In 2002, Fukaya [19] proposed a remarkable explanation of mirror symmetry detailing the Strominger–Yau–Zaslow (SYZ) conjecture [47] by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi–Yau manifold $\check {X}$ and the multivalued Morse theory on the base $\check {B}$ of an SYZ fibration $\check {p}\colon \check {X}\to \check {B}$, and the other between deformation theory of the mirror X and the same multivalued Morse theory on $\check {B}$. In this paper, we prove a reformulation of the main conjecture in Fukaya’s second correspondence, where multivalued Morse theory on the base $\check {B}$ is replaced by tropical geometry on the Legendre dual B. In the proof, we apply techniques of asymptotic analysis developed in [7, 9] to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi–Yau log variety introduced in [8]. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semiflat part $X_{\mathrm {sf}} \subset X$ allows us to extract consistent scattering diagrams from appropriate Maurer–Cartan solutions.

Building upon the classification by Lacini, we determine the isomorphism classes of log del Pezzo surfaces of rank one over an algebraically closed field of characteristic five either which are not log liftable over the ring of Witt vectors or whose singularities are not feasible in characteristic zero. We also show that such a surface is always constructed from the Du Val del Pezzo surface of Dynkin type $2[2^4]$. Furthermore, We show that the Kawamata–Viehweg vanishing theorem for ample $\mathbb {Z}$-Weil divisors holds for log del Pezzo surfaces of rank one in characteristic five if those singularities are feasible in characteristic zero.

We analyze infinitesimal deformations of morphisms of locally free sheaves on a smooth projective variety X over an algebraically closed field of characteristic zero. In particular, we describe a differential graded Lie algebra controlling the deformation problem. As an application, we study infinitesimal deformations of the pairs given by a locally free sheaf and a subspace of its sections with a view toward Brill-Noether theory.

The goal of this paper is to describe certain nonlinear topological obstructions for the existence of first-order smoothings of mildly singular Calabi–Yau varieties of dimension at least $4$. For nodal Calabi–Yau threefolds, a necessary and sufficient linear topological condition for the existence of a first-order smoothing was first given in [Fri86]. Subsequently, Rollenske–Thomas [RT09] generalized this picture to nodal Calabi–Yau varieties of odd dimension by finding a necessary nonlinear topological condition for the existence of a first-order smoothing. In a complementary direction, in [FL22a], the linear necessary and sufficient conditions of [Fri86] were extended to Calabi–Yau varieties in every dimension with $1$-liminal singularities (which are exactly the ordinary double points in dimension $3$ but not in higher dimensions). In this paper, we give a common formulation of all of these previous results by establishing analogues of the nonlinear topological conditions of [RT09] for Calabi–Yau varieties with weighted homogeneous k-liminal hypersurface singularities, a broad class of singularities that includes ordinary double points in odd dimensions.

Let $X$ and $Y$ be compact hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of length $n$ subschemes of a $K3$ surface. A class in $H^{p,p}(X\times Y,{\mathbb {Q}})$ is an analytic correspondence, if it belongs to the subring generated by Chern classes of coherent analytic sheaves. Let $f:H^2(X,{\mathbb {Q}})\rightarrow H^2(Y,{\mathbb {Q}})$ be a rational Hodge isometry with respect to the Beauville–Bogomolov–Fujiki pairings. We prove that $f$ is induced by an analytic correspondence. We furthermore lift $f$ to an analytic correspondence $\tilde {f}: H^*(X,{\mathbb {Q}})[2n]\rightarrow H^*(Y,{\mathbb {Q}})[2n]$, which is a Hodge isometry with respect to the Mukai pairings and which preserves the gradings up to sign. When $X$ and $Y$ are projective, the correspondences $f$ and $\tilde {f}$ are algebraic.

We give a detailed proof that locally Noetherian moduli stacks of sections carry canonical obstruction theories. As part of the argument, we construct a dualising sheaf and trace map, in the lisse-étale topology, for families of tame twisted curves when the base stack is locally Noetherian.

We prove that actions of complex reductive Lie groups on a holomorphic vector bundle over a complex compact manifold are locally extendable to its local moduli space.

We prove the existence of a smoothing for a toroidal crossing space under mild assumptions. By linking log structures with infinitesimal deformations, the result receives a very compact form for normal crossing spaces. The main approach is to study log structures that are incoherent on a subspace of codimension 2 and prove a Hodge–de Rham degeneration theorem for such log spaces that also settles a conjecture by Danilov. We show that the homotopy equivalence between Maurer–Cartan solutions and deformations combined with Batalin–Vilkovisky theory can be used to obtain smoothings. The construction of new Calabi–Yau and Fano manifolds as well as Frobenius manifold structures on moduli spaces provides potential applications.

