Seminario de Geometría y Topología Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid

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1 Seminario de Geometría y Topología Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid Cohomological aspects in complex non-kähler geometry Daniele Angella Istituto Nazionale di Alta Matematica and Dipartimento di Matematica, Università di Pisa November 19, 2013

2 Introduction aim Consider, e.g., a complex manifold:

3 Introduction aim Consider, e.g., a complex manifold: several cohomological invariants can be dened (de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies).

4 Introduction aim Consider, e.g., a complex manifold: several cohomological invariants can be dened (de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies). For compact Kähler mfds, they are all isomorphic,...

5 Introduction aim Consider, e.g., a complex manifold: several cohomological invariants can be dened (de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies). For compact Kähler mfds, they are all isomorphic, but Bott-Chern cohom may give further inform for non-kähler.

6 Introduction aim Consider, e.g., a complex manifold: several cohomological invariants can be dened (de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies). For compact Kähler mfds, they are all isomorphic, but Bott-Chern cohom may give further inform for non-kähler. We are interested in studying how they are related each other...

7 Introduction aim Consider, e.g., a complex manifold: several cohomological invariants can be dened (de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies). For compact Kähler mfds, they are all isomorphic, but Bott-Chern cohom may give further inform for non-kähler. We are interested in studying how they are related each other by investigating explicit examples...

8 Introduction aim Consider, e.g., a complex manifold: several cohomological invariants can be dened (de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies). For compact Kähler mfds, they are all isomorphic, but Bott-Chern cohom may give further inform for non-kähler. We are interested in studying how they are related each other by investigating explicit examples whose cohom is encoded in a suitable cohomological-model.

9 Introduction aim Consider, e.g., a complex manifold: several cohomological invariants can be dened (de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies). For compact Kähler mfds, they are all isomorphic, but Bott-Chern cohom may give further inform for non-kähler. We are interested in studying how they are related each other by investigating explicit examples whose cohom is encoded in a suitable cohomological-model. We focus on property all cohomol are isomorphic ( -Lemma)...

10 Introduction aim Consider, e.g., a complex manifold: several cohomological invariants can be dened (de Rham, Dolbeault, Bott-Chern, Aeppli cohomologies). For compact Kähler mfds, they are all isomorphic, but Bott-Chern cohom may give further inform for non-kähler. We are interested in studying how they are related each other by investigating explicit examples whose cohom is encoded in a suitable cohomological-model. We focus on property all cohomol are isomorphic ( -Lemma) by studying its behaviour for deformations of cplx struct.

11 Cohomologies of a complex manifold, i the double complex of forms, i X complex manifold

12 Cohomologies of a complex manifold, i the double complex of forms, i X complex manifold, i.e., locally modelled on C n

13 Cohomologies of a complex manifold, i the double complex of forms, i X complex manifold, i.e., locally modelled on C n The tangent spaces have a linear complex structure

14 Cohomologies of a complex manifold, i the double complex of forms, i X complex manifold, i.e., locally modelled on C n The tangent spaces have a linear complex structure, yielding the bundle decomposition TX C = T 1,0 X T 0,1 X.

15 Cohomologies of a complex manifold, i the double complex of forms, i X complex manifold, i.e., locally modelled on C n The tangent spaces have a linear complex structure, yielding the bundle decomposition It moves to TX C = T 1,0 X T 0,1 X. X C = p+q=k p,q X.

16 Cohomologies of a complex manifold, i the double complex of forms, i X complex manifold, i.e., locally modelled on C n The tangent spaces have a linear complex structure, yielding the bundle decomposition It moves to TX C = T 1,0 X T 0,1 X. X C = p+q=k p,q X. The exterior dierential d: X +1 X splits as d = + :, X +1, X, +1 X.

17 Cohomologies of a complex manifold, i the double complex of forms, i X complex manifold, i.e., locally modelled on C n The tangent spaces have a linear complex structure, yielding the bundle decomposition It moves to TX C = T 1,0 X T 0,1 X. X C = p+q=k p,q X. The exterior dierential d: X +1 X splits as d = + :, X +1, X, +1 X. By d 2 = 0, it follows 2 = 2 = + = 0

18 Cohomologies of a complex manifold, i the double complex of forms, i X complex manifold, i.e., locally modelled on C n The tangent spaces have a linear complex structure, yielding the bundle decomposition It moves to TX C = T 1,0 X T 0,1 X. X C = p+q=k p,q X. The exterior dierential d: X +1 X splits as d = + :, X +1, X, +1 X. By d 2 = 0, it follows 2 = 2 = + = 0, i.e., (, X,, ) double complex.

