Adriano Tomassini

ON THE MASLOV INDEX OF LAGRANGIAN SUBMANIFOLDS OF GENERALIZED CALABI–YAU MANIFOLDS

We characterize the special Lagrangian submanifolds of a generalized Calabi–Yau manifold, with vanishing Maslov class. Then, we carefully describe several examples, including a non-Kahler generalized Calabi–Yau manifold foliated by special Lagrangian submanifolds.

ON THE COHOMOLOGY OF ALMOST-COMPLEX MANIFOLDS

Following [T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.17(4) (2009) 651–683], we continue to study the link between the cohomology of an almost-complex manifold and its almost-complex structure. In particular, we apply the same argument in [T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.17(4) (2009) 651–683] and the results obtained by [D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math.36(1) (1976) 225–255] to study the cone of semi-Kahler structures on a compact semi-Kahler manifold.

CONTACT CALABI-YAU MANIFOLDS AND SPECIAL LEGENDRIAN SUBMANIFOLDS

We consider a generalization of Calabi-Yau structures in the context of Sasakian manifolds. We study deformations of a special class of Legendrian submanifolds and classify invariant contact Calabi-Yau structures on 5-dimensional nilmanifolds. Finally we generalize to codimension r.

On formality of some symplectic manifolds

Let (M, κ) be a compact symplectic manifold of dimension 2n; we prove that, if there exists a κ-calibrated complex structure J such that, with respect to the Hermitian structure g J , the Nijenhuis tensor A J satisfies, for 0<p<n, ‖A‖<K(n,p)μr+1(J) (where r = n − p and r (J) is the smallest positive eigenvalue of Δ g J, acting on r + 1-forms), then (M, κ) satisfies the hard Lefschetz condition. Note that this implies that the differentiable Gerstenhaber-Batalin-Vilkovisky algebra (Λ*(M),d⋆,d) is integrable (i.e., the dd⋆-lemma holds). Consequently, the formality of the deRham complexand the existence of a structure of (formal) Frobenius manifold on C [[H * (M)]] follow at once; an example shows that the given estimate is sharp.

On Bott-Chern cohomology of compact complex surfaces

We study Bott-Chern cohomology on compact complex non-Kähler surfaces. In particular, we compute such a cohomology for compact complex surfaces in class

GENERALIZED G2-MANIFOLDS AND SU(3)-STRUCTURES

\text {VII}

Balanced Hermitian metrics from SU(2)-structures

VII and for compact complex surfaces diffeomorphic to solvmanifolds.

Curvature homogeneous metrics on principal fibre bundles

Given a closed symplectic manifold, we study when the Lefschetz decomposition induced by the sl(2;R)-representation yields a decomposition of the de Rham cohomology. In particular, this holds always true for the second de Rham cohomology group, or if the symplectic manifold satisfies the Hard Lefschetz Condition.