Symplectic 4-manifolds with non-positive Kodaira dimension

Symplectic Kodaira dimensionκ = −∞

κ = 0

Symplectic 4-manifolds with non-positive Kodairadimension

Tian-Jun Li

University of Minnesota

July 31, 2013

Tian-Jun Li Symplectic 4-manifolds with non-positive Kodaira dimension


κ = 0

1 Symplectic Kodaira dimension

2 κ = −∞ClassificationSome open problems

3 κ = 0Known symplectic CY surfacesHomological classification of symplectic CY surfacesProgress towards smooth classification

SpeculationFibred sums, fibrationsSurgery



κ = 0

Recall κ (= κs)

(M, ω) is said to be (symplectically) minimal if it doesn’t containany symplectic sphere with self intersection -1.For a minimal symplectic 4−manifold (M, ω) its Kodaira dimension

κ(M, ω) =

−∞ if Kω · [ω] < 0 or K 2

ω < 0,0 if Kω · [ω] = 0 and K 2

ω = 0,1 if Kω · [ω] > 0 and K 2

ω = 0,2 if Kω · [ω] > 0 and K 2

ω > 0.

The Kodaira dimension of a non-minimal manifold is defined to bethat of any of its minimal models.

Seiberg-Witten theory andexistence and almost uniqueness of symplectic minimal model areneeded to show that κ(M, ω) is well defined.



κ = 0

Properties of κ

κ(M, ω) agrees with the usual Kod dim for a Kahler surface

κ(M, ω) does not depend on the choice of ω, and hence is anoriented diffeomorphism invariant.

We will denote it by κ(M).



κ = 0

Additivity

If (M, ω) is a finite cover of (M, ω), then κ(M, ω) = κ(M, ω)

For a surface bundle Σ→ (M, ω)→ B,κ(M, ω) = κ(Σ) + κ(B) if we define

κ(S2) = −∞, κ(T 2) = 0, κ(Σg≥2) = 1

Additivity is conjectured to hold for circle bundles or mapping toriwith respect to the Kodaira dimension of 3-manifolds introducedby Weiyi Zhang.



κ = 0

ClassificationSome open problems

κ = −∞ I. Smooth classification

Liu: κ(M, ω) = −∞ if and only if M is rational or ruled

(Rational manifolds) CP2#kCP2 and S2 × S2;

(Ruled manifolds) S2−bundle #kCP2,

These manifolds have b+ = 1 and have Kahler structuresAll toric surfaces have κ = −∞



κ = 0


Characterizations

Existence of an embedded symplectic sphere withnon-negative self-intersection

Existence of an embedded SMOOTH sphere withnon-negative self-intersection

Existence of a smooth circle action with fixed point

Uniruled: there is a non-trivial genus 0 GW invariant with apoint constraint

A longstanding conjecture in algebraic geometry: A smoothprojective variety is uniruled if and only if it has Kodaira dimension−∞.



κ = 0


κ = −∞ II. Symplectic structures

Let Ω be the space of symplectic formsmoduli space of symplectic structures

M = Ω/Diff +

L-Liu, McDuff-LaLonde: M described by C(M)/D(M)C(M) = [ω]|ω ∈ Ω symplectic coneD(M) =Image (Diff + → AutH2) geometric automorphism groupL-Liu: C(M) = ξ ∈ H2|ξ2 > 0, ξ(E ) 6= 0 for any E ∈ ECasini-Panov: On non-minimal irrational ruled manifold, thereexists non-Kahler symplectic structuresNot known for rational manifolds with large Euler number.If every symplectic structure on a rational manifold is Kahler, thenthe longstanding Nagata conjecture on singularities of plane curvesis a consequence



κ = 0




M = Ω/Diff +

L-Liu, McDuff-LaLonde: M described by C(M)/D(M)C(M) = [ω]|ω ∈ Ω symplectic coneD(M) =Image (Diff + → AutH2) geometric automorphism group

L-Liu: C(M) = ξ ∈ H2|ξ2 > 0, ξ(E ) 6= 0 for any E ∈ ECasini-Panov: On non-minimal irrational ruled manifold, thereexists non-Kahler symplectic structuresNot known for rational manifolds with large Euler number.If every symplectic structure on a rational manifold is Kahler, thenthe longstanding Nagata conjecture on singularities of plane curvesis a consequence