19 Cohomologies of a complex manifold, ii the double complex of forms, ii q + 2 q + 1 (, X,, ) double complex q q 1 q 2 p 2 p 1 p p + 1 p + 2

20 Cohomologies of a complex manifold, iii Dolbeault cohomology q + 2 q + 1 H, ker (X ) := im q q 1 q 2 p 2 p 1 p p + 1 p + 2

21 Cohomologies of a complex manifold, iv Bott-Chern cohomology q + 2 q + 1 H, ker ker BC (X ) := im q q 1 q 2 p 2 p 1 p p + 1 p + 2 R. Bott, S. S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math. 114 (1965), no. 1,

22 Cohomologies of a complex manifold, v Aeppli cohomology q + 2 q + 1 H, ker A (X ) := im + im q q 1 q 2 p 2 p 1 p p + 1 p + 2 A. Aeppli, On the cohomology structure of Stein manifolds, Proc. Conf. Complex Analysis (Minneapolis, Minn., 1964), Springer, Berlin, 1965, pp

23 Cohomologies of a complex manifold, vi why Bott-Chern and Aeppli cohomologies On cplx mfds, there are natural maps H, (X ) H, BC (X ) H dr(x ; C) H, A (X ) H, (X )

24 Cohomologies of a complex manifold, vi why Bott-Chern and Aeppli cohomologies On cplx mfds, there are natural maps H, (X ) H, BC (X ) H dr(x ; C) H, A (X ) While, on compact Kähler mfds H, (X )

25 Cohomologies of a complex manifold, vi why Bott-Chern and Aeppli cohomologies On cplx mfds, there are natural maps H, (X ) H, BC (X ) H dr(x ; C) H, A (X ) H, (X ) While, on compact Kähler mfds, they are all isomorphisms, because of the -Lemma,...

26 Cohomologies of a complex manifold, vi why Bott-Chern and Aeppli cohomologies On cplx mfds, there are natural maps H, (X ) H, BC (X ) H dr(x ; C) H, A (X ) H, (X ) While, on compact Kähler mfds, they are all isomorphisms, because of the -Lemma, Bott-Chern cohomology may supply further informations on the geometry of non-kähler manifolds.

27 Cohomologies of a complex manifold, vi why Bott-Chern and Aeppli cohomologies On cplx mfds, there are natural maps H, (X ) H, BC (X ) H dr(x ; C) H, A (X ) H, (X ) While, on compact Kähler mfds, they are all isomorphisms, because of the -Lemma, Bott-Chern cohomology may supply further informations on the geometry of non-kähler manifolds. Interest in Bott-Chern cohomology arises from: cycles on algebraic or analytic mfds; special metrics on complex mfds; cohomology theory; Strominger's equations in string theory;...

28 Cohomologies of a complex manifold, vii inequality à la Frölicher for the Bott-Chern cohomology and -Lemma For a compact cplx mfd, one has the Frölicher inequality dim C H p,q (X ) dim C H k dr (X ; C). p+q=k, A. Tomassini, On the -Lemma and Bott-Chern cohomology, Invent. Math. 192 (2013), no. 1, 7181.

29 Cohomologies of a complex manifold, vii inequality à la Frölicher for the Bott-Chern cohomology and -Lemma For a compact cplx mfd, one has the Frölicher inequality dim C H p,q (X ) dim C H k dr (X ; C). p+q=k As for Bott-Chern cohom: Thm (, A. Tomassini) X cpt cplx mfd. The following inequality à la Frölicher holds: ( dimc H p,q BC (X ) + dim C H p,q A (X )) 2 dim C H k dr (X ; C). p+q=k, A. Tomassini, On the -Lemma and Bott-Chern cohomology, Invent. Math. 192 (2013), no. 1, 7181.

30 Cohomologies of a complex manifold, vii inequality à la Frölicher for the Bott-Chern cohomology and -Lemma For a compact cplx mfd, one has the Frölicher inequality dim C H p,q (X ) dim C H k dr (X ; C). p+q=k As for Bott-Chern cohom: Thm (, A. Tomassini) X cpt cplx mfd. The following inequality à la Frölicher holds: ( dimc H p,q BC (X ) + dim C H p,q A (X )) 2 dim C H k dr (X ; C). p+q=k Furthermore, the equality characterizes the -Lemma, namely, the property that H, BC (X ) H dr (X ; C)., A. Tomassini, On the -Lemma and Bott-Chern cohomology, Invent. Math. 192 (2013), no. 1, 7181.

