Moduli space of $J$-holomorphic subvarieties

Zhang, Weiyi

doi:10.48550/arxiv.1601.07855

Cited by 4 publications

(15 citation statements)

References 28 publications

(123 reference statements)

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“…The answer to Question A.1 was known to be true for irrational ruled surfaces [52], and there are even many integrable complex structures on rational surfaces where the statement does not hold [51,52].…”

Section: Further Discussion and Open Problemsmentioning

confidence: 99%

“…When k ≥ 8, even the statement of Question A.1 is not true, see [52]. The argument of Theorem A.4 could be used to prove other uniqueness results of subvarieties.…”

Section: Further Discussion and Open Problemsmentioning

confidence: 99%

“…First, we need a lemma. This is an adaption of Lemma 2.5 in [52]. Here, we no longer assume e is J-nef and Θ ∈ M e is connected.…”

Section: Bounding κ J By Symplectic Kodaira Dimensionmentioning

confidence: 91%

“…As noticed in [51,52] as a somewhat surprising result, there are many integrable complex structures on rational surfaces where this statement does not hold. The link of Theorem 1.9 with this paper is its consequence, Proposition A.5, which says that there is a unique J-holomorphic subvariety in any positive combination of exceptional curve classes for any tamed almost complex structure on a non rational or ruled symplectic 4-manifold.…”

Section: Introductionmentioning

confidence: 95%

“…When e is a spherical class, we have shown in [52] that, for any almost complex structure, M e behaves like linear system in algebraic geometry. However, for a general class e and a general almost complex structure, the moduli space is very complicated.…”

Section: Bounding κ J By Symplectic Kodaira Dimensionmentioning

confidence: 99%

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Kodaira dimensions of almost complex manifolds I

Chen¹,

Zhang²

2018

Preprint

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In this paper, we introduce the plurigenera, Kodaira dimensions and more generally Iitaka dimensions on compact almost complex manifolds. These are equal to the original notions when the almost complex structure is integrable. The definitions are based on the Hodge theory on almost complex manifolds. An alternative interpretation using pseudoholomorphic maps leads to fundamental geometric consequences.We show that the plurigenera and the Kodaira dimension as well as the irregularity are birational invariants in almost complex category, at least in dimension 4, where a birational morphism is defined to be a degree one pseudoholomorphic map. However, they are no longer deformation invariants, even in dimension 4 or under tameness assumption. For tamed almost complex 4-manifolds, our Kodaira dimension is bounded above by the symplectic Kodaira dimension.By studying pluricanonical maps, we are able to show the almost complex structures that achieve the top Kodaira dimension are integrable. On the other hand, the Kodaira dimension could take all the other possible integer values for non-integrable almost complex structures.When we apply our invariants to the standard almost complex structure on the six sphere S 6 , we show the Kodaira dimension is 0 and all the plurigenera are 1, which is different from the data of a hypothetical complex structure.In the appendix, we show the uniqueness of subvarieties in exceptional curve classes for irrational symplectic 4-manifolds, which was a question of the second author.

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Section: Further Discussion and Open Problemsmentioning

confidence: 99%

“…When k ≥ 8, even the statement of Question A.1 is not true, see [52]. The argument of Theorem A.4 could be used to prove other uniqueness results of subvarieties.…”

Section: Further Discussion and Open Problemsmentioning

confidence: 99%

“…First, we need a lemma. This is an adaption of Lemma 2.5 in [52]. Here, we no longer assume e is J-nef and Θ ∈ M e is connected.…”

Section: Bounding κ J By Symplectic Kodaira Dimensionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 95%

Section: Bounding κ J By Symplectic Kodaira Dimensionmentioning

confidence: 99%

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Kodaira dimensions of almost complex manifolds I

Chen¹,

Zhang²

2018

Preprint

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S1‐invariant symplectic hypersurfaces in dimension 6 and the Fano condition

Lindsay

Panov

2019

Journal of Topology

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We prove that any symplectic Fano 6‐manifold M with a Hamiltonian S1‐action is simply connected and satisfies c1c2false(Mfalse)=24. This is done by showing that the fixed submanifold Mtrueprefixmin⊆M on which the Hamiltonian attains its minimum is diffeomorphic to either a del Pezzo surface, a 2‐sphere or a point. In the case when dim(Mtrueprefixmin)=4, we use the fact that symplectic Fano 4‐manifolds are symplectomorphic to del Pezzo surfaces. The case when dim(Mtrueprefixmin)=2 involves a study of six‐dimensional Hamiltonian S1‐manifolds with Mmin diffeomorphic to a surface of positive genus. By exploiting an analogy with the algebro‐geometric situation we construct in each such 6‐manifold an S1‐invariant symplectic hypersurface F(M) playing the role of a smooth fibre of a hypothetical Mori fibration over Mmin. This relies upon applying Seiberg–Witten theory to the resolution of symplectic 4‐orbifolds occurring as the reduced spaces of M.

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Kodaira dimensions of almost complex manifolds II

Chen¹,

Zhang²

2020

Preprint

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Moduli space of $J$-holomorphic subvarieties

Cited by 4 publications

References 28 publications

Kodaira dimensions of almost complex manifolds I

Kodaira dimensions of almost complex manifolds I

S1‐invariant symplectic hypersurfaces in dimension 6 and the Fano condition

Kodaira dimensions of almost complex manifolds II

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