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Motivation Tamed and calibrated On some cohomological properties almost complex structures Symplectic cones C pure and full of almost complex manifolds almost complex structures Calibrated and 4-dimensional case Example of non C pure

  1. Symplectic cones Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full C ( M ) : symplectic cone of M , i.e. the image of the space of almost complex symplectic forms on M compatible with the orientation by the structures Calibrated and projection ω �→ [ ω ] ∈ H 2 ( M , R ) . 4-dimensional case Example of non C∞ pure almost complex structure T. J. Li e W. Zhang studied the following subcones of C ( M ) : the Pure and full almost J -tamed symplectic cone complex structures Main result Sketch of the proof K t [ ω ] ∈ H 2 ( M , R ) | ω is tamed by J � � J ( M ) = Link with Hard Lefschetz condition Sketch of the Proof Integrable case and the J -compatible symplectic cone Examples Nakamura manifold K c [ ω ] ∈ H 2 ( M , R ) | ω is compatible with J � � J ( M ) = . Families in dimension six References For almost-Kähler manifolds ( M , J , ω ) , the cone K c J ( M ) � = ∅ and if J is integrable K c J ( M ) coincides with the Kähler cone. 5

  2. Symplectic cones Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full C ( M ) : symplectic cone of M , i.e. the image of the space of almost complex symplectic forms on M compatible with the orientation by the structures Calibrated and projection ω �→ [ ω ] ∈ H 2 ( M , R ) . 4-dimensional case Example of non C∞ pure almost complex structure T. J. Li e W. Zhang studied the following subcones of C ( M ) : the Pure and full almost J -tamed symplectic cone complex structures Main result Sketch of the proof K t [ ω ] ∈ H 2 ( M , R ) | ω is tamed by J � � J ( M ) = Link with Hard Lefschetz condition Sketch of the Proof Integrable case and the J -compatible symplectic cone Examples Nakamura manifold K c [ ω ] ∈ H 2 ( M , R ) | ω is compatible with J � � J ( M ) = . Families in dimension six References For almost-Kähler manifolds ( M , J , ω ) , the cone K c J ( M ) � = ∅ and if J is integrable K c J ( M ) coincides with the Kähler cone. 5

  3. Symplectic cones Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full C ( M ) : symplectic cone of M , i.e. the image of the space of almost complex symplectic forms on M compatible with the orientation by the structures Calibrated and projection ω �→ [ ω ] ∈ H 2 ( M , R ) . 4-dimensional case Example of non C∞ pure almost complex structure T. J. Li e W. Zhang studied the following subcones of C ( M ) : the Pure and full almost J -tamed symplectic cone complex structures Main result Sketch of the proof K t [ ω ] ∈ H 2 ( M , R ) | ω is tamed by J � � J ( M ) = Link with Hard Lefschetz condition Sketch of the Proof Integrable case and the J -compatible symplectic cone Examples Nakamura manifold K c [ ω ] ∈ H 2 ( M , R ) | ω is compatible with J � � J ( M ) = . Families in dimension six References For almost-Kähler manifolds ( M , J , ω ) , the cone K c J ( M ) � = ∅ and if J is integrable K c J ( M ) coincides with the Kähler cone. 5

  4. Symplectic cones Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full C ( M ) : symplectic cone of M , i.e. the image of the space of almost complex symplectic forms on M compatible with the orientation by the structures Calibrated and projection ω �→ [ ω ] ∈ H 2 ( M , R ) . 4-dimensional case Example of non C∞ pure almost complex structure T. J. Li e W. Zhang studied the following subcones of C ( M ) : the Pure and full almost J -tamed symplectic cone complex structures Main result Sketch of the proof K t [ ω ] ∈ H 2 ( M , R ) | ω is tamed by J � � J ( M ) = Link with Hard Lefschetz condition Sketch of the Proof Integrable case and the J -compatible symplectic cone Examples Nakamura manifold K c [ ω ] ∈ H 2 ( M , R ) | ω is compatible with J � � J ( M ) = . Families in dimension six References For almost-Kähler manifolds ( M , J , ω ) , the cone K c J ( M ) � = ∅ and if J is integrable K c J ( M ) coincides with the Kähler cone. 5

  5. Motivation Tamed and calibrated almost complex structures Theorem (Li, Zhang) Symplectic cones C∞ pure and full If J is integrable and K c J ( M ) � = ∅ , one has almost complex structures Calibrated and � � ( H 2 , 0 ∂ ( M ) ⊕ H 0 , 2 4-dimensional case K t J ( M ) = K c ∂ ( M )) ∩ H 2 ( M , R ) J ( M ) + , Example of non C∞ pure almost complex structure � � H 1 , 1 K t ∂ ( M ) ∩ H 2 ( M , R ) = K c J ( M ) ∩ J ( M ) . Pure and full almost complex structures Main result Sketch of the proof Link with Hard Lefschetz condition Sketch of the Proof Problem Integrable case Find a relation between K t J ( M ) and K c J ( M ) in the case that J is Examples Nakamura manifold non integrable, related to the question by Donaldson for n = 2 : Families in dimension six if K t J ( M ) � = ∅ for some J, then K c J ( M ) � = ∅ as well? References To solve this problem Li and Zhang introduced the analogous of the previous (real) Dolbeault groups for general almost complex manifolds ( M , J ) . 6

  6. Motivation Tamed and calibrated almost complex structures Theorem (Li, Zhang) Symplectic cones C∞ pure and full If J is integrable and K c J ( M ) � = ∅ , one has almost complex structures Calibrated and � � ( H 2 , 0 ∂ ( M ) ⊕ H 0 , 2 4-dimensional case K t J ( M ) = K c ∂ ( M )) ∩ H 2 ( M , R ) J ( M ) + , Example of non C∞ pure almost complex structure � � H 1 , 1 K t ∂ ( M ) ∩ H 2 ( M , R ) = K c J ( M ) ∩ J ( M ) . Pure and full almost complex structures Main result Sketch of the proof Link with Hard Lefschetz condition Sketch of the Proof Problem Integrable case Find a relation between K t J ( M ) and K c J ( M ) in the case that J is Examples Nakamura manifold non integrable, related to the question by Donaldson for n = 2 : Families in dimension six if K t J ( M ) � = ∅ for some J, then K c J ( M ) � = ∅ as well? References To solve this problem Li and Zhang introduced the analogous of the previous (real) Dolbeault groups for general almost complex manifolds ( M , J ) . 6

  7. Motivation Tamed and calibrated almost complex structures Theorem (Li, Zhang) Symplectic cones C∞ pure and full If J is integrable and K c J ( M ) � = ∅ , one has almost complex structures Calibrated and � � ( H 2 , 0 ∂ ( M ) ⊕ H 0 , 2 4-dimensional case K t J ( M ) = K c ∂ ( M )) ∩ H 2 ( M , R ) J ( M ) + , Example of non C∞ pure almost complex structure � � H 1 , 1 K t ∂ ( M ) ∩ H 2 ( M , R ) = K c J ( M ) ∩ J ( M ) . Pure and full almost complex structures Main result Sketch of the proof Link with Hard Lefschetz condition Sketch of the Proof Problem Integrable case Find a relation between K t J ( M ) and K c J ( M ) in the case that J is Examples Nakamura manifold non integrable, related to the question by Donaldson for n = 2 : Families in dimension six if K t J ( M ) � = ∅ for some J, then K c J ( M ) � = ∅ as well? References To solve this problem Li and Zhang introduced the analogous of the previous (real) Dolbeault groups for general almost complex manifolds ( M , J ) . 6

  8. C ∞ pure and full almost complex structures Motivation Tamed and calibrated almost complex structures On ( M , J ) for the space Ω k ( M ) R of real smooth differential Symplectic cones C∞ pure and full k -forms one has: almost complex structures � Ω p , q Calibrated and Ω k ( M ) R = J ( M ) R , 4-dimensional case Example of non C∞ pure almost complex structure p + q = k Pure and full almost complex structures where Main result Sketch of the proof Ω p , q α ∈ Ω p , q J ( M ) ⊕ Ω q , p Link with Hard Lefschetz � � J ( M ) R = J ( M ) | α = α . condition Sketch of the Proof Integrable case Examples Nakamura manifold S : a finite set of pairs of integers. Let Families in dimension six References � Z p , q � B p , q Z S B S J = , J = J , J ( p , q ) ∈ S ( p , q ) ∈ S where Z p , q and B p , q are the spaces of real d -closed (resp. J J d -exacts) ( p , q ) -forms. 7

  9. C ∞ pure and full almost complex structures Motivation Tamed and calibrated almost complex structures On ( M , J ) for the space Ω k ( M ) R of real smooth differential Symplectic cones C∞ pure and full k -forms one has: almost complex structures � Ω p , q Calibrated and Ω k ( M ) R = J ( M ) R , 4-dimensional case Example of non C∞ pure almost complex structure p + q = k Pure and full almost complex structures where Main result Sketch of the proof Ω p , q α ∈ Ω p , q J ( M ) ⊕ Ω q , p Link with Hard Lefschetz � � J ( M ) R = J ( M ) | α = α . condition Sketch of the Proof Integrable case Examples Nakamura manifold S : a finite set of pairs of integers. Let Families in dimension six References � Z p , q � B p , q Z S B S J = , J = J , J ( p , q ) ∈ S ( p , q ) ∈ S where Z p , q and B p , q are the spaces of real d -closed (resp. J J d -exacts) ( p , q ) -forms. 7

  10. C ∞ pure and full almost complex structures Motivation Tamed and calibrated almost complex structures On ( M , J ) for the space Ω k ( M ) R of real smooth differential Symplectic cones C∞ pure and full k -forms one has: almost complex structures � Ω p , q Calibrated and Ω k ( M ) R = J ( M ) R , 4-dimensional case Example of non C∞ pure almost complex structure p + q = k Pure and full almost complex structures where Main result Sketch of the proof Ω p , q α ∈ Ω p , q J ( M ) ⊕ Ω q , p Link with Hard Lefschetz � � J ( M ) R = J ( M ) | α = α . condition Sketch of the Proof Integrable case Examples Nakamura manifold S : a finite set of pairs of integers. Let Families in dimension six References � Z p , q � B p , q Z S B S J = , J = J , J ( p , q ) ∈ S ( p , q ) ∈ S where Z p , q and B p , q are the spaces of real d -closed (resp. J J d -exacts) ( p , q ) -forms. 7

  11. Motivation Tamed and calibrated almost complex structures Symplectic cones There is a natural map C∞ pure and full almost complex structures ρ S : Z S J / B S J → Z S J / B , Calibrated and 4-dimensional case Example of non C∞ pure where B is the space of d -exact forms. almost complex structure Pure and full almost complex structures We will write ρ S ( Z S J / B S J ) as Z S J / B S J . Main result Sketch of the proof Define Link with Hard Lefschetz condition = Z S � � Sketch of the Proof H S [ α ] | α ∈ Z S J J ( M ) R = B . Integrable case J Examples Nakamura manifold Families in dimension six Then References J ( M ) R + H ( 2 , 0 ) , ( 0 , 2 ) H 1 , 1 ( M ) R ⊆ H 2 ( M , R ) . J 8

  12. Motivation Tamed and calibrated almost complex structures Symplectic cones There is a natural map C∞ pure and full almost complex structures ρ S : Z S J / B S J → Z S J / B , Calibrated and 4-dimensional case Example of non C∞ pure where B is the space of d -exact forms. almost complex structure Pure and full almost complex structures We will write ρ S ( Z S J / B S J ) as Z S J / B S J . Main result Sketch of the proof Define Link with Hard Lefschetz condition = Z S � � Sketch of the Proof H S [ α ] | α ∈ Z S J J ( M ) R = B . Integrable case J Examples Nakamura manifold Families in dimension six Then References J ( M ) R + H ( 2 , 0 ) , ( 0 , 2 ) H 1 , 1 ( M ) R ⊆ H 2 ( M , R ) . J 8