31 Cohomological models, i models and computability We would like to compute H { H dr, H, H, H BC, H A } for a compact complex manifold X. M. Schweitzer, Autour de la cohomologie de Bott-Chern, arxiv: v1 [math.ag].

32 Cohomological models, i models and computability We would like to compute H { } H dr, H, H, H BC, H A for a compact complex manifold X. Hodge theory assures nite-dimensionality (Schweitzer). M. Schweitzer, Autour de la cohomologie de Bott-Chern, arxiv: v1 [math.ag].

33 Cohomological models, i models and computability We would like to compute H { } H dr, H, H, H BC, H A for a compact complex manifold X. Hodge theory assures nite-dimensionality (Schweitzer). It may turn out useful to restrict to a H -model, that is, a subobject ι: ( M,,, ) (, X,, ) giving the same H -cohomology (i.e., H (ι) isom). M. Schweitzer, Autour de la cohomologie de Bott-Chern, arxiv: v1 [math.ag].

34 Cohomological models, i models and computability We would like to compute H { } H dr, H, H, H BC, H A for a compact complex manifold X. Hodge theory assures nite-dimensionality (Schweitzer). It may turn out useful to restrict to a H -model, that is, a subobject ι: ( M,,, ) (, X,, ) giving the same H -cohomology (i.e., H (ι) isom). We are interested in H -computable cplx mfds, that is, admitting a nite-dimensional H -model (in the correct category). M. Schweitzer, Autour de la cohomologie de Bott-Chern, arxiv: v1 [math.ag].

35 Cohomological models, ii from Dolbeault to de Rham Note that: H -computable mfds are H dr -computable, too. A. Frölicher, Relations between the cohomology groups of Dolbeault and topological invariants, Proc. Nat. Acad. Sci. U.S.A. 41 (1955),

36 Cohomological models, ii from Dolbeault to de Rham Note that: H -computable mfds are H dr -computable, too. In fact, any H -model is a H dr -model, too, because of the spectral sequence induced by the double complex structure. A. Frölicher, Relations between the cohomology groups of Dolbeault and topological invariants, Proc. Nat. Acad. Sci. U.S.A. 41 (1955),

37 Cohomological models, iii from Dolbeault to Bott-Chern, i surjectivity Prop (, Kasuya) X cplx mfd. Consider ι: ( M,,, ) (, X,, )., H. Kasuya, Bott-Chern cohomology of solvmanifolds, arxiv: v3 [math.dg].

38 Cohomological models, iii from Dolbeault to Bott-Chern, i surjectivity Prop (, Kasuya) X cplx mfd. Consider ι: ( M,,, ) (, X,, ). If ι is a H -model, ι is a H -model, and ker d Mp,q im d ker d p,q X im d surj,, H. Kasuya, Bott-Chern cohomology of solvmanifolds, arxiv: v3 [math.dg].

39 Cohomological models, iii from Dolbeault to Bott-Chern, i surjectivity Prop (, Kasuya) X cplx mfd. Consider ι: ( M,,, ) (, X,, ). If ι is a H -model, ι is a H -model, and ker d Mp,q im d then H BC (ι) surjective. ker d p,q X im d surj,, H. Kasuya, Bott-Chern cohomology of solvmanifolds, arxiv: v3 [math.dg].

40 Cohomological models, iii from Dolbeault to Bott-Chern, i surjectivity Prop (, Kasuya) X cplx mfd. Consider ι: ( M,,, ) (, X,, ). If ι is a H -model, ι is a H -model, and ker d Mp,q im d then H BC (ι) surjective. ker d p,q X im d surj, Proof. 0 im d M p,q im H p,q BC (M, ) ker d M p,q im d 0 0 im d p,q X im H p,q BC (X ) ker d p,q X im d 0, H. Kasuya, Bott-Chern cohomology of solvmanifolds, arxiv: v3 [math.dg].

41 Cohomological models, iv from Dolbeault to Bott-Chern, ii injectivity As for injectivity, use Hodge theory: Prop (, H. Kasuya) X cplx mfd. Consider ι: ( M,,, ) (, X,, ).

