We prove a version of equivariant split generation of Fukaya category when a symplectic manifold admits a free action of a finite group <i>G</i>. Combining this with some generalizations of Seidel's algebraic frameworks from [35], we obtain new cases of homological mirror symmetry for some symplectic tori with non-split symplectic forms, which we call <i>special isogenous tori</i>. This extends the work of AbouzaidSmith [2]. We also show that derived Fukaya categories are complete invariants of special isogenous tori.

We study the algebraic dimension a(X) of a compact hyperk ¨ahler manifold of dimension 2n. We show that a(X) is at most n unless X is projective. If a compact K¨ahler manifold with algebraic dimension 0 and Kodaira dimension 0 has a minimal model, then only the values 0, n and 2n are possible. In case of middle dimension, the algebraic reduction is holomorphic Lagrangian. If n = 2, then - without any assumptions - the algebraic dimension only takes the values 0, 2 and 4. The paper also gives structure results for ”generalised hyperk¨ahler” manifolds and studies nef lines bundles.

We count the number$S$($x$) of quadruples $ {\left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right)} \in \mathbb{Z}^{4} $ for which $$ p = x^{2}_{1} + x^{2}_{2} + x^{2}_{3} + x^{2}_{4} \leqslant x $$ is a prime number and satisfying the determinant condition:$x$_{1}$x$_{4}−$x$_{2}$x$_{3}= 1. By means of the sieve, one shows easily the upper bound$S$($x$) ≪$x$/log$x$. Under a hypothesis about prime numbers, which is stronger than the Bombieri–Vinogradov theorem but is weaker than the Elliott–Halberstam conjecture, we prove that this order is correct, that is$S$($x$) ≫$x$/log$x$.

We introduce a semi-algebraic structure on the set of all isotopy classes of non-separating simple closed curves in any compact oriented surface and show that the structure is finitely generated. As a consequence, we produce a natural finite dimensional linear representation of the mapping class group of the surface. Applications to the Teichmller space, Thurston's measured lamination space, the harmonic Beltrami differentials, and the first cohomology groups of the surface are discussed.

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