Given a degenerate Calabi–Yau variety $X$ equipped with local deformation data, we construct an almost differential graded Batalin–Vilkovisky algebra $PV^{\ast,\ast}(X)$, producing a singular version of the extended Kodaira–Spencer differential graded Lie algebra in the Calabi–Yau setting. Assuming Hodge-to-de Rham degeneracy and a local condition that guarantees freeness of the Hodge bundle, we prove a Bogomolov–Tian–Todorov–type unobstructedness theorem for smoothing of singular Calabi–Yau varieties. In particular, this provides a unified proof for the existence of smoothing of both $d$-semistable log smooth Calabi–Yau varieties (as studied by Friedman [$\href{https://doi.org/10.2307/2006955}{22}$] and Kawamata–Namikawa [$\href{ https://doi.org/10.1007/BF01231538}{41}$]) and maximally degenerate Calabi–Yau varieties (as studied by Kontsevich–Soibelman $[\href{ https://link.springer.com/chapter/10.1007/0-8176-4467-9_9}{45}$] and Gross–Siebert [$\href{ http://doi.org/10.4007/annals.2011.174.3.1}{30}$]). We also demonstrate how our construction yields a logarithmic Frobenius manifold structure on a formal neighborhood of $X$ in the extended moduli space by applying the technique of Barannikov–Kontsevich [$\href{https://doi.org/10.1155/S1073792898000166}{2}$, $\href{https://doi.org/10.48550/arXiv.math/9903124}{1}$].

We study the $SU(3)$ Toda system with singular sources\begin{equation*}\begin{cases}\Delta u+2e^{u}-e^{v}=4\pi \sum _{k=0}^{m} n_{1,k}\delta _{p_{k}}\quad \text{ on }\; E_{\tau},\\\Delta v+2e^{v}-e^{u}=4\pi \sum _{k=0}^{m} n_{2,k}\delta _{p_{k}}\quad \text{ on }\; E_{\tau},\end{cases}\end{equation*}where $E_{\tau}:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$ with $\operatorname{Im}\tau \gt 0$ is a flat torus, $\delta _{p_{k}}$ is the Dirac measure at $p_{k}$, and $n_{i,k}\in \mathbb{Z}_{\geq 0}$ satisfy $\sum _{k}n_{1,k}\not \equiv \sum _{k} n_{2,k} \mod 3$. This is known as the non-critical case and it follows from a general existence result of [$\href{ https://doi.org/10.1016/j.aim.2015.07.036}{3}$] that solutions always exist. In this paper we prove that

(i) The system has at most\begin{equation*}\frac{1}{3\times 2^{m+1}}\prod _{k=0}^{m}(n_{1,k}+1)(n_{2,k}+1)(n_{1,k}+n_{2,k}+2)\in \mathbb{N}\end{equation*}solutions. We have several examples to indicate that this upper bound should be sharp. Our proof presents a nice combination of the apriori estimates from analysis and the classical Bézout theorem from algebraic geometry.

(ii) For $m=0$ and $p_{0}=0$, the system has even solutions if and only if at least one of $\{n_{1,0}, n_{2,0}\}$ is even. Furthermore, if $n_{1,0}$ is odd, $n_{2,0}$ is even and $n_{1,0}\lt n_{2,0}$, then except for finitely many $\tau $’s modulo $SL(2,\mathbb{Z})$ action, the system has exactly $\frac{n_{1,0}+1}{2}$ even solutions.

Differently from [$\href{ https://doi.org/10.1016/j.aim.2015.07.036}{3}$], our proofs are based on the integrability of the Toda system, and also imply a general non-existence result for even solutions of the Toda system with four singular sources.

We combine Freedman’s topology with Eliashberg’s holomorphic theory to construct Stein neighborhood systems in complex surfaces, and use these to study various notions of convexity and concavity. Every tame, topologically embedded $2$-complex $K$ in a complex surface, after $C^0$-small topological ambient isotopy, is the intersection of an uncountable nested family of Stein regular neighborhoods that are all topologically ambiently isotopic rel $K$, but frequently realize uncountably many diffeomorphism types. These arise from the Cantor set levels of a topological mapping cylinder. The boundaries of the neighborhoods are $3$-manifolds that are only topologically embedded, but still satisfy a notion of pseudoconvexity. Such $3$-manifolds share some basic properties of hypersurfaces that are strictly pseudoconvex in the usual smooth sense, but they are far more common. The complementary notion of topological pseudoconcavity is realized by uncountably many diffeomorphism types homeomorphic to $\mathbb{R}^4$.

We prove that any smooth harmonic map from $S^3$ into $S^2$ of Morse index less or equal than $4$ has to be an harmonic morphism, that is the successive composition of an isometry of $S^3$, the Hopf fibration and an holomorphic map from $\mathbb{C}P^1$ into itself.

We classify the volume preserving stable hypersurfaces in the real projective space $\mathbb{RP}^n$. As a consequence, the solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces $\mathbb{RP}^k \subset \mathbb{RP}^n$ (starting with points). This confirms a conjecture of Burago and Zalgaller from 1988 and extends to higher dimensions previous result of M. Ritoré and A. Ros on $\mathbb{RP}^3$. We also derive an Willmore type inequality for antipodal invariant hypersurfaces in $\mathbb{S}^n$.