We compute the co-multiplication of the algebraic Morava K-theory for split orthogonal groups. This allows us to compute the decomposition of the Morava motives of generic maximal orthogonal Grassmannians and to compute a Morava K-theory analogue of the J-invariant in terms of the ordinary (Chow) J-invariant.

We prove the coherence of multiplier submodule sheaves associated with Griffiths semi-positive singular hermitian metrics over holomorphic vector bundles on complex manifolds which have no nontrivial subvarieties, such as generic complex tori.

We prove a functional transcendence theorem for the integrals of algebraic forms in families of algebraic varieties. This allows us to prove a geometric version of André’s generalization of the Grothendieck period conjecture, which we state using the formalism of Nori motives.

More precisely, we prove a version of the Ax–Schanuel conjecture for the comparison between the flat and algebraic coordinates of an arbitrary admissible graded polarizable variation of integral mixed Hodge structures. This can be seen as a generalization of the recent Ax–Schanuel theorems of [13, 18] for mixed period maps.

We prove the conjecture of Franceschini and Lorenzini [‘Fat points of $\mathbb P^n$ whose support is contained in a linear proper subspace’, J. Pure and Appl. Algebra 160 (2001), 169–182] about the regularity index of fat points of $\mathbb P^n$ whose support is contained in a linear proper subspace.

A real variety whose real locus achieves the Smith–Thom equality is called maximal. This paper introduces new constructions of maximal real varieties, by using moduli spaces of geometric objects. We establish the maximality of the following real varieties:

• – moduli spaces of stable vector bundles of coprime rank and degree over a maximal real curve (recovering Brugallé–Schaffhauser’s theorem with a short new proof), which extends to moduli spaces of parabolic vector bundles;

• – moduli spaces of stable Higgs bundles of coprime rank and degree over a maximal real curve, providing maximal hyper-Kähler manifolds in every even dimension;

• – if a real variety has maximal Hilbert square, then the variety and its Hilbert cube are maximal, which happens for all maximal real cubic 3-folds, but never for maximal real cubic 4-folds;

• – punctual Hilbert schemes on a maximal real surface with vanishing first $\mathbb {F}_2$-Betti number and connected real locus, such as $\mathbb {R}$-rational maximal real surfaces and some generalized Dolgachev surfaces;

• – moduli spaces of stable sheaves on an $\mathbb {R}$-rational maximal Poisson surface (e.g. the real projective plane).

Let $\mathcal {X}\to \mathbb {D}$ be a flat family of projective complex 3-folds over a disc $\mathbb {D}$ with smooth total space $\mathcal {X}$ and smooth general fibre $\mathcal {X}_t,$ and whose special fiber $\mathcal {X}_0$ has double normal crossing singularities, in particular, $\mathcal {X}_0=A\cup B$, with A, B smooth threefolds intersecting transversally along a smooth surface $R=A\cap B.$ In this paper, we first study the limit singularities of a $\delta $-nodal surface in the general fibre $S_t\subset \mathcal {X}_t$, when $S_t$ tends to the central fibre in such a way its $\delta $ nodes tend to distinct points in R. The result is that the limit surface $S_0$ is in general the union $S_0=S_A\cup S_B$, with $S_A\subset A$, $S_B\subset B$ smooth surfaces, intersecting on R along a $\delta $-nodal curve $C=S_A\cap R=S_B\cap B$. Then we prove that, under suitable conditions, a surface $S_0=S_A\cup S_B$ as above indeed deforms to a $\delta $-nodal surface in the general fibre of $\mathcal {X}\to \mathbb {D}$. As applications, we prove that there are regular irreducible components of the Severi variety of degree d surfaces with $\delta $ nodes in $\mathbb {P}^3$, for every $\delta \leqslant {d-1\choose 2}$ and of the Severi variety of complete intersection $\delta $-nodal surfaces of type $(d,h)$, with $d\geqslant h-1$ in $\mathbb {P}^4$, for every $\delta \leqslant {{d+3}\choose 3}-{{d-h+1}\choose 3}-1.$

We define kappa classes on moduli spaces of Kollár-Shepherd-Barron-Alexeev (KSBA)-stable varieties and pairs, generalizing the Miller–Morita–Mumford classes on moduli of curves, and computing them in some cases where the virtual fundamental class is known to exist, including Burniat and Campedelli surfaces. For Campedelli surfaces, an intermediate step is finding the Chow (same as cohomology) ring of the GIT quotient $(\mathbb {P}^2)^7//SL(3)$.