κ = 0




M = Ω/Diff +

L-Liu, McDuff-LaLonde: M described by C(M)/D(M)C(M) = [ω]|ω ∈ Ω symplectic coneD(M) =Image (Diff + → AutH2) geometric automorphism groupL-Liu: C(M) = ξ ∈ H2|ξ2 > 0, ξ(E ) 6= 0 for any E ∈ E

Casini-Panov: On non-minimal irrational ruled manifold, thereexists non-Kahler symplectic structuresNot known for rational manifolds with large Euler number.If every symplectic structure on a rational manifold is Kahler, thenthe longstanding Nagata conjecture on singularities of plane curvesis a consequence



κ = 0




M = Ω/Diff +

L-Liu, McDuff-LaLonde: M described by C(M)/D(M)C(M) = [ω]|ω ∈ Ω symplectic coneD(M) =Image (Diff + → AutH2) geometric automorphism groupL-Liu: C(M) = ξ ∈ H2|ξ2 > 0, ξ(E ) 6= 0 for any E ∈ ECasini-Panov: On non-minimal irrational ruled manifold, thereexists non-Kahler symplectic structures

Not known for rational manifolds with large Euler number.If every symplectic structure on a rational manifold is Kahler, thenthe longstanding Nagata conjecture on singularities of plane curvesis a consequence



κ = 0




M = Ω/Diff +

L-Liu, McDuff-LaLonde: M described by C(M)/D(M)C(M) = [ω]|ω ∈ Ω symplectic coneD(M) =Image (Diff + → AutH2) geometric automorphism groupL-Liu: C(M) = ξ ∈ H2|ξ2 > 0, ξ(E ) 6= 0 for any E ∈ ECasini-Panov: On non-minimal irrational ruled manifold, thereexists non-Kahler symplectic structuresNot known for rational manifolds with large Euler number.If every symplectic structure on a rational manifold is Kahler, thenthe longstanding Nagata conjecture on singularities of plane curvesis a consequence



κ = 0


Symp group of rational manifolds–Monotone case

Symp(M, ω) Infinite dimensional groupGoal: Homotopy type

Monotone: Kω = −[ω](Gromov, Lalonde-Pinnsonnault, Seidel, Evans)

S2 × S2, Symp' SO(3)× SO(3) o Z2

CP2, Symp' PU(3)

CP2#CP2, Symp' U(2)

CP2#2CP2, Symp' T 2 o Z2

CP2#3CP2, Symp' T 2

CP2#4CP2, Symp' pt

CP2#5CP2, Symp' Diff +(S2, 5) (∞ components)

CP2#kCP2 with k = 6, 7, 8?



κ = 0


Symplectomorphism group–General case

Arbitrary symplectic formGoal: Calculate the homotopy groupsMcDuff, Abreu, Lalonde, Pinnsonnault, etc

Symplectic mapping class group, ie. π0Homology action of SMC generated by Lagrangian Dehn twists(L+Wu).Homology trivial subgroup of SMC is trivial for up to 4 blow ups.(McDuff-Abreu, Lalonde+Pinnsonnault+Anjos, J. Li+L+Wu)



κ = 0




Symplectic mapping class group, ie. π0

Homology action of SMC generated by Lagrangian Dehn twists(L+Wu).Homology trivial subgroup of SMC is trivial for up to 4 blow ups.(McDuff-Abreu, Lalonde+Pinnsonnault+Anjos, J. Li+L+Wu)



κ = 0




Symplectic mapping class group, ie. π0Homology action of SMC generated by Lagrangian Dehn twists(L+Wu).

Homology trivial subgroup of SMC is trivial for up to 4 blow ups.(McDuff-Abreu, Lalonde+Pinnsonnault+Anjos, J. Li+L+Wu)



κ = 0




Symplectic mapping class group, ie. π0Homology action of SMC generated by Lagrangian Dehn twists(L+Wu).Homology trivial subgroup of SMC is trivial for up to 4 blow ups.(McDuff-Abreu, Lalonde+Pinnsonnault+Anjos, J. Li+L+Wu)



κ = 0


Symplectic surfaces

Given a homology class A, find the conditions for the existence ofsymplectic surfaces representing A.

Well understood when A · A is ‘big’ via Seiberg-Witten theory.Eg. For up to 8 blowups of CP2, every class of non-negativesquare is represented by a connected symplectic surface withrespect to some symplectic form.

