The aim of this work is to construct examples of pairs whose logarithmic cotangent bundles have strong positivity properties. These examples are constructed from any smooth $n$-dimensional complex projective varieties by considering the sum of at least $n$ general sufficiently ample hypersurfaces.

Let <i>L</i> be a holomorphic line bundle over a compact Khler manifold <i>X</i>. Motivated by mirror symmetry, we study the deformed HermitianYangMills equation on <i>L</i>, which is the line bundle analogue of the special Lagrangian equation in the case that <i>X</i> is CalabiYau. We show that this equation is the Euler-Lagrange equation for a positive functional, and that solutions are unique global minimizers. We provide a necessary and sufficient criterion for existence in the case that <i>X</i> is a Khler surface. For the higher dimensional cases, we introduce a line bundle version of the Lagrangian mean curvature flow, and prove convergence when <i>L</i> is ample and <i>X</i> has non-negative orthogonal bisectional curvature.

We study analytic integrable deformations of the germ of a holomorphic foliation given by df=0 at the origin 0∈Cn,n≥3. We consider the case where f is a germ of an irreducible and reduced holomorphic function. Our central hypotheses is that, outside of a dimension ≤n−3 analytic subset Y⊂X, the analytic hypersurface Xf:(f=0) has only normal crossings singularities. We then prove that, as germs, such deformations also exhibit a holomorphic first integral, depending analytically on the parameter of the deformation. This applies to the study of integrable germs writing as ω=df+fη where f is quasi-homogeneous. Under the same hypotheses for Xf:(f=0) we prove that ω also admits a holomorphic first integral. Finally, we conclude that an integrable germ ω=adf+fη admits a holomorphic first integral provided that: (i) Xf:(f=0) is irreducible with an isolated singularity at the origin 0∈Cn,n≥3; (ii) the algebraic multiplicities of ω and f at the origin satisfy ν(ω)=ν(df). In the case of an isolated singularity for (f=0) the writing ω=adf+fη is always assured so that we conclude the existence of a holomorphic first integral. Some questions related to Relative Cohomology are naturally considered and not all of them answered.

For families of continuous plurisubharmonic functions we show that, in a local sense, separately bounded above implies bounded above.

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