  13. Motivation Tamed and calibrated almost complex structures Symplectic cones There is a natural map C∞ pure and full almost complex structures ρ S : Z S J / B S J → Z S J / B , Calibrated and 4-dimensional case Example of non C∞ pure where B is the space of d -exact forms. almost complex structure Pure and full almost complex structures We will write ρ S ( Z S J / B S J ) as Z S J / B S J . Main result Sketch of the proof Define Link with Hard Lefschetz condition = Z S � � Sketch of the Proof H S [ α ] | α ∈ Z S J J ( M ) R = B . Integrable case J Examples Nakamura manifold Families in dimension six Then References J ( M ) R + H ( 2 , 0 ) , ( 0 , 2 ) H 1 , 1 ( M ) R ⊆ H 2 ( M , R ) . J 8

  14. Motivation Tamed and calibrated almost complex structures Symplectic cones There is a natural map C∞ pure and full almost complex structures ρ S : Z S J / B S J → Z S J / B , Calibrated and 4-dimensional case Example of non C∞ pure where B is the space of d -exact forms. almost complex structure Pure and full almost complex structures We will write ρ S ( Z S J / B S J ) as Z S J / B S J . Main result Sketch of the proof Define Link with Hard Lefschetz condition = Z S � � Sketch of the Proof H S [ α ] | α ∈ Z S J J ( M ) R = B . Integrable case J Examples Nakamura manifold Families in dimension six Then References J ( M ) R + H ( 2 , 0 ) , ( 0 , 2 ) H 1 , 1 ( M ) R ⊆ H 2 ( M , R ) . J 8

  15. Motivation Tamed and calibrated almost complex structures Symplectic cones There is a natural map C∞ pure and full almost complex structures ρ S : Z S J / B S J → Z S J / B , Calibrated and 4-dimensional case Example of non C∞ pure where B is the space of d -exact forms. almost complex structure Pure and full almost complex structures We will write ρ S ( Z S J / B S J ) as Z S J / B S J . Main result Sketch of the proof Define Link with Hard Lefschetz condition = Z S � � Sketch of the Proof H S [ α ] | α ∈ Z S J J ( M ) R = B . Integrable case J Examples Nakamura manifold Families in dimension six Then References J ( M ) R + H ( 2 , 0 ) , ( 0 , 2 ) H 1 , 1 ( M ) R ⊆ H 2 ( M , R ) . J 8

  16. Motivation Tamed and calibrated Definition (Li, Zhang) almost complex structures Symplectic cones J is C ∞ pure and full if and only if C∞ pure and full almost complex structures J ( M ) R ⊕ H ( 2 , 0 ) , ( 0 , 2 ) H 2 ( M , R ) = H 1 , 1 ( M ) R . Calibrated and 4-dimensional case J Example of non C∞ pure almost complex structure • J is C ∞ pure if and only if H 1 , 1 J ( M ) R ∩ H ( 2 , 0 ) , ( 0 , 2 ) ( M ) R = { 0 } . Pure and full almost J complex structures • J is C ∞ full if and only if Main result Sketch of the proof Link with Hard Lefschetz J ( M ) R + H ( 2 , 0 ) , ( 0 , 2 ) H 2 ( M , R ) = H 1 , 1 condition ( M ) R . Sketch of the Proof J Integrable case Examples Nakamura manifold Families in dimension six Theorem (Li, Zhang) References If J is a C ∞ full almost complex structure and K c J ( M ) � = ∅ , then J ( M ) + H ( 2 , 0 ) , ( 0 , 2 ) K t J ( M ) = K c ( M ) R . J 9

  17. Motivation Tamed and calibrated Definition (Li, Zhang) almost complex structures Symplectic cones J is C ∞ pure and full if and only if C∞ pure and full almost complex structures J ( M ) R ⊕ H ( 2 , 0 ) , ( 0 , 2 ) H 2 ( M , R ) = H 1 , 1 ( M ) R . Calibrated and 4-dimensional case J Example of non C∞ pure almost complex structure • J is C ∞ pure if and only if H 1 , 1 J ( M ) R ∩ H ( 2 , 0 ) , ( 0 , 2 ) ( M ) R = { 0 } . Pure and full almost J complex structures • J is C ∞ full if and only if Main result Sketch of the proof Link with Hard Lefschetz J ( M ) R + H ( 2 , 0 ) , ( 0 , 2 ) H 2 ( M , R ) = H 1 , 1 condition ( M ) R . Sketch of the Proof J Integrable case Examples Nakamura manifold Families in dimension six Theorem (Li, Zhang) References If J is a C ∞ full almost complex structure and K c J ( M ) � = ∅ , then J ( M ) + H ( 2 , 0 ) , ( 0 , 2 ) K t J ( M ) = K c ( M ) R . J 9

  18. Motivation Tamed and calibrated Definition (Li, Zhang) almost complex structures Symplectic cones J is C ∞ pure and full if and only if C∞ pure and full almost complex structures J ( M ) R ⊕ H ( 2 , 0 ) , ( 0 , 2 ) H 2 ( M , R ) = H 1 , 1 ( M ) R . Calibrated and 4-dimensional case J Example of non C∞ pure almost complex structure • J is C ∞ pure if and only if H 1 , 1 J ( M ) R ∩ H ( 2 , 0 ) , ( 0 , 2 ) ( M ) R = { 0 } . Pure and full almost J complex structures • J is C ∞ full if and only if Main result Sketch of the proof Link with Hard Lefschetz J ( M ) R + H ( 2 , 0 ) , ( 0 , 2 ) H 2 ( M , R ) = H 1 , 1 condition ( M ) R . Sketch of the Proof J Integrable case Examples Nakamura manifold Families in dimension six Theorem (Li, Zhang) References If J is a C ∞ full almost complex structure and K c J ( M ) � = ∅ , then J ( M ) + H ( 2 , 0 ) , ( 0 , 2 ) K t J ( M ) = K c ( M ) R . J 9

  19. Calibrated and 4 -dimensional case Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex Proposition (–, Tomassini) structures Calibrated and 4-dimensional case Let ω be a symplectic form on a compact manifold M 2 n . If J is Example of non C∞ pure almost complex structure an almost complex structure on M 2 n calibrated by ω , then J is Pure and full almost C ∞ pure. complex structures Main result Sketch of the proof Link with Hard Lefschetz Theorem (Draghici, Li, Zhang) condition Sketch of the Proof On a compact manifold M 4 of real dimension 4 any almost Integrable case Examples complex structure is C ∞ pure and full. Nakamura manifold Families in dimension six References Problem Does the previous property hold in higher dimension? 10

  20. Calibrated and 4 -dimensional case Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex Proposition (–, Tomassini) structures Calibrated and 4-dimensional case Let ω be a symplectic form on a compact manifold M 2 n . If J is Example of non C∞ pure almost complex structure an almost complex structure on M 2 n calibrated by ω , then J is Pure and full almost C ∞ pure. complex structures Main result Sketch of the proof Link with Hard Lefschetz Theorem (Draghici, Li, Zhang) condition Sketch of the Proof On a compact manifold M 4 of real dimension 4 any almost Integrable case Examples complex structure is C ∞ pure and full. Nakamura manifold Families in dimension six References Problem Does the previous property hold in higher dimension? 10

  21. Calibrated and 4 -dimensional case Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex Proposition (–, Tomassini) structures Calibrated and 4-dimensional case Let ω be a symplectic form on a compact manifold M 2 n . If J is Example of non C∞ pure almost complex structure an almost complex structure on M 2 n calibrated by ω , then J is Pure and full almost C ∞ pure. complex structures Main result Sketch of the proof Link with Hard Lefschetz Theorem (Draghici, Li, Zhang) condition Sketch of the Proof On a compact manifold M 4 of real dimension 4 any almost Integrable case Examples complex structure is C ∞ pure and full. Nakamura manifold Families in dimension six References Problem Does the previous property hold in higher dimension? 10

  22. Calibrated and 4 -dimensional case Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex Proposition (–, Tomassini) structures Calibrated and 4-dimensional case Let ω be a symplectic form on a compact manifold M 2 n . If J is Example of non C∞ pure almost complex structure an almost complex structure on M 2 n calibrated by ω , then J is Pure and full almost C ∞ pure. complex structures Main result Sketch of the proof Link with Hard Lefschetz Theorem (Draghici, Li, Zhang) condition Sketch of the Proof On a compact manifold M 4 of real dimension 4 any almost Integrable case Examples complex structure is C ∞ pure and full. Nakamura manifold Families in dimension six References Problem Does the previous property hold in higher dimension? 10

  23. Example of non C ∞ pure almost complex structure Motivation Tamed and calibrated A compact manifold of real dimension 6 may admit non C ∞ almost complex structures Symplectic cones pure almost complex structures. C∞ pure and full almost complex structures Example Calibrated and 4-dimensional case Consider the nilmanifold M 6 , compact quotient of the Lie group: Example of non C∞ pure almost complex structure Pure and full almost de j = 0 ,  j = 1 , . . . , 4 , complex structures  Main result  de 5 = e 12 , Sketch of the proof Link with Hard Lefschetz de 6 = e 13 . condition   Sketch of the Proof Integrable case Examples The left-invariant almost complex structure on M 6 , defined by Nakamura manifold Families in dimension six η 1 = e 1 + ie 2 , η 2 = e 3 + ie 4 , η 3 = e 5 + ie 6 , References is not C ∞ pure, since one has that [ Re ( η 1 ∧ η 2 )] = [ e 13 + e 24 ] = [ e 24 ] = [ Re ( η 1 ∧ η 2 )] = [ e 13 − e 24 ] . 11

  24. Example of non C ∞ pure almost complex structure Motivation Tamed and calibrated A compact manifold of real dimension 6 may admit non C ∞ almost complex structures Symplectic cones pure almost complex structures. C∞ pure and full almost complex structures Example Calibrated and 4-dimensional case Consider the nilmanifold M 6 , compact quotient of the Lie group: Example of non C∞ pure almost complex structure Pure and full almost de j = 0 ,  j = 1 , . . . , 4 , complex structures  Main result  de 5 = e 12 , Sketch of the proof Link with Hard Lefschetz de 6 = e 13 . condition   Sketch of the Proof Integrable case Examples The left-invariant almost complex structure on M 6 , defined by Nakamura manifold Families in dimension six η 1 = e 1 + ie 2 , η 2 = e 3 + ie 4 , η 3 = e 5 + ie 6 , References is not C ∞ pure, since one has that [ Re ( η 1 ∧ η 2 )] = [ e 13 + e 24 ] = [ e 24 ] = [ Re ( η 1 ∧ η 2 )] = [ e 13 − e 24 ] . 11

