Algebraic Gromov’s Ellipticity of Cubic Hypersurfaces

We study structures of embedded projective manifolds swept out by cubic varieties. We show if an embedded projective manifold is swept out by high-dimensional smooth cubic hypersurfaces, then it admits an extremal contraction which is a linear projective bundle or a cubic fibration. As an application, we give a characterization of smooth cubic hypersurfaces. We also classify embedded projective manifolds of dimension at most five swept out by copies of the Segre threefold ℙ1 × ℙ2. In the course of the proof, we classify projective manifolds of dimension five swept out by planes.

Let $Y$ be a smooth hypersurface in a projective irreducible holomorphic symplectic manifold X of dimension 2n. The characteristic foliation $F$ is the kernel of the symplectic form restricted to Y. In this article, we prove that a generic leaf of the characteristic foliation is dense in Y if Y has positive Beauville–Bogomolov–Fujiki square.

Let Y be a smooth projective algebraic surface over ℂ, and T(Y) the kernel of the Albanese map CH0(Y)deg0 Alb(Y). It was first proven by D. Mumford that if the genus Pg(Y) > 0, then T(Y) is 'infinite dimensional'. One would like to have a better idea about the structure of T(Y). For example, if Y is dominated by a product of curves E1 × E2, such as an abelian or a Kummer surface, then one can easily construct an abelian variety B and a surjective 'regular' homomorphism B⊗z2 T(Y). A similar story holds for the case where Y is the Fano surface of lines on a smooth cubic hypersurface in P4. This implies a sort of boundedness result for T(Y). It is natural to ask if this is the case for any smooth projective algebraic surface Y ? Partial results have been attained in this direction by the author [Illinois. J. Math. 35 (2), 1991]. In this paper, we show that the answer to this question is in general no. Furthermore, we generalize this question to the case of the Chow group of k—cycles on any projective algebraic manifold X, and arrive at, from a conjectural standpoint, necessary and sufficient cohomological conditions on X for which the question can be answered affirmatively.

We prove in this note the following result: Theorem .− A smooth complex projective hypersurface of dimension ≥ 2 and degree ≥ 3 admits no endomorphism of degree > 1 . Since the case of quadrics is treated in [PS], this settles the question of endomorphisms of hypersurfaces. We prove the theorem in Section 1, using a simple but efficient trick devised by Amerik, Rovinsky and Van de Ven [ARV]. In Section 2 we collect some general results on endomorphisms of projective manifolds; we classify in particular the Del Pezzo surfaces which admit an endomorphism of degree > 1 .

We investigate the relation between the Hodge theory of a smooth subcanonical $n$-dimensional projective variety $X$ and the deformation theory of the affine cone $A_X$ over $X$. We start by identifying $H^{n-1,1}_{\mathrm{prim}}(X)$ as a distinguished graded component of the module of first order deformations of $A_X$, and later on we show how to identify the whole primitive cohomology of $X$ as a distinguished graded component of the Hochschild cohomology module of the punctured affine cone over $X$. In the particular case of a projective smooth hypersurface $X$ we recover Griffiths' isomorphism between the primitive cohomology of $X$ and certain distinguished graded components of the Milnor algebra of a polynomial defining $X$. The main result of the article can be effectively exploited to compute Hodge numbers of smooth subcanonical projective varieties. We provide a few example computation, as well a SINGULAR code, for Fano and Calabi-Yau threefolds.

Let $M$ be a complex projective Fano manifold whose Picard group is isomorphic to $\mathbb{Z}$ and the tangent bundle $TM$ is semistable. Let $Z \subset M$ be a smooth hypersurface of degree strictly greater than degree($TM$)$(\mathrm{dim}_{\mathbb{C}} Z-1)/(2\mathrm{dim}_{\mathbb{C}} Z-1)$ and satisfying the condition that the inclusion of $Z$ in $M$ gives an isomorphism of Picard groups. We prove that the tangent bundle of $Z$ is stable. A similar result is proved also for smooth complete intersections in $M$. The main ingredient in the proof of it is a vanishing result for the top cohomology of the twisted holomorphic differential forms on $Z$.

Let X be a projective algebraic manifold of dimension n (over C), CH1(X) the Chow group of algebraic cycles of codimension l on X, modulo rational equivalence, and A1(X) ⊂ CH1(X) the subgroup of cycles algebraically equivalent to zero. We say that A1(X) is finite dimensional if there exists a (possibly reducible) smooth curve T and a cycle z∈CH1(Γ × X) such that z*:A1(Γ)‐A1(X) is surjective. There is the well known Abel‐Jacobi map λ1:A1(X)‐J(X), where J(X) is the lth Lieberman Jacobian. It is easy to show that A1(X)→J(X) equation image A1(X) finite dimensional. Now set equation image with corresponding map A*(X)→J(X). Also define Level equation image . In a recent book by the author, there was stated the following conjecture: equation image where it was also shown that (⟹) in (**) is a consequence of the General Hodge Conjecture (GHC). In this present paper, we prove A*(X) finite dimensional ⇔︁ Level (H*(X)) ≤ 1 for a special (albeit significant) class of smooth hypersurfaces. We make use of the family of k‐planes on X, where equation image ([…] = greatest integer function) and d = deg X; moreover the essential technical ingredients are the Lefschetz theorems for cohomology and an analogue for Chow groups of hypersurfaces. These ingredients in turn imply very special cases of the GHC for our choice of hypersurfaces X. Some applications to the Griffiths group, vanishing results, and (universal) algebraic representatives for certain Chow groups are given.