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This paper is devoted to the study of the Cauchy problem in$C$^{∞}and in the Gevrey classes for some second order degenerate hyperbolic equations with time dependent coefficients and lower order terms satisfying a suitable Levi condition.

Let X and Y be two complex manifolds, let$D$⊂$X$and$G$⊂$Y$be two nonempty open sets, let$A$(resp.$B$) be an open subset of ∂$D$(resp. ∂$G$), and let$W$be the 2-fold cross (($D$∪$A$)×$B$)∪($A$×($B$∪$G$)). Under a geometric condition on the boundary sets$A$and$B$, we show that every function locally bounded, separately continuous on$W$, continuous on$A$×$B$, and separately holomorphic on ($A$×$G$)∪($D$×$B$) “extends” to a function continuous on a “domain of holomorphy” $\widehat{W}$ and holomorphic on the interior of $\widehat{W}$ .

The face numbers of simplicial complexes without missing faces of dimension larger than$i$are studied. It is shown that among all such ($d$−1)-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the componentwise minimal$f$-vector; and moreover, among all such 2-Cohen–Macaulay (2-CM) complexes, the same sphere has the componentwise minimal$h$-vector. It is also verified that the$l$-skeleton of a flag ($d$−1)-dimensional 2-CM complex is 2($d$−$l$)-CM, while the$l$-skeleton of a flag piecewise linear ($d$−1)-sphere is 2($d$−$l$)-homotopy CM. In addition, tight lower bounds on the face numbers of 2-CM balanced complexes in terms of their dimension and the number of vertices are established.

We prove that the punctured generalized conifolds and punctured orbifolded conifolds are mirror symmetric under the SYZ program with quantum corrections. This mathematically confirms the gauge-theoretic prediction by Aganagic–Karch–L¨ust–Miemiec, and also provides a supportive evidence to Morrison’s conjecture that geometric transitions are reversed under mirror symmetry.