  25. Example of non C ∞ pure almost complex structure Motivation Tamed and calibrated A compact manifold of real dimension 6 may admit non C ∞ almost complex structures Symplectic cones pure almost complex structures. C∞ pure and full almost complex structures Example Calibrated and 4-dimensional case Consider the nilmanifold M 6 , compact quotient of the Lie group: Example of non C∞ pure almost complex structure Pure and full almost de j = 0 ,  j = 1 , . . . , 4 , complex structures  Main result  de 5 = e 12 , Sketch of the proof Link with Hard Lefschetz de 6 = e 13 . condition   Sketch of the Proof Integrable case Examples The left-invariant almost complex structure on M 6 , defined by Nakamura manifold Families in dimension six η 1 = e 1 + ie 2 , η 2 = e 3 + ie 4 , η 3 = e 5 + ie 6 , References is not C ∞ pure, since one has that [ Re ( η 1 ∧ η 2 )] = [ e 13 + e 24 ] = [ e 24 ] = [ Re ( η 1 ∧ η 2 )] = [ e 13 − e 24 ] . 11

  26. Example of non C ∞ pure almost complex structure Motivation Tamed and calibrated A compact manifold of real dimension 6 may admit non C ∞ almost complex structures Symplectic cones pure almost complex structures. C∞ pure and full almost complex structures Example Calibrated and 4-dimensional case Consider the nilmanifold M 6 , compact quotient of the Lie group: Example of non C∞ pure almost complex structure Pure and full almost de j = 0 ,  j = 1 , . . . , 4 , complex structures  Main result  de 5 = e 12 , Sketch of the proof Link with Hard Lefschetz de 6 = e 13 . condition   Sketch of the Proof Integrable case Examples The left-invariant almost complex structure on M 6 , defined by Nakamura manifold Families in dimension six η 1 = e 1 + ie 2 , η 2 = e 3 + ie 4 , η 3 = e 5 + ie 6 , References is not C ∞ pure, since one has that [ Re ( η 1 ∧ η 2 )] = [ e 13 + e 24 ] = [ e 24 ] = [ Re ( η 1 ∧ η 2 )] = [ e 13 − e 24 ] . 11

  27. Pure and full almost complex structures Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex ( M , J ) (almost) complex manifold of (real) dimension 2 n . structures Calibrated and 4-dimensional case Example of non C∞ pure E k ( M ) the space of k -currents on M , i.e. the topological dual of almost complex structure Ω 2 n − k ( M ) . Pure and full almost complex structures Since the smooth k -forms can be considered as Main result Sketch of the proof ( 2 n − k ) -currents, then Link with Hard Lefschetz condition Sketch of the Proof H k ( M , R ) ∼ = H 2 n − k ( M , R ) , Integrable case Examples Nakamura manifold where H k ( M , R ) is the k -th de Rham homology group. Families in dimension six References • A k -current is a boundary if and only if it vanishes on the space of closed k -forms. 12

  28. Pure and full almost complex structures Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex ( M , J ) (almost) complex manifold of (real) dimension 2 n . structures Calibrated and 4-dimensional case Example of non C∞ pure E k ( M ) the space of k -currents on M , i.e. the topological dual of almost complex structure Ω 2 n − k ( M ) . Pure and full almost complex structures Since the smooth k -forms can be considered as Main result Sketch of the proof ( 2 n − k ) -currents, then Link with Hard Lefschetz condition Sketch of the Proof H k ( M , R ) ∼ = H 2 n − k ( M , R ) , Integrable case Examples Nakamura manifold where H k ( M , R ) is the k -th de Rham homology group. Families in dimension six References • A k -current is a boundary if and only if it vanishes on the space of closed k -forms. 12

  29. Pure and full almost complex structures Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex ( M , J ) (almost) complex manifold of (real) dimension 2 n . structures Calibrated and 4-dimensional case Example of non C∞ pure E k ( M ) the space of k -currents on M , i.e. the topological dual of almost complex structure Ω 2 n − k ( M ) . Pure and full almost complex structures Since the smooth k -forms can be considered as Main result Sketch of the proof ( 2 n − k ) -currents, then Link with Hard Lefschetz condition Sketch of the Proof H k ( M , R ) ∼ = H 2 n − k ( M , R ) , Integrable case Examples Nakamura manifold where H k ( M , R ) is the k -th de Rham homology group. Families in dimension six References • A k -current is a boundary if and only if it vanishes on the space of closed k -forms. 12

  30. Pure and full almost complex structures Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex ( M , J ) (almost) complex manifold of (real) dimension 2 n . structures Calibrated and 4-dimensional case Example of non C∞ pure E k ( M ) the space of k -currents on M , i.e. the topological dual of almost complex structure Ω 2 n − k ( M ) . Pure and full almost complex structures Since the smooth k -forms can be considered as Main result Sketch of the proof ( 2 n − k ) -currents, then Link with Hard Lefschetz condition Sketch of the Proof H k ( M , R ) ∼ = H 2 n − k ( M , R ) , Integrable case Examples Nakamura manifold where H k ( M , R ) is the k -th de Rham homology group. Families in dimension six References • A k -current is a boundary if and only if it vanishes on the space of closed k -forms. 12

  31. Pure and full almost complex structures Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex ( M , J ) (almost) complex manifold of (real) dimension 2 n . structures Calibrated and 4-dimensional case Example of non C∞ pure E k ( M ) the space of k -currents on M , i.e. the topological dual of almost complex structure Ω 2 n − k ( M ) . Pure and full almost complex structures Since the smooth k -forms can be considered as Main result Sketch of the proof ( 2 n − k ) -currents, then Link with Hard Lefschetz condition Sketch of the Proof H k ( M , R ) ∼ = H 2 n − k ( M , R ) , Integrable case Examples Nakamura manifold where H k ( M , R ) is the k -th de Rham homology group. Families in dimension six References • A k -current is a boundary if and only if it vanishes on the space of closed k -forms. 12

  32. On ( M , J ) for the space of real k -currents E k ( M ) R one has: Motivation Tamed and calibrated almost complex structures Symplectic cones � E J E k ( M ) R = p , q ( M ) R , C∞ pure and full almost complex p + q = k structures Calibrated and 4-dimensional case where E J p , q ( M ) R is the space of real k -currents of bidimension Example of non C∞ pure almost complex structure ( p , q ) . Pure and full almost complex structures S : a finite set of pairs of integers. Let Main result Sketch of the proof Link with Hard Lefschetz � � condition Z J Z J B J B J S = p , q , S = p , q , Sketch of the Proof Integrable case ( p , q ) ∈ S ( p , q ) ∈ S Examples Nakamura manifold where Z J p , q and B J p , q are the space of real d -closed (resp. Families in dimension six References boundary) currents of bidimension ( p , q ) . Define = Z J H J [ α ] | α ∈ Z J S � � S ( M ) R = B , S where B denotes the space of currents which are boundaries. 13

  33. On ( M , J ) for the space of real k -currents E k ( M ) R one has: Motivation Tamed and calibrated almost complex structures Symplectic cones � E J E k ( M ) R = p , q ( M ) R , C∞ pure and full almost complex p + q = k structures Calibrated and 4-dimensional case where E J p , q ( M ) R is the space of real k -currents of bidimension Example of non C∞ pure almost complex structure ( p , q ) . Pure and full almost complex structures S : a finite set of pairs of integers. Let Main result Sketch of the proof Link with Hard Lefschetz � � condition Z J Z J B J B J S = p , q , S = p , q , Sketch of the Proof Integrable case ( p , q ) ∈ S ( p , q ) ∈ S Examples Nakamura manifold where Z J p , q and B J p , q are the space of real d -closed (resp. Families in dimension six References boundary) currents of bidimension ( p , q ) . Define = Z J H J [ α ] | α ∈ Z J S � � S ( M ) R = B , S where B denotes the space of currents which are boundaries. 13

  34. On ( M , J ) for the space of real k -currents E k ( M ) R one has: Motivation Tamed and calibrated almost complex structures Symplectic cones � E J E k ( M ) R = p , q ( M ) R , C∞ pure and full almost complex p + q = k structures Calibrated and 4-dimensional case where E J p , q ( M ) R is the space of real k -currents of bidimension Example of non C∞ pure almost complex structure ( p , q ) . Pure and full almost complex structures S : a finite set of pairs of integers. Let Main result Sketch of the proof Link with Hard Lefschetz � � condition Z J Z J B J B J S = p , q , S = p , q , Sketch of the Proof Integrable case ( p , q ) ∈ S ( p , q ) ∈ S Examples Nakamura manifold where Z J p , q and B J p , q are the space of real d -closed (resp. Families in dimension six References boundary) currents of bidimension ( p , q ) . Define = Z J H J [ α ] | α ∈ Z J S � � S ( M ) R = B , S where B denotes the space of currents which are boundaries. 13

  35. On ( M , J ) for the space of real k -currents E k ( M ) R one has: Motivation Tamed and calibrated almost complex structures Symplectic cones � E J E k ( M ) R = p , q ( M ) R , C∞ pure and full almost complex p + q = k structures Calibrated and 4-dimensional case where E J p , q ( M ) R is the space of real k -currents of bidimension Example of non C∞ pure almost complex structure ( p , q ) . Pure and full almost complex structures S : a finite set of pairs of integers. Let Main result Sketch of the proof Link with Hard Lefschetz � � condition Z J Z J B J B J S = p , q , S = p , q , Sketch of the Proof Integrable case ( p , q ) ∈ S ( p , q ) ∈ S Examples Nakamura manifold where Z J p , q and B J p , q are the space of real d -closed (resp. Families in dimension six References boundary) currents of bidimension ( p , q ) . Define = Z J H J [ α ] | α ∈ Z J S � � S ( M ) R = B , S where B denotes the space of currents which are boundaries. 13

  36. Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex structures Definition (Li, Zhang) Calibrated and 4-dimensional case Example of non C∞ pure An almost complex structure J is pure if almost complex structure H J 1 , 1 ( M ) R ∩ H J ( 2 , 0 ) , ( 0 , 2 ) ( M ) R = { 0 } or equivalently if Pure and full almost complex structures π 1 , 1 B 2 ∩ Z J 1 , 1 = B J 1 , 1 . Main result Sketch of the proof J is full if H 2 ( M , R ) = H J 1 , 1 ( M ) R + H J ( 2 , 0 ) , ( 0 , 2 ) ( M ) R . Link with Hard Lefschetz condition Sketch of the Proof Integrable case Examples Problem Nakamura manifold Relation between C ∞ pure and full and pure and full? Families in dimension six References 14

  37. Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex structures Definition (Li, Zhang) Calibrated and 4-dimensional case Example of non C∞ pure An almost complex structure J is pure if almost complex structure H J 1 , 1 ( M ) R ∩ H J ( 2 , 0 ) , ( 0 , 2 ) ( M ) R = { 0 } or equivalently if Pure and full almost complex structures π 1 , 1 B 2 ∩ Z J 1 , 1 = B J 1 , 1 . Main result Sketch of the proof J is full if H 2 ( M , R ) = H J 1 , 1 ( M ) R + H J ( 2 , 0 ) , ( 0 , 2 ) ( M ) R . Link with Hard Lefschetz condition Sketch of the Proof Integrable case Examples Problem Nakamura manifold Relation between C ∞ pure and full and pure and full? Families in dimension six References 14

  38. Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex structures Definition (Li, Zhang) Calibrated and 4-dimensional case Example of non C∞ pure An almost complex structure J is pure if almost complex structure H J 1 , 1 ( M ) R ∩ H J ( 2 , 0 ) , ( 0 , 2 ) ( M ) R = { 0 } or equivalently if Pure and full almost complex structures π 1 , 1 B 2 ∩ Z J 1 , 1 = B J 1 , 1 . Main result Sketch of the proof J is full if H 2 ( M , R ) = H J 1 , 1 ( M ) R + H J ( 2 , 0 ) , ( 0 , 2 ) ( M ) R . Link with Hard Lefschetz condition Sketch of the Proof Integrable case Examples Problem Nakamura manifold Relation between C ∞ pure and full and pure and full? Families in dimension six References 14