Scalar relative invariants play an important role in the theory of group actions on a manifold as their zero sets are invariant hypersurfaces. Relative invariants are central in many applications, where they often are treated locally since an invariant hypersurface may not be a locus of a single function. Our aim is to establish a global theory of relative invariants.

For a Lie algebra ${\mathfrak g}$ of holomorphic vector fields on a complex manifold M, any holomorphic ${\mathfrak g}$-invariant hypersurface is given in terms of a ${\mathfrak g}$-invariant divisor. This generalizes the classical notion of scalar relative ${\mathfrak g}$-invariant. Any ${\mathfrak g}$-invariant divisor gives rise to a ${\mathfrak g}$-equivariant line bundle, and a large part of this paper is therefore devoted to the investigation of the group $\mathrm {Pic}_{\mathfrak g}(M)$ of ${\mathfrak g}$-equivariant line bundles. We give a cohomological description of $\mathrm {Pic}_{\mathfrak g}(M)$ in terms of a double complex interpolating the Chevalley-Eilenberg complex for ${\mathfrak g}$ with the Čech complex of the sheaf of holomorphic functions on M.

We also obtain results about polynomial divisors on affine bundles and jet bundles. This has applications to the theory of differential invariants. Those were actively studied in relation to invariant differential equations, but the description of multipliers (or weights) of relative differential invariants was an open problem. We derive a characterization of them with our general theory. Examples, including projective geometry of curves and second-order ODEs, not only illustrate the developed machinery but also give another approach and rigorously justify some classical computations. At the end, we briefly discuss generalizations of this theory.

We examine the maximum dimension of a linear system of plane cubic curves whose $\mathbb {F}_q$-members are all geometrically irreducible. Computational evidence suggests that such a system has a maximum (projective) dimension of $3$. As a step towards the conjecture, we prove that there exists a three-dimensional linear system $\mathcal {L}$ with at most one geometrically reducible $\mathbb {F}_q$-member.

We study hyperbolicity properties of the moduli space of polarized abelian varieties (also known as the Siegel modular variety) in characteristic p. Our method uses the plethysm operation for Schur functors as a key ingredient and requires a new positivity notion for vector bundles in characteristic p called $(\varphi,D)$-ampleness. Generalizing what was known for the Hodge line bundle, we also show that many automorphic vector bundles on the Siegel modular variety are $(\varphi,D)$-ample.

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.

In this paper we prove a new generic vanishing theorem for $X$ a complete homogeneous variety with respect to an action of a connected algebraic group. Let $A, B_0\subset X$ be locally closed affine subvarieties, and assume that $B_0$ is smooth and pure dimensional. Let ${\mathcal {P}}$ be a perverse sheaf on $A$ and let $B=g B_0$ be a generic translate of $B_0$. Then our theorem implies $(-1)^{\operatorname {codim} B}\chi (A\cap B, {\mathcal {P}}|_{A\cap B})\geq 0$. As an application, we prove in full generality a positivity conjecture about the signed Euler characteristic of generic triple intersections of Schubert cells. Such Euler characteristics are known to be the structure constants for the multiplication of the Segre–Schwartz–MacPherson classes of these Schubert cells.

We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth, irreducible, and non-degenerate curve of degree d and genus g in $\mathbb{P}^r.$ In this article, we study $\mathcal{H}_{15,g,5}$ for every possible genus g and determine when it is irreducible. We also study the moduli map $\mathcal{H}_{15,g,5}\rightarrow\mathcal{M}_g$ and several key properties such as gonality of a general element as well as characterizing smooth elements of each component.

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 $(E,\theta )$.

We provide an explicit formula for all primary genus-zero $r$-spin invariants. Our formula is piecewise polynomial in the monodromies at each marked point and in $r$. To deduce the structure of these invariants, we use a tropical realisation of the corresponding cohomological field theories. We observe that the collection of all Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) relations is equivalent to the relations deduced from the fact that genus-zero tropical CohFT cycles are balanced.

We compute the integral Chow ring of the moduli stack of smooth elliptic curves with n marked points for $3\leq n\leq 10$.

Given a connected reductive algebraic group G over an algebraically closed field, we investigate the Picard group of the moduli stack of principal G-bundles over an arbitrary family of smooth curves.

We study the Abel-Jacobi image of the Ceresa cycle $W_{k, e}-W_{k, e}^-$, where $W_{k, e}$ is the image of the k-th symmetric product of a curve X with a base point e on its Jacobian variety. For certain Fermat quotient curves of genus g, we prove that for any choice of the base point and $k \leq g-2$, the Abel-Jacobi image of the Ceresa cycle is non-torsion. In particular, these cycles are non-torsion modulo rational equivalence.