scholarly journals A polar dual to the momentum of toric Fano manifolds

2021 ◽  
Vol 8 (1) ◽  
pp. 230-246
Author(s):  
Yuji Sano
Keyword(s):  
Fano Manifold ◽  
Fano Manifolds ◽  
Fano Polytope

Abstract We introduce an invariant on the Fano polytope of a toric Fano manifold as a polar dual counterpart to the momentum of its polar dual polytope. Moreover, we prove that if the momentum of the polar dual polytope is equal to zero, then the dual invariant on a Fano polytope vanishes.

scholarly journals On the limit behavior of metrics in the continuity method for the Kähler–Einstein problem on a toric Fano manifold

2012 ◽  
Vol 148 (6) ◽  
pp. 1985-2003 ◽  
Author(s):  
Chi Li
Keyword(s):  
Ricci Curvature ◽  
Lower Bounds ◽  
Limit Behavior ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Continuity Method ◽  
Type Singularity ◽  
Moment Polytope ◽  
The Moment ◽  
Monge Ampere

AbstractThis work is a continuation of the author’s previous paper [Greatest lower bounds on the Ricci curvature of toric Fano manifolds, Adv. Math. 226 (2011), 4921–4932]. On any toric Fano manifold, we discuss the behavior of the limit metric of a sequence of metrics which are solutions to a continuity family of complex Monge–Ampère equations in the Kähler–Einstein problem. We show that the limit metric satisfies a singular complex Monge–Ampère equation. This gives a conic-type singularity for the limit metric. Information on conic-type singularities can be read off from the geometry of the moment polytope.

scholarly journals Gravitational Quantum Cohomology

1997 ◽  
Vol 12 (09) ◽  
pp. 1743-1782 ◽  
Author(s):  
Tohru Eguchi ◽  
Kentaro Hori ◽  
Chuan-Sheng Xiong
Keyword(s):  
Quantum Cohomology ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Hamiltonian Structures ◽  
Rational Surfaces ◽  
Algebraic Torus ◽  
Lax Operator ◽  
Dimensional Gravity ◽  
New Type ◽  
First Chern Class

We discuss how the theory of quantum cohomology may be generalized to "gravitational quantum cohomology" by studying topological σ models coupled to two-dimensional gravity. We first consider σ models defined on a general Fano manifold M (manifold with a positive first Chern class) and derive new recursion relations for its two-point functions. We then derive bi-Hamiltonian structures of the theories and show that they are completely integrable at least at the level of genus 0. We next consider the subspace of the phase space where only a marginal perturbation (with a parameter t) is turned on and construct Lax operators (superpotentials) L whose residue integrals reproduce correlation functions. In the case of M = CP N the Lax operator is given by [Formula: see text] and agrees with the potential of the affine Toda theory of the A N type. We also obtain Lax operators for various Fano manifolds; Grassmannians, rational surfaces, etc. In these examples the number of variables of the Lax operators is the same as the dimension of the original manifold. Our result shows that Fano manifolds exhibit a new type of mirror phenomenon where mirror partner is a noncompact Calabi–Yau manifold of the type of an algebraic torus C *N equipped with a specific superpotential.

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 The Mukai conjecture for log Fano manifolds

2014 ◽  
Vol 12 (1) ◽  
Author(s):  
Kento Fujita
Keyword(s):  
Irreducible Component ◽  
Minimal Model ◽  
Model Program ◽  
Minimal Model Program ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Picard Groups ◽  
Restriction Homomorphism

AbstractFor a log Fano manifold (X,D) with D ≠ 0 and of the log Fano pseudoindex ≥2, we prove that the restriction homomorphism Pic(X) → Pic(D 1) of Picard groups is injective for any irreducible component D 1 ⊂ D. The strategy of our proof is to run a certain minimal model program and is similar to Casagrande’s argument. As a corollary, we prove that the Mukai conjecture (resp. the generalized Mukai conjecture) implies the log Mukai conjecture (resp. the log generalized Mukai conjecture).