42 Cohomological models, iv from Dolbeault to Bott-Chern, ii injectivity As for injectivity, use Hodge theory: Prop (, H. Kasuya) X cplx mfd. Consider ι: ( M,,, ) (, X,, ). If dim M, < +, and M, is closed under and,

43 Cohomological models, iv from Dolbeault to Bott-Chern, ii injectivity As for injectivity, use Hodge theory: Prop (, H. Kasuya) X cplx mfd. Consider ι: ( M,,, ) (, X,, ). If dim M, < +, and M, is closed under and, then H BC (ι) injective.

44 Nilmanifolds, i de Rham cohomology G connected simply-connected nilpotent Lie group Γ discrete co-compact subgroup X = Γ\G nilmanifold K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. (2) 59 (1954), no. 3,

45 Nilmanifolds, i de Rham cohomology G connected simply-connected nilpotent Lie group Γ discrete co-compact subgroup X = Γ\G nilmanifold Thm (Nomizu) X = Γ\G nilmanifold. The inclusion of left-invariant forms, is a H dr -model. ι: ( g, d) ( X, d), K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. (2) 59 (1954), no. 3,

46 Nilmanifolds, i de Rham cohomology G connected simply-connected nilpotent Lie group Γ discrete co-compact subgroup X = Γ\G nilmanifold Thm (Nomizu) X = Γ\G nilmanifold. The inclusion of left-invariant forms, is a H dr -model. ι: ( g, d) ( X, d), T j 0 T j 1 T j k X = Γ\G X 1. X k T j k+1 K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. (2) 59 (1954), no. 3,

47 Nilmanifolds, ii Dolbeault cohomology Thm (Sakane, Cordero, Fernández, Gray, Ugarte, Console, Fino, Rollenske) X = Γ\G nilmfd with a left-inv suitable cplx struct. The inclusion of left-invariant forms, is a H -model. ι: (, (g C),, ) (, X,, ), S. Console, A. Fino, Dolbeault cohomology of compact nilmanifolds, Transform. Groups 6 (2001), no. 2, S. Rollenske, Dolbeault cohomology of nilmanifolds with left-invariant complex structure, in W. Ebeling, K. Hulek, K. Smoczyk (eds.), Complex and Dierential Geometry: Conference held at Leibniz Universität Hannover, September 14 18, 2009, Springer Proceedings in Mathematics 8, Springer, 2011,

48 Nilmanifolds, ii Dolbeault cohomology Thm (Sakane, Cordero, Fernández, Gray, Ugarte, Console, Fino, Rollenske) X = Γ\G nilmfd with a left-inv suitable cplx struct. The inclusion of left-invariant forms, is a H -model. ι: (, (g C),, ) (, X,, ), It is conjectured that any left-inv cplx structure on nilmanifolds is suitable (Rollenske). S. Console, A. Fino, Dolbeault cohomology of compact nilmanifolds, Transform. Groups 6 (2001), no. 2, S. Rollenske, Dolbeault cohomology of nilmanifolds with left-invariant complex structure, in W. Ebeling, K. Hulek, K. Smoczyk (eds.), Complex and Dierential Geometry: Conference held at Leibniz Universität Hannover, September 14 18, 2009, Springer Proceedings in Mathematics 8, Springer, 2011,

49 Nilmanifolds, iii Bott-Chern cohomology Thm () X = Γ\G nilmfd with a left-inv suitable cplx struct. The inclusion of left-invariant forms, is a H BC -model. ι: (, (g C),, ) (, X,, ),, The cohomologies of the Iwasawa manifold and of its small deformations, J. Geom. Anal. 23 (2013), no. 3,

50 Nilmanifolds, iii Bott-Chern cohomology Thm () X = Γ\G nilmfd with a left-inv suitable cplx struct. The inclusion of left-invariant forms, is a H BC -model. ι: (, (g C),, ) (, X,, ), Summarizing: nilmanifolds with suitable left-inv cplx struct are cohomologicallycomputable, ( by means of the nite-dim sub-complex of left-inv forms,, g C,, )., The cohomologies of the Iwasawa manifold and of its small deformations, J. Geom. Anal. 23 (2013), no. 3,

51 Nilmanifolds, iv Iwasawa manifold I 3, i denition Iwasawa manifold: I 3 := (Z [i]) 3 1 z 1 z z ( ) GL C 3

52 Nilmanifolds, iv Iwasawa manifold I 3, i denition Iwasawa manifold: I 3 := (Z [i]) 3 1 z 1 z z ( ) GL C 3 holomorphically-parallelizable nilmanifold

