If X is a smooth projective complex threefold, the Hodge conjecture holds for degree 4 rational Hodge classes on X. Kollar gave examples where it does not hold for integral Hodge classes of degree 4, that is integral Hodge classes need not be classes of algebraic 1-cycles with integral coefficients. We show however that the Hodge conjecture holds for integral Hodge classes of degree 4 on a uniruled or a Calabi-Yau threefold.
We prove the Jordan property for groups of bimeromorphic selfmaps of three-dimensional compact Kähler varieties of non-negative Kodaira dimension and positive irregularity.
In this paper we show that the Chern numbers of a smooth Mori fibre space in dimension three are bounded in terms of the underlying topological manifold. We also generalise a theorem of Cascini and the second named author on the boundedness of Chern numbers of certain threefolds to the case of negative Kodaira dimension.
We show that polarized endomorphisms of rationally connected threefolds with at worst terminal singularities are equivariantly built up from those on Q-Fano threefolds, Gorenstein log del Pezzo surfaces and P^1. Similar results are obtained for polarized endomorphisms of uniruled threefolds and fourfolds. As a consequence, we show that every smooth Fano threefold with a polarized endomorphism of degree > 1, is rational.
We shall reproof formulas for the Hodge numbers of Calabi-Yau threefolds of Borcea-Voisin type constructed by A. Cattaneo and A. Garbagnati, using the orbifold cohomology formula and the orbifold Euler characteristic.
We prove that the product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira dimension zero for which the integral Hodge conjecture fails, and the first examples of non-algebraic torsion cohomology classes of degree 4 on smooth projective complex threefolds.
This paper studies the Hilbert scheme of a curve on a complete-intersection K-trivial threefold, in the case in which the curve is unobstructed in the ambient variety in which the threefold lives. The basic result is that the obstruction theory of the curve is completely determined by the scheme-theoretic Abel-Jacobi mapping. Several applications of this fact are given.
Batyrev has defined the stringy E-function for complex varieties with at most log terminal singularities. It is a rational function in two variables if the singularities are Gorenstein. Furthermore, if the variety is projective and its stringy E-function is a polynomial, Batyrev defined its stringy Hodge numbers essentially as the coefficients of this E-function, generalizing the usual notion of Hodge numbers of a nonsingular projective variety. He conjectured that they are nonnegative. We prove this for a class of `mild' isolated singularities (the allowed singularities depend on the dimensio…
In this note, we consider crepant resolutions of the quotient varieties of smooth quintic threefolds by Gorenstein group actions. We compute their Hodge numbers via McKay correspondence. In this way, we find some new pairs $(h^{1,1},\;h^{2,1})$ of Hodge numbers of Calabi-Yau threefolds.
We correct the definitions and descriptions of the integral structures in [U14]. The previous flat basis in [ibid] is characterized by the Frobenius solutions and integral in the first approximation by mean of the graded quotients of monodromy filtration, but not integral in the strict sense. In this article, we use the integral structure of Iritani in [I11] for A-model. Using this precise version, we study open mirror symmetry for quintic threefolds through log mixed Hodge theory, especially the recent result on Neron models for admissible normal functions with non-torsion extensions in the j…
For each smooth Fano threefold $X$ with Picard number 1 we consider a weak Landau--Ginzburg model, that is a fibration over $\mathbb C^1$ given by a certain Laurent polynomial. In the spirit of L. Katzarkov's program we prove that the number of irreducible components of the central fiber of its compactification is $h^{1,2}(X)+1$. In particular, it does not depend on the compactification. The question of dependence on the model is open; however we produce examples of different weak Landau--Ginzburg models for the same variety with the same number of components of the central fiber.
We study nodal complete intersection threefolds of type $(2,4)$ in $\PP^5$ which contain an Enriques surface in its Fano embedding. We completely determine Calabi-Yau birational models of a generic such threefold. These models have Hodge numbers $(h^{11},h^{12})=(2,32)$. We also describe Calabi-Yau varieties with Hodge numbers equal to $(2,26)$, $(23,5)$ and $(31,1)$. The last two pairs of Hodge numbers are, to the best of our knowledge, new.
