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Volume spécial en l'honneur de Claire Voisin
Ce volume est constitué d'articles écrits en l'honneur de Claire Voisin.
1. Finiteness for self-dual classes in integral variations of Hodge structure
We generalize the finiteness theorem for the locus of Hodge classes with fixed self-intersection number, due to Cattani, Deligne, and Kaplan, from Hodge classes to self-dual classes. The proof uses the definability of period mappings in the o-minimal structure $\mathbb{R}_{\mathrm{an},\exp}$.
2. Smooth subvarieties of Jacobians
We give new examples of algebraic integral cohomology classes on smooth projective complex varieties that are not integral linear combinations of classes of smooth subvarieties. Some of our examples have dimension 6, the lowest possible. The classes that we consider are minimal cohomology classes on Jacobians of very general curves. Our main tool is complex cobordism.
3. The cotangent bundle of K3 surfaces of degree two
K3 surfaces have been studied from many points of view, but the positivity of the cotangent bundle is not well understood. In this paper we explore the surprisingly rich geometry of the projectivised cotangent bundle of a very general polarised K3 surface $S$ of degree two. In particular, we describe the geometry of a surface $D_S \subset \mathbb{P}(\Omega_S)$ that plays a similar role to the surface of bitangents for a quartic in $\mathbb{P}^3$.
4. Rigidity of projective symmetric manifolds of Picard number 1 associated to composition algebras
To each complex composition algebra $\mathbb{A}$, there associates a projective symmetric manifold $X(\mathbb{A})$ of Picard number one, which is just 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 these varieties are rigid, namely for any smooth family of projective manifolds over a connected base, if one fiber is isomorphic to $X(\mathbb{A})$, then every fiber is isomorphic to $X(\mathbb{A})$.
5. Chow groups of surfaces of lines in cubic fourfolds
The surface of lines in a cubic fourfold intersecting a fixed line splits motivically into two parts, one of which resembles a K3 surface. We define the analogue of the Beauville-Voisin class and study the push-forward map to the Fano variety of all lines with respect to the natural splitting of the Bloch-Beilinson filtration introduced by Mingmin Shen and Charles Vial.
6. Perverse-Hodge complexes for Lagrangian fibrations
Perverse-Hodge complexes are objects in the derived category of coherent sheaves obtained from Hodge modules associated with Saito's decomposition theorem. We study perverse-Hodge complexes for Lagrangian fibrations and propose a symmetry between them. This conjectural symmetry categorifies the "Perverse = Hodge" identity of the authors and specializes to Matsushita's theorem on the higher direct images of the structure sheaf. We verify our conjecture in several cases by making connections with variations of Hodge structures, Hilbert schemes, and Looijenga-Lunts-Verbitsky Lie algebras.
7. K-stability for varieties with a big anticanonical class
We extend the algebraic K-stability theory to projective klt pairs with a big anticanonical class. While in general such a pair could behave pathologically, it is observed in this note that K-semistability condition will force them to have a klt anticanonical model, whose stability property is the same as the original pair.
8. Remarks on the geometry of the variety of planes of a cubic fivefold
This note presents some properties of the variety of planes $F_2(X)\subset G(3,7)$ of a cubic $5$-fold $X\subset \mathbb P^6$. A cotangent bundle exact sequence 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 Gauss map of $F_2(X)$ is then proven to be an embedding. The last section is devoted to the relation between the variety of osculating planes of a cubic $4$-fold and the variety of planes of the associated cyclic cubic $5$-fold.
9. The second fundamental form of the moduli space of cubic threefolds in $\mathcal A_5$
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 prove that the image of this second fundamental form, which is known to be non-trivial, is contained in the kernel of a suitable multiplication map. Some ingredients are: the conic bundle structure of cubic threefolds, Prym theory, Gaussian maps and Jacobian ideals.
10. Measures of association between algebraic varieties, II: self-correspondences
Following a suggestion of Jordan Ellenberg, we study measures of complexity for self-correspondences of some classes of varieties. We also answer a question of Rhyd concerning curves sitting in the square of a very general hyperelliptic curve.
11. On algebraically coisotropic submanifolds of holomorphic symplectic manifolds
We investigate algebraically coisotropic submanifolds $X$ in a holomorphic symplectic projective manifold $M$. Motivated by our results in the hypersurface 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 $Z\subset N$ is Lagrangian? We prove that this is indeed the case when $M$ is an abelian variety, and give some partial answer when the canonical bundle $K_X$ is semi-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 that Lagrangian submanifolds do not exist on a sufficiently general Abelian variety, in contrast to the case when $M$ is irreducible hyperkähler.