53 Nilmanifolds, iv Iwasawa manifold I 3, i denition Iwasawa manifold: I 3 := (Z [i]) 3 1 z 1 z z ( ) GL C 3 holomorphically-parallelizable nilmanifold left-inv co-frame for ( T 1,0 I 3 ) : { ϕ 1 := d z 1, ϕ 2 := d z 2, ϕ 3 := d z 3 z 1 d z 2}

54 Nilmanifolds, iv Iwasawa manifold I 3, i denition Iwasawa manifold: I 3 := (Z [i]) 3 1 z 1 z z ( ) GL C 3 holomorphically-parallelizable nilmanifold left-inv co-frame for ( T 1,0 I 3 ) : { ϕ 1 := d z 1, ϕ 2 := d z 2, ϕ 3 := d z 3 z 1 d z 2} structure equations: d ϕ = 0 d ϕ 2 = 0 d ϕ 3 = ϕ 1 ϕ 2

55 Nilmanifolds, v Iwasawa manifold I 3, ii double complex of left-invariant forms

56 Nilmanifolds, vi Iwasawa manifold I 3, iii Dolbeault cohomology (p, q) dim C H p,q (0, 0) 1 (1, 0) 3 (0, 1) 2 (2, 0) 3 (1, 1) 6 (0, 2) 2 (3, 0) 1 (2, 1) 6 (1, 2) 6 (0, 3) 1 (3, 1) 2 (2, 2) 6 (1, 3) 3 (3, 2) 2 (2, 3) 3 (3, 3) 1

57 Nilmanifolds, vii Iwasawa manifold I 3, iv Bott-Chern cohomology (p, q) dim C H p,q BC (I3) (0, 0) 1 (1, 0) 2 (0, 1) 2 (2, 0) 3 (1, 1) 4 (0, 2) 3 (3, 0) 1 (2, 1) 6 (1, 2) 6 (0, 3) 1 (3, 1) 2 (2, 2) 8 (1, 3) 2 (3, 2) 3 (2, 3) 3 (3, 3) 1

58 Nilmanifolds, viii 6-dimensional nilmanifolds More in general: any left-invariant complex structure on a 6-dim nilmfd admits a nite-dim cohomological-model (except, perhaps, h 7 ) cohomol classication of 6-dim nilmfds with left-inv cplx struct., M. G. Franzini, F. A. Rossi, Degree of non-kählerianity for 6-dimensional nilmanifolds, arxiv: [math.dg]. A. Latorre, L. Ugarte, R. Villacampa, On the Bott-Chern cohomology and balanced Hermitian nilmanifolds, arxiv: v1 [math.dg].

59 Solvmanifolds, i denition and motivations G connected simply-connected solvable Lie group Γ discrete co-compact subgroup X = Γ\G solvmanifold Ch. Benson, C. S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), no. 4, K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, H. Kasuya, Hodge symmetry and decomposition on non-kähler solvmanifolds, arxiv: v5 [math.dg].

60 Solvmanifolds, i denition and motivations G connected simply-connected solvable Lie group Γ discrete co-compact subgroup X = Γ\G solvmanifold While nilmanifolds (except tori) are non-kähler, non-formal, non- -Lemma, non-hlc,... (Benson and Gordon, Hasegawa) Ch. Benson, C. S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), no. 4, K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, H. Kasuya, Hodge symmetry and decomposition on non-kähler solvmanifolds, arxiv: v5 [math.dg].

61 Solvmanifolds, i denition and motivations G connected simply-connected solvable Lie group Γ discrete co-compact subgroup X = Γ\G solvmanifold While nilmanifolds (except tori) are non-kähler, non-formal, non- -Lemma, non-hlc,... (Benson and Gordon, Hasegawa)... there exist non-kähler solvmanifolds satisfying -Lemma Ch. Benson, C. S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), no. 4, K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, H. Kasuya, Hodge symmetry and decomposition on non-kähler solvmanifolds, arxiv: v5 [math.dg]. (Kasuya).

