The goal of this paper is first of all to propose a strategy to attack the generalized Hodge conjecture for coniveau 2 complete intersections, and secondly to state a conjecture concerning the cones of effective cycle classes in intermediate dimensions. Our main results show that the generalized Hodge conjecture for coniveau 2 complete intersections would follow from a particular case of this effectiveness conjecture.
Let $X$ be a smooth complex projective variety with trivial Chow groups. (By trivial, we mean that the cycle class is injective.) We show (assuming the Lefschetz standard conjecture) that if the vanishing cohomology of a general complete intersection $Y$ of ample hypersurfaces in $X$ has geometric coniveau $\geq c$, then the Chow groups of cycles of dimension $\leq c-1$ of $Y$ are trivial. The generalized Bloch conjecture for $Y$ is this statement with "geometric coniveau" replaced by "Hodge coniveau".
The Bloch-Beilinson-Murre conjectures predict the existence of a descending filtration on Chow groups of smooth projective varieties which is functorial with respect to the action of correspondences and whose graded parts depend solely on the topology -- i.e. the cohomology -- of $X$. In this paper, we wish to explore, at the cost of having to assume general conjectures about algebraic cycles, how the coniveau filtration on the cohomology of $X$ has an incidence on the Chow groups of $X$. However, by keeping such assumptions minimal, we are able to prove some of these conjectures either in low…
We study Calabi-Yau manifolds which are complete intersections of hypersurfaces of multidegree $1$ in an $m$-fold product of $n$-dimensional projective spaces. Using the theory of Coxeter groups, we show that the birational automorphism group of such a Calabi-Yau manifold $X$ is infinite and a free product of copies of $\mathbb{Z}$ . Moreover, we give an explicit description of the boundary of the movable cone $\overline{\operatorname{Mov}}(X)$. In the end, we consider examples for the general and non-general case and picture the movable cone and the fundamental domain for the action of $\oper…
A curve on a projective variety is called movable if it belongs to an algebraic family of curves covering the variety. We consider when the cone of movable curves can be characterized without existence statements of covering families by studying the complete intersection cone on a family of blow-ups of complex projective space, including the moduli space of stable six-pointed rational curves, $\bar{M}_{0,6}$, and the permutohedral or Losev-Manin moduli space of four-pointed rational curves. Our main result is that the movable and complete intersection cones coincide for the toric members of th…
In this paper we give a geometric characterization of the cones of toric varieties that are complete intersections. In particular, we prove that the class of complete intersection cones is the smallest class of cones which is closed under direct sum and contains all simplex cones. Further, we show that the number of the extreme rays of such a cone, which is less than or equal to $2n-2$, is exactly $2n-2$ if and only if the cone is a bipyramidal cone, where $n>1$ is the dimension of the cone. Finally, we characterize all toric varieties whose associated cones are complete intersection cones.
Let $C({\bf n})$ be a complete intersection monomial curve in the 4-dimensional affine space. In this paper we study the complete intersection property of the monomial curve $C({\bf n}+w{\bf v})$, where $w>0$ is an integer and ${\bf v} \in \mathbb{N}^{4}$. Also we investigate the Cohen-Macaulayness of the tangent cone of $C({\bf n}+w{\bf v})$.
We show how to equip the cone complexes of toroidal embeddings with additional structure that allows to define a balancing condition for weighted subcomplexes. We then proceed to develop the foundations of an intersection theory on cone complexes including push-forwards, intersections with tropical divisors, and rational equivalence. These constructions are shown to have an algebraic interpretation: Ulirsch's tropicalizations of subvarieties of toroidal embeddings carry natural multiplicities making them tropical cycles, and the induced tropicalization map for cycles respects push-forwards, in…
We study mirror symmetry of complete intersection Calabi-Yau manifolds which have birational automorphisms of infinite order. We observe that movable cones in birational geometry are transformed, under mirror symmetry, to the monodromy nilpotent cones which are naturally glued together.
For any conical symplectic resolution, we give a conjecture relating the intersection cohomology of the singular cone to the quantum cohomology of its resolution. We prove this conjecture for hypertoric varieties, recovering the ring structure on hypertoric intersection cohomology that was originally constructed by Braden and the second author.
