We consider log Calabi-Yau surfaces $(Y, D)$ with singular boundary. In each deformation type, there is a distinguished surface $(Y_e,D_e)$ such that the mixed Hodge structure on $H_2(Y \setminus D)$ is split. We prove that (1) the action of the automorphism group of $(Y_e,D_e)$ on its nef effective cone admits a rational polyhedral fundamental domain; and (2) the action of the monodromy group on the nef effective cone of a very general surface in the deformation type admits a rational polyhedral fundamental domain. These statements can be viewed as versions of the Morrison cone conjecture for log Calabi–Yau surfaces. In addition, if the number of components of D is no greater than six, we show that the nef cone of $Y_e$ is rational polyhedral and describe it explicitly. This provides infinite series of new examples of Mori Dream Spaces.

We classify nef vector bundles on a smooth hyperquadric of dimension three with first Chern class two over an algebraically closed field of characteristic zero. In particular, we see that they are globally generated.

We construct families of non-toric ${\mathbb {Q}}$-factorial terminal Fano (${\mathbb {Q}}$-Fano) threefolds of codimension $\geq 20$ corresponding to 54 mutation classes of rigid maximally mutable Laurent polynomials. From the point of view of mirror symmetry, they are the highest codimension (non-toric) ${\mathbb {Q}}$-Fano varieties for which we can currently establish the Fano/Landau–Ginzburg correspondence. We construct 46 additional ${\mathbb {Q}}$-Fano threefolds with codimensions of new examples ranging between 19 and 5. Some of these varieties are presented as toric complete intersections, and others as Pfaffian varieties.

A famous theorem of Shokurov states that a general anticanonical divisor of a smooth Fano threefold is a smooth K3 surface. This is quite surprising since there are several examples where the base locus of the anticanonical system has codimension two. In this paper, we show that for four-dimensional Fano manifolds the behaviour is completely opposite: if the base locus is a normal surface, and hence has codimension two, all the anticanonical divisors are singular.

We exhibit large families of K3 surfaces with real multiplication, both abstractly, using lattice theory, the Torelli theorem and the surjectivity of the period map, as well as explicitly, using dihedral covers and isogenies.

Let $(X, \Delta )$ be a klt threefold pair with nef anti-log canonical divisor $-(K_X+\Delta )$. We show that $\kappa (X, -(K_X+\Delta ))\geq 0$. To do so, we prove a more general equivariant non-vanishing result for anti-log canonical bundles, which is valid in any dimension.

Let $E/\mathbb {Q}(T)$ be a nonisotrivial elliptic curve of rank r. A theorem due to Silverman [‘Heights and the specialization map for families of abelian varieties’, J. reine angew. Math. 342 (1983), 197–211] implies that the rank $r_t$ of the specialisation $E_t/\mathbb {Q}$ is at least r for all but finitely many $t \in \mathbb {Q}$. Moreover, it is conjectured that $r_t \leq r+2$, except for a set of density $0$. When $E/\mathbb {Q}(T)$ has a torsion point of order $2$, under an assumption on the discriminant of a Weierstrass equation for $E/\mathbb {Q}(T)$, we produce an upper bound for $r_t$ that is valid for infinitely many t. We also present two examples of nonisotrivial elliptic curves $E/\mathbb {Q}(T)$ such that $r_t \leq r+1$ for infinitely many t.

In this paper, we study the positivity property of the tangent bundle $T_X$ of a Fano threefold X with Picard number $2$. We determine the bigness of the tangent bundle of the whole $36$ deformation types. Our result shows that $T_X$ is big if and only if $(-K_X)^3\ge 34$. As a corollary, we prove that the tangent bundle is not big when X has a standard conic bundle structure with non-empty discriminant. Our main methods are to produce irreducible effective divisors on ${\mathbb {P}}(T_X)$ constructed from the total dual VMRT associated to a family of rational curves. Additionally, we present some criteria to determine the bigness of $T_X$.

Works by O’Grady allow to associate with a two-dimensional Gushel–Mukai (GM) variety, which is a K3 surface, a double Eisenbud–Popescu–Walter (EPW) sextic. We characterize the $K3$ surfaces whose associated double EPW sextic is smooth. As a consequence, we are able to produce symplectic actions on some families of smooth double EPW sextics which are hyper-Kähler manifolds.

We also provide bounds for the automorphism group of GM varieties in dimension 2 and higher.

The blow-up of the anticanonical base point on a del Pezzo surface S of degree 1 gives rise to a rational elliptic surface $\mathscr {E}$ with only irreducible fibers. The sections of minimal height of $\mathscr {E}$ are in correspondence with the $240$ exceptional curves on S. A natural question arises when studying the configuration of these curves: if a point on S is contained in “many” exceptional curves, is it torsion on its fiber on $\mathscr {E}$? In 2005, Kuwata proved for the analogous question on del Pezzo surfaces of degree $2$, where there are 56 exceptional curves, that if “many” equals $4$ or more, the answer is yes. In this paper, we prove that for del Pezzo surfaces of degree 1, the answer is yes if ‘many’ equals $9$ or more. Moreover, we give counterexamples where a non-torsion point lies in the intersection of $7$ exceptional curves. We give partial results for the still open case of 8 intersecting exceptional curves.