62 Solvmanifolds, ii de Rham cohomology Thm (Hattori) X = Γ\G solvmanifold. If G is completely-solvable, then the inclusion of left-inv forms, ι: g X, is a H dr -model. A. Hattori, Spectral sequence in the de Rham cohomology of bre bundles, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), no. 1960, P. de Bartolomeis, A. Tomassini, On solvable generalized Calabi-Yau manifolds, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 5, H. Kasuya, Minimal models, formality and hard Lefschetz properties of solvmanifolds with local systems, J. Dier. Geom. 93, (2013),

63 Solvmanifolds, ii de Rham cohomology Thm (Hattori) X = Γ\G solvmanifold. If G is completely-solvable, then the inclusion of left-inv forms, ι: g X, is a H dr -model. In general, de Rham cohom depends on Γ (de Bartolomeis, Tomassini): A. Hattori, Spectral sequence in the de Rham cohomology of bre bundles, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), no. 1960, P. de Bartolomeis, A. Tomassini, On solvable generalized Calabi-Yau manifolds, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 5, H. Kasuya, Minimal models, formality and hard Lefschetz properties of solvmanifolds with local systems, J. Dier. Geom. 93, (2013),

64 Solvmanifolds, ii de Rham cohomology Thm (Hattori) X = Γ\G solvmanifold. If G is completely-solvable, then the inclusion of left-inv forms, ι: g X, is a H dr -model. In general, de Rham cohom depends on Γ (de Bartolomeis, Tomassini): Thm (Kasuya) X = Γ\G solvmfd. There exists a nite-dim H dr -model (A Γ, d) ( X C, d). A. Hattori, Spectral sequence in the de Rham cohomology of bre bundles, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), no. 1960, P. de Bartolomeis, A. Tomassini, On solvable generalized Calabi-Yau manifolds, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 5, H. Kasuya, Minimal models, formality and hard Lefschetz properties of solvmanifolds with local systems, J. Dier. Geom. 93, (2013),

65 Solvmanifolds, iii Dolbeault cohomology and Bott-Chern cohomology X = Γ\G solvmfd with left-inv cplx struct of one of the following types: holomorphically-parallelizable; splitting-type: i.e., G = C n φ N, with N nilpotent with left-inv cplx struct, φ semi-simple, holomorphic at every time; assume Γ \N has left-inv forms as H -model. H. Kasuya, De Rham and Dolbeault cohomology of solvmanifolds with local systems, arxiv: v3 [math.dg]. H. Kasuya, Techniques of computations of Dolbeault cohomology of solvmanifolds, Math. Z. 273 (2013), no. 1-2,

66 Solvmanifolds, iii Dolbeault cohomology and Bott-Chern cohomology X = Γ\G solvmfd with left-inv cplx struct of one of the following types: holomorphically-parallelizable; splitting-type: i.e., G = C n φ N, with N nilpotent with left-inv cplx struct, φ semi-simple, holomorphic at every time; assume Γ \N has left-inv forms as H -model. Thm (Kasuya;, Kasuya) Then there exist: a nite-dim H -model ( B, Γ,, ) (, X,, ) ; H. Kasuya, De Rham and Dolbeault cohomology of solvmanifolds with local systems, arxiv: v3 [math.dg]. H. Kasuya, Techniques of computations of Dolbeault cohomology of solvmanifolds, Math. Z. 273 (2013), no. 1-2,

67 Solvmanifolds, iii Dolbeault cohomology and Bott-Chern cohomology X = Γ\G solvmfd with left-inv cplx struct of one of the following types: holomorphically-parallelizable; splitting-type: i.e., G = C n φ N, with N nilpotent with left-inv cplx struct, φ semi-simple, holomorphic at every time; assume Γ \N has left-inv forms as H -model. Thm (Kasuya;, Kasuya) Then there exist: a nite-dim H -model ( B, Γ,, ) (, X,, ) ; a nite-dim H BC -model ( C, Γ,, ) (, X,, ). H. Kasuya, De Rham and Dolbeault cohomology of solvmanifolds with local systems, arxiv: v3 [math.dg]. H. Kasuya, Techniques of computations of Dolbeault cohomology of solvmanifolds, Math. Z. 273 (2013), no. 1-2,

68 Solvmanifolds, iv cohomological models for deformations By using the spectral theory for families of elliptic dierential operators: Thm (, Kasuya) Consider a cohomologically-computable cplx manifold. Then suitable small deformations are still cohomologically-computable., H. Kasuya, Cohomologies of deformations of solvmanifolds and closedness of some properties, arxiv: v1 [math.cv].

