The method of intersection spaces associates cell-complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that Poincaré duality holds between the ordinary rational cohomology groups of the cell-complexes associated to complementary perversities. The cohomology of these intersection spaces defines a cohomology theory HI for singular spaces, which is not isomorphic to intersection cohomology IH. Mirror symmetry tends to interchange IH and HI. The theory IH can be tied to type IIA string theory, while HI can be tied to IIB theory. For pseudomanifolds with stratification depth 1 and flat link bundles, the present paper provides a de Rham-theoretic description of the theory HI by a complex of global smooth differential forms on the top stratum. We prove that the wedge product of forms introduces a perversity-internal cup product on HI, for every perversity. Flat link bundles arise for example in foliated stratified spaces and in reductive Borel–Serre compactifications of locally symmetric spaces. A precise topological definition of the notion of a stratified foliation is given.

For a $\mathbb{Q}$-Fano 3-fold $X$ on which $K_X$ is a canonical divisor, we investigate the geometry induced from the linear system $\lvert -mK_X \rvert$ and prove that the anti-$m$-canonical map $\varphi - m$ is birational onto its image for all $m \geq 39$. By a weak $\mathbb{Q}$-Fano 3-fold $X$ we mean a projective one with at worst terminal singularities on which $-K_X$ is $\mathbb{Q}$-Cartier, nef and big. For weak $\mathbb{Q}$-Fano 3-folds, we prove that $\varphi - m$ is birational onto its image for all $m \geq 97$.

Let $\mathfrak{C}_{2,1}$ be the class of connected 5-dimensional CR-hypersurfaces that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In our recent article, we proved that the CR-structures in $\mathfrak{C}_{2,1}$ are reducible to $\mathfrak{so}(3, 2)$-valued absolute parallelisms. In the present paper, we apply this result to study tube hypersurfaces in $\mathbb{C}^3$ that belong to $\mathfrak{C}_{2,1}$ and whose CR-curvature identically vanishes. By explicitly solving the zero CR-curvature equations up to affine equivalence, we show that every such hypersurface is affinely equivalent to an open subset of the tube $M_0$ over the future light cone $\lbrace (x_1, x_2, x_3) \in \mathbb{R}^3 \: \vert \: x^2_1 + x^2_2 - x^2_3= 0 , x_3 \gt 0 \rbrace$. Thus, if a tube hypersurface in the class $\mathfrak{C}_{2,1}$ locally looks like a piece of $M_0$ from the point of view of CR-geometry, then from the point of view of affine geometry it (globally) looks like a piece of $M_0$ as well. This rigidity result is in stark contrast to the Levi nondegenerate case, where the CR-geometric and affine-geometric classifications significantly differ.

The paper is devoted to studying the Dirichlet energy of moving frames on $2$-dimensional tori immersed in the Euclidean $3 \leq m$-dimensional space. This functional, called frame energy, is naturally linked to the Willmore energy of the immersion and on the conformal structure of the abstract underlying surface. As the first result, a Willmore-conjecture type lower bound is established: namely for every torus immersed in $\mathbb{R}^m, m \geq 3$, and any moving frame on it, the frame energy is at least $2\pi^2$ and equality holds if and only if $m \geq 4$, the immersion is the standard Clifford torus (up to rotations and dilations), and the frame is the flat one. Smoothness of the critical points of the frame energy is proved after the discovery of hidden conservation laws and, as application, the minimization of the frame energy in regular homotopy classes of immersed tori in $\mathbb{R}^3$ is performed.