Uniqueness up to isotopy



κ = 0


Lagrangian surfaces

Existence of Lagrangian spheres

Necessary conditions:ω · A = 0, A · A = −2, A represented by a smooth sphere.

L+Wu, Shevchishin: Also sufficient

For Homologous Lag spheres,Uniqueness up to Lagrangian isotopy for small rational manifolds(Hind, Evans, L+Wu).Uniqueness up to smooth isotopy (L+Wu).



κ = 0


Lagrangian surfaces

Existence of Lagrangian spheresNecessary conditions:ω · A = 0, A · A = −2, A represented by a smooth sphere.





κ = 0


Lagrangian surfaces



For Homologous Lag spheres,Uniqueness up to Lagrangian isotopy for small rational manifolds(Hind, Evans, L+Wu).

Uniqueness up to smooth isotopy (L+Wu).



κ = 0


Lagrangian surfaces






κ = 0

Known symplectic CY surfacesHomological classification of symplectic CY surfacesProgress towards smooth classification

Known symplectic CY surfaces I. Smooth structures

κ = 0 and minimal ⇔ Kω is a torsion class

Known manifolds with torsion Kω

1. Kahler surfaces with κ = 0:K3, Enriques surface, complex tori, Hyperelliptic surfaces2. T 2−bundles over T 2

all admit symplectic structure with κ = 0 (Thurston+Geiges)

classified up to diffeomorphism (Sakamoto-Fukuhara)

geometric manifolds modeled on: Nil3 ×R, Nil4, Sol3 ×R, R4

(Ue)

Sol manifolds

include T 4 and hyperelliptic surfaces

All fibred by T 2

fibers could be singular, multiple, or homologous trivial



κ = 0



κ = 0 and minimal ⇔ Kω is a torsion classKnown manifolds with torsion Kω

1. Kahler surfaces with κ = 0:K3, Enriques surface, complex tori, Hyperelliptic surfaces

2. T 2−bundles over T 2




(Ue)

Sol manifolds


All fibred by T 2




κ = 0



κ = 0 and minimal ⇔ Kω is a torsion classKnown manifolds with torsion Kω

1. Kahler surfaces with κ = 0:K3, Enriques surface, complex tori, Hyperelliptic surfaces2. T 2−bundles over T 2




(Ue)

Sol manifolds


All fibred by T 2




κ = 0


Known symplectic CY surfaces II. Symplectic structures

C(M) is the cone P of classes with positive square

There are winding families of symplectic forms

When b+ > 1, P is connected and πb+−1(P) = Z A windingfamily is a Sb+−1 family of symplectic forms φ : Sb+−1 → Ω whichrepresents a generator of πb+−1(P) = ZExamples:Hyperkahler family on K3 and T 4

S1 family on T 2−bundle over T 2 with b+ = 2 (including theKodaira-Thurston manifold)Question: ω/Diff + = C(M)/D(M)?For Enriques and T 2−bundle over T 2 with b+ = 1, eachconnected component of M is described this wayIf true when b+ > 1, then every symplectic form lies in somewinding family



κ = 0





When b+ > 1, P is connected and πb+−1(P) = Z A windingfamily is a Sb+−1 family of symplectic forms φ : Sb+−1 → Ω whichrepresents a generator of πb+−1(P) = Z

Examples:Hyperkahler family on K3 and T 4




κ = 0






S1 family on T 2−bundle over T 2 with b+ = 2 (including theKodaira-Thurston manifold)

Question: ω/Diff + = C(M)/D(M)?For Enriques and T 2−bundle over T 2 with b+ = 1, eachconnected component of M is described this wayIf true when b+ > 1, then every symplectic form lies in somewinding family



κ = 0









κ = 0


Betti number bounds

Theorem

(L, Bauer)b+(M) ≤ 3 if κ(M, ω) = 0

Consequently,

b1(M) ≤ 4

Euler number ≥ 0

Symplectic Noether type inequality holdsb+ ≤ 3 + |comp(Kω)|holds when κ = 0

vb1(M) ≤ 4, where vb1(M) is the supremum of b1(M) amongall finite covers M.



