Almost Kähler Forms on Rational 4-Manifolds

Li, Tian Jun; Zhang, Weiyi

doi:10.1353/ajm.2015.0037

Cited by 9 publications

(86 citation statements)

References 32 publications

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“…Let

S

be the set of homology classes which are represented by smoothly embedded spheres. We define

S_{K_{J}}^{+} = false{e \in S false| g_{J} (e) = 0, e^{2} > 0 false} .

Using this notation,

{scriptE}_{K_{J}} = false{e \in S false| g_{J} (e) = 0, e^{2} = - 1 false} .

By [, Proposition 5.20],

P_{K_{J}} = {scriptS}_{K_{J}}^{+}

where the latter is the open cone spanned by

S_{K_{J}}^{+}

(and furthermore equals to the almost Kähler cone when

J

is good generic). By [, Lemma 5.24 (2)], each face

F_{E_{k}}

P_{M_{k}, K}

corresponding to

E_{k}

is naturally identified with

P_{M_{k - 1}, K}

.…”

Section: The Cone Theoremmentioning

confidence: 99%

“…We define

S_{K_{J}}^{+} = false{e \in S false| g_{J} (e) = 0, e^{2} > 0 false} .

Using this notation,

{scriptE}_{K_{J}} = false{e \in S false| g_{J} (e) = 0, e^{2} = - 1 false} .

By [, Proposition 5.20],

P_{K_{J}} = {scriptS}_{K_{J}}^{+}

where the latter is the open cone spanned by

S_{K_{J}}^{+}

(and furthermore equals to the almost Kähler cone when

J

is good generic). By [, Lemma 5.24 (2)], each face

F_{E_{k}}

P_{M_{k}, K}

corresponding to

E_{k}

is naturally identified with

P_{M_{k - 1}, K}

. Here

M_{k} = C P^{2} # k \bar{C P^{2}}

for

k \neq 1

, and

M_{1}

might be

S^{2} \times S^{2}

C P^{2} # \bar{C P^{2}}

depending on whether the set of classes in

E_{M_{k}, K}

orthogonal to

E_{k}

is empty or not.…”

Section: The Cone Theoremmentioning

confidence: 99%

“…Usually, to prove a geometric result for a general (non‐generic) tamed almost complex structure is far more delicate than the generic case, since we have to deal with very general subvarieties. The techniques developed in enable us to have a fairly clear structural picture for subvarieties in a

J

‐nef class. In this paper, we would develop more techniques to work with a general tamed almost complex structure.…”

Section: Introductionmentioning

confidence: 99%

“…The most well‐known examples are rational surfaces

C P^{2} # k \bar{C P^{2}}

when

k ⩽ 9

and

S^{2} \times S^{2}

. The cases of

S^{2}

bundles over

S^{2}

are treated in , for example.…”

Section: Introductionmentioning

confidence: 99%

“…The first is using Taubes' subvarieties‐current‐form technique . However, to argue it for an arbitrary

J

, we have to use spherical subvarieties as in . It was successfully used to affirm Question for

S^{2}

bundles over

S^{2}

.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The curve cone of almost complex 4-manifolds

Zhang

2017

Proc. London Math. Soc.

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In this paper, we study the curve cone of an almost complex 4‐manifold which is tamed by a symplectic form. In particular, we prove the cone theorem as in Mori theory for all such manifolds using the Seiberg–Witten theory. For small rational surfaces and minimal ruled surfaces, we study the configuration of negative curves. We define abstract configuration of negative curves, which records the homological and intersection information of curves. Combinatorial blowdown is the main tool to study these configurations. As an application of our investigation of the curve cone, we prove the Nakai–Moishezon type duality for all almost Kähler structures on CP2#kCP2¯ with k⩽9 and minimal ruled surfaces with a negative curve. This is proved using a version of Gram–Schmidt orthogonalization process for the J‐tamed symplectic inflation.

show abstract

“…Let

S

be the set of homology classes which are represented by smoothly embedded spheres. We define

S_{K_{J}}^{+} = false{e \in S false| g_{J} (e) = 0, e^{2} > 0 false} .

