Normal Crossing ◽  
Simple Normal Crossing

AbstractThis paper classifies Poisson structures with the reduced simple normal crossing divisor on a product of Fano varieties of Picard number 1. The characterization of even-dimensional projective spaces from the viewpoint of Poisson structures is given by Lima and Pereira. In this paper, we generalize the characterization of projective spaces to any dimension.

scholarly journals Simple normal crossing Fano varieties and log Fano manifolds

2014 ◽  
Vol 214 ◽  
pp. 95-123
Author(s):  
Kento Fujita
Keyword(s):  
Fano Varieties ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Normal Crossing ◽  
Fano Index ◽  
Simple Normal Crossing

AbstractA projective log variety (X, D) is called alog Fano manifoldifXis smooth and ifDis a reduced simple normal crossing divisor onΧwith − (KΧ+D) ample. Then-dimensional log Fano manifolds (X, D) with nonzeroDare classified in this article when the log Fano indexrof (X, D) satisfies eitherr≥n/2withρ(X) ≥ 2 orr≥n− 2. This result is a partial generalization of the classification of logarithmic Fano 3-folds by Maeda.

scholarly journals Simple normal crossing Fano varieties and log Fano manifolds

2014 ◽  
Vol 214 ◽  
pp. 95-123 ◽  
Author(s):  
Kento Fujita
Keyword(s):  
Fano Varieties ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Normal Crossing ◽  
Fano Index ◽  
Simple Normal Crossing

AbstractA projective log variety (X, D) is called a log Fano manifold if X is smooth and if D is a reduced simple normal crossing divisor on Χ with − (KΧ + D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this article when the log Fano index r of (X, D) satisfies either r ≥ n/2 with ρ(X) ≥ 2 or r ≥ n − 2. This result is a partial generalization of the classification of logarithmic Fano 3-folds by Maeda.

scholarly journals Higher codimensional alpha invariants and characterization of projective spaces

2019 ◽  
Vol 31 (02) ◽  
pp. 2050012
Author(s):  
Ziwen Zhu
Keyword(s):  
Lower Bound ◽  
Projective Spaces ◽  
Fano Varieties ◽  
Fano Manifolds ◽  
Arbitrary Codimension ◽  
Definition Of

We generalize the definition of alpha invariant to arbitrary codimension. We also give a lower bound of these alpha invariants for K-semistable [Formula: see text]-Fano varieties and show that we can characterize projective spaces among all K-semistable Fano manifolds in terms of higher codimensional alpha invariants. Our results demonstrate the relation between alpha invariants of any codimension and volumes of Fano manifolds in the characterization of projective spaces.

scholarly journals ON THE MODIFIED FUTAKI INVARIANT OF COMPLETE INTERSECTIONS IN PROJECTIVE SPACES

2016 ◽  
Vol 222 (1) ◽  
pp. 186-209
Author(s):  
RYOSUKE TAKAHASHI
Keyword(s):  
Vector Field ◽  
Recent Work ◽  
Projective Spaces ◽  
Ricci Soliton ◽  
Complete Intersections ◽  
Necessary Condition ◽  
Ricci Solitons ◽  
Fano Varieties ◽  
Holomorphic Vector Field ◽  
Fano Manifold

Let $M$ be a Fano manifold. We call a Kähler metric ${\it\omega}\in c_{1}(M)$ a Kähler–Ricci soliton if it satisfies the equation $\text{Ric}({\it\omega})-{\it\omega}=L_{V}{\it\omega}$ for some holomorphic vector field $V$ on $M$. It is known that a necessary condition for the existence of Kähler–Ricci solitons is the vanishing of the modified Futaki invariant introduced by Tian and Zhu. In a recent work of Berman and Nyström, it was generalized for (possibly singular) Fano varieties, and the notion of algebrogeometric stability of the pair $(M,V)$ was introduced. In this paper, we propose a method of computing the modified Futaki invariant for Fano complete intersections in projective spaces.

A Characterization of Products of Projective Spaces

2006 ◽  
Vol 49 (2) ◽  
pp. 270-280 ◽  
Author(s):  
Gianluca Occhetta
Keyword(s):  
Projective Spaces ◽  
Rational Curves

AbstractWe give a characterization of products of projective spaces using unsplit covering families of rational curves.

scholarly journals Asymptotically conical Calabi-Yau metrics with cone singularities along a compact divisor

2019 ◽  
Author(s):  
Martin de Borbon ◽  
Cristiano Spotti
Keyword(s):  
Finite Subgroup ◽  
Minimal Resolution ◽  
Gravitational Instantons ◽  
Normal Crossing ◽  
Three Sphere ◽  
Asymptotically Locally Euclidean ◽  
Quotient Singularities ◽  
Simple Normal Crossing

Abstract We construct Asymptotically Locally Euclidean (ALE) and, more generally, asymptotically conical Calabi–Yau metrics with cone singularities along a compact simple normal crossing divisor. In particular, this includes the case of the minimal resolution of 2D quotient singularities for any finite subgroup $\Gamma \subset U(2)$ acting freely on the three-sphere, hence generalizing Kronheimer’s construction of smooth ALE gravitational instantons.

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