International Journal of Mathematics

We give a bound on which singularities may appear on Kollár–Shepherd-Barron–Alexeev stable surfaces for a wide range of topological invariants and use this result to describe all stable numerical quintic surfaces (KSBA-stable surfaces with $K^2=\chi =5$) whose unique non-Du Val singularity is a Wahl singularity. We then extend the deformation theory of Horikawa to the log setting in order to describe the boundary divisor of the moduli space ${\overline{{ℳ}}}_{5,5}$ corresponding to these surfaces. Quintic surfaces are the simplest examples of surfaces of general type and the question of describing their moduli is a long-standing question in algebraic geometry.

We study knots of order $2$ in the grope filtration $\{{{𝒢}}_h\}$ and the solvable filtration $\{{{ℱ}}_h\}$ of the knot concordance group. We show that, for any integer $n\ge 4$, there are knots generating a ${\mathbb{Z}}_2^{\infty }$ subgroup of ${{𝒢}}_n/{{𝒢}}_{n.5}$. Considering the solvable filtration, our knots generate a ${\mathbb{Z}}_2^{\infty }$ subgroup of ${{ℱ}}_n/{{ℱ}}_{n.5}$ $(n\ge 2)$ distinct from the subgroup generated by the previously known $2$-torsion knots of Cochran, Harvey, and Leidy. We also present a result on the $2$-torsion part in the Cochran, Harvey, and Leidy’s primary decomposition of the solvable filtration.

We derive a formula of the greatest lower bounds on Ricci curvature of Fano homogeneous toric bundles. A criteria for the ampleness of line bundles over general homogeneous toric bundles is also obtained.

In this paper, an equality between the Hochs–Mathai type index and the Atiyah–Patodi–Singer type index is established when the manifold and the group action are both non-compact, which generalizes a result of Ma and Zhang for compact group actions. As a technical preparation, a problem concerning the Fredholm property of the global elliptic boundary value problems of the Atiyah–Patodi–Singer type on a non-compact manifold is studied.

Given a compact Lagrangian submanifold $L$ of a symplectic manifold $(M,\omega )$, Fukaya, Oh, Ohta and Ono construct a filtered $A_{\infty }$-algebra ${ℱ}(L)$, on the cohomology of $L$, which we call the Fukaya algebra of $L$. In this paper, we describe the Fukaya algebra of a product of two Lagrangians submanifolds $L_1\times L_2$. Namely, we show that ${ℱ}(L_1\times L_2)$ is quasi-isomorphic to ${ℱ}(L_1){\otimes }_{\infty }{ℱ}(L_2)$, where ${\otimes }_{\infty }$ is the tensor product of filtered $A_{\infty }$-algebras defined in [L. Amorim, Tensor product of filtered $A_{\infty }$-algebras, J. Pure Appl. Algebra 220(12) (2016) 3984–4016]. As a corollary of this quasi-isomorphism, we obtain a description of the bounding cochains on ${ℱ}(L_1\times L_2)$ and of the Floer cohomology of $L_1\times L_2$.

We show that the indefinite complex space form ${ℂ}^{r,s}$ is not a relative to the indefinite complex space form $ℂ{\mathbb{P}}_l^N(b)$ or $ℂ{ℍ}_m^N(b)$. We further study whether two Fubini–Study spaces are relatives or not.

We study when there exists a dense holomorphic curve in a space of holomorphic maps from a Stein space. We first show that for any bounded convex domain $\mathrm{\Omega }⋐{ℂ}^n$ and any connected complex manifold $Y$, the space ${𝒪}(\mathrm{\Omega },Y)$ contains a dense holomorphic disc. Our second result states that $Y$ is an Oka manifold if and only if for any Stein space $X$ there exists a dense entire curve in every path component of ${𝒪}(X,Y)$.

In the second half of this paper, we apply the above results to the theory of universal functions. It is proved that for any bounded convex domain $\mathrm{\Omega }⋐{ℂ}^n$, any fixed-point-free automorphism of $\mathrm{\Omega }$ and any connected complex manifold $Y$, there exists a universal map $\mathrm{\Omega }\to Y$. We also characterize Oka manifolds by the existence of universal maps.