We establish the real integral Hodge conjecture for 1-cycles on various classes of uniruled threefolds (conic bundles, Fano threefolds with no real point, some del Pezzo fibrations) and on conic bundles over higher-dimensional bases which themselves satisfy the real integral Hodge conjecture for 1-cycles. In addition, we show that rationally connected threefolds over non-archimedean real closed fields do not satisfy the real integral Hodge conjecture in general and that over such fields, Bröcker's EPT theorem remains true for simply connected surfaces of geometric genus zero but fails for some…
In this article we prove a fundamental inequality between Artin-Mazur heights and Yobuko heights of certain proper log smooth schemes of Cartier type over a fine log scheme whose underlying scheme is the spectrum of a perfect field $κ$ of characteristic $p>0$. We also prove that the cohomologies of Witt-sheaves of them are finitely generated ${\cal W}(κ)$-modules if the Yobuko heights of them are finite. As an application, we prove that the $p$-primary torsion parts of the Chow groups of codimension $2$ of proper smooth threefolds over $κ$ is of finite cotype if the Yobuko heights of them are …
We study type III contractions of Calabi-Yau threefolds containing a ruled surface over a smooth curve. We discuss the conditions necessary for the image threefold to by smoothable. We describe the change in Hodge numbers caused by this contraction and smoothing deformation. A generalization of a fomula for calculating Hodge numbers of hypersurfaces in $\mathbb{P}^4$ with ordinary double and triple points is presented. We use these results to construct new Calabi-Yau threefolds of Picard rank two arising from a family of quintic threefolds containing a cone.
Let $X$ be a non-uniruled compact Kähler space of dimension 3. We show that the group of bimeromorphic automorphisms of $X$ is Jordan. More generally, the same result holds for any compact Kähler space admitting a quasi-minimal model.
In this paper, we represent the Hodge metric in terms of the Weil-Petersson metric and its Ricci curvature on the moduli spaces of polarized Calabi-Yau threefolds.
We formulate the "real integral Hodge conjecture", a version of the integral Hodge conjecture for real varieties, and raise the question of its validity for cycles of dimension 1 on uniruled and Calabi-Yau threefolds and on rationally connected varieties. We relate it to the problem of determining the image of the Borel-Haefliger cycle class map for 1-cycles, with the problem of deciding whether a real variety with no real point contains a curve of even geometric genus and with the problem of computing the torsion of the Chow group of 1-cycles of real threefolds. New results about these proble…
We consider algebraic manifolds $Y$ of dimension 3 over $\Bbb{C}$ with $H^i(Y, Ω^j_Y)=0$ for all $j\geq 0$ and $i>0$. Let $X$ be a smooth completion of $Y$ with $D=X-Y$, an effective divisor on $X$ with normal crossings. If the $D$-dimension of $X$ is not zero, then $Y$ is a fibre space over a smooth affine curve $C$ (i.e., we have a surjective morphism from $Y$ to $C$ such that general fibre is smooth and irreducible) such that every fibre satisfies the same vanishing condition. If an irreducible smooth fibre is not affine, then the Kodaira dimension of $X$ is $-\infty$ and the $D$-dimension …
We study the Hodge numbers of Landau-Ginzburg models as defined by Katzarkov, Kontsevich and Pantev. First we show that these numbers can be computed using ordinary mixed Hodge theory, then we give a concrete recipe for computing these numbers for the Landau-Ginzburg mirrors of Fano threefolds. We finish by proving that for a crepant resolution of a Gorenstein toric Fano threefold $X$ there is a natural LG mirror $(Y,\mathsf{w})$ so that $h^{p,q}(X) = f^{3-q,p}(Y,\mathsf{w})$.
The Hodge numbers of generic elliptically fibered Calabi-Yau threefolds over toric base surfaces fill out the "shield" structure previously identified by Kreuzer and Skarke. The connectivity structure of these spaces and bounds on the Hodge numbers are illuminated by considerations from F-theory and the minimal model program. In particular, there is a rigorous bound on the Hodge number h_{21} <= 491 for any elliptically fibered Calabi-Yau threefold. The threefolds with the largest known Hodge numbers are associated with a sequence of blow-ups of toric bases beginning with the Hirzebruch surfac…
In this paper, we give an expression and some estimates of the curvature tensor of the Hodge metric over the moduli space of a polarized Calabi-Yau threefold. The symmetricity of the Yukawa coupling is also studied. In the last section of this paper, an extra restriction of the limiting Hodge structure for the degeneration of Calabi-Yau threefolds is given.
