We formulate a concrete geometric approximation hypothesis (Hypothesis~BB) asserting that codimension-$2$ Hodge classes on a smooth projective threefold can be realized as specializations of families whose general members are complete-intersection curves. We prove that Hypothesis~BB implies the (rational) Hodge conjecture for the threefold. We then give deformation-theoretic sufficient criteria (cohomology-vanishing and surjectivity conditions) which imply Hypothesis~BB, and we prove these criteria hold for the class of a line on a \emph{general} quintic threefold containing that line. We further formulate and prove several propositions showing that, under natural Noether--Lefschetz and unobstructedness hypotheses, Hypothesis~BB holds \emph{generically} in families of Calabi--Yau and Fano threefolds; these propositions reduce the problem to checkable conditions (normal-bundle cohomology, surjectivity of restriction maps). Finally, we include a Macaulay2 appendix with scripts to verify the required cohomology and splitting conditions in explicit examples.
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 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.
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 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 …
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…
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.
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.
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 $h^{p,q}(X) = f^{3-q,p}(Y,\mathsf{w})$.
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.
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 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 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 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…
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…
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 nonnegativity of stringy Hodge numbers of a projective variety with Gorenstein canonical singularities, which was conjectured by Batyrev. We prove that the $(p,1)$-stringy Hodge numbers are nonnegative, and for threefolds we obtain new results about the stringy Hodge diamond, which hold even when the stringy $E$-function is not a polynomial. We also use the Decomposition Theorem and mixed Hodge theory to prove Batyrev's conjecture for a class of fourfolds.
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…
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…
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…
Borcea-Voisin threefolds provided some of the first examples of mirror pairs in the Hodge-theoretic sense, but their mirror symmetry at the quantum level have not previously been shown. We prove a Givental-style quantum mirror theorem for certain Borcea-Voisin threefolds: by means of certain birational models, we show that their Gromov-Witten J-functions are related by a mirror map to solutions of the multi-parameter Picard-Fuchs equations coming from the variation of Hodge structures of their mirror partners.
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 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 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.
We compute the integral homology (including torsion), the topological K-theory, and the Hodge structure on cohomology of Calabi-Yau threefold hypersurfaces and complete intersections in Gorenstein toric Fano varieties. The methods are purely topological.
We develop a combinatorial approach to the construction of general smooth compact base surfaces that support elliptic Calabi-Yau threefolds. This extends previous analyses that have relied on toric or semi-toric structure. The resulting algorithm is used to construct all classes of such base surfaces $S$ with $h^{1, 1} (S) < 8$ and all base surfaces over which there is an elliptically fibered Calabi-Yau threefold $X$ with Hodge number $h^{2, 1} (X) \geq 150$. These two sets can be used todescribe all 6D F-theory models that have fewer than seven tensor multiplets or more than 150 neutral scala…
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 the mixed Hodge structure on the third homology group of a threefold which is the double cover of projective three-space ramified over a quartic surface with a double conic. We deal with the Torelli problem for such threefolds.
The class in the Brauer group of a quaternion algebra over a field is 2-torsion. We study the following question: Which 2-torsion elements of the Brauer group of a complex function field are representable by quaternion algebras? Using intersection theory to show that a certain cohomology class (on a smooth projective model) is the class of an algebraic cycle, we arrive at an obstruction, defined on a subgroup of the 2-torsion of the Brauer group, to representability by quaternion algebras. For the function fields of some complex threefolds, the obstruction map is computed and found to be nontr…
In this note we initiate a program to obtain global descriptions of Calabi-Yau moduli spaces, to calculate their Picard group, and to identify within that group the Hodge line bundle, and the closely-related Bagger-Witten line bundle. We do this here for several Calabi-Yau's obtained in [DW09] as crepant resolutions of the orbifold quotient of the product of three elliptic curves. In particular we verify in these cases a recent claim of [GHKSST16] by noting that a power of the Hodge line bundle is trivial -- even though in most of these cases the Picard group is infinite.
We give a Hodge-theoretic criterion for a Calabi--Yau variety to have finite Weil--Petersson distance on higher dimensional bases up to a set of codimension $\geq 2$. The main tool is variation of Hodge structures and variation of mixed Hodge structures. We also give a description on the codimension 2 locus for the moduli space of Calabi--Yau threefolds. We prove that the points lying on exactly one finite and one infinite divisor have infinite Weil--Petersson distance along angular slices. Finally, by giving a classification of the dominant term of the candidates of the Weil--Petersson potent…
Limiting Hodge Structures for Cubic Hypersurfaces of Secant Type
Mathematical School of Science, Xiamen University（厦门大学数学科学学院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
In this paper we describe a connection between realizations of the action of the motivic Galois group on the motivic fundamental groups of Gaussian and Eisenstein elliptic curves punctured at the $\mathfrak{p}$-torsion points, $π_1^{\rm Mot}(E-E[\mathfrak{p}],v_0)$, and the geometry of the Bianchi hyperbolic threefolds $Γ_1(\mathfrak{p})\setminus\mathbb{H}^3$, where $Γ_1(\mathfrak{p})$ is a congruence subgroup of ${\rm GL}_2({\rm End}(E))$. The first instance of such a connection was found by A.Goncharov (arXiv:math/0510310). In particular, we study the Hodge realization of the image of the ab…
This is a short review of recent constructions of new Calabi-Yau threefolds with small Hodge numbers and/or non-trivial fundamental group, which are of particular interest for model-building in the context of heterotic string theory. The two main tools are topological transitions and taking quotients by actions of discrete groups. Both of these techniques can produce new manifolds from existing ones, and they have been used to bring many new specimens to the previously sparse corner of the Calabi-Yau zoo where both Hodge numbers are small. Two new manifolds are also obtained here from hypercon…
We prove a representation-theoretic version of Borisov-Batyrev mirror symmetry, and use it to construct infinitely many new pairs of orbifolds with mirror Hodge diamonds, with respect to the usual Hodge structure on singular complex cohomology. We conjecture that the corresponding orbifold Hodge diamonds are also mirror. When $X$ is the Fermat quintic in $¶^4$, and $\tilde{X}^*$ is a $\Sym_5$-equivariant, toric resolution of its mirror $X^*$, we deduce that for any subgroup $Γ$ of the alternating group $A_5$, the $Γ$-Hilbert schemes $Γ$-$\Hilb(X)$ and $Γ$-$\Hilb(\tilde{X}^*)$ are smooth Calabi…