κ = 0


If M is minimal, then it has the same Q−cohomology ring asK3, Enriques surface or a T 2−bundle over T 2

In fact, Z−homology K3, Z−homology Enriques

The following table list possible homological invariants of κ = 0manifolds:

b1 b2 b+ χ σ known manifolds

0 22 3 24 -16 K3

0 10 1 12 -8 Enriques surface

4 6 3 0 0 4-torus

3 4 2 0 0 T 2−bundles over T 2




κ = 0


Virtual 1st Betti number and fundamental group

G a SCY group if G = π1(M) for some SCY surface (M, ω).If (M, ω) is a SCY surface, then any finite cover is also SCY.If b1(G ) = 0 then G = 1 or Z2 and the corresponding SCY surfaceunique up to homeomorphism.

If G = 1, homeomorphic to K3 (apply Freedman)

If G = Z2, homeo. to Enriques (apply Hambleton+Kreck)

If b1(G ) > 0 then 2 ≤ vb1(M) ≤ 4, χ(M) = σ(M) = 0

Friedl+Vidussi: Suppose G is SCY with b1(G ) > 0.

If H2(G ,Z[G ]) = 0 then the corresponding SCY surfaces arehomotopic to K (G , 1) ( unique up to homotopy)

If G = π1 of a (Infra)solvable manifold, then thecorresponding SCY surfaces are unique up tohomeomorphism. In particular, for all known examples, π1determines the homeomorphism type.



κ = 0


Idea of proof

If b+(M) > 1 then Kω = 0 in H2(M;Z)

⇒ Symplectic Spinc structure has a spin reduction⇒ The corresponding SW equation is Pin(2)−equivariantPin(2) = z , ι|z ∈ S1, ι2 = −1, zι = ιz1 −→ S1 −→ Pin(2) −→ Z2 −→ 1Via Furuta’s Pin(2)−equivariant finite dimensional approximationto the SW equation⇒ SW (Kω) even if b+ > 3Now invoke Taubes’ SW (Kω) = ±1



κ = 0


Idea of proof

If b+(M) > 1 then Kω = 0 in H2(M;Z)⇒ Symplectic Spinc structure has a spin reduction

⇒ The corresponding SW equation is Pin(2)−equivariantPin(2) = z , ι|z ∈ S1, ι2 = −1, zι = ιz1 −→ S1 −→ Pin(2) −→ Z2 −→ 1Via Furuta’s Pin(2)−equivariant finite dimensional approximationto the SW equation⇒ SW (Kω) even if b+ > 3Now invoke Taubes’ SW (Kω) = ±1



κ = 0


Idea of proof

If b+(M) > 1 then Kω = 0 in H2(M;Z)⇒ Symplectic Spinc structure has a spin reduction⇒ The corresponding SW equation is Pin(2)−equivariant

Pin(2) = z , ι|z ∈ S1, ι2 = −1, zι = ιz1 −→ S1 −→ Pin(2) −→ Z2 −→ 1Via Furuta’s Pin(2)−equivariant finite dimensional approximationto the SW equation⇒ SW (Kω) even if b+ > 3Now invoke Taubes’ SW (Kω) = ±1



κ = 0


Idea of proof

If b+(M) > 1 then Kω = 0 in H2(M;Z)⇒ Symplectic Spinc structure has a spin reduction⇒ The corresponding SW equation is Pin(2)−equivariantPin(2) = z , ι|z ∈ S1, ι2 = −1, zι = ιz1 −→ S1 −→ Pin(2) −→ Z2 −→ 1

Via Furuta’s Pin(2)−equivariant finite dimensional approximationto the SW equation⇒ SW (Kω) even if b+ > 3Now invoke Taubes’ SW (Kω) = ±1



κ = 0


Idea of proof

If b+(M) > 1 then Kω = 0 in H2(M;Z)⇒ Symplectic Spinc structure has a spin reduction⇒ The corresponding SW equation is Pin(2)−equivariantPin(2) = z , ι|z ∈ S1, ι2 = −1, zι = ιz1 −→ S1 −→ Pin(2) −→ Z2 −→ 1Via Furuta’s Pin(2)−equivariant finite dimensional approximationto the SW equation