Let X be a subscheme of a reduced scheme Y. Then Y has a flat "degeneration to the normal cone" C_X Y of X, and this degeneration plays a key step in Fulton and MacPherson's "basic construction" in intersection theory. The intersection product has a canonical refinement as a sum over the components of C_X Y, for X and Y depending on the given intersection problem. The cone C_X Y is usually not reduced, which leads to the appearance of multiplicities in intersection formulae. We describe a variant of this degeneration, due essentially to Samuel, Rees, and Nagata, in which Y flatly degenerates t…
We show that the dual of the Cayley cone, associated to a Minkowski sum decomposition of a reflexive polytope, contains a reflexive polytope admitting a nef-partition. This nef-partition corresponds to a Calabi-Yau complete intersection in a Gorenstein Fano toric variety degenerating to an ample Calabi-Yau hypersurface in another Fano toric variety. Using the Batyrev-Borisov mirror symmetry construction, we found the mirror contraction of a Calabi-Yau complete intersection to the mirror of the ample Calabi-Yau hypersurface.
The goal of this note is to show that in the case of transversal intersections the "true local terms" appearing in the Lefschetz trace formula equal to the "naive local terms". To prove the result we extend the method of [Va], where the case of contracting correspondences is treated. Our new ingredients are the observation of Verdier that specialization of any etale sheaf to the normal cone is monodromic and the assertion that in some cases local terms are "constant in families". As an application, we get a generalization of the Deligne-Lusztig trace formula.
The existence of a vector field on a compact Kaehler manifold with nonempty zero locus and the properties of this zero locus strongly influence the geometry of the manifold. For example, J. Wahl proved that the existence of a vector field vanishing on an ample divisor of a projective normal variety X implies that X is a cone over this divisor. If X is smooth, X will be isomorphic to the n-dimensional projective space. This paper is a first attempt to generalize Wahl's theorem to higher codimensions: Given a complex smooth projective threefold X and a vector field on X vanishing on an irreducib…
As shown in a previous paper, certain naturally arising cones of holomorphic vector bundle sections over the main component $\ov\M_{1,k}^0(¶,d)$ of the moduli space of stable genus-one holomorphic maps into $¶$ have a well-defined euler class. In this paper, we extend this result to moduli spaces of perturbed, in a restricted way, $J$-holomorphic maps. We show that euler classes of such cones relate the reduced genus-one Gromov-Witten invariants of complete intersections to the corresponding GW-invariants of the ambient projective space. As a consequence, the standard genus-one GW-invariants o…
We compute the equivariant Hirzebruch class of the quadric cone in C^n degenerating it to an intersection to hyperplanes. The difference of the Hirzebruch classes (measured by the Milnor class) turns out to be the Hirzebruch class of the complement of the quadratic cone of smaller dimension.
Let $C$ be a Gorenstein non complete intersection monomial curve in the 4-dimensional affine space. In this paper we study the minimal number of generators of the tangent cone of $C$. Special attention will be paid to the case where $C$ has Cohen-Macaulay tangent cone at the origin.
Let $C$ be a smooth curve which is complete intersection of a quadric and a degree $k>2$ surface in $\mathbb{P}^3$ and let $C^{(2)}$ be its second symmetric power. In this paper we study the finite generation of the extended canonical ring $R(Δ,K) := \bigoplus_{(a,b)\in\mathbb{Z}^2}H^0(C^{(2)},aΔ+bK)$, where $Δ$ is the image of the diagonal and $K$ is the canonical divisor. We first show that $R(Δ,K)$ is finitely generated if and only if the difference of the two $g_k^1$ on $C$ is torsion non-trivial and then show that this holds on an analytically dense locus of the moduli space of such curve…
We present a class of homogeneous ideals which are generated by monomials and binomials of degree two and are set-theoretic complete intersections. This class includes certain reducible varieties of minimal degree and, in particular, the presentation ideals of the fiber cone algebras of monomial varieties of codimension two.
We prove that if a smooth variety with non-positive canonical class can be embedded into a weighted projective space of dimension $n$ as a well formed complete intersection and it is not an intersection with a linear cone therein, then the weights of the weighted projective space do not exceed $n+1$. Based on this bound we classify all smooth Fano complete intersections of dimensions $4$ and $5$, and compute their invariants.
We prove the Morrison-Kawamata cone conjecture for klt Calabi-Yau pairs in dimension 2. That is, for a large class of rational surfaces as well as K3 surfaces and abelian surfaces, the action of the automorphism group of the surface on the convex cone of ample divisors has a rational polyhedral fundamental domain. More concretely: there many be infinitely many curves with negative self-intersection on the surface, but all such curves fall into finitely many orbits under the automorphism group of the surface. The proof uses the geometry of groups acting on hyperbolic space. We deduce a characte…
A coordinate cone in R^n is an intersection of some coordinate hyperplanes and open coordinate half-spaces. A semi-monotone set is a defnable in an o-minimal structure over the reals, open bounded subset of R^n such that its intersection with any translation of any coordinate cone is connected. This can be viewed as a generalization of the convexity property. Semi-monotone sets have a number of interesting geometric and combinatorial properties. The main result of the paper is that every semi-monotone set is a topological regular cell.