We give a mathematically precise statement of the SYZ conjecture between mirror space pairs and prove it for any toric Calabi-Yau manifold with the Gross Lagrangian fibration. To date, it is the first time we realize the SYZ proposal with singular fibers beyond the topological level. The dual singular fibration is explicitly written and proved to be compatible with the family Floer mirror construction. Moreover, we discover that the Maurer-Cartan set of a singular Lagrangian is only a strict subset of the corresponding dual singular fiber. This responds negatively to the previous expectation and leads to new perspectives of SYZ singularities. As extra evidence, we also check some computations for a well-known folklore conjecture for the Landau-Ginzburg model.

We establish a McKay correspondence for finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_2$ in positive characteristic. As an application, we obtain a McKay correspondence for all rational double point singularities in characteristic $p\geq 7$. We discuss linearly reductive quotient singularities and canonical lifts over the ring of Witt vectors. In dimension 2, we establish simultaneous resolutions of singularities of these canonical lifts via G-Hilbert schemes. In the appendix, we discuss several approaches towards the notion of conjugacy classes for finite group schemes: This is an ingredient in McKay correspondences, but also of independent interest.

Fujino gave a proof for the semi-ampleness of the moduli part in the canonical bundle formula in the case when the general fibers are K3 surfaces or abelian varieties. We show a similar statement when the general fibers are primitive symplectic varieties. This answers a question of Fujino raised in the same article. Moreover, using the structure theory of varieties with trivial first Chern class, we reduce the question of semi-ampleness in the case of families of K-trivial varieties to a question when the general fibers satisfy a slightly weaker Calabi–Yau condition.

Using $L^2$-methods, we prove a vanishing theorem for tame harmonic bundles over quasi-compact Kähler manifolds in a very general setting. As a special case, we give a completely new proof of the Kodaira-type vanishing theorems for Higgs bundles due to Arapura. To prove our vanishing theorem, we construct a fine resolution of the Dolbeault complex for tame harmonic bundles via the complex of sheaves of $L^2$-forms, and we establish the Hörmander $L^2$-estimate and solve $(\bar {\partial }_E+\theta )$-equations for Higgs bundles $(E,\theta )$.

Let $X$ be an $n$-dimensional (smooth) intersection of two quadrics, and let ${T^{\rm{*}}}X$ be its cotangent bundle. We show that the algebra of symmetric tensors on $X$ is a polynomial algebra in $n$ variables. The corresponding map ${\rm{\Phi }}:{T^{\rm{*}}}X \to {\mathbb{C}^n}$ is a Lagrangian fibration, which admits an explicit geometric description; its general fiber is a Zariski open subset of an abelian variety, which is a quotient of a hyperelliptic Jacobian by a $2$-torsion subgroup. In dimension $3$, ${\rm{\Phi }}$ is the Hitchin fibration of the moduli space of rank $2$ bundles with fixed determinant on a curve of genus $2$.

We show the properness of the moduli stack of stable surfaces over $\mathbb{Z}\left[ {1/30} \right]$, assuming the locally-stable reduction conjecture for stable surfaces. This relies on a local Kawamata–Viehweg vanishing theorem for 3-dimensional log canonical singularities at closed point of characteristic $p \ne 2,3$ and $5$, which are not log canonical centres.

We compute odd-degree genus 1 quasimap and Gromov–Witten invariants of moduli spaces of Higgs ${\rm{S}}{{\rm{L}}_2}$-bundles on a curve of genus $g \geqslant 2$. We also compute certain invariants for all prime ranks. This proves some parts of the author’s conjectures on quasimap invariants of moduli spaces of Higgs bundles. More generally, our methods provide a computation scheme for genus 1 quasimap and Gromov–Witten invariants in the case when degrees of maps are coprime to the rank. This requires an analysis of the localisation formula for certain Quot schemes parametrising higher-rank quotients on an elliptic curve. Invariants for degrees that are not coprime to the rank exhibit a very different structure for a reason that we explain.

KSB stability holds at codimension $1$ points trivially, and it is quite well understood at codimension $2$ points because we have a complete classification of $2$-dimensional slc singularities. We show that it is automatic in codimension $3$.

We prove that the period mapping is dominant for elliptic surfaces over an elliptic curve with $12$ nodal fibers, and that its degree is larger than $1$. This settles the final case of infinitesimal Torelli for a generic elliptic surface.

Mirror symmetry for a semistable degeneration of a Calabi–Yau manifold was first investigated by Doran–Harder–Thompson when the degenerate fiber is a union of two quasi-Fano manifolds. They proposed a topological construction of a mirror Calabi–Yau by gluing of two Landau–Ginzburg models that are mirror to those Fano manifolds. We extend this construction to a general type semistable degeneration where the dual boundary complex of the degenerate fiber is the standard N-simplex. Since each component in the degenerate fiber comes with the simple normal crossing anticanonical divisor, one needs the notion of a hybrid Landau–Ginzburg model – a multipotential analogue of classical Landau–Ginzburg models. We show that these hybrid Landau–Ginzburg models can be glued to be a topological mirror candidate for the nearby Calabi–Yau, which also exhibits the structure of a Calabi–Yau fibration over $\mathbb P^N$. Furthermore, it is predicted that the perverse Leray filtration associated to this fibration is mirror to the monodromy weight filtration on the degeneration side [12]. We explain how this can be deduced from the original mirror P=W conjecture [18].