To a one-parameter family of Calabi-Yau threefolds, we can associate the extended period map by the log Hodge theory of Kato and Usui. In the present paper, we study the image of a maximally unipotent monodromy point under the extended period map. As an application, we prove the generic Torelli theorem for a large class of one-parameter families of Calabi-Yau threefolds.
It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi-Yau threefolds coming from eight planes in $\mathbb{P}^3$ does {\em not} have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is natura…
We study the Variations of mixed Hodge structures (VMHS) associated to a pencil ${\cal X}$ (parametrised by an open set $B \subset {\Bbb P}^1$) of equisingular hypersurfaces of degree $d$ in ${\Bbb P}^{4}$ with exactly $m$ ordinary double points as singularities as well as the variations of Hodge structures (VHS) associated to the desingularization of this family $ \widetilde{\cal X}$. The case where exactly $l \le m $ of those double points are in algebraic general position (short:agp) is studied in detail and determine the possible limiting mixed Hodge structures (LMHS) associated to each of…
We present a list of Calabi-Yau threefolds known to us, and with holonomy groups that are precisely SU(3), rather than a subgroup, with small Hodge numbers, which we understand to be those manifolds with height $(h^{1,1}+h^{2,1})\le 24$. With the completion of a project to compute the Hodge numbers of free quotients of complete intersection Calabi-Yau threefolds, most of which were computed in Refs. [1-3] and the remainder in Ref. [4], many new points have been added to the tip of the Hodge plot, updating the reviews by Davies and Candelas in Refs. [1, 5]. In view of this and other recent cons…
We use Batyrev-Borisov's formula for the generating function of stringy Hodge numbers of Calabi-Yau varieties realized as complete intersections in toric varieties in order to get closed form expressions for Hodge numbers of Calabi-Yau threefolds in five-dimensional ambient spaces. These expressions involve counts of lattice points on faces of associated Cayley polytopes. Using the same techniques, similar expressions may be obtained for higher dimensional varieties realized as complete intersections of two hypersurfaces.
We analyze the structure of the algebraic manifolds $Y$ of dimension 3 with $H^i(Y, Ω^j_Y)=0$ for all $j\geq 0$, $i>0$ and $h^0(Y, {\mathcal{O}}_Y) > 1$, by showing the deformation invariant of some open surfaces. Secondly, we show when a smooth threefold with nonconstant regular functions satisfies the vanishing Hodge cohomology. As an application, we prove the existence of nonaffine and nonproduct threefolds $Y$ with this property by constructing a family of a certain type of open surfaces parametrized by the affine curve $\C-\{0\}$ such that the corresponding smooth completion $X$ has Kodai…
We study Kustin--Miller unprojections of Calabi--Yau threefolds. As an application we work out the geometric properties of Calabi--Yau threefolds defined as linear sections of determinantal varieties. We compute their Hodge numbers and describe the morphisms corresponding to the faces of their Kähler--Mori cone.