12. Categorical absorptions of singularities and degenerations
We introduce the notion of categorical absorption of singularities: an operation that removes from the derived category of a singular variety a small admissible subcategory responsible for singularity and leaves a smooth and proper category. We construct (under appropriate assumptions) a categorical absorption for a projective variety $X$ with isolated ordinary double points. We further show that for any smoothing $\mathcal{X}/B$ of $X$ over a smooth curve $B$, the smooth part of the derived category of $X$ extends to a smooth and proper over $B$ family of triangulated subcategories in the fibers of $\mathcal{X}$.
13. On the dual positive cones and the algebraicity of a compact Kähler manifold
We investigate the algebraicity of compact Kähler manifolds admitting a positive rational Hodge class of bidimension $(1,1)$. We prove that if the dual Kähler cone of a compact Kähler manifold $X$ contains a rational class as an interior point, then its Albanese variety is projective. As a consequence, we answer the Oguiso--Peternell problem for Ricci-flat compact Kähler manifolds. We also study related algebraicity problems for threefolds.
14. Cohomology classes of complex approximable algebras
Huayi Chen introduces the notion of an approximable graded algebra, which he uses to prove a Fujita-type theorem in the arithmetic setting, and asked if any such 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, who subsequently proved that any such algebra is associated to an infinite Weil divisor. In this paper, we show that over the complex numbers, this infinite Weil divisor necessarily has finite cohomology class.
15. Difference varieties and the Green-Lazarsfeld Secant Conjecture
The Green-Lazarsfeld Secant Conjecture is a generalization of Green's Conjecture on syzygies of canonical curves to the cases of arbitrary line bundles. We establish the Green-Lazarsfeld Secant Conjecture for curves of genus g in all the divisorial case, that is, when the line bundles that fail to be (p+1)-very ample form a divisor in the Jacobian of the curve.
16. Extensions of curves with high degree with respect to the genus
We classify linearly normal surfaces $S \subset \mathbf{P}^{r+1}$ of degree $d$ such that $4g-4 \leq d \leq 4g+4$, where $g>1$ is the sectional genus (it is a classical result that for larger $d$ there are only cones). We apply this to the study of the extension theory of pluricanonical curves and genus $3$ curves, whenever they verify Property $N_2$, using and slightly expanding the theory of integration of ribbons of the authors and E.~Sernesi. We compute the corank of the relevant Gaussian maps, and we show that all ribbons over such curves are integrable, and thus there exists a universal extension. We carry out a similar program for linearly normal hyperelliptic curves of degree $d\geq 2g+3$. We classify surfaces having such a curve $C$ as a hyperplane section, compute the corank of the relevant Gaussian maps, and prove that all ribbons over $C$ are integrable if and only if $d=2g+3$. In the latter case we obtain the existence of a universal extension.
17. Algebraic cycles on Gushel-Mukai varieties
We study algebraic cycles on complex Gushel-Mukai (GM) varieties. We prove the generalised Hodge conjecture, the (motivated) Mumford-Tate conjecture, and the generalised Tate conjecture for all GM varieties. We compute all integral Chow groups of GM varieties, except for the only two infinite-dimensional cases (1-cycles on GM fourfolds and 2-cycles on GM sixfolds). We prove that if two GM varieties are generalised partners or generalised duals, their rational Chow motives in middle degree are isomorphic.
18. Deformations of some local Calabi-Yau manifolds
We study deformations of certain crepant resolutions of isolated rational Gorenstein singularities. After a general discussion of the deformation theory, we specialize to dimension $3$ and consider examples which are good (log) resolutions as well as the case of small resolutions. We obtain some partial results on the classification of canonical threefold singularities that admit good crepant resolutions. Finally, we study a noncrepant example, the blowup of a small resolution whose exceptional set is a smooth curve.
19. Rank 4 stable vector bundles on hyperkähler fourfolds of Kummer type
We partially extend to hyperkähler fourfolds of Kummer type the results that we have proved regarding stable rigid vector bundles on hyperkähler (HK) varieties of type $K3^{[n]}$. Let $(M,h)$ be a general polarized HK fourfold of Kummer type such that $q_M(h)\equiv -6\pmod{16}$ and the divisibility of $h$ is $2$, or $q_M(h)\equiv -6\pmod{144}$ and the divisibility of $h$ is $6$. We show that there exists a unique (up to isomorphism) slope stable vector bundle $\cal F$ on $M$ such that $r({\cal F})=4$, $ c_1({\cal F})=h$, $\Delta({\cal F})=c_2(M)$. Moreover $\cal F$ is rigid. One of our motivations is the desire to describe explicitly a locally complete family of polarized HK fourfolds of Kummer type.