⇒ SW (Kω) even if b+ > 3Now invoke Taubes’ SW (Kω) = ±1



κ = 0


Idea of proof

If b+(M) > 1 then Kω = 0 in H2(M;Z)⇒ Symplectic Spinc structure has a spin reduction⇒ The corresponding SW equation is Pin(2)−equivariantPin(2) = z , ι|z ∈ S1, ι2 = −1, zι = ιz1 −→ S1 −→ Pin(2) −→ Z2 −→ 1Via Furuta’s Pin(2)−equivariant finite dimensional approximationto the SW equation⇒ SW (Kω) even if b+ > 3

Now invoke Taubes’ SW (Kω) = ±1



κ = 0


Idea of proof

If b+(M) > 1 then Kω = 0 in H2(M;Z)⇒ Symplectic Spinc structure has a spin reduction⇒ The corresponding SW equation is Pin(2)−equivariantPin(2) = z , ι|z ∈ S1, ι2 = −1, zι = ιz1 −→ S1 −→ Pin(2) −→ Z2 −→ 1Via Furuta’s Pin(2)−equivariant finite dimensional approximationto the SW equation⇒ SW (Kω) even if b+ > 3Now invoke Taubes’ SW (Kω) = ±1



κ = 0


Speculation on smooth classification

Speculation on smooth classification:If κ(M, ω) = 0 and (M, ω) minimal,then M is diffeomorphic to K3, Enriques surface or a T 2−bundleover T 2

If (M, ω) has Kω = 0, (M, ω) is called a symplectic Calabi-Yausurface.In this case, the speculation is that M is diffeomorphic to K3 or aT 2−bundle over T 2

By the homological classification, a Degree 4 symplectichypersurface in CP3 is a homology K3.Is it diffeomorphic to K3?



κ = 0


Existence of a symplectic torus

Basically, we need to show that if κ(M, ω) = 0 then M isT 2−fibred.

Suppose a homology K3 has a winding family,L-Liu’s parametrized SW theory ⇒ there is an embeddedsymplectic torus for some symplectic form in the winding family

Existence of symplectic torus can be proved in some otherhomology types



κ = 0


Fibred sum

Positive genus sumsUsher: If (M, ω) with κ = 0 is a non-trivial positive genus fibredsum, then the summands have κ = −∞ (rational or ruled), andthe sum is along tori representing −Kω. If (M, ω) is minimal, thenM is difeomorphic to K3, Enriques surface or a T 2−bundle overT 2 with b1 = 2.

Ex: K 3 = E (1)#f E (1)Enriques =E (1)#f (S2 × T 2)The sum of two S2−bundles over T 2 along bi-sections give rise toT 2−bundles over T 2

Genus 0 sumsDorfmeister: If (M, ω) is minimal and has κ = 0 is a non-trivialgenus 0 fibred sum, then M is diffeomorphic to Enriques surface

Ex: The rational blow down of CP2#CP2

is Enriques surface



κ = 0


Lefschetz fibrations

If (M, ω) with κ = 0 is a surface bundle Σ→ M → B, thenκ(M, ω) = κ(Σ) + κ(B)⇒ Σ and B are T 2

Lefschez fibrationsSmith: If (M, ω) with κ = 0 is a genus g Lefschetz fibration withsingular fibers, then g = 1 by the adjunction formulaMoishezon+Matsumoto ⇒ M is K3If allowing multiple fibers, then M is Enriques surface



κ = 0




Lefschez fibrationsSmith: If (M, ω) with κ = 0 is a genus g Lefschetz fibration withsingular fibers, then g = 1 by the adjunction formulaMoishezon+Matsumoto ⇒ M is K3

If allowing multiple fibers, then M is Enriques surface



κ = 0




Lefschez fibrationsSmith: If (M, ω) with κ = 0 is a genus g Lefschetz fibration withsingular fibers, then g = 1 by the adjunction formulaMoishezon+Matsumoto ⇒ M is K3If allowing multiple fibers, then M is Enriques surface



κ = 0


Circle bundles

Friedl-Vidussi: If S1 → (M, ω)→ Y has Kω = 0 and is a circlebundle, then Y is a T 2−bundle.

