Related papers: A universal mirror to $(\mathbb{P}^2, \Omega)$ as …

Every indefinite binary form occurs as the Picard lattice of some K3-surface. The group of its isometries, or automorphs, coincides with the automorphism group of the K3-surface, but only up to finite groups. The classical theory of…

We analyse the global symmetry structure of two-dimensional Non-Linear Sigma Models with Wess-Zumino term. When the target space has a compact isometry without fixed points, the theory has a pair of (group-like) global symmetries and many…

The goal of this paper is to propose a theory of mirror symmetry for varieties of general type. Using Landau-Ginzburg mirrors as motivation, we describe the mirror of a hypersurface of general type (and more generally varieties of…

We describe the mirror of the Z orbifold as a representation of a class of generalized Calabi-Yau manifolds that can be realized as manifolds of dimension five and seven. Despite their dimension these correspond to superconformal theories…

We consider K3 surfaces that possess certain automorphisms of prime order p>2 and we present, for these surfaces, a correspondence between the mirror symmetry of Berglund-Huebsch-Chiodo-Ruan and that for lattice polarized K3 surfaces…

We prove the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\geq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or the total space of an orbifiber bundle over $\mathbb{S}^1$…

Via the formulation of (quantum) Hikita conjecture with coefficients in a characteristic $p$ field, we explain an arithmetic aspect of the theory of 3D mirror symmetry. Namely, we propose that the action of Steenrod-type operations and…

An automorphism $u$ of a vector space is called unipotent of index $2$ whenever $(u-\mathrm{id})^2=0$. Let $b$ be a non-degenerate symmetric or skewsymmetric bilinear form on a vector space $V$ over a field $\mathbb{F}$ of characteristic…

A perfect focus telescope is one in which all rays parallel to the axis meet at a point and give equal magnification there. It is shown that these two conditions define the shapes of both primary and secondary mirrors. Apart from scale, the…

Let $\Omega \subset \mathbb{C}^n$ be a domain whose Bergman space contains all holomorphic monomials. We derive sufficient conditions for $\Omega$ to be Reinhardt, complete Reinhardt, circular or Hartogs in terms of the orthogonality…

For certain K3 surfaces, there are two constructions of mirror symmetry that are very different. The first, known as BHK mirror symmetry, comes from the Landau-Ginzburg model for the K3 surface; the other, known as LPK3 mirror symmetry, is…

We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an…

For all punctured Riemann surfaces arising as mirror curves of toric Calabi--Yau threefolds, we show that their symplectic cohomology is isomorphic to the compactly supported Hochschild cohomology of the noncommutative Landau--Ginzburg…

For any compact Lie group $G$ and any $n$ we construct a smooth $G$-manifold $U_n(G)$ such that any smooth $n$-dimensional $G$-manifold can be embedded in $U_n(G)$ with a trivial normal bundle. Furthermore, we show that such embeddings are…

Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide…

Let $\Omega\subset\mathbb{C}^n$ be a bounded domain with smooth boundary, whose Bergman projection $B$ maps the Sobolev space $H^{k_{1}}(\Omega)$ (continuously) into $H^{k_{2}}(\Omega)$. We establish two smoothing results: (i) the full…

We give a Riemannian structure to the set $\Sigma$ of positive invertible unitized Hilbert-Schmidt operators, by means of the trace inner product. This metric makes of $\Sigma$ a nonpositively curved, simply connected and metrically…

We use a relative Fourier-Mukai transform on elliptic K3 surfaces $X$ to describe mirror symmetry. The action of this Fourier-Mukai transform on the cohomology ring of $X$ reproduces relative T-duality and provides an infinitesimal isometry…

We study the canonical Poisson structure on the loop space of the super-double-twisted-torus and its quantization. As a consequence we obtain a rigorous construction of mirror symmetry as an intertwiner of the N=2 super-conformal structures…

We construct functors sending torus-equivariant quasi-coherent sheaves on toric schemes over the sphere spectrum to constructible sheaves of spectra on real vector spaces. This provides a spectral lift of the toric homolgoical mirror…