  39. Main result Motivation If a 2-form ω on M 2 n is not necessarily closed but it is only Tamed and calibrated almost complex structures Symplectic cones non-degenerate, ( M 2 n , ω ) is called almost symplectic . C∞ pure and full almost complex structures Theorem (–, Tomassini) Calibrated and 4-dimensional case Let ( M 2 n , ω ) be an almost symplectic compact manifold and J Example of non C∞ pure almost complex structure be a C ∞ pure and full almost complex structure calibrated by ω . Pure and full almost complex structures Then J is pure. Main result If, in addition, either n = 2 or any class in H 1 , 1 J ( M 2 n ) R Sketch of the proof Link with Hard Lefschetz (H ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R resp.) has a harmonic representative in Z 1 , 1 condition Sketch of the Proof J J Integrable case ( Z ( 2 , 0 ) , ( 0 , 2 ) resp.) with respect to the metric induced by ω and J, J Examples then J is pure and full. Nakamura manifold Families in dimension six References Remark • In order to get the pureness of J , it is enough to assume that J is C ∞ full. • If n = 2, then by previous Theorem any almost complex structure J is pure and full. 15

  40. Main result Motivation If a 2-form ω on M 2 n is not necessarily closed but it is only Tamed and calibrated almost complex structures Symplectic cones non-degenerate, ( M 2 n , ω ) is called almost symplectic . C∞ pure and full almost complex structures Theorem (–, Tomassini) Calibrated and 4-dimensional case Let ( M 2 n , ω ) be an almost symplectic compact manifold and J Example of non C∞ pure almost complex structure be a C ∞ pure and full almost complex structure calibrated by ω . Pure and full almost complex structures Then J is pure. Main result If, in addition, either n = 2 or any class in H 1 , 1 J ( M 2 n ) R Sketch of the proof Link with Hard Lefschetz (H ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R resp.) has a harmonic representative in Z 1 , 1 condition Sketch of the Proof J J Integrable case ( Z ( 2 , 0 ) , ( 0 , 2 ) resp.) with respect to the metric induced by ω and J, J Examples then J is pure and full. Nakamura manifold Families in dimension six References Remark • In order to get the pureness of J , it is enough to assume that J is C ∞ full. • If n = 2, then by previous Theorem any almost complex structure J is pure and full. 15

  41. Main result Motivation If a 2-form ω on M 2 n is not necessarily closed but it is only Tamed and calibrated almost complex structures Symplectic cones non-degenerate, ( M 2 n , ω ) is called almost symplectic . C∞ pure and full almost complex structures Theorem (–, Tomassini) Calibrated and 4-dimensional case Let ( M 2 n , ω ) be an almost symplectic compact manifold and J Example of non C∞ pure almost complex structure be a C ∞ pure and full almost complex structure calibrated by ω . Pure and full almost complex structures Then J is pure. Main result If, in addition, either n = 2 or any class in H 1 , 1 J ( M 2 n ) R Sketch of the proof Link with Hard Lefschetz (H ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R resp.) has a harmonic representative in Z 1 , 1 condition Sketch of the Proof J J Integrable case ( Z ( 2 , 0 ) , ( 0 , 2 ) resp.) with respect to the metric induced by ω and J, J Examples then J is pure and full. Nakamura manifold Families in dimension six References Remark • In order to get the pureness of J , it is enough to assume that J is C ∞ full. • If n = 2, then by previous Theorem any almost complex structure J is pure and full. 15

  42. Main result Motivation If a 2-form ω on M 2 n is not necessarily closed but it is only Tamed and calibrated almost complex structures Symplectic cones non-degenerate, ( M 2 n , ω ) is called almost symplectic . C∞ pure and full almost complex structures Theorem (–, Tomassini) Calibrated and 4-dimensional case Let ( M 2 n , ω ) be an almost symplectic compact manifold and J Example of non C∞ pure almost complex structure be a C ∞ pure and full almost complex structure calibrated by ω . Pure and full almost complex structures Then J is pure. Main result If, in addition, either n = 2 or any class in H 1 , 1 J ( M 2 n ) R Sketch of the proof Link with Hard Lefschetz (H ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R resp.) has a harmonic representative in Z 1 , 1 condition Sketch of the Proof J J Integrable case ( Z ( 2 , 0 ) , ( 0 , 2 ) resp.) with respect to the metric induced by ω and J, J Examples then J is pure and full. Nakamura manifold Families in dimension six References Remark • In order to get the pureness of J , it is enough to assume that J is C ∞ full. • If n = 2, then by previous Theorem any almost complex structure J is pure and full. 15

  43. Sketch of the proof Motivation Tamed and calibrated almost complex structures We start to prove that J is pure, i.e. π 1 , 1 B 2 ∩ Z J 1 , 1 = B J Symplectic cones 1 , 1 . C∞ pure and full Let T ∈ π 1 , 1 B 2 ∩ Z J almost complex 1 , 1 ⇒ T = π 1 , 1 dS , where S is a real structures 3-current and d ( π 1 , 1 dS ) = 0. Calibrated and 4-dimensional case Example of non C∞ pure We have to show that T = π 1 , 1 dS is a boundary, i.e. that almost complex structure T ( α ) = 0, for any closed real 2-form α . Pure and full almost complex structures Main result If α is exact, then ( π 1 , 1 dS )( α ) = 0. Sketch of the proof If [ α ] � = 0 ∈ H 2 ( M 2 n , R ) , since J is C ∞ pure and full, we have Link with Hard Lefschetz condition Sketch of the Proof Integrable case J , α 2 ∈ Z ( 2 , 0 ) , ( 0 , 2 ) α = α 1 + α 2 + d γ, with α 1 ∈ Z 1 , 1 . Examples J Nakamura manifold Families in dimension six References Then T ( α ) = ( π 1 , 1 dS )( α ) = ( π 1 , 1 dS )( α 1 + α 2 ) = ( dS )( α 1 ) = 0 . 16

  44. Sketch of the proof Motivation Tamed and calibrated almost complex structures We start to prove that J is pure, i.e. π 1 , 1 B 2 ∩ Z J 1 , 1 = B J Symplectic cones 1 , 1 . C∞ pure and full Let T ∈ π 1 , 1 B 2 ∩ Z J almost complex 1 , 1 ⇒ T = π 1 , 1 dS , where S is a real structures 3-current and d ( π 1 , 1 dS ) = 0. Calibrated and 4-dimensional case Example of non C∞ pure We have to show that T = π 1 , 1 dS is a boundary, i.e. that almost complex structure T ( α ) = 0, for any closed real 2-form α . Pure and full almost complex structures Main result If α is exact, then ( π 1 , 1 dS )( α ) = 0. Sketch of the proof If [ α ] � = 0 ∈ H 2 ( M 2 n , R ) , since J is C ∞ pure and full, we have Link with Hard Lefschetz condition Sketch of the Proof Integrable case J , α 2 ∈ Z ( 2 , 0 ) , ( 0 , 2 ) α = α 1 + α 2 + d γ, with α 1 ∈ Z 1 , 1 . Examples J Nakamura manifold Families in dimension six References Then T ( α ) = ( π 1 , 1 dS )( α ) = ( π 1 , 1 dS )( α 1 + α 2 ) = ( dS )( α 1 ) = 0 . 16

  45. Sketch of the proof Motivation Tamed and calibrated almost complex structures We start to prove that J is pure, i.e. π 1 , 1 B 2 ∩ Z J 1 , 1 = B J Symplectic cones 1 , 1 . C∞ pure and full Let T ∈ π 1 , 1 B 2 ∩ Z J almost complex 1 , 1 ⇒ T = π 1 , 1 dS , where S is a real structures 3-current and d ( π 1 , 1 dS ) = 0. Calibrated and 4-dimensional case Example of non C∞ pure We have to show that T = π 1 , 1 dS is a boundary, i.e. that almost complex structure T ( α ) = 0, for any closed real 2-form α . Pure and full almost complex structures Main result If α is exact, then ( π 1 , 1 dS )( α ) = 0. Sketch of the proof If [ α ] � = 0 ∈ H 2 ( M 2 n , R ) , since J is C ∞ pure and full, we have Link with Hard Lefschetz condition Sketch of the Proof Integrable case J , α 2 ∈ Z ( 2 , 0 ) , ( 0 , 2 ) α = α 1 + α 2 + d γ, with α 1 ∈ Z 1 , 1 . Examples J Nakamura manifold Families in dimension six References Then T ( α ) = ( π 1 , 1 dS )( α ) = ( π 1 , 1 dS )( α 1 + α 2 ) = ( dS )( α 1 ) = 0 . 16

  46. Sketch of the proof Motivation Tamed and calibrated almost complex structures We start to prove that J is pure, i.e. π 1 , 1 B 2 ∩ Z J 1 , 1 = B J Symplectic cones 1 , 1 . C∞ pure and full Let T ∈ π 1 , 1 B 2 ∩ Z J almost complex 1 , 1 ⇒ T = π 1 , 1 dS , where S is a real structures 3-current and d ( π 1 , 1 dS ) = 0. Calibrated and 4-dimensional case Example of non C∞ pure We have to show that T = π 1 , 1 dS is a boundary, i.e. that almost complex structure T ( α ) = 0, for any closed real 2-form α . Pure and full almost complex structures Main result If α is exact, then ( π 1 , 1 dS )( α ) = 0. Sketch of the proof If [ α ] � = 0 ∈ H 2 ( M 2 n , R ) , since J is C ∞ pure and full, we have Link with Hard Lefschetz condition Sketch of the Proof Integrable case J , α 2 ∈ Z ( 2 , 0 ) , ( 0 , 2 ) α = α 1 + α 2 + d γ, with α 1 ∈ Z 1 , 1 . Examples J Nakamura manifold Families in dimension six References Then T ( α ) = ( π 1 , 1 dS )( α ) = ( π 1 , 1 dS )( α 1 + α 2 ) = ( dS )( α 1 ) = 0 . 16

  47. Sketch of the proof Motivation Tamed and calibrated almost complex structures We start to prove that J is pure, i.e. π 1 , 1 B 2 ∩ Z J 1 , 1 = B J Symplectic cones 1 , 1 . C∞ pure and full Let T ∈ π 1 , 1 B 2 ∩ Z J almost complex 1 , 1 ⇒ T = π 1 , 1 dS , where S is a real structures 3-current and d ( π 1 , 1 dS ) = 0. Calibrated and 4-dimensional case Example of non C∞ pure We have to show that T = π 1 , 1 dS is a boundary, i.e. that almost complex structure T ( α ) = 0, for any closed real 2-form α . Pure and full almost complex structures Main result If α is exact, then ( π 1 , 1 dS )( α ) = 0. Sketch of the proof If [ α ] � = 0 ∈ H 2 ( M 2 n , R ) , since J is C ∞ pure and full, we have Link with Hard Lefschetz condition Sketch of the Proof Integrable case J , α 2 ∈ Z ( 2 , 0 ) , ( 0 , 2 ) α = α 1 + α 2 + d γ, with α 1 ∈ Z 1 , 1 . Examples J Nakamura manifold Families in dimension six References Then T ( α ) = ( π 1 , 1 dS )( α ) = ( π 1 , 1 dS )( α 1 + α 2 ) = ( dS )( α 1 ) = 0 . 16