Calabi-Yau manifolds are important objects in algebraic geometry and in theoretical physics. A hypothesis of mirror symmetry is that Calabi-Yau manifolds of dimension 3 come in pairs, with the Hodge numbers of one manifold mirroring the Hodge numbers of the other. In this thesis, Calabi-Yau manifolds are constructed by smoothing Stanley-Reisner schemes of triangulations of 3-spheres. The triangulations of the 3-sphere with 7 or 8 vertices have been constructed and classified by Grünbaum and Sreedharan (1967). In this thesis we characterize the Calabi-Yau manifolds obtained from the triangulati…
Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this paper the existence of families of 1-cycles in Y for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. When Y itself is rationally connected, we relate this property to the existence of an integral homological decomposition of the diagonal. We also study this property for cubic threefolds, completing the work of Iliev-Ma…
Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this paper the existence of families of 1-cycles in Y for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. When Y itself is rationally connected, we relate this property to the existence of an integral homological decomposition of the diagonal. We also study this property for cubic threefolds, completing the work of Iliev-Ma…
We carry out a systematic analysis of Calabi-Yau threefolds that are elliptically fibered with section ("EFS") and have a large Hodge number h^{2, 1}. EFS Calabi-Yau threefolds live in a single connected space, with regions of moduli space associated with different topologies connected through transitions that can be understood in terms of singular Weierstrass models. We determine the complete set of such threefolds that have h^{2, 1} >= 350 by tuning coefficients in Weierstrass models over Hirzebruch surfaces. The resulting set of Hodge numbers includes those of all known Calabi-Yau threefold…
We compare the sets of Calabi-Yau threefolds with large Hodge numbers that are constructed using toric hypersurface methods with those can be constructed as elliptic fibrations using Weierstrass model techniques motivated by F-theory. There is a close correspondence between the structure of "tops" in the toric polytope construction and Tate form tunings of Weierstrass models for elliptic fibrations. We find that all of the Hodge number pairs ($h^{1, 1},h^{2, 1}$) with $h^{1,1}$ or $h^{2, 1}\geq 240$ that are associated with threefolds in the Kreuzer-Skarke database can be realized explicitly b…
We study holomorphic 2-forms on projective (or compact Kaehler) threefolds not of general type and prove that in almost all cases the 2-form is created by some standard process. This means roughly that every 2-form is induced by a meromorphic map to a surface, a torus or a symplectic manifold. If the 2-form has only finitely many zeroes, more precise results hold. Finally we prove that compact Kaehler threefolds with negative Kodaira dimension are uniruled unles they are simple, i.e. there is no positive dimensional compact subvariety through the general point of the manifold (these simple thr…
We construct a gauged linear sigma model with two non-birational Kälher phases which we prove to be derived equivalent, $\mathbb{L}$-equivalent, deformation equivalent and Hodge equivalent. This provides a new counterexample to the birational Torelli problem which admits a simple GLSM interpretation.
The Grassmannian Gr(2,5) is embedded in $\Bbb{P}^9$ via the Plücker embedding. The intersection of two general PGL(10)-translates of Gr(2,5) is a Calabi-Yau 3-fold X, and the intersection of the projective duals of the two translates is another Calabi-Yau 3-fold Y, deformation equivalent to X. Applying results of Kuznetsov and Jiang-Leung-Xie shows that X and Y are derived equivalent, which by a result of Addington implies that their third cohomology groups are isomorphic as polarised Hodge structures. We show that X and Y provide counterexamples to a certain "birational" Torelli statement for…
This proceedings note introduces aspects of the authors' work relating mirror symmetry and integral variations of Hodge structure. The emphasis is on their classification of the integral variations of Hodge structure which can underly families of Calabi-Yau threefolds over the thrice-punctured sphere with b^3 = 4, or equivalently h^{2,1} = 1, and the related issues of geometric realization of these variations. The presentation parallels that of the first author's talk at the BIRS workshop.
We prove a formula for the Hodge numbers of square-free divisors of Calabi-Yau threefold hypersurfaces in toric varieties. Euclidean branes wrapping divisors affect the vacuum structure of Calabi-Yau compactifications of type IIB string theory, M-theory, and F-theory. Determining the nonperturbative couplings due to Euclidean branes on a divisor $D$ requires counting fermion zero modes, which depend on the Hodge numbers $h^i({\cal{O}}_D)$. Suppose that $X$ is a smooth Calabi-Yau threefold hypersurface in a toric variety $V$, and let $D$ be the restriction to $X$ of a square-free divisor of $V$…
We study the moduli space of pairs $(X,H)$ consisting of a cubic threefold $X$ and a hyperplane $H$ in $\mathbb P^4$. The interest in this moduli comes from two sources: the study of certain weighted hypersurfaces whose middle cohomology admit Hodge structures of $K3$ type and, on the other hand, the study of the singularity $O_{16}$ (the cone over a cubic surface). In this paper, we give a Hodge theoretic construction of the moduli space of cubic pairs by relating $(X,H)$ to certain "lattice polarized" cubic fourfolds $Y$. A period map for the pairs $(X,H)$ is then defined using the periods o…
Suppose that $X$ is a smooth, projective threefold over $\mathbb C$ and that $ϕ: X \to X$ is an automorphism of positive entropy. We show that one of the following must hold, after replacing $ϕ$ by an iterate: i) the canonical class of $X$ is numerically trivial; ii) $ϕ$ is imprimitive; iii) $ϕ$ is not dynamically minimal. As a consequence, we show that if a smooth threefold $M$ does not admit a primitive automorphism of positive entropy, then no variety constructed by a sequence of smooth blow-ups of $M$ can admit a primitive automorphism of positive entropy. In explaining why the method does…
The hypothetical existence of a good theory of mixed motives predicts many deep phenomena related to algebraic cycles. One of these, a generalization of Bloch's conjecture says that "small Hodge diamonds" go with "small Chow groups". Voisin's method (which produces examples with small Chow groups) is analyzed carefully to widen its applicability. A threefold of general type without 1- and 2-forms is exhibited for which this extension yields Bloch's generalized conjecture.