69 Solvmanifolds, iv cohomological models for deformations By using the spectral theory for families of elliptic dierential operators: Thm (, Kasuya) Consider a cohomologically-computable cplx manifold. Then suitable small deformations are still cohomologically-computable. As a corollary, one recovers: Thm (Console, Fino) On a nilmanifold, the set of cplx structures having left-inv forms as H -model is open in the space of left-inv cplx structures., H. Kasuya, Cohomologies of deformations of solvmanifolds and closedness of some properties, arxiv: v1 [math.cv].

70 -Lemma and deformations, i -Lemma and deformations A compact cplx mfd X satises the -Lemma i the natural map H, BC (X ) H dr (X ; C) is an isomorphism: it is a cohomological-decomposition property.

71 -Lemma and deformations, i -Lemma and deformations A compact cplx mfd X satises the -Lemma i the natural map H, BC (X ) H dr (X ; C) is an isomorphism: it is a cohomological-decomposition property. Aim: to study the behaviour of -Lemma under deformations of the cplx structure.

72 -Lemma and deformations, i -Lemma and deformations A compact cplx mfd X satises the -Lemma i the natural map H, BC (X ) H dr (X ; C) is an isomorphism: it is a cohomological-decomposition property. Aim: to study the behaviour of -Lemma under deformations of the cplx structure. i.e., consider a family {X t} t (0,ε) of cpt cplx mfds:

73 -Lemma and deformations, i -Lemma and deformations A compact cplx mfd X satises the -Lemma i the natural map H, BC (X ) H dr (X ; C) is an isomorphism: it is a cohomological-decomposition property. Aim: to study the behaviour of -Lemma under deformations of the cplx structure. i.e., consider a family {X t} t (0,ε) of cpt cplx mfds: if X 0 satises the -Lemma, then does X t for t small? (openness)

74 -Lemma and deformations, i -Lemma and deformations A compact cplx mfd X satises the -Lemma i the natural map H, BC (X ) H dr (X ; C) is an isomorphism: it is a cohomological-decomposition property. Aim: to study the behaviour of -Lemma under deformations of the cplx structure. i.e., consider a family {X t} t (0,ε) of cpt cplx mfds: if X 0 satises the -Lemma, then does X t for t small? (openness) if X t satises the -Lemma for any t 0, then does X 0? (closedness)

75 -Lemma and deformations, ii Iwasawa manifold, i summary of cohomologies I 3 dr BC A (0, 0) (1, 0) (0, 1) (2, 0) (1, 1) (0, 2) (3, 0) (2, 1) (1, 2) (0, 3) (3, 1) (2, 2) (1, 3) (3, 2) (2, 3) (3, 3) Iwasawa manifold

76 -Lemma and deformations, ii Iwasawa manifold, i summary of cohomologies Iwasawa manifold I 3 dr BC A (0, 0) (1, 0) (0, 1) (2, 0) (1, 1) (0, 2) (3, 0) (2, 1) (1, 2) (0, 3) (3, 1) (2, 2) (1, 3) (3, 2) (2, 3) (3, 3) I 3 dr BC A

77 -Lemma and deformations, iii Iwasawa manifold, ii remarks on the cohomologies Iwasawa manifold I 3 dr BC A h k b k (Frölicher inequality) but, in general, hbc k is not greater than or equal to h k or b k. Anyway, note that h k BC + hk A 2 b k in the example and this is a general result

78 -Lemma and deformations, iv inequality à la Frölicher for Bott-Chern, i an heuristic argument Dolbeault cohomology cares only about horizontal arrows, as Bott-Chern cares only about ingoing arrows, and, dually, Aeppli cares only about outgoing arrows

79 -Lemma and deformations, iv inequality à la Frölicher for Bott-Chern, i an heuristic argument Dolbeault cohomology cares only about horizontal arrows, as Bott-Chern cares only about ingoing arrows, and, dually, Aeppli cares only about outgoing arrows

80 -Lemma and deformations, iv inequality à la Frölicher for Bott-Chern, i an heuristic argument Dolbeault cohomology cares only about horizontal arrows, as Bott-Chern cares only about ingoing arrows, and, dually, Aeppli cares only about outgoing arrows

81 -Lemma and deformations, iv inequality à la Frölicher for Bott-Chern, i an heuristic argument Dolbeault cohomology cares only about horizontal arrows, as Bott-Chern cares only about ingoing arrows, and, dually, Aeppli cares only about outgoing arrows. Since {ingoing} + {outgoing} {horizontal} + {vertical} 0 one gets:

82 -Lemma and deformations, iv inequality à la Frölicher for Bott-Chern, i an heuristic argument Dolbeault cohomology cares only about horizontal arrows, as Bott-Chern cares only about ingoing arrows, and, dually, Aeppli cares only about outgoing arrows. Since {ingoing} + {outgoing} {horizontal} + {vertical} 0 one gets: Thm (, A. Tomassini) X cpt cplx mfd. The following inequality à la Frölicher holds: ( dimc H p,q BC (X ) + dim C H p,q A (X )) 2 dim C H k dr (X ; C). p+q=k Furthermore, the equality characterizes the -Lemma.