By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira–Spencer’s local stability theorem of Kähler structures. We also obtain two new local stability theorems, one of balanced structures on an n-dimensional balanced manifold with the $(n-1,n)$th mild $\partial \overline {\partial }$-lemma by power series method and the other one on p-Kähler structures with the deformation invariance of $(p,p)$-Bott–Chern numbers.

Let $\mathcal {F}$ be a polystable sheaf on a smooth minimal projective surface of Kodaira dimension 0. Then the differential graded (DG) Lie algebra $R\operatorname {Hom}(\mathcal {F},\mathcal {F})$ of derived endomorphisms of $\mathcal {F}$ is formal. The proof is based on the study of equivariant $L_{\infty }$ minimal models of DG Lie algebras equipped with a cyclic structure of degree 2 which is non-degenerate in cohomology, and does not rely (even for K3 surfaces) on previous results on the same subject.

By use of a natural extension map and a power series method, we obtain a local stability theorem for $p$-Kähler structures with the $(p,p+1)$th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$-lemma under small differentiable deformations.

We analyse infinitesimal deformations of pairs $(X,{\mathcal{F}})$ with ${\mathcal{F}}$ a coherent sheaf on a smooth projective variety $X$ over an algebraically closed field of characteristic 0. We describe a differential graded Lie algebra controlling the deformation problem, and we prove an analog of a Mukai–Artamkin theorem about the trace map.

This paper concerns the classification of isogeny classes of $p$-divisible groups with saturated Newton polygons. Let $S$ be a normal Noetherian scheme in positive characteristic $p$ with a prime Weil divisor $D$. Let ${\mathcal{X}}$ be a $p$-divisible group over $S$ whose geometric fibers over $S\setminus D$ (resp. over $D$) have the same Newton polygon. Assume that the Newton polygon of ${\mathcal{X}}_{D}$ is saturated in that of ${\mathcal{X}}_{S\setminus D}$. Our main result (Corollary 1.1) says that ${\mathcal{X}}$ is isogenous to a $p$-divisible group over $S$ whose geometric fibers are all minimal. As an application, we give a geometric proof of the unpolarized analogue of Oort’s conjecture (Oort, J. Amer. Math. Soc. 17(2) (2004), 267–296; 6.9).

An affine symplectic singularity $X$ with a good $\mathbf{C}^{\ast }$-action is called a conical symplectic variety. In this paper we prove the following theorem. For fixed positive integers $N$ and $d$, there are only a finite number of conical symplectic varieties of dimension $2d$ with maximal weights $N$, up to an isomorphism. To prove the main theorem, we first relate a conical symplectic variety with a log Fano Kawamata log terminal (klt) pair, which has a contact structure. By the boundedness result for log Fano klt pairs with fixed Cartier index, we prove that conical symplectic varieties of a fixed dimension and with a fixed maximal weight form a bounded family. Next we prove the rigidity of conical symplectic varieties by using Poisson deformations.

To study infinitesimal deformation problems with cohomology constraints, we introduce and study cohomology jump functors for differential graded Lie algebra (DGLA) pairs. We apply this to local systems, vector bundles, Higgs bundles, and representations of fundamental groups. The results obtained describe the analytic germs of the cohomology jump loci inside the corresponding moduli space, extending previous results of Goldman–Millson, Green–Lazarsfeld, Nadel, Simpson, Dimca–Papadima, and of the second author.

We study the moduli space of a product of stable varieties over the field of complex numbers, as defined via the minimal model program. Our main results are: (a) taking products gives a well-defined morphism from the product of moduli spaces of stable varieties to the moduli space of a product of stable varieties; (b) this map is always finite étale; and (c) this map very often is an isomorphism. Our results generalize and complete the work of Van Opstall in dimension $1$. The local results rely on a study of the cotangent complex using some derived algebro-geometric methods, while the global ones use some differential-geometric input.

We show that the Hilbert scheme, that parameterizes all ideals with the same Hilbert function over a Clements–Lindström ring W, is connected. More precisely, we prove that every graded ideal is connected by a sequence of deformations to the lex-plus-powers ideal with the same Hilbert function. This is an analogue of Hartshorne’s theorem that Grothendieck’s Hilbert scheme is connected. We also prove a conjecture by Gasharov, Hibi, and Peeva that the lex ideal attains maximal Betti numbers among all graded ideals in W with a fixed Hilbert function.

Let k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A4 in characteristic two. This proves one direction of a strong form of the Oort conjecture.