  48. Sketch of the proof Motivation Tamed and calibrated almost complex structures We start to prove that J is pure, i.e. π 1 , 1 B 2 ∩ Z J 1 , 1 = B J Symplectic cones 1 , 1 . C∞ pure and full Let T ∈ π 1 , 1 B 2 ∩ Z J almost complex 1 , 1 ⇒ T = π 1 , 1 dS , where S is a real structures 3-current and d ( π 1 , 1 dS ) = 0. Calibrated and 4-dimensional case Example of non C∞ pure We have to show that T = π 1 , 1 dS is a boundary, i.e. that almost complex structure T ( α ) = 0, for any closed real 2-form α . Pure and full almost complex structures Main result If α is exact, then ( π 1 , 1 dS )( α ) = 0. Sketch of the proof If [ α ] � = 0 ∈ H 2 ( M 2 n , R ) , since J is C ∞ pure and full, we have Link with Hard Lefschetz condition Sketch of the Proof Integrable case J , α 2 ∈ Z ( 2 , 0 ) , ( 0 , 2 ) α = α 1 + α 2 + d γ, with α 1 ∈ Z 1 , 1 . Examples J Nakamura manifold Families in dimension six References Then T ( α ) = ( π 1 , 1 dS )( α ) = ( π 1 , 1 dS )( α 1 + α 2 ) = ( dS )( α 1 ) = 0 . 16

  49. Motivation Tamed and calibrated almost complex structures • If n = 2, let [ T ] ∈ H 2 ( M 4 , R ) ; then ∃ a smooth closed 2-form α Symplectic cones C∞ pure and full such that [ T ] = [ α ] . almost complex structures Since J is C ∞ full, we have that [ α ] = [ α 1 ] + [ α 2 ] , with α 1 ∈ Z 1 , 1 Calibrated and J 4-dimensional case Example of non C∞ pure and α 2 ∈ Z ( 2 , 0 ) , ( 0 , 2 ) . almost complex structure J Pure and full almost • If n > 2, let [ T ] ∈ H 2 ( M 2 n , R ) , then ∃ a smooth harmonic complex structures ( 2 n − 2 ) -form β such that [ T ] = [ β ] . Main result Sketch of the proof The 2-form γ = ∗ β defines [ γ ] ∈ H 2 ( M 2 n , R ) . By the Link with Hard Lefschetz condition Sketch of the Proof assumption, ∃ real harmonic forms γ 1 ∈ Ω 1 , 1 J ( M 2 n ) R and Integrable case γ 2 ∈ Ω ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R such that [ γ ] = [ γ 1 ] + [ γ 2 ] . Examples J Nakamura manifold Families in dimension six The ( 2 n − 2 ) -forms β 1 = ∗ γ 1 and β 2 = ∗ γ 2 then can be viewed References as elements respectively of Z J 1 , 1 and Z J ( 2 , 0 ) , ( 0 , 2 ) = ⇒ [ T ] = [ β 1 ] + [ β 2 ] . � 17

  50. Motivation Tamed and calibrated almost complex structures • If n = 2, let [ T ] ∈ H 2 ( M 4 , R ) ; then ∃ a smooth closed 2-form α Symplectic cones C∞ pure and full such that [ T ] = [ α ] . almost complex structures Since J is C ∞ full, we have that [ α ] = [ α 1 ] + [ α 2 ] , with α 1 ∈ Z 1 , 1 Calibrated and J 4-dimensional case Example of non C∞ pure and α 2 ∈ Z ( 2 , 0 ) , ( 0 , 2 ) . almost complex structure J Pure and full almost • If n > 2, let [ T ] ∈ H 2 ( M 2 n , R ) , then ∃ a smooth harmonic complex structures ( 2 n − 2 ) -form β such that [ T ] = [ β ] . Main result Sketch of the proof The 2-form γ = ∗ β defines [ γ ] ∈ H 2 ( M 2 n , R ) . By the Link with Hard Lefschetz condition Sketch of the Proof assumption, ∃ real harmonic forms γ 1 ∈ Ω 1 , 1 J ( M 2 n ) R and Integrable case γ 2 ∈ Ω ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R such that [ γ ] = [ γ 1 ] + [ γ 2 ] . Examples J Nakamura manifold Families in dimension six The ( 2 n − 2 ) -forms β 1 = ∗ γ 1 and β 2 = ∗ γ 2 then can be viewed References as elements respectively of Z J 1 , 1 and Z J ( 2 , 0 ) , ( 0 , 2 ) = ⇒ [ T ] = [ β 1 ] + [ β 2 ] . � 17

  51. Motivation Tamed and calibrated almost complex structures • If n = 2, let [ T ] ∈ H 2 ( M 4 , R ) ; then ∃ a smooth closed 2-form α Symplectic cones C∞ pure and full such that [ T ] = [ α ] . almost complex structures Since J is C ∞ full, we have that [ α ] = [ α 1 ] + [ α 2 ] , with α 1 ∈ Z 1 , 1 Calibrated and J 4-dimensional case Example of non C∞ pure and α 2 ∈ Z ( 2 , 0 ) , ( 0 , 2 ) . almost complex structure J Pure and full almost • If n > 2, let [ T ] ∈ H 2 ( M 2 n , R ) , then ∃ a smooth harmonic complex structures ( 2 n − 2 ) -form β such that [ T ] = [ β ] . Main result Sketch of the proof The 2-form γ = ∗ β defines [ γ ] ∈ H 2 ( M 2 n , R ) . By the Link with Hard Lefschetz condition Sketch of the Proof assumption, ∃ real harmonic forms γ 1 ∈ Ω 1 , 1 J ( M 2 n ) R and Integrable case γ 2 ∈ Ω ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R such that [ γ ] = [ γ 1 ] + [ γ 2 ] . Examples J Nakamura manifold Families in dimension six The ( 2 n − 2 ) -forms β 1 = ∗ γ 1 and β 2 = ∗ γ 2 then can be viewed References as elements respectively of Z J 1 , 1 and Z J ( 2 , 0 ) , ( 0 , 2 ) = ⇒ [ T ] = [ β 1 ] + [ β 2 ] . � 17

  52. Motivation Tamed and calibrated almost complex structures • If n = 2, let [ T ] ∈ H 2 ( M 4 , R ) ; then ∃ a smooth closed 2-form α Symplectic cones C∞ pure and full such that [ T ] = [ α ] . almost complex structures Since J is C ∞ full, we have that [ α ] = [ α 1 ] + [ α 2 ] , with α 1 ∈ Z 1 , 1 Calibrated and J 4-dimensional case Example of non C∞ pure and α 2 ∈ Z ( 2 , 0 ) , ( 0 , 2 ) . almost complex structure J Pure and full almost • If n > 2, let [ T ] ∈ H 2 ( M 2 n , R ) , then ∃ a smooth harmonic complex structures ( 2 n − 2 ) -form β such that [ T ] = [ β ] . Main result Sketch of the proof The 2-form γ = ∗ β defines [ γ ] ∈ H 2 ( M 2 n , R ) . By the Link with Hard Lefschetz condition Sketch of the Proof assumption, ∃ real harmonic forms γ 1 ∈ Ω 1 , 1 J ( M 2 n ) R and Integrable case γ 2 ∈ Ω ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R such that [ γ ] = [ γ 1 ] + [ γ 2 ] . Examples J Nakamura manifold Families in dimension six The ( 2 n − 2 ) -forms β 1 = ∗ γ 1 and β 2 = ∗ γ 2 then can be viewed References as elements respectively of Z J 1 , 1 and Z J ( 2 , 0 ) , ( 0 , 2 ) = ⇒ [ T ] = [ β 1 ] + [ β 2 ] . � 17

  53. Motivation Tamed and calibrated almost complex structures • If n = 2, let [ T ] ∈ H 2 ( M 4 , R ) ; then ∃ a smooth closed 2-form α Symplectic cones C∞ pure and full such that [ T ] = [ α ] . almost complex structures Since J is C ∞ full, we have that [ α ] = [ α 1 ] + [ α 2 ] , with α 1 ∈ Z 1 , 1 Calibrated and J 4-dimensional case Example of non C∞ pure and α 2 ∈ Z ( 2 , 0 ) , ( 0 , 2 ) . almost complex structure J Pure and full almost • If n > 2, let [ T ] ∈ H 2 ( M 2 n , R ) , then ∃ a smooth harmonic complex structures ( 2 n − 2 ) -form β such that [ T ] = [ β ] . Main result Sketch of the proof The 2-form γ = ∗ β defines [ γ ] ∈ H 2 ( M 2 n , R ) . By the Link with Hard Lefschetz condition Sketch of the Proof assumption, ∃ real harmonic forms γ 1 ∈ Ω 1 , 1 J ( M 2 n ) R and Integrable case γ 2 ∈ Ω ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R such that [ γ ] = [ γ 1 ] + [ γ 2 ] . Examples J Nakamura manifold Families in dimension six The ( 2 n − 2 ) -forms β 1 = ∗ γ 1 and β 2 = ∗ γ 2 then can be viewed References as elements respectively of Z J 1 , 1 and Z J ( 2 , 0 ) , ( 0 , 2 ) = ⇒ [ T ] = [ β 1 ] + [ β 2 ] . � 17

  54. Motivation Tamed and calibrated almost complex structures • If n = 2, let [ T ] ∈ H 2 ( M 4 , R ) ; then ∃ a smooth closed 2-form α Symplectic cones C∞ pure and full such that [ T ] = [ α ] . almost complex structures Since J is C ∞ full, we have that [ α ] = [ α 1 ] + [ α 2 ] , with α 1 ∈ Z 1 , 1 Calibrated and J 4-dimensional case Example of non C∞ pure and α 2 ∈ Z ( 2 , 0 ) , ( 0 , 2 ) . almost complex structure J Pure and full almost • If n > 2, let [ T ] ∈ H 2 ( M 2 n , R ) , then ∃ a smooth harmonic complex structures ( 2 n − 2 ) -form β such that [ T ] = [ β ] . Main result Sketch of the proof The 2-form γ = ∗ β defines [ γ ] ∈ H 2 ( M 2 n , R ) . By the Link with Hard Lefschetz condition Sketch of the Proof assumption, ∃ real harmonic forms γ 1 ∈ Ω 1 , 1 J ( M 2 n ) R and Integrable case γ 2 ∈ Ω ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R such that [ γ ] = [ γ 1 ] + [ γ 2 ] . Examples J Nakamura manifold Families in dimension six The ( 2 n − 2 ) -forms β 1 = ∗ γ 1 and β 2 = ∗ γ 2 then can be viewed References as elements respectively of Z J 1 , 1 and Z J ( 2 , 0 ) , ( 0 , 2 ) = ⇒ [ T ] = [ β 1 ] + [ β 2 ] . � 17

  55. Link with Hard Lefschetz condition Motivation Tamed and calibrated almost complex structures A symplectic manifold ( M 2 n , ω ) satisfies the Hard Lefschetz Symplectic cones C∞ pure and full condition if : almost complex structures ω k : Ω n − k ( M 2 n ) → Ω n + k ( M 2 n ) , α �→ ω k ∧ α Calibrated and 4-dimensional case Example of non C∞ pure almost complex structure induce isomorphisms in cohomology. Pure and full almost complex structures Main result Theorem (–, Tomassini) Sketch of the proof Link with Hard Lefschetz condition Let ( M 2 n , ω ) be a compact symplectic manifold which satisfies Sketch of the Proof Hard Lefschetz condition and J be a C ∞ pure and full almost Integrable case Examples complex structure calibrated by ω . Then J is pure and full. Nakamura manifold Families in dimension six References Problem Find for n > 2 an example of compact symplectic manifold ( M 2 n , ω ) which satisfies Hard Lefschetz condition and with an non pure and full almost complex structure calibrated by ω . 18

