## The Gromov width of 4-dimensional tori.(English)Zbl 1277.57024

A symplectic manifold is a $$2n$$-dimensional manifold $$M$$ together with a closed 2-form $$\omega$$ such that $$\omega^n$$ is non-zero everywhere. For example, $$\omega_0:=dx_1\wedge dy_1+dx_2\wedge dy_2$$ is a symplectic structure on $$\mathbb{R}^4$$. Other examples are given by the quotient of $$({\mathbb R}^4,\omega_0)$$ by a suitable lattice $$\Lambda$$. Such a quotient manifold is necessarily a 4-torus, and the resulting symplectic form is called a linear symplectic form on a 4-torus. It is not known whether every symplectic form on a 4-torus is symplectomorphic to such a form.
Given $$a>0$$ we consider the open ball of capacity $$a$$, i.e. we consider $B^4(a):=\{ z\in \mathbb{C}^2\,|\, \pi(|z_1|^2+|z_2|^2|)<a\}$ as a subset of the symplectic space $$({\mathbb R}^4,\omega_0)$$. The ball filling number of a finite volume symplectic manifold $$(M,\omega)$$ is defined as $p(M,\omega):=\text{sup} \frac{ \text{Vol}(B^4(a))}{\text{Vol}(M,\omega)},$ where the supremum is taken over all balls $$B^4(a)$$ that symplectically embed into $$(M,\omega)$$. If $$p(M,\omega)<1$$, then one says that there is a filling obstruction, while if $$p(M,\omega)=1$$, one says that $$(M,\omega)$$ admits a full filling by one ball.
There are symplectic manifolds which have a filling obstruction. But the main theorem of the paper says that a 4-torus with a linear symplectic form admits a full filling by one ball.

### MSC:

 57R17 Symplectic and contact topology in high or arbitrary dimension 57R40 Embeddings in differential topology 32J27 Compact Kähler manifolds: generalizations, classification
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### References:

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