Relations among fundamental invariants play an important role in algebraic geometry. It is known that an $n$-dimensional variety of general type whose image of its canonical map is of maximal dimension, satisfies $\textrm{Vol} \geq 2 (p_{g} - n)$. In this article, we investigate the very interesting extremal situation of varieties with $\textrm{Vol} = 2(p_{g} - n)$, which we call Horikawa varieties for they are natural higher dimensional analogues of Horikawa surfaces.

We obtain a structure theorem for Horikawa varieties and explore their pluriregularity. We use this to prove optimal results on projective normality of pluricanonical linear systems. We study the fundamental groups of Horikawa varieties, showing that they are simply connected. We prove results on deformations of Horikawa varieties, whose implications on the moduli space make them the higher dimensional analogue of curves of genus 2.

Even though there are infinitely many families of Horikawa varieties in any given dimension $n$, we show that when the image of the canonical map is singular, the geometric genus of the Horikawa varieties is bounded by $n + 4$.

We prove categorical systolic inequalities for the derived categories of 2-Calabi–Yau Ginzburg dg algebras associated to ADE quivers and explore their symplecto-geometric aspects.

This paper studies the first passage percolation model on crystal lattices, which is a generalization of that on the cubic lattice. Here, each edge of the graph induced by a crystal lattice is assigned a random passage time, and consideration is given to the behavior of the percolation region $B(t)$, which consists of those vertices that can be reached from the origin within a time $t > 0$. Our first result is the shape theorem, stating that the normalized region $B(t)/t$ converges to some deterministic one, called the limit shape. The second result is the monotonicity of the limit shapes under covering maps. In particular, this provides insight into the limit shape of the cubic first passage percolation model.

The notion of quasi-log schemes was first introduced by Florin Ambro in his epoch-making paper: Quasi-log varieties. In this paper, we establish the basepoint-free theorem of Reid–Fukuda type for quasi-log schemes in full generality. Roughly speaking, it means that all the results for quasi-log schemes claimed in Ambro's paper hold true. The proof is Kawamata's X-method with the aid of the theory of basic slc-trivial fibrations. For the reader's convenience, we make many comments on the theory of quasi-log schemes in order to make it more accessible.

The moduli space of spatial polygons is known as a symplectic manifold equipped with both Kähler and real polarizations. In this paper, associated to the Kähler and real polarizations, morphisms of operads $\mathsf{f}_{\mathsf{K}\ddot{\mathsf{a}}\mathsf{h}}$ and $\mathsf{f}_{\mathsf{re}}$ are constructed by using the quantum Hilbert spaces $\mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}}$ and $\mathscr{H}_\mathrm{re}$, respectively. Moreover, the relationship between the two morphisms of operads $\mathsf{f}_{\mathsf{K}\ddot{\mathsf{a}}\mathsf{h}}$ and $\mathsf{f}_{\mathsf{re}}$ is studied and then the equality $\dim \mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}} = \dim \mathscr{H}_\mathrm{re}$ is proved in general setting. This operadic framework is regarded as a development of the recurrence relation method by Kamiyama (2000) for proving $\dim \mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}} = \dim \mathscr{H}_\mathrm{re}$ in a special case.

It is known that if the Gaussian curvature function along each meridian on a surface of revolution $(\mathbb{R}^{2}, dr^{2} + m(r)^{2} d\theta^{2})$ is decreasing, then the cut locus of each point of $\theta^{-1} (0)$ is empty or a subarc of the opposite meridian $\theta^{-1} (\pi)$. Such a surface is called a von Mangoldt's surface of revolution. A surface of revolution $(\mathbb{R}^{2}, dr^{2} + m(r)^{2} d\theta^{2})$ is called a generalized von Mangoldt surface of revolution if the cut locus of each point of $\theta^{-1} (0)$ is empty or a subarc of the opposite meridian $\theta^{-1} (\pi)$.

For example, the surface of revolution $(\mathbb{R}^{2}, dr^{2} + m_0(r)^{2} d\theta^{2})$, where $m_{0}(x) = x/(1 + x^{2})$, has the same cut locus structure as above and the cut locus of each point in $r^{-1}((0, \infty))$ is nonempty. Note that the Gaussian curvature function is not decreasing along a meridian for this surface. In this article, we give sufficient conditions for a surface of revolution $(\mathbb{R}^{2}, dr^{2} + m(r)^{2} d\theta^{2})$ to be a generalized von Mangoldt surface of revolution. Moreover, we prove that for any surface of revolution with finite total curvature $c$, there exists a generalized von Mangoldt surface of revolution with the same total curvature $c$ such that the Gaussian curvature function along a meridian is not monotone on $[a, \infty)$ for any $a > 0$.

We revisit the localization formulas of cohomology intersection numbers associated to a logarithmic connection. The main contribution of this paper is threefold: we prove the localization formula of the cohomology intersection number of logarithmic forms in terms of residue of a connection; we prove that the leading term of the Laurent expansion of the cohomology intersection number is Grothendieck residue when the connection is hypergeometric and we prove that the leading term of stringy integral discussed by Arkani-Hamed, He and Lam is nothing but the self-cohomology intersection number of the canonical form.

We study the distribution of extreme values of automorphic $L$-functions in a family of holomorphic cusp forms with prime levels. First, we prove an asymptotic formula for a certain density function closely related to this value-distribution. Then we apply it to estimate the probabilities of large deviations for values of automorphic $L$-functions.

We study Hadamard variations with respect to general domain perturbations, particularly for the Neumann boundary condition. They are derived from new Liouville's formulae concerning the transformation of volume and area integrals. Then, relations to several geometric quantities are discussed; differential forms and the second fundamental form on the boundary.

It is very well known that Hopf real hypersurfaces in the complex projective space can be locally characterized as tubes over complex submanifolds. This also holds true for some, but not all, Hopf real hypersurfaces in the complex hyperbolic space. The main goal of this paper is to show, in a unified way, how to construct Hopf real hypersurfaces in the complex hyperbolic space from a horizontal submanifold in one of the three twistor spaces of the indefinite complex 2-plane Grassmannian with respect to the natural para-quaternionic Kähler structure. We also identify these twistor spaces with the sets of circles in totally geodesic complex hyperbolic lines in the complex hyperbolic space. As an application, we describe all classical Hopf examples. We also solve the remarkable and long-standing problem of the existence of Hopf real hypersurfaces in the complex hyperbolic space, different from the horosphere, such that the associated principal curvature is 2. We exhibit a method to obtain plenty of them.

In this paper, we consider the summability of formal solutions with singularities (such as logarithmic singularities, functional power singularities, etc.) of nonlinear partial differential equations in the complex domain. The main result is as follows: when a formal solution with singularities is given, under appropriate assumptions related to the formal solution, the equation has a true solution that admits the given formal solution as an asymptotic expansion. The proof is done by constructing a new formal solution that is equivalent to the given formal solution in the asymptotic sense and is multisummable in a suitable direction. The assumptions are stated in terms of the Newton polygon associated with the given formal solution.