Using this notation,

{scriptE}_{K_{J}} = false{e \in S false| g_{J} (e) = 0, e^{2} = - 1 false} .

By [, Proposition 5.20],

P_{K_{J}} = {scriptS}_{K_{J}}^{+}

where the latter is the open cone spanned by

S_{K_{J}}^{+}

(and furthermore equals to the almost Kähler cone when

J

is good generic). By [, Lemma 5.24 (2)], each face

F_{E_{k}}

P_{M_{k}, K}

corresponding to

E_{k}

is naturally identified with

P_{M_{k - 1}, K}

.…”

Section: The Cone Theoremmentioning

confidence: 99%

“…We define

S_{K_{J}}^{+} = false{e \in S false| g_{J} (e) = 0, e^{2} > 0 false} .

Using this notation,

{scriptE}_{K_{J}} = false{e \in S false| g_{J} (e) = 0, e^{2} = - 1 false} .

By [, Proposition 5.20],

P_{K_{J}} = {scriptS}_{K_{J}}^{+}

where the latter is the open cone spanned by

S_{K_{J}}^{+}

(and furthermore equals to the almost Kähler cone when

J

is good generic). By [, Lemma 5.24 (2)], each face

F_{E_{k}}

P_{M_{k}, K}

corresponding to

E_{k}

is naturally identified with

P_{M_{k - 1}, K}

. Here

M_{k} = C P^{2} # k \bar{C P^{2}}

for

k \neq 1

, and

M_{1}

might be

S^{2} \times S^{2}

C P^{2} # \bar{C P^{2}}

depending on whether the set of classes in

E_{M_{k}, K}

orthogonal to

E_{k}

is empty or not.…”

Section: The Cone Theoremmentioning

confidence: 99%

J

‐nef class. In this paper, we would develop more techniques to work with a general tamed almost complex structure.…”

Section: Introductionmentioning

confidence: 99%

“…The most well‐known examples are rational surfaces

C P^{2} # k \bar{C P^{2}}

when

k ⩽ 9

and

S^{2} \times S^{2}

. The cases of

S^{2}

bundles over

S^{2}

are treated in , for example.…”

Section: Introductionmentioning

confidence: 99%

“…The first is using Taubes' subvarieties‐current‐form technique . However, to argue it for an arbitrary

J

, we have to use spherical subvarieties as in . It was successfully used to affirm Question for

S^{2}

bundles over

S^{2}

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The curve cone of almost complex 4-manifolds

Zhang

2017

Proc. London Math. Soc.

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Moduli space of J-holomorphic subvarieties

Zhang

2021

Sel. Math. New Ser.

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We study the moduli space of J-holomorphic subvarieties in a 4-dimensional symplectic manifold. For an arbitrary tamed almost complex structure, we show that the moduli space of a sphere class is formed by a family of linear system structures as in algebraic geometry. Among the applications, we show various uniqueness results of J-holomorphic subvarieties, e.g. for the fiber and exceptional classes in irrational ruled surfaces. On the other hand, non-uniqueness and other exotic phenomena of subvarieties in complex rational surfaces are explored. In particular, connected subvarieties in an exceptional class with higher genus components are constructed. The moduli space of tori is also discussed, and leads to an extension of the elliptic curve theory.

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$$C^{2,\alpha }$$ C 2 , α estimates for nonlinear elliptic equations in complex and almost complex geometry

et al. 2014

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We describe how to use the perturbation theory of Caffarelli to prove Evans-Krylov type C 2,α estimates for solutions of nonlinear elliptic equations in complex geometry, assuming a bound on the Laplacian of the solution. Our results can be used to replace the various Evans-Krylov type arguments in the complex geometry literature with a sharper and more unified approach. In addition, our methods extend to almost-complex manifolds, and we use this to obtain a new local estimate for an equation of Donaldson.

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Almost Kähler Forms on Rational 4-Manifolds

Cited by 9 publications

References 32 publications

The curve cone of almost complex 4-manifolds

The curve cone of almost complex 4-manifolds

Moduli space of J-holomorphic subvarieties

$$C^{2,\alpha }$$ C 2 , α estimates for nonlinear elliptic equations in complex and almost complex geometry

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