  56. Link with Hard Lefschetz condition Motivation Tamed and calibrated almost complex structures A symplectic manifold ( M 2 n , ω ) satisfies the Hard Lefschetz Symplectic cones C∞ pure and full condition if : almost complex structures ω k : Ω n − k ( M 2 n ) → Ω n + k ( M 2 n ) , α �→ ω k ∧ α Calibrated and 4-dimensional case Example of non C∞ pure almost complex structure induce isomorphisms in cohomology. Pure and full almost complex structures Main result Theorem (–, Tomassini) Sketch of the proof Link with Hard Lefschetz condition Let ( M 2 n , ω ) be a compact symplectic manifold which satisfies Sketch of the Proof Hard Lefschetz condition and J be a C ∞ pure and full almost Integrable case Examples complex structure calibrated by ω . Then J is pure and full. Nakamura manifold Families in dimension six References Problem Find for n > 2 an example of compact symplectic manifold ( M 2 n , ω ) which satisfies Hard Lefschetz condition and with an non pure and full almost complex structure calibrated by ω . 18

  57. Link with Hard Lefschetz condition Motivation Tamed and calibrated almost complex structures A symplectic manifold ( M 2 n , ω ) satisfies the Hard Lefschetz Symplectic cones C∞ pure and full condition if : almost complex structures ω k : Ω n − k ( M 2 n ) → Ω n + k ( M 2 n ) , α �→ ω k ∧ α Calibrated and 4-dimensional case Example of non C∞ pure almost complex structure induce isomorphisms in cohomology. Pure and full almost complex structures Main result Theorem (–, Tomassini) Sketch of the proof Link with Hard Lefschetz condition Let ( M 2 n , ω ) be a compact symplectic manifold which satisfies Sketch of the Proof Hard Lefschetz condition and J be a C ∞ pure and full almost Integrable case Examples complex structure calibrated by ω . Then J is pure and full. Nakamura manifold Families in dimension six References Problem Find for n > 2 an example of compact symplectic manifold ( M 2 n , ω ) which satisfies Hard Lefschetz condition and with an non pure and full almost complex structure calibrated by ω . 18

  58. Link with Hard Lefschetz condition Motivation Tamed and calibrated almost complex structures A symplectic manifold ( M 2 n , ω ) satisfies the Hard Lefschetz Symplectic cones C∞ pure and full condition if : almost complex structures ω k : Ω n − k ( M 2 n ) → Ω n + k ( M 2 n ) , α �→ ω k ∧ α Calibrated and 4-dimensional case Example of non C∞ pure almost complex structure induce isomorphisms in cohomology. Pure and full almost complex structures Main result Theorem (–, Tomassini) Sketch of the proof Link with Hard Lefschetz condition Let ( M 2 n , ω ) be a compact symplectic manifold which satisfies Sketch of the Proof Hard Lefschetz condition and J be a C ∞ pure and full almost Integrable case Examples complex structure calibrated by ω . Then J is pure and full. Nakamura manifold Families in dimension six References Problem Find for n > 2 an example of compact symplectic manifold ( M 2 n , ω ) which satisfies Hard Lefschetz condition and with an non pure and full almost complex structure calibrated by ω . 18

  59. Sketch of the Proof Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex • If n = 2 the result follows by the last Theorem. structures Calibrated and 4-dimensional case • If n > 2 J is pure. We have to show that Example of non C∞ pure almost complex structure H 2 ( M 2 n , R ) = H J 1 , 1 ( M 2 n ) R ⊕ H J ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R . Pure and full almost complex structures Main result Sketch of the proof Link with Hard Lefschetz condition Let a = [ T ] ∈ H 2 ( M 2 n , R ) . Then a = [ α ] , where α ∈ Ω 2 n − 2 ( M 2 n ) Sketch of the Proof Integrable case is d -closed. Examples Nakamura manifold HL condition ⇒ ∃ b ∈ H 2 ( M 2 n , R ) , b = [ β ] such that Families in dimension six a = b ∪ [ ω ] n − 2 , i.e. References [ β ∧ ω n − 2 ] = [ α ] . 19

  60. Sketch of the Proof Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex • If n = 2 the result follows by the last Theorem. structures Calibrated and 4-dimensional case • If n > 2 J is pure. We have to show that Example of non C∞ pure almost complex structure H 2 ( M 2 n , R ) = H J 1 , 1 ( M 2 n ) R ⊕ H J ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R . Pure and full almost complex structures Main result Sketch of the proof Link with Hard Lefschetz condition Let a = [ T ] ∈ H 2 ( M 2 n , R ) . Then a = [ α ] , where α ∈ Ω 2 n − 2 ( M 2 n ) Sketch of the Proof Integrable case is d -closed. Examples Nakamura manifold HL condition ⇒ ∃ b ∈ H 2 ( M 2 n , R ) , b = [ β ] such that Families in dimension six a = b ∪ [ ω ] n − 2 , i.e. References [ β ∧ ω n − 2 ] = [ α ] . 19

  61. Sketch of the Proof Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex • If n = 2 the result follows by the last Theorem. structures Calibrated and 4-dimensional case • If n > 2 J is pure. We have to show that Example of non C∞ pure almost complex structure H 2 ( M 2 n , R ) = H J 1 , 1 ( M 2 n ) R ⊕ H J ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R . Pure and full almost complex structures Main result Sketch of the proof Link with Hard Lefschetz condition Let a = [ T ] ∈ H 2 ( M 2 n , R ) . Then a = [ α ] , where α ∈ Ω 2 n − 2 ( M 2 n ) Sketch of the Proof Integrable case is d -closed. Examples Nakamura manifold HL condition ⇒ ∃ b ∈ H 2 ( M 2 n , R ) , b = [ β ] such that Families in dimension six a = b ∪ [ ω ] n − 2 , i.e. References [ β ∧ ω n − 2 ] = [ α ] . 19

  62. Sketch of the Proof Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex • If n = 2 the result follows by the last Theorem. structures Calibrated and 4-dimensional case • If n > 2 J is pure. We have to show that Example of non C∞ pure almost complex structure H 2 ( M 2 n , R ) = H J 1 , 1 ( M 2 n ) R ⊕ H J ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R . Pure and full almost complex structures Main result Sketch of the proof Link with Hard Lefschetz condition Let a = [ T ] ∈ H 2 ( M 2 n , R ) . Then a = [ α ] , where α ∈ Ω 2 n − 2 ( M 2 n ) Sketch of the Proof Integrable case is d -closed. Examples Nakamura manifold HL condition ⇒ ∃ b ∈ H 2 ( M 2 n , R ) , b = [ β ] such that Families in dimension six a = b ∪ [ ω ] n − 2 , i.e. References [ β ∧ ω n − 2 ] = [ α ] . 19

  63. Sketch of the Proof Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex • If n = 2 the result follows by the last Theorem. structures Calibrated and 4-dimensional case • If n > 2 J is pure. We have to show that Example of non C∞ pure almost complex structure H 2 ( M 2 n , R ) = H J 1 , 1 ( M 2 n ) R ⊕ H J ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R . Pure and full almost complex structures Main result Sketch of the proof Link with Hard Lefschetz condition Let a = [ T ] ∈ H 2 ( M 2 n , R ) . Then a = [ α ] , where α ∈ Ω 2 n − 2 ( M 2 n ) Sketch of the Proof Integrable case is d -closed. Examples Nakamura manifold HL condition ⇒ ∃ b ∈ H 2 ( M 2 n , R ) , b = [ β ] such that Families in dimension six a = b ∪ [ ω ] n − 2 , i.e. References [ β ∧ ω n − 2 ] = [ α ] . 19

  64. Motivation J is C ∞ pure and full ⇒ Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full [ β ] = [ ϕ ] + [ ψ ] , almost complex structures Calibrated and [ ϕ ] ∈ H 1 , 1 J ( M 2 n ) R , [ ψ ] ∈ H ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R . 4-dimensional case Example of non C∞ pure J almost complex structure Then Pure and full almost complex structures a = [ T ] = [ β ∧ ω n − 2 ] = [ ϕ ∧ ω n − 2 ] + [ ψ ∧ ω n − 2 ] . Main result Sketch of the proof Link with Hard Lefschetz condition Sketch of the Proof Integrable case Since ϕ , ψ are real 2-forms of type ( 1 , 1 ) , ( 2 , 0 ) + ( 0 , 2 ) respectively and ω n − 2 is a real form of type ( n − 2 , n − 2 ) ⇒ Examples Nakamura manifold Families in dimension six R ∈ H J 1 , 1 ( M 2 n ) R , S ∈ H J ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R . References a = [ T ] = [ R ] + [ S ] , = ⇒ J is pure and full. � 20

  65. Motivation J is C ∞ pure and full ⇒ Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full [ β ] = [ ϕ ] + [ ψ ] , almost complex structures Calibrated and [ ϕ ] ∈ H 1 , 1 J ( M 2 n ) R , [ ψ ] ∈ H ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R . 4-dimensional case Example of non C∞ pure J almost complex structure Then Pure and full almost complex structures a = [ T ] = [ β ∧ ω n − 2 ] = [ ϕ ∧ ω n − 2 ] + [ ψ ∧ ω n − 2 ] . Main result Sketch of the proof Link with Hard Lefschetz condition Sketch of the Proof Integrable case Since ϕ , ψ are real 2-forms of type ( 1 , 1 ) , ( 2 , 0 ) + ( 0 , 2 ) respectively and ω n − 2 is a real form of type ( n − 2 , n − 2 ) ⇒ Examples Nakamura manifold Families in dimension six R ∈ H J 1 , 1 ( M 2 n ) R , S ∈ H J ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R . References a = [ T ] = [ R ] + [ S ] , = ⇒ J is pure and full. � 20

  66. Motivation J is C ∞ pure and full ⇒ Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full [ β ] = [ ϕ ] + [ ψ ] , almost complex structures Calibrated and [ ϕ ] ∈ H 1 , 1 J ( M 2 n ) R , [ ψ ] ∈ H ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R . 4-dimensional case Example of non C∞ pure J almost complex structure Then Pure and full almost complex structures a = [ T ] = [ β ∧ ω n − 2 ] = [ ϕ ∧ ω n − 2 ] + [ ψ ∧ ω n − 2 ] . Main result Sketch of the proof Link with Hard Lefschetz condition Sketch of the Proof Integrable case Since ϕ , ψ are real 2-forms of type ( 1 , 1 ) , ( 2 , 0 ) + ( 0 , 2 ) respectively and ω n − 2 is a real form of type ( n − 2 , n − 2 ) ⇒ Examples Nakamura manifold Families in dimension six R ∈ H J 1 , 1 ( M 2 n ) R , S ∈ H J ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R . References a = [ T ] = [ R ] + [ S ] , = ⇒ J is pure and full. � 20

  67. Motivation J is C ∞ pure and full ⇒ Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full [ β ] = [ ϕ ] + [ ψ ] , almost complex structures Calibrated and [ ϕ ] ∈ H 1 , 1 J ( M 2 n ) R , [ ψ ] ∈ H ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R . 4-dimensional case Example of non C∞ pure J almost complex structure Then Pure and full almost complex structures a = [ T ] = [ β ∧ ω n − 2 ] = [ ϕ ∧ ω n − 2 ] + [ ψ ∧ ω n − 2 ] . Main result Sketch of the proof Link with Hard Lefschetz condition Sketch of the Proof Integrable case Since ϕ , ψ are real 2-forms of type ( 1 , 1 ) , ( 2 , 0 ) + ( 0 , 2 ) respectively and ω n − 2 is a real form of type ( n − 2 , n − 2 ) ⇒ Examples Nakamura manifold Families in dimension six R ∈ H J 1 , 1 ( M 2 n ) R , S ∈ H J ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R . References a = [ T ] = [ R ] + [ S ] , = ⇒ J is pure and full. � 20