We systematically analyze the fibration structure of toric hypersurface Calabi-Yau threefolds with large and small Hodge numbers. We show that there are only four such Calabi-Yau threefolds with $h^{1, 1} \geq 140$ or $h^{2, 1} \geq 140$ that do not have manifest elliptic or genus one fibers arising from a fibration of the associated 4D polytope. There is a genus one fibration whenever either Hodge number is 150 or greater, and an elliptic fibration when either Hodge number is 228 or greater. We find that for small $h^{1, 1}$ the fraction of polytopes in the KS database that do not have a genu…
Let X be the toric variety (P^1)^4 associated with its four-dimensional polytope. Consider a resolution of the singular Fano variety associated with the dual polytope of X. Generically, anti-canonical sections Y of X and anticanonical sections of the resolution are mirror partners in the sense of Batyrev. Our main result is the following: the Hodge-theoretic mirror of the quotient Z associated to a maximal admissible pair (Y,G) in X is not a quotient associated to an admissible pair in the resolution. Nevertheless, it is possible to construct a mirror orbifold of Z by means of a quotient of a …
We prove that there exists a pencil of Enriques surfaces defined over $\mathbb{Q}$ with non-algebraic integral Hodge classes of non-torsion type. This gives the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question of Murre on the universality of the Abel-Jacobi maps in codimension three.
We construct examples of primitive contractions of Calabi--Yau threefolds with exceptional locus being $ \mathbb{P}^1 \times \mathbb{P}^1$, $\mathbb{P}^2$, and smooth del Pezzo surfaces of degrees $\leq 5$. We describe the images of these primitive contractions and find their smoothing families. In particular, we give a method to compute the Hodge numbers of a generic fiber of the smoothing family of each Calabi--Yau threefold with one isolated singularity obtained after a primitive contraction of type II. As an application, we get examples of natural conifold transitions between some families…
We use $L^2$-Higgs cohomology to determine the Hodge numbers of the parabolic cohomology $H^1(\bar S, j_*\V)$, where the local system $\V$ arises from the third primitive cohomology of family of Calabi-Yau threefolds over a curve $\bar S$. The method gives a way to predict the presence of algebraic 2-cycles in the total space of the family and is applied to some examples.
For projective conifold transitions between Calabi-Yau threefolds $X$ and $Y$, with $X$ close to $Y$ in the moduli, we show that the combined information provided by the $A$ model (Gromov--Witten theory in all genera) and $B$ model (variation of Hodge structures) on $X$, linked along the vanishing cycles, determines the corresponding combined information on $Y$. Similar result holds in the reverse direction when linked with the exceptional curves.
Building upon the Bloch-Kato conjecture in Milnor K-theory, we relate the third unramified cohomology group with Q/Z coefficients with a group which measures the failure of the integral Hodge conjecture in degree 4. As a first consequence, a geometric theorem of the second-named author implies that the third unramified cohomology group with Q/Z coefficients vanishes on all uniruled threefolds. As a second consequence, a 1989 example by Ojanguren and the first named author implies that the integral Hodge conjecture in degree 4 fails for unirational varieties of dimension at least 6. For certain…
We study threefolds fibred by Kummer surfaces associated to products of elliptic curves, that arise as resolved quotients of threefolds fibred by certain lattice polarized K3 surfaces under a fibrewise Nikulin involution. We present a general construction for such surfaces, before specializing our results to study Calabi-Yau threefolds arising as resolved quotients of threefolds fibred by mirror quartic K3 surfaces. Finally, we give some geometric properties of the Calabi-Yau threefolds that we have constructed, including expressions for Hodge numbers.
