Calabi-Yau hypersurfaces in products of semi-ample surfaces

We study Calabi-Yau manifolds that are embedded as hypersurfaces in products of semi-ample complex surfaces. We classify the deformation classes of the latter and thereby achieve a classification of the Calabi-Yau manifolds that are constructed in this way. Complementing the results in the existing literature, we obtain the complete Hodge diamond for all Calabi-Yau hypersurfaces in products of semi-ample surfaces.     

Cauchy–Schwarz-type inequalities on Kähler manifolds

In this note we investigate Cauchy–Schwarz-type inequalities for cohomology elements on compact Kähler manifolds, which can be viewed as generalizations of a classical case. We obtain, as a corollary, some Chern number inequalities when the Hodge numbers of Kähler manifolds satisfy certain restrictions. The same argument can also be applied to compact quaternion-Kähler manifolds with positive scalar curvature to obtain a similar result.     

On the number of complete intersection Calabi-Yau manifolds

The intersection numbers and the action of the Pontryagin class on the integral cohomology are used to distinguish between the many CICY manifolds that have the same Hodge numbers. It is shown by examining manifolds embedded in fewer than six projective spaces that at least 2590 of the manifolds are distinct.     

Searching for K3 fibrations

We present two methods for studying fibrations of Calabi-Yau manifolds embedded in toric varieties described by single weight systems. We analyze 184 026 such spaces and identify among them the 124 701 which are K3 fibrations. As some of the weights give rise to two or three distinct types of fibrations, the total number we find is 167 406. With our methods one can also study elliptic fibrations of 3-folds and K3 surfaces. We also calculate the Hodge numbers of the 3-folds obtaining more than three times as many as were previously known.     

Non-Commutative Periods and Mirror Symmetry¶in Higher Dimensions

We study an analog for higher-dimensional Calabi–Yau manifolds of the standard predictions of Mirror Symmetry. We introduce periods associated with “non-commutative” deformations of Calabi–Yau manifolds. These periods define a map on the moduli space of such deformations which is a local isomorphism. Using these non-commutative periods we introduce invariants of variations of semi-infinite generalized Hodge structures living over the moduli space ℳ. It is shown that the generating function of such invariants satisfies the system of WDVV-equations exactly as in the case of Gromov–Witten invariants. We prove that the total collection of rational Gromov–Witten invariants of complete intersection Calabi–Yau manifold can be identified with the collection of invariants of variations of generalized (semi-infinite) Hodge structures attached to the mirror variety. The basic technical tool utilized is the deformation theory. Received: 6 April 2000 / Accepted: 15 January 2002     

An Abundance of K3 Fibrations from Polyhedra with Interchangeable Parts

Even a cursory inspection of the Hodge plot associated with Calabi-Yau threefolds that are hypersurfaces in toric varieties reveals striking structures. These patterns correspond to webs of elliptic-K3 fibrations whose mirror images are also elliptic-K3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along slices corresponding to the K3 fibers. Any two half-polytopes over a given slice can be combined into a reflexive polytope. This fact, together with a remarkable relation on the additivity of Hodge numbers, explains much of the structure of the observed patterns.     

Calabi-Yau 4-folds and toric fibrations

We present a general scheme for identifying fibrations in the framework of toric geometry and provide a large list of weights for Calabi-Yau 4-folds. We find 914 164 weights with degree d ≤ 150 whose maximal Newton polyhedra are reflexive and 525 572 weights with degree d ≤ 4000 that give rise to weighted projective spaces such that the polynomial defining a hypersurface of trivial canonical class is transversal. We compute all Hodge numbers, using Batyrev's formulas (derived by toric methods) for the first and Vafa's formulas (obtained by counting of Ramond ground states in N = 2 LG models) for the latter class, checking their consistency for the 109 308 weights in the overlap. Fibrations of k-folds, including the elliptic case, manifest themselves in the N lattice in the following simple way: The polyhedron corresponding to the fiber is a subpolyhedron of that corresponding to the k-fold, whereas the fan determining the base is a linear projection of the fan corresponding to the k-fold.     

Exact Computation of the Special Geometry for Calabi–Yau Hypersurfaces of Fermat Type

JETP Letters - We continue to develop our method for effectively computing the special Kähler geometry on the moduli space of Calabi–Yau manifolds. We generalize it to all polynomial...     

Virasoro constraints and polynomial recursion for the linear Hodge integrals

The Hodge tau-function is a generating function for the linear Hodge integrals. It is also a tau-function of the KP hierarchy. In this paper, we first present the Virasoro constraints for the Hodge tau-function in the explicit form of the Virasoro equations. The expression of our Virasoro constraints is simply a linear combination of the Virasoro operators, where the coefficients are restored from a power series for the Lambert W function. Then, using this result, we deduce a simple version of the Virasoro constraints for the linear Hodge partition function, where the coefficients are restored from the Gamma function. Finally, we establish the equivalence relation between the Virasoro constraints and polynomial recursion formula for the linear Hodge integrals.     

Variational equations on mixed Riemannian–Lorentzian metrics

A class of elliptic–hyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes in this way is the extended projective disc, which is Riemannian at ordinary points, Lorentzian at ideal points, and singular on the absolute. Harmonic fields on such a metric can be interpreted as the hodograph image of extremal surfaces in Minkowski 3-space. This suggests an approach to generalized Plateau problems in three-dimensional space-time via Hodge theory on the extended projective disc. Analogous variational problems arise on Riemannian–Lorentzian flow metrics in fiber bundles (twisted nonlinear Hodge equations), and on certain Riemannian–Lorentzian manifolds which occur in relativity and quantum cosmology. The examples surveyed come with natural gauge theories and Hodge dualities. This paper is mainly a review, but some technical extensions are proven.     

A new approach for controlling chaotic dynamical systems

The chaos control method proposed by Ott and his coworkers and now called the OGY method has attracted much attention. However, in some applications this technique requires a very long time until the control applies and it is not so effective. In this Letter, we present a new chaos control method in which this problem is improved. The main difference from the OGY method is the use of nonlinear approximations for chaotic dynamical systems and stable manifolds of the target points. We give an example for the Hénon map to demonstrate the effectiveness of the present method. Our method is also shown to be robust to modeling errors like the OGY method.     

Covariant background field calculation of ultraviolet counterterms for the nonlinear sigma model in three dimensions

Based on the covariant background field method, we calculate the ultraviolet counterterms up to two-loop order and discuss the renormalizability of the three-dimensional non-linear sigma models with arbitrary Riemannian manifolds as target spaces. We investigate the bosonic model and its supersymmetric extension. We show that at the one-loop level these models are renormalizable and even finite when the manifolds are Ricci-flat. However, at the two-loop order, we find non-renormalizable counterterms in all cases considered, so the renormalizability and finiteness of such models are completely lost in this order.     

Exact wave functions of two-electron quantum rings

We demonstrate that the Schr?dinger equation for two electrons on a ring, which is the usual paradigm to model quantum rings, is solvable in closed form for particular values