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.

We study the classes of modules which are generated by a silting module. In the case of either hereditary or perfect rings, it is proved that these are exactly the torsion T such that the regular module has a special T-preenvelope. In particular, every torsion-enveloping class in Mod-R are of the form Gen(T) for a minimal silting module T. For the dual case, we obtain for general rings that the covering torsion-free classes of modules are exactly the classes of the form Cogen(T), where T is a cosilting module.

It is known that if the special automorphism group SAut(X) of a quasiaffine variety X of dimension at least 2 acts transitively on X, then this action is infinitely transitive. In this paper we question whether this is the only possibility for the automorphism group Aut(X) to act infinitely transitively on X. We show that this is the case, provided X admits a nontrivial Ga or Gm-action. Moreover, 2-transitivity of the automorphism group implies infinite transitivity.

For the free surface problem of the highly subsonic heat-conducting inviscid flow in 2-D and 3-D, a priori estimates for geometric quantities of free surfaces, such as the second fundamental form and the injectivity radius of the normal exponential map, and the Sobolev norms of fluid variables are proved by investigating the coupling of the boundary geometry and the interior solutions. An interesting feature for the free surface problem studied in this paper is the loss of one more derivative than the problem of incompressible Euler equations for which a geometric approach was introduced by Christodoulou and Lindblad in [11]. Due to the loss of one more derivative and loss of symmetry of equations, the geometric approach in [11] needs to be substantially developed by exploring the interaction of large variation of temperature, heat-conduction, non-zero divergence of the fluid velocity and the evolution of free surfaces.

We prove that every lattice in a product of higher-rank simple Lie groups or higher-rank simple algebraic groups over local fields has Vincent Lafforgue’s strong property (T). Over non-Archimedean local fields, we also prove that they have strong Banach property (T) with respect to all Banach spaces with non-trivial type, whereas in general we obtain such a result with additional hypotheses on the Banach spaces. The novelty is that we deal with non-co-compact lattices, such as SLn(Z) for n⩾3. To do so, we introduce a stronger form of strong property (T) which allows us to deal with more general objects than group representations on Banach spaces that we call two-step representations, namely families indexed by a group of operators between different Banach spaces that we can compose only once. We prove that higher-rank groups have this property and that this property passes to undistorted lattices. Full Text (PDF format)