• Hermitian symmetric space
• Gromov width
• Symplectic geometry
• Symplectic atlas

Our work fits in the context of symplectic geometry and, in particular, the aim of this work is to compute minimal symplectic atlases for classical Hermitian symmetric spaces of compact type.

A symplectic manifold (M,\omega) is a 2n-dimensional manifold M equipped with a closed and nondegenerate 2-form \omega. The basic example of symplectic manifold is R^{2n} equipped with the standard symplectic form \omega_{0}=\sum_{j}dx_{j}\wedge dy_{j}$. The first interesting result about symplectic geometry is that for all p in M there is a symplectic embedding f of the 2n-dimensional ball equipped with the standard symplectic form (B^{2n}(r),\omega_{0}) in (M,\omega) such that f(0)=p. This result gave rise to the introduction of an important symplectic invariant c_{G} called Gromov width: it is \pi(r)^2 where r is the maximum radius of a ball which can be symplectically embedded in M.

In [1] Rudyak-Schlenk introduced the invariant:
S_{B}(M,\omega)which is the minimal number of symplectic charts needed to cover (M,\omega). An immediate lower bound for S_{B}(M,\omega) is \lambda(M,\omega):=max{\Gamma(M,\omega);B(M)} where B(M) is the number of charts of a minimal (not necessarily symplectic) atlas and
\Gamma(M,\omega):=\lfloor{Vol(M,\omega)n!}/{c_{G}(M,\omega)^{n}}\rfloor + 1;
the braket \lfloor x \rfloor denoting the maximal integer smaller than or equal to x.
Their main result about minimal atlases in [1] is the following
Theorem 1

i) If \lambda(M,\omega)>= 2n+1$ then S_{B}(M,\omega)=\lambda(M,\omega).
ii) If \lambda(M,\omega) < 2n+1 then n+1<=\lambda(M,\omega)<=S_{B}(M,\omega)<=2n+1.

It is then clear that problem of computing the invariant S_{B}(M,\omega) is strictly related to the knowledge of the Gromov width of (M,\omega).

If we consider a 2n-dimensional projective variety M in CP^{d} the result can be presented in terms of the degree of the embedding F:M\rightarrow CP^{d}. Indeed such a manifold is Kaehler when equipped with the restriction \omega_{M} of the Fubini-Study form \omega_{FS} of CP^{d}. Moreover the volume of such a manifold is related to the volume of CP^{n} by the formula Vol(M)=deg(F)Vol(CP^{n}). Thus we get the following:
Corollary 2
If (M,\omega_{M}) is a projectively induced Kaehler manifold and {deg(F)\pi^{n}}/{c_{G}(M,\omega_{M})^{n}}>= 2n then S_{B}(M,\omega_{M})=deg(F)+1.

In particular this corollary implies that for a projectively induced Kaehler manifold (M,\omega_{M}) it is sufficient to
know c_{G}(M,\omega_{M}) and the degree of the embedding in order to compute the invariant S_{B}(M,\omega_{M}).
Unfortunately computing the Gromov width of a symplectic manifold is usually a very delicate problem. However in [2] Loi-Mossa-Zuddas calculated the Gromov width of Hermitian symmetric spaces of compact type. In this thesis, using the above results, we prove the following
Theorem 3
Let (M,\omega) be an irreducible compact Hermitian symmetric spaces of type I,II or III. Then S_{B}(M,\omega_{M})=deg(F)+1 when the dimension of M is sufficiently large.

Moreover, using the work of Loi-Mossa-Zuddas, we are able to extend this result to product of these spaces.
Unfortunately the irreducible compact domain of type IV Q_{n} does not satisfy the hypothesis of corollary 2 thus we cannot compute S_{B} using the same arguments.
Nevertheless, in the last part of the thesis, we provide an explicit construction of a full symplectic embedding of Q_{n}, namely a collection of symplectic embeddings f_{i}:B^{2n}(1)\rightarrow Q_{n} such that the closures of the images of these embeddings cover M.


References
[1] Y.B. Rudyak, F. Schlenk, Minimal atlases of closed symplectic manifolds, Commun. Contemp. Math 9 (2007), no. 6, 811-855.
[2] A. Loi, R. Mossa, F. Zuddas, Symplectic capacities of Hermitian symmetric spaces of compact and non compact type, to appear in Journal of Symplectic Geometry.