Paolo de Bartolomeis

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 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.

DEFORMATIONS OF LEVI FLAT STRUCTURES IN SMOOTH MANIFOLDS

We define a complex whose cohomology group of order 1 contains the infinitesimal deformations of a Levi flat structure on a smooth manifold. In the case of real analytic Levi flat structures, this cohomology group is the product of the d-bar cohomology group of order 1 of tangent vector fields to the Levi structure and the cohomology group of order 1 of the associated DGLA.

On the Obstruction of the Deformation Theory in the DGLA of Graded Derivations

In a recent paper, the authors studied the deformation theory in the DGLA of graded derivations \(\mathcal{D}^{{\ast}}\left (M\right )\) of differential forms on M. They proved the existence of canonical solutions \(e_{\Phi }\) of Maurer-Cartan equation depending on a vector valued differential form \(\Phi\) and gave a classification of these canonical solutions by their type: a canonical solution \(e_{\Phi }\) is of finite type r if \(\Phi ^{r}\left [\Phi,\Phi \right ]_{\mathcal{F}\mathcal{N}} = 0\) and \(r =\min \left \{\,j \in \mathbb{N}: \Phi ^{\,j}\left [\Phi,\Phi \right ]_{\mathcal{F}\mathcal{N}} = 0\ \right \}\), where \(\left [\cdot,\cdot \right ]_{\mathcal{F}\mathcal{N}}\) is the Frolicher-Nijenhuis bracket. In this paper it is shown that the deformation theory in the DGLA of graded derivations is not obstructed, but it is level-wise obstructed.

SOME RESULTS ON GENERALIZED CALABI–YAU MANIFOLDS

We consider generalized Calabi–Yau manifolds and we give a formula for the Maslov class of a Lagrangian submanifold of a generalized Calabi–Yau manifold. In particular, we characterize the Lagrangian submanifolds with vanishing Maslov class. In the 6-dimensional case, we refine our definition. Finally, we construct some examples.

ℤ2 and ℤ-Deformation Theory for Holomorphic and Symplectic Manifolds

We present and investigate, within the general frame of deformation theory, new ℤ2-constructions for generalized moduli spaces of holomorphic and symplectic structures.