(Torsion Euler class case can be reduced to their solution toTaubes’ conjecture)

In the non-torsion case, the first step is to use the properties ofsum of coefficients of Alexander polynomial to showVirtual Betti number of Y with coefficient Z or Fp is bounded by 3Non-torsion Euler class⇒ b1(Y ) > 1McCarthy+Bowdin: Y is irreducibleSo Y is Haken with b1 > 1



κ = 0


Circle bundles



In the non-torsion case, the first step is to use the properties ofsum of coefficients of Alexander polynomial to showVirtual Betti number of Y with coefficient Z or Fp is bounded by 3

Non-torsion Euler class⇒ b1(Y ) > 1McCarthy+Bowdin: Y is irreducibleSo Y is Haken with b1 > 1



κ = 0


Circle bundles



In the non-torsion case, the first step is to use the properties ofsum of coefficients of Alexander polynomial to showVirtual Betti number of Y with coefficient Z or Fp is bounded by 3Non-torsion Euler class⇒ b1(Y ) > 1McCarthy+Bowdin: Y is irreducibleSo Y is Haken with b1 > 1



κ = 0


If Y has non-trivial JSJ decomposition, or Seifeit fibred, thenit contains an incompressible torus that is not a fiber

vb1(Y ) ≤ 3, together with Kojima, Luecke ⇒ a finite cover Yis a T 2−bundleb1(Y ) ≥ 2 ⇒ Y is itself a T 2−bundle

Y is not hyperbolic(vb1(Y ) should be ∞ if b1(Y ) ≥ 1)Lubotzky alternative+ Lackenby ⇒ vb1(Y ,Fp) =∞

Question: whether the Fp−virtual Betti number of a symplecticCalabi-Yau surface is bounded by 4.



κ = 0


If Y has non-trivial JSJ decomposition, or Seifeit fibred, thenit contains an incompressible torus that is not a fibervb1(Y ) ≤ 3, together with Kojima, Luecke ⇒ a finite cover Yis a T 2−bundle

b1(Y ) ≥ 2 ⇒ Y is itself a T 2−bundle





κ = 0


If Y has non-trivial JSJ decomposition, or Seifeit fibred, thenit contains an incompressible torus that is not a fibervb1(Y ) ≤ 3, together with Kojima, Luecke ⇒ a finite cover Yis a T 2−bundleb1(Y ) ≥ 2 ⇒ Y is itself a T 2−bundle





κ = 0


Mapping tori

L-Ni:3-manifolds torus bundleSeifert fibred spacesirreducible with nontrivial JSJ decomposition⇒ vb1(M) =∞hyperbolic



κ = 0


Given a Lagrangian torus L in a symplectic 4-manifold, there is asymplectic surgery along L, depending on an integer k and anembedded curve in L.It is called a Luttinger surgery. Topologically, it is a generalized logtransform of multiplicity 1 and auxiliary multiplicity kHo-L:Luttinger surgery preserves the symplectic Kodaira dimension κIn particular, it is a symplectic CY surgery

The following table list possible homological invariants of κ = 0manifolds:



κ = 0


Given a Lagrangian torus L in a symplectic 4-manifold, there is asymplectic surgery along L, depending on an integer k and anembedded curve in L.It is called a Luttinger surgery. Topologically, it is a generalized logtransform of multiplicity 1 and auxiliary multiplicity kHo-L:Luttinger surgery preserves the symplectic Kodaira dimension κIn particular, it is a symplectic CY surgeryThe following table list possible homological invariants of κ = 0manifolds:



κ = 0



0 22 3 24 -16 K3


4 6 3 0 0 4-torus



Luttinger surgery changes b2 at most by 2⇒ Luttinger surgery preserves

ZZ−homology K3Q−homology T 2−bundle over T 2

This implies some framing constraints on Lag tori in κ = 0manifoldsQuestion: Can we obtain any T 2−bundle over T 2 from T 4 viaLuttinger surgeries?True for all Lag T 2−bundles, including all T 2−bundles over T 2

with b1 ≥ 3.



κ = 0



0 22 3 24 -16 K3


4 6 3 0 0 4-torus



Luttinger surgery changes b2 at most by 2⇒ Luttinger surgery preserves

ZZ−homology K3Q−homology T 2−bundle over T 2

This implies some framing constraints on Lag tori in κ = 0manifoldsQuestion: Can we obtain any T 2−bundle over T 2 from T 4 viaLuttinger surgeries?True for all Lag T 2−bundles, including all T 2−bundles over T 2

with b1 ≥ 3.Tian-Jun Li Symplectic 4-manifolds with non-positive Kodaira dimension

Symplectic 4-manifolds with non-positive Kodaira dimension

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