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12
GromovWitten invariants and pseudo symplectic capacities
, 2001
"... We define the concept of pseudo symplectic capacities that is a mild generalization of that of the symplectic capacities. In particular a typical pseudo symplectic capacity, whose special case is a variant of the HoferZehnder symplectic capacity, is constructed and estimated in terms of the Gromov ..."
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Cited by 18 (6 self)
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We define the concept of pseudo symplectic capacities that is a mild generalization of that of the symplectic capacities. In particular a typical pseudo symplectic capacity, whose special case is a variant of the HoferZehnder symplectic capacity, is constructed and estimated in terms of the GromovWitten invariants. An example is also given to illustrate that using the pseudo symplectic capacity may get better estimation result than doing the HoferZehnder capacity. Among various potential applications of these estimations three typical applications are given. The first one is to derive some general nonsqueezing theorems that generalize and unite many past versions. The second is to give an alternate generalization of LalondeMcDuff theorem on length minimizing Hamiltonian paths to a general closed symplectic manifold. In the third application we give the new symplectic packing obstructions for a wider class of symplectic manifolds.
Height zeta functions of toric varieties
, 1996
"... 2. Algebraic tori................................................ 6 ..."
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Cited by 14 (0 self)
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2. Algebraic tori................................................ 6
BIRATIONAL COBORDISM INVARIANCE OF UNIRULED SYMPLECTIC MANIFOLDS
, 2006
"... 2. Birational cobordism 3 2.1. Coupling form and linear deformations 4 2.2. Blowup and blowdown 6 ..."
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Cited by 10 (1 self)
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2. Birational cobordism 3 2.1. Coupling form and linear deformations 4 2.2. Blowup and blowdown 6
Hyperelliptic and trigonal Fano threefolds
, 2004
"... We classify Fano 3folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano 3folds with canonical Gorenstein singularities which are not an intersection of quadrics. ..."
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Cited by 5 (0 self)
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We classify Fano 3folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano 3folds with canonical Gorenstein singularities which are not an intersection of quadrics.
NONRATIONAL DEL PEZZO FIBRATIONS
, 2007
"... Abstract. Let X be a general divisor in 3M + nL  on the rational scroll Proj( ⊕ 4 i=1O P 1(di)), where di and n are integers, M is the tautological line bundle, L is a fibre of the natural projection to P 1, and d1 � · · · � d4 = 0. We prove that X is rational ⇐ ⇒ d1 = 0 and n = 1. 1. Introduc ..."
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Cited by 3 (1 self)
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Abstract. Let X be a general divisor in 3M + nL  on the rational scroll Proj( ⊕ 4 i=1O P 1(di)), where di and n are integers, M is the tautological line bundle, L is a fibre of the natural projection to P 1, and d1 � · · · � d4 = 0. We prove that X is rational ⇐ ⇒ d1 = 0 and n = 1. 1. Introduction. The rationality problem for threefolds 1 splits in three cases: conic bundles, del Pezzo fibrations, and Fano threefolds. The cases of conic bundles and Fano threefolds are well studied. Let ψ: X → P 1 be a fibration into del Pezzo surfaces of degree k � 1 such that X is smooth and rkPic(X) = 2. Then X is rational if k � 5. The following result is due to [1] and [12].
Birationally superrigid cyclic triple spaces
 Izv. Math
"... Abstract. We prove the birational superrigidity and the nonrationality of a cyclic triple cover of P 2n branched over a nodal hypersurface of degree 3n for n ≥ 2. In particular, the obtained result solves the problem of the birational superrigidity of smooth cyclic triple spaces. We also consider ce ..."
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Cited by 3 (2 self)
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Abstract. We prove the birational superrigidity and the nonrationality of a cyclic triple cover of P 2n branched over a nodal hypersurface of degree 3n for n ≥ 2. In particular, the obtained result solves the problem of the birational superrigidity of smooth cyclic triple spaces. We also consider certain relevant problems. 1. Introduction. The problem of the rationality of an algebraic variety 1 is one of the most interesting problems in algebraic geometry. Global holomorphic differential forms are natural birational invariants of a smooth algebraic variety that solve the problem ot the rationality of algebraic curves and surfaces (see [205], [100]). However, even in threedimensional
HYPERELLIPTIC AND TRIGONAL FANO THREEFOLDS
, 2005
"... Abstract. We classify Fano 3folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano 3folds with canonical Gorenstein singularities which are not intersections of quadrics. We al ..."
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Abstract. We classify Fano 3folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano 3folds with canonical Gorenstein singularities which are not intersections of quadrics. We also study the rationality questions for most of these varieties. Consider a Fano threefold X with canonical Gorenstein singularities 1 (see [45], [112], [85], [39], [40]). Suppose that the anticanonical linear system  − KX  is base point free. It is well known that such varieties are divided into three classes. 1) Hyperelliptic varieties (that is, the morphism ϕ −KX  is not an embedding).
Do uniruled sixmanifolds contain Sol Lagrangian submanifolds?
, 2010
"... We prove using symplectic field theory that if the suspension of a hyperbolic diffeomorphism of the twotorus Lagrangian embeds in a closed uniruled symplectic sixmanifold, then its image contains the boundary of a symplectic disc with vanishing Maslov index. This prevents such a Lagrangian submani ..."
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We prove using symplectic field theory that if the suspension of a hyperbolic diffeomorphism of the twotorus Lagrangian embeds in a closed uniruled symplectic sixmanifold, then its image contains the boundary of a symplectic disc with vanishing Maslov index. This prevents such a Lagrangian submanifold to be monotone, for instance the real locus of a smooth real Fano manifold. It also prevents any Sol manifold to be in the real locus of an orientable real Del Pezzo fibration over a curve, confirmingan expectation of J. Kollár. Finally, it constraints Hamiltonian diffeomorphisms of uniruled symplectic fourmanifolds.