scholarly journals Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions I

2019 ◽  
Vol 30 (06) ◽  
pp. 1950032 ◽  
Author(s):  
Yunhyung Cho
Keyword(s):  
Symplectic Manifold ◽  
Symplectic Manifolds ◽  
Complete List ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Circle Actions ◽  
Isolated Point

Let [Formula: see text] be a six-dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian [Formula: see text]-action. We show that if the minimal (or maximal) fixed component of the action is an isolated point, then [Formula: see text] is [Formula: see text]-equivariantly symplectomorphic to some Kähler Fano manifold [Formula: see text] with a certain holomorphic [Formula: see text]-action. We also give a complete list of all such Fano manifolds and describe all semifree [Formula: see text]-actions on them specifically.

On the examples of Nill and Paffenholz

2015 ◽  
Vol 26 (04) ◽  
pp. 1540007
Author(s):  
Yasuhiro Nakagawa
Keyword(s):  
Fano Manifold ◽  
Fano Manifolds ◽  
Kähler Metric ◽  
Higher Dimensional ◽  
Kahler Metric

In 1999, Batyrev and Selivanova introduced the notion of "symmetric toric Fano manifolds" and proved that every symmetric toric Fano manifold admits an Einstein–Kähler metric ([1]). For n ≦ 4, every Einstein–Kähler toric Fano n-fold is symmetric. Hence, it was conjectured that every Einstein–Kähler toric Fano manifold is symmetric. However, in [6], Nill and Paffenholz constructed the counter-examples of this conjecture. In this paper, we shall construct higher-dimensional generalizations of their examples and also discuss the Chow unstability of these generalizations.

scholarly journals Pluricomplex Green's functions and Fano manifolds

2019 ◽  
Vol Volume 3 ◽  
Author(s):  
Nicholas McCleerey ◽  
Valentino Tosatti
Keyword(s):  
Green's Functions ◽  
Einstein Metrics ◽  
Green’S Functions ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Continuity Method ◽  
Homogeneous Complex ◽  
Analytic Singularities ◽  
Kähler Einstein Metrics ◽  
Monge Ampere

We show that if a Fano manifold does not admit Kahler-Einstein metrics then the Kahler potentials along the continuity method subconverge to a function with analytic singularities along a subvariety which solves the homogeneous complex Monge-Ampere equation on its complement, confirming an expectation of Tian-Yau. Comment: EpiGA Volume 3 (2019), Article Nr. 9

scholarly journals Greatest lower bounds on the Ricci curvature of Fano manifolds

2010 ◽  
Vol 147 (1) ◽  
pp. 319-331 ◽  
Author(s):  
Gábor Székelyhidi
Keyword(s):  
Ricci Curvature ◽  
Lower Bounds ◽  
Einstein Metrics ◽  
Upper Bounds ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Existence Time ◽  
Kähler Einstein Metrics ◽  
Bounded Below ◽  
Kahler Metric

AbstractOn a Fano manifoldMwe study the supremum of the possibletsuch that there is a Kähler metricω∈c1(M) with Ricci curvature bounded below byt. This is shown to be the same as the maximum existence time of Aubin’s continuity path for finding Kähler–Einstein metrics. We show that onP2blown up in one point this supremum is 6/7, and we give upper bounds for other manifolds.

scholarly journals Four-dimensional Fano toric complete intersections

2015 ◽  
Vol 471 (2175) ◽  
pp. 20140704 ◽  
Author(s):  
T. Coates ◽  
A. Kasprzyk ◽  
T. Prince
Keyword(s):  
Complete Intersection ◽  
Complete Intersections ◽  
Fano Manifold ◽  
Fano Manifolds

We find at least 527 new four-dimensional Fano manifolds, each of which is a complete intersection in a smooth toric Fano manifold.

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