Related papers: Deformations of Frobenius structures on Hurwitz sp…
Motivated by DeVleming's work on moduli of surfaces in $\mathbb{P}^3$ and Chen-Hu-Jiang's work on moduli of threefolds with volume $2$ and geometric genus $4$, we study the deformation of pairs of $\mathbb{P}^3$ and hypersurfaces using the…
In this paper, we investigate the moduli of surfaces of general type admitting genus 2 fibrations with irregularity q = g_b + 1, where g_b >= 2 is the genus of the base. We prove that smooth fibrations are parametrized by a unique component…
Flat structure was introduced by K. Saito and his collaborators at the end of 1970's. Independently the WDVV equation arose from the 2D topological field theory. B. Dubrovin unified these two notions as Frobenius manifold structure. In this…
Calabi--Hirzebruch manifolds are higher-dimensional generalizations of both the football and Hirzebruch surfaces. We construct a family of Kahler--Einstein edge metrics singular along two disjoint divisors on the Calabi--Hirzebruch…
We study threefolds of general type constructed as $\mathbb{Z}_2^s$-covers of weighted projective spaces with a particular focus on their invariants, deformation theory, and the behavior of the $m$-canonical map. For the invariants, we…
The general construction of self-adjoint configuration space representations of the Heisenberg algebra over an arbitrary manifold is considered. All such inequivalent representations are parametrised in terms of the topology classes of flat…
There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface.…
We consider the Dubrovin--Frobenius manifold of rank $2$ whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces,…
We identify two Frobenius manifolds obtained from two different differential Gerstenhaber-Batalin-Vilkovisky algebras on a compact Kaehler manifold. One is constructed on the Dolbeault cohomology, and the other on the de Rham cohomology.…
Quantum deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by the quantum central systems which has a geometrical meaning of vanishing…
We show that the Frobenius manifold associated to the pair of a cusp singularity and it's canonical primitive form is isomorphic to the one constructed from the Gromov--Witten theory for an orbifold projective line with at most three…
This paper review one construction of Frobenius manifolds (and slightly weaker structures). It splits it into several steps and discusses the freedom and the constraints in these steps. The steps pass through holomorphic bundles with…
We consider in this paper an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are…
Let $(N,g_{0})$ be a Kahler-Einstein surface with the first Chern class negative and assume that there exists a branched Lagrangian minimal surfaces with respect to the metric $g_{0}$. We show that when the Kahler-Einstein metric is changed…
For a closed K\"{a}hler manifold with a Hamiltonian action of a connected compact Lie group by holomorphic isometries, we construct a formal Frobenius manifold structure on the equivariant cohomology by exploiting a natural DGBV algebra…
We consider a deformation of N=1 supersymmetric gauge theories in four dimensions, which we call the C-deformation, where the gluino field satisfies a Clifford-like algebra dictated by a self-dual two-form, instead of the standard…
The well known Hurwitz upper bound states that a closed Riemann surface $S$ of genus $g \geq 2$ has at most $84(g-1)$ conformal automorphisms. If $S$ has exactly $84(g-1)$ conformal automorphisms, then it is called a Hurwitz curve. The…
We introduce a variety $\hat{G}_2$ parameterizing isotropic five-spaces of a general degenerate four-form in a seven dimensional vector space. It is in a natural way a degeneration of the variety $G_2$, the adjoint variety of the simple Lie…
The Riemann hierarchy is the simplest example of rank one, ($1$+$1$)-dimensional integrable system of nonlinear evolutionary PDEs. It corresponds to the dispersionless limit of the Korteweg-de Vries hierarchy. In the language of formal…
Sub-Riemannian structures on odd-dimensional spheres respecting the Hopf fibration naturally appear in quantum mechanics. We study the curvature maps for such a sub-Riemannian structure and express them using the Riemannian curvature tensor…