This volume consists in articles written in honour of Claire Voisin.

We generalize the finiteness theorem for the locus of Hodge classes withfixed self-intersection number, due to Cattani, Deligne, and Kaplan, from Hodgeclasses to self-dual classes. The proof uses the definability of periodmappings in the o-minimal structure $\mathbb{R}_{\mathrm{an},\exp}$.

We give new examples of algebraic integral cohomology classes on smoothprojective complex varieties that are not integral linear combinations ofclasses of smooth subvarieties. Some of our examples have dimension 6, thelowest possible. The classes that we consider are minimal cohomology classes onJacobians of very general curves. Our main tool is complex cobordism.

K3 surfaces have been studied from many points of view, but the positivity ofthe cotangent bundle is not well understood. In this paper we explore thesurprisingly rich geometry of the projectivised cotangent bundle of a verygeneral polarised K3 surface $S$ of degree two. In particular, we describe thegeometry of a surface $D_S \subset \mathbb{P}(\Omega_S)$ that plays a similarrole to the surface of bitangents for a quartic in $\mathbb{P}^3$.

To each complex composition algebra $\mathbb{A}$, there associates aprojective symmetric manifold $X(\mathbb{A})$ of Picard number one, which isjust a smooth hyperplane section of the following varieties ${\rm Lag}(3,6),{\rm Gr}(3,6), \mathbb{S}_6, E_7/P_7.$ In this paper, it is proven that thesevarieties are rigid, namely for any smooth family of projective manifolds overa connected base, if one fiber is isomorphic to $X(\mathbb{A})$, then everyfiber is isomorphic to $X(\mathbb{A})$.

The surface of lines in a cubic fourfold intersecting a fixed line splitsmotivically into two parts, one of which resembles a K3 surface. We define theanalogue of the Beauville-Voisin class and study the push-forward map to theFano variety of all lines with respect to the natural splitting of theBloch-Beilinson filtration introduced by Mingmin Shen and Charles Vial.

Perverse-Hodge complexes are objects in the derived category of coherentsheaves obtained from Hodge modules associated with Saito's decompositiontheorem. We study perverse-Hodge complexes for Lagrangian fibrations andpropose a symmetry between them. This conjectural symmetry categorifies the"Perverse = Hodge" identity of the authors and specializes to Matsushita'stheorem on the higher direct images of the structure sheaf. We verify ourconjecture in several cases by making connections with variations of Hodgestructures, Hilbert schemes, and Looijenga-Lunts-Verbitsky Lie algebras.

We extend the algebraic K-stability theory to projective klt pairs with a biganticanonical class. While in general such a pair could behave pathologically,it is observed in this note that K-semistability condition will force them tohave a klt anticanonical model, whose stability property is the same as theoriginal pair.

This note presents some properties of the variety of planes $F_2(X)\subsetG(3,7)$ of a cubic $5$-fold $X\subset \mathbb P^6$. A cotangent bundle exactsequence is first derived from the remark made by Iliev and Manivel that$F_2(X)$ sits as a Lagrangian subvariety of the variety of lines of a cubic$4$-fold, which is a hyperplane section of $X$. Using the sequence, the Gaussmap of $F_2(X)$ is then proven to be an embedding. The last section is devotedto the relation between the variety of osculating planes of a cubic $4$-foldand the variety of planes of the associated cyclic cubic $5$-fold.

We study the second fundamental form of the Siegel metric in $\mathcal A_5$restricted to the locus of intermediate Jacobians of cubic threefolds. We provethat the image of this second fundamental form, which is known to benon-trivial, is contained in the kernel of a suitable multiplication map. Someingredients are: the conic bundle structure of cubic threefolds, Prym theory,Gaussian maps and Jacobian ideals.

Following a suggestion of Jordan Ellenberg, we study measures of complexityfor self-correspondences of some classes of varieties. We also answer aquestion of Rhyd concerning curves sitting in the square of a very generalhyperelliptic curve.

We investigate algebraically coisotropic submanifolds $X$ in a holomorphicsymplectic projective manifold $M$. Motivated by our results in thehypersurface case, we raise the following question: when $X$ is not uniruled,is it true that up to a finite étale cover, the pair $(X,M)$ is a product$(Z\times Y, N\times Y)$ where $N, Y$ are holomorphic symplectic and $Z\subsetN$ is Lagrangian? We prove that this is indeed the case when $M$ is an abelianvariety, and give some partial answer when the canonical bundle $K_X$ issemi-ample. In particular, when $K_X$ is nef and big, $X$ is Lagrangian in $M$(in fact this also holds without nefness assumption). We also remark thatLagrangian submanifolds do not exist on a sufficiently general Abelian variety,in contrast to the case when $M$ is irreducible hyperkähler.

We introduce the notion of categorical absorption of singularities: anoperation that removes from the derived category of a singular variety a smalladmissible subcategory responsible for singularity and leaves a smooth andproper category. We construct (under appropriate assumptions) a categoricalabsorption for a projective variety $X$ with isolated ordinary double points.We further show that for any smoothing $\mathcal{X}/B$ of $X$ over a smoothcurve $B$, the smooth part of the derived category of $X$ extends to a smoothand proper over $B$ family of triangulated subcategories in the fibers of$\mathcal{X}$.

We investigate the algebraicity of compact Kähler manifolds admitting apositive rational Hodge class of bidimension $(1,1)$. We prove that if the dualKähler cone of a compact Kähler manifold $X$ contains a rational class asan interior point, then its Albanese variety is projective. As a consequence,we answer the Oguiso--Peternell problem for Ricci-flat compact Kählermanifolds. We also study related algebraicity problems for threefolds.

Huayi Chen introduces the notion of an approximable graded algebra, which heuses to prove a Fujita-type theorem in the arithmetic setting, and asked if anysuch algebra is the graded ring of a big line bundle on a projective variety.This was proved to be false in a previous paper of the author's, whosubsequently proved that any such algebra is associated to an infinite Weildivisor. In this paper, we show that over the complex numbers, this infiniteWeil divisor necessarily has finite cohomology class.