83 -Lemma and deformations, v inequality à la Frölicher for Bott-Chern, ii openness of -Lemma By Hodge theory, dim C H p,q BC and dim C H p,q A are upper-semi-continuous for deformations of the complex structure.

84 -Lemma and deformations, v inequality à la Frölicher for Bott-Chern, ii openness of -Lemma By Hodge theory, dim C H p,q BC and dim C H p,q A are upper-semi-continuous for deformations of the complex structure. Hence the equality p+q=k ( dimc H p,q BC (X ) + dim C H p,q A (X )) = 2 dim C H k dr (X ; C) is stable for small deformations.

85 -Lemma and deformations, v inequality à la Frölicher for Bott-Chern, ii openness of -Lemma By Hodge theory, dim C H p,q BC and dim C H p,q A are upper-semi-continuous for deformations of the complex structure. Hence the equality p+q=k ( dimc H p,q BC (X ) + dim C H p,q A (X )) = 2 dim C H k dr (X ; C) is stable for small deformations. It follows: Cor (Voisin; Wu; Tomasiello;, A. Tomassini) The property of satisfying the -Lemma is open under deformations.

86 -Lemma and deformations, vi Nakamura manifold and deformations, i The Lie group C φ C 2 dove φ(z) = ( e z 0 0 e z ). admits a lattice: the quotient is called Nakamura manifold.

87 -Lemma and deformations, vi Nakamura manifold and deformations, i The Lie group C φ C 2 dove φ(z) = ( e z 0 0 e z ). admits a lattice: the quotient is called Nakamura manifold. Consider the small deformations in the direction t z 1 d z 1.

88 -Lemma and deformations, vi Nakamura manifold and deformations, i The Lie group C φ C 2 dove φ(z) = ( e z 0 0 e z ). admits a lattice: the quotient is called Nakamura manifold. Consider the small deformations in the direction t z 1 d z 1. the previous theorems furnish nite-dim sub-complexes to compute Dolbeault and Bott-Chern cohomologies

89 -Lemma and deformations, vii Nakamura manifold and deformations, ii non-closedness of -Lemma dim C H, Nakamura deformations dr BC dr BC (0, 0) (1, 0) (0, 1) (2, 0) (1, 1) (0, 2) (3, 0) (2, 1) (1, 2) (0, 3) (3, 1) (2, 2) (1, 3) (3, 2) (2, 3) (3, 3)

90 -Lemma and deformations, vii Nakamura manifold and deformations, ii non-closedness of -Lemma dim C H, Nakamura deformations dr BC dr BC (0, 0) (1, 0) (0, 1) (2, 0) (1, 1) (0, 2) (3, 0) (2, 1) (1, 2) (0, 3) (3, 1) (2, 2) (1, 3) (3, 2) (2, 3) (3, 3) Thm (, Kasuya) The property of satisfying the -Lemma is not closed under deformations.

91 Future work Dicult to see. Always in motion is the future. For the future:, A. Tomassini., S. Calamai., H. Kasuya., J. Raissy. construct a cohomological-model for (generalized-)cplx mfds, maybe including geometric informations; (Sullivan formality, Dolbeault homotopy theory,... ) compute it for an enlarged class of cohom-computable cplx mfds... (López de Medrano and Verjovsky mfds, cpt cplx surf,... )... in order to investigate geometric properties. (contractions of Fujiki class C or Mo ²hezon,... )

92 Joint work with: Adriano Tomassini, Hisashi Kasuya, Federico A. Rossi, Maria Giovanna Franzini, Simone Calamai. And with the essential contribution of: Serena, Maria Beatrice e Luca, Andrea, Maria Rosaria, Matteo, Jasmin, Carlo, Junyan, Michele, Chiara, Simone, Eridano, Laura, Paolo, Marco, Cristiano, Daniele, Matteo,...