  68. Motivation J is C ∞ pure and full ⇒ Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full [ β ] = [ ϕ ] + [ ψ ] , almost complex structures Calibrated and [ ϕ ] ∈ H 1 , 1 J ( M 2 n ) R , [ ψ ] ∈ H ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R . 4-dimensional case Example of non C∞ pure J almost complex structure Then Pure and full almost complex structures a = [ T ] = [ β ∧ ω n − 2 ] = [ ϕ ∧ ω n − 2 ] + [ ψ ∧ ω n − 2 ] . Main result Sketch of the proof Link with Hard Lefschetz condition Sketch of the Proof Integrable case Since ϕ , ψ are real 2-forms of type ( 1 , 1 ) , ( 2 , 0 ) + ( 0 , 2 ) respectively and ω n − 2 is a real form of type ( n − 2 , n − 2 ) ⇒ Examples Nakamura manifold Families in dimension six R ∈ H J 1 , 1 ( M 2 n ) R , S ∈ H J ( 2 , 0 ) , ( 0 , 2 ) ( M 2 n ) R . References a = [ T ] = [ R ] + [ S ] , = ⇒ J is pure and full. � 20

  69. Integrable case Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex structures If J is integrable, in general it is not necessarily ( C ∞ ) pure and Calibrated and full. 4-dimensional case Example of non C∞ pure almost complex structure If J is an integrable almost complex structure and the Frölicher Pure and full almost spectral sequence degenerates at E 1 , then J is pure and full complex structures Main result [Li, Zhang]. Sketch of the proof Link with Hard Lefschetz condition Theorem (–, Tomassini) Sketch of the Proof Integrable case If ( M = Γ \ G , J ) is a complex parallelizable manifold and Examples H 2 ( M , R ) ∼ = H 2 ( g ) , then J is C ∞ full and it is pure. Nakamura manifold Families in dimension six References ⇒ Let ( M , J ) be a complex parallelizable nilmanifold. Then J is C ∞ full and it is pure. 21

  70. Integrable case Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex structures If J is integrable, in general it is not necessarily ( C ∞ ) pure and Calibrated and full. 4-dimensional case Example of non C∞ pure almost complex structure If J is an integrable almost complex structure and the Frölicher Pure and full almost spectral sequence degenerates at E 1 , then J is pure and full complex structures Main result [Li, Zhang]. Sketch of the proof Link with Hard Lefschetz condition Theorem (–, Tomassini) Sketch of the Proof Integrable case If ( M = Γ \ G , J ) is a complex parallelizable manifold and Examples H 2 ( M , R ) ∼ = H 2 ( g ) , then J is C ∞ full and it is pure. Nakamura manifold Families in dimension six References ⇒ Let ( M , J ) be a complex parallelizable nilmanifold. Then J is C ∞ full and it is pure. 21

  71. Integrable case Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex structures If J is integrable, in general it is not necessarily ( C ∞ ) pure and Calibrated and full. 4-dimensional case Example of non C∞ pure almost complex structure If J is an integrable almost complex structure and the Frölicher Pure and full almost spectral sequence degenerates at E 1 , then J is pure and full complex structures Main result [Li, Zhang]. Sketch of the proof Link with Hard Lefschetz condition Theorem (–, Tomassini) Sketch of the Proof Integrable case If ( M = Γ \ G , J ) is a complex parallelizable manifold and Examples H 2 ( M , R ) ∼ = H 2 ( g ) , then J is C ∞ full and it is pure. Nakamura manifold Families in dimension six References ⇒ Let ( M , J ) be a complex parallelizable nilmanifold. Then J is C ∞ full and it is pure. 21

  72. Integrable case Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex structures If J is integrable, in general it is not necessarily ( C ∞ ) pure and Calibrated and full. 4-dimensional case Example of non C∞ pure almost complex structure If J is an integrable almost complex structure and the Frölicher Pure and full almost spectral sequence degenerates at E 1 , then J is pure and full complex structures Main result [Li, Zhang]. Sketch of the proof Link with Hard Lefschetz condition Theorem (–, Tomassini) Sketch of the Proof Integrable case If ( M = Γ \ G , J ) is a complex parallelizable manifold and Examples H 2 ( M , R ) ∼ = H 2 ( g ) , then J is C ∞ full and it is pure. Nakamura manifold Families in dimension six References ⇒ Let ( M , J ) be a complex parallelizable nilmanifold. Then J is C ∞ full and it is pure. 21

  73. Nakamura manifold Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex structures Let G be the solvable Lie group with structure equations Calibrated and 4-dimensional case Example of non C∞ pure ( 0 , e 12 − e 45 , − e 13 + e 46 , 0 , e 15 − e 24 , − e 16 + e 34 ) . almost complex structure Pure and full almost complex structures Main result Sketch of the proof G ∼ = ( C 3 , ∗ ) , with ∗ defined in terms of the coordinates Link with Hard Lefschetz condition z j = x j + ix 3 + j by Sketch of the Proof Integrable case Examples t ( z 1 , z 2 , z 3 ) ∗ t ( w 1 , w 2 , w 3 ) = t ( z 1 + w 1 , e − w 1 z 2 + w 2 , e w 1 z 3 + w 3 ) . Nakamura manifold Families in dimension six References The Nakamura manifold is the compact quotient X 6 = Γ \ G . 22

  74. Nakamura manifold Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex structures Let G be the solvable Lie group with structure equations Calibrated and 4-dimensional case Example of non C∞ pure ( 0 , e 12 − e 45 , − e 13 + e 46 , 0 , e 15 − e 24 , − e 16 + e 34 ) . almost complex structure Pure and full almost complex structures Main result Sketch of the proof G ∼ = ( C 3 , ∗ ) , with ∗ defined in terms of the coordinates Link with Hard Lefschetz condition z j = x j + ix 3 + j by Sketch of the Proof Integrable case Examples t ( z 1 , z 2 , z 3 ) ∗ t ( w 1 , w 2 , w 3 ) = t ( z 1 + w 1 , e − w 1 z 2 + w 2 , e w 1 z 3 + w 3 ) . Nakamura manifold Families in dimension six References The Nakamura manifold is the compact quotient X 6 = Γ \ G . 22

  75. Nakamura manifold Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex structures Let G be the solvable Lie group with structure equations Calibrated and 4-dimensional case Example of non C∞ pure ( 0 , e 12 − e 45 , − e 13 + e 46 , 0 , e 15 − e 24 , − e 16 + e 34 ) . almost complex structure Pure and full almost complex structures Main result Sketch of the proof G ∼ = ( C 3 , ∗ ) , with ∗ defined in terms of the coordinates Link with Hard Lefschetz condition z j = x j + ix 3 + j by Sketch of the Proof Integrable case Examples t ( z 1 , z 2 , z 3 ) ∗ t ( w 1 , w 2 , w 3 ) = t ( z 1 + w 1 , e − w 1 z 2 + w 2 , e w 1 z 3 + w 3 ) . Nakamura manifold Families in dimension six References The Nakamura manifold is the compact quotient X 6 = Γ \ G . 22

  76. Nakamura manifold Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full almost complex structures Let G be the solvable Lie group with structure equations Calibrated and 4-dimensional case Example of non C∞ pure ( 0 , e 12 − e 45 , − e 13 + e 46 , 0 , e 15 − e 24 , − e 16 + e 34 ) . almost complex structure Pure and full almost complex structures Main result Sketch of the proof G ∼ = ( C 3 , ∗ ) , with ∗ defined in terms of the coordinates Link with Hard Lefschetz condition z j = x j + ix 3 + j by Sketch of the Proof Integrable case Examples t ( z 1 , z 2 , z 3 ) ∗ t ( w 1 , w 2 , w 3 ) = t ( z 1 + w 1 , e − w 1 z 2 + w 2 , e w 1 z 3 + w 3 ) . Nakamura manifold Families in dimension six References The Nakamura manifold is the compact quotient X 6 = Γ \ G . 22

  77. Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full By de Bartolomeis-Tomassini we have almost complex structures Calibrated and R < [ e 14 ] , [ e 26 − e 35 ] , [ e 23 − e 56 ] , H 2 ( X 6 , R ) 4-dimensional case = Example of non C∞ pure [ cos ( 2 x 4 )( e 23 + e 56 ) − sin ( 2 x 4 )( e 26 + e 35 )] , almost complex structure Pure and full almost complex structures [ sin ( 2 x 4 )( e 23 + e 56 ) − cos ( 2 x 4 )( e 26 + e 35 )] > . Main result Sketch of the proof Link with Hard Lefschetz condition Sketch of the Proof • X 6 has a left-invariant J defined by: Integrable case Examples η 1 = e 1 + ie 4 , η 2 = e 3 + ie 5 , η 3 = e 6 + ie 2 Nakamura manifold Families in dimension six References calibrated by ω = e 14 + e 35 + e 62 . 23

  78. Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full By de Bartolomeis-Tomassini we have almost complex structures Calibrated and R < [ e 14 ] , [ e 26 − e 35 ] , [ e 23 − e 56 ] , H 2 ( X 6 , R ) 4-dimensional case = Example of non C∞ pure [ cos ( 2 x 4 )( e 23 + e 56 ) − sin ( 2 x 4 )( e 26 + e 35 )] , almost complex structure Pure and full almost complex structures [ sin ( 2 x 4 )( e 23 + e 56 ) − cos ( 2 x 4 )( e 26 + e 35 )] > . Main result Sketch of the proof Link with Hard Lefschetz condition Sketch of the Proof • X 6 has a left-invariant J defined by: Integrable case Examples η 1 = e 1 + ie 4 , η 2 = e 3 + ie 5 , η 3 = e 6 + ie 2 Nakamura manifold Families in dimension six References calibrated by ω = e 14 + e 35 + e 62 . 23

  79. Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full By de Bartolomeis-Tomassini we have almost complex structures Calibrated and R < [ e 14 ] , [ e 26 − e 35 ] , [ e 23 − e 56 ] , H 2 ( X 6 , R ) 4-dimensional case = Example of non C∞ pure [ cos ( 2 x 4 )( e 23 + e 56 ) − sin ( 2 x 4 )( e 26 + e 35 )] , almost complex structure Pure and full almost complex structures [ sin ( 2 x 4 )( e 23 + e 56 ) − cos ( 2 x 4 )( e 26 + e 35 )] > . Main result Sketch of the proof Link with Hard Lefschetz condition Sketch of the Proof • X 6 has a left-invariant J defined by: Integrable case Examples η 1 = e 1 + ie 4 , η 2 = e 3 + ie 5 , η 3 = e 6 + ie 2 Nakamura manifold Families in dimension six References calibrated by ω = e 14 + e 35 + e 62 . 23

  80. Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full The harmonic forms almost complex structures Calibrated and e 14 , e 26 − e 35 , cos ( 2 x 4 )( e 23 + e 56 ) − sin ( 2 x 4 )( e 26 + e 35 ) , 4-dimensional case Example of non C∞ pure sin ( 2 x 4 )( e 23 + e 56 ) − cos ( 2 x 4 )( e 26 + e 35 ) almost complex structure Pure and full almost complex structures are all of type ( 1 , 1 ) and e 23 − e 56 is of type ( 2 , 0 ) ⇒ Main result Sketch of the proof J is pure and full. Link with Hard Lefschetz condition Sketch of the Proof • X 6 admits the pure and full bi-invariant complex structure ˜ J : Integrable case Examples η 1 = e 1 + ie 4 , η 2 = e 2 + ie 5 , η 3 = e 3 + ie 6 . Nakamura manifold ˜ ˜ ˜ Families in dimension six References 24

  81. Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full The harmonic forms almost complex structures Calibrated and e 14 , e 26 − e 35 , cos ( 2 x 4 )( e 23 + e 56 ) − sin ( 2 x 4 )( e 26 + e 35 ) , 4-dimensional case Example of non C∞ pure sin ( 2 x 4 )( e 23 + e 56 ) − cos ( 2 x 4 )( e 26 + e 35 ) almost complex structure Pure and full almost complex structures are all of type ( 1 , 1 ) and e 23 − e 56 is of type ( 2 , 0 ) ⇒ Main result Sketch of the proof J is pure and full. Link with Hard Lefschetz condition Sketch of the Proof • X 6 admits the pure and full bi-invariant complex structure ˜ J : Integrable case Examples η 1 = e 1 + ie 4 , η 2 = e 2 + ie 5 , η 3 = e 3 + ie 6 . Nakamura manifold ˜ ˜ ˜ Families in dimension six References 24

  82. Motivation Tamed and calibrated almost complex structures Symplectic cones C∞ pure and full The harmonic forms almost complex structures Calibrated and e 14 , e 26 − e 35 , cos ( 2 x 4 )( e 23 + e 56 ) − sin ( 2 x 4 )( e 26 + e 35 ) , 4-dimensional case Example of non C∞ pure sin ( 2 x 4 )( e 23 + e 56 ) − cos ( 2 x 4 )( e 26 + e 35 ) almost complex structure Pure and full almost complex structures are all of type ( 1 , 1 ) and e 23 − e 56 is of type ( 2 , 0 ) ⇒ Main result Sketch of the proof J is pure and full. Link with Hard Lefschetz condition Sketch of the Proof • X 6 admits the pure and full bi-invariant complex structure ˜ J : Integrable case Examples η 1 = e 1 + ie 4 , η 2 = e 2 + ie 5 , η 3 = e 3 + ie 6 . Nakamura manifold ˜ ˜ ˜ Families in dimension six References 24

  83. Families in dimension six Motivation Tamed and calibrated almost complex structures Symplectic cones Consider the completely solvable Lie algebra C∞ pure and full s = sol ( 3 ) ⊕ sol ( 3 ) with structure equations almost complex structures Calibrated and ( 0 , − f 12 , f 34 , 0 , f 15 , f 46 ) . 4-dimensional case Example of non C∞ pure almost complex structure Pure and full almost complex structures S admits a compact quotient M 6 = Γ \ S [Fernandez-Gray]. Main result Sketch of the proof Link with Hard Lefschetz By Hattori’s Theorem condition Sketch of the Proof Integrable case H 2 ( M 6 , R ) ∼ = H ∗ ( s ) = R < [ f 14 ] , [ f 25 ] , [ f 36 ] > . Examples Nakamura manifold Families in dimension six References J 0 defined by the ( 1 , 0 ) -forms ϕ 1 = f 1 + if 4 , ϕ 2 = f 2 + if 5 , ϕ 3 = f 3 + if 6 . is almost-Kähler with respect to ω = f 14 + f 25 + f 36 . 25

  84. Families in dimension six Motivation Tamed and calibrated almost complex structures Symplectic cones Consider the completely solvable Lie algebra C∞ pure and full s = sol ( 3 ) ⊕ sol ( 3 ) with structure equations almost complex structures Calibrated and ( 0 , − f 12 , f 34 , 0 , f 15 , f 46 ) . 4-dimensional case Example of non C∞ pure almost complex structure Pure and full almost complex structures S admits a compact quotient M 6 = Γ \ S [Fernandez-Gray]. Main result Sketch of the proof Link with Hard Lefschetz By Hattori’s Theorem condition Sketch of the Proof Integrable case H 2 ( M 6 , R ) ∼ = H ∗ ( s ) = R < [ f 14 ] , [ f 25 ] , [ f 36 ] > . Examples Nakamura manifold Families in dimension six References J 0 defined by the ( 1 , 0 ) -forms ϕ 1 = f 1 + if 4 , ϕ 2 = f 2 + if 5 , ϕ 3 = f 3 + if 6 . is almost-Kähler with respect to ω = f 14 + f 25 + f 36 . 25

  85. Families in dimension six Motivation Tamed and calibrated almost complex structures Symplectic cones Consider the completely solvable Lie algebra C∞ pure and full s = sol ( 3 ) ⊕ sol ( 3 ) with structure equations almost complex structures Calibrated and ( 0 , − f 12 , f 34 , 0 , f 15 , f 46 ) . 4-dimensional case Example of non C∞ pure almost complex structure Pure and full almost complex structures S admits a compact quotient M 6 = Γ \ S [Fernandez-Gray]. Main result Sketch of the proof Link with Hard Lefschetz By Hattori’s Theorem condition Sketch of the Proof Integrable case H 2 ( M 6 , R ) ∼ = H ∗ ( s ) = R < [ f 14 ] , [ f 25 ] , [ f 36 ] > . Examples Nakamura manifold Families in dimension six References J 0 defined by the ( 1 , 0 ) -forms ϕ 1 = f 1 + if 4 , ϕ 2 = f 2 + if 5 , ϕ 3 = f 3 + if 6 . is almost-Kähler with respect to ω = f 14 + f 25 + f 36 . 25

  86. Families in dimension six Motivation Tamed and calibrated almost complex structures Symplectic cones Consider the completely solvable Lie algebra C∞ pure and full s = sol ( 3 ) ⊕ sol ( 3 ) with structure equations almost complex structures Calibrated and ( 0 , − f 12 , f 34 , 0 , f 15 , f 46 ) . 4-dimensional case Example of non C∞ pure almost complex structure Pure and full almost complex structures S admits a compact quotient M 6 = Γ \ S [Fernandez-Gray]. Main result Sketch of the proof Link with Hard Lefschetz By Hattori’s Theorem condition Sketch of the Proof Integrable case H 2 ( M 6 , R ) ∼ = H ∗ ( s ) = R < [ f 14 ] , [ f 25 ] , [ f 36 ] > . Examples Nakamura manifold Families in dimension six References J 0 defined by the ( 1 , 0 ) -forms ϕ 1 = f 1 + if 4 , ϕ 2 = f 2 + if 5 , ϕ 3 = f 3 + if 6 . is almost-Kähler with respect to ω = f 14 + f 25 + f 36 . 25

  87. Motivation Tamed and calibrated almost complex structures Symplectic cones ( M 6 , J 0 , ω ) satisfies the Hard Lefschetz condition [Fernandez, C∞ pure and full almost complex Munoz] and H 2 ( M 6 , R ) = H 1 , 1 J 0 ( M ) R . structures Calibrated and 4-dimensional case Example of non C∞ pure Define the family of almost complex structure almost complex structure Pure and full almost J t = ( I + L t ) J 0 ( I + L t ) − 1 complex structures Main result Sketch of the proof with respect to the basis ( f 1 , . . . , f 6 ) , where Link with Hard Lefschetz condition Sketch of the Proof � 0 � 0 Integrable case � � − I tI 6 t 2 < 1 . J 0 = , L t = , Examples I 0 tI 0 Nakamura manifold Families in dimension six References Then, J t is a family of ω -calibrated almost complex structures. 26

  88. Motivation Tamed and calibrated almost complex structures Symplectic cones ( M 6 , J 0 , ω ) satisfies the Hard Lefschetz condition [Fernandez, C∞ pure and full almost complex Munoz] and H 2 ( M 6 , R ) = H 1 , 1 J 0 ( M ) R . structures Calibrated and 4-dimensional case Example of non C∞ pure Define the family of almost complex structure almost complex structure Pure and full almost J t = ( I + L t ) J 0 ( I + L t ) − 1 complex structures Main result Sketch of the proof with respect to the basis ( f 1 , . . . , f 6 ) , where Link with Hard Lefschetz condition Sketch of the Proof � 0 � 0 Integrable case � � − I tI 6 t 2 < 1 . J 0 = , L t = , Examples I 0 tI 0 Nakamura manifold Families in dimension six References Then, J t is a family of ω -calibrated almost complex structures. 26

  89. Motivation Tamed and calibrated almost complex structures Symplectic cones ( M 6 , J 0 , ω ) satisfies the Hard Lefschetz condition [Fernandez, C∞ pure and full almost complex Munoz] and H 2 ( M 6 , R ) = H 1 , 1 J 0 ( M ) R . structures Calibrated and 4-dimensional case Example of non C∞ pure Define the family of almost complex structure almost complex structure Pure and full almost J t = ( I + L t ) J 0 ( I + L t ) − 1 complex structures Main result Sketch of the proof with respect to the basis ( f 1 , . . . , f 6 ) , where Link with Hard Lefschetz condition Sketch of the Proof � 0 � 0 Integrable case � � − I tI 6 t 2 < 1 . J 0 = , L t = , Examples I 0 tI 0 Nakamura manifold Families in dimension six References Then, J t is a family of ω -calibrated almost complex structures. 26

  90. Motivation Tamed and calibrated almost complex structures Symplectic cones ( M 6 , J 0 , ω ) satisfies the Hard Lefschetz condition [Fernandez, C∞ pure and full almost complex Munoz] and H 2 ( M 6 , R ) = H 1 , 1 J 0 ( M ) R . structures Calibrated and 4-dimensional case Example of non C∞ pure Define the family of almost complex structure almost complex structure Pure and full almost J t = ( I + L t ) J 0 ( I + L t ) − 1 complex structures Main result Sketch of the proof with respect to the basis ( f 1 , . . . , f 6 ) , where Link with Hard Lefschetz condition Sketch of the Proof � 0 � 0 Integrable case � � − I tI 6 t 2 < 1 . J 0 = , L t = , Examples I 0 tI 0 Nakamura manifold Families in dimension six References Then, J t is a family of ω -calibrated almost complex structures. 26

  91. ⇒ Any J t is C ∞ pure. Motivation = Tamed and calibrated almost complex structures A basis of ( 1 , 0 ) -forms for J t is Symplectic cones C∞ pure and full almost complex t = f 1 + i � ( 1 − t 2 ) f 1 + 1 + t 2 1 − t 2 f 4 � structures ϕ 1 2 t , Calibrated and 4-dimensional case Example of non C∞ pure t = f 2 + i � ( 1 − t 2 ) f 2 + 1 + t 2 1 − t 2 f 5 � almost complex structure ϕ 2 2 t , Pure and full almost complex structures Main result t = f 3 + i � ( 1 − t 2 ) f 3 + 1 + t 2 1 − t 2 f 6 � ϕ 3 2 t . Sketch of the proof Link with Hard Lefschetz condition Sketch of the Proof Then J t is also C ∞ full. Integrable case Examples J t is actually pure and full, since ϕ 1 t ∧ ϕ 1 t , ϕ 2 t ∧ ϕ 2 t , ϕ 3 t ∧ ϕ 3 t are Nakamura manifold Families in dimension six harmonic. References The family ˜ J t associated to the basis of ( 1 , 0 ) -forms t = f 1 + i − 2 tf 2 + f 4 � t = f 2 + if 5 , ˜ t = f 3 + if 6 ϕ 1 ϕ 2 ϕ 3 � ˜ , ˜ is a family of pure and full ω -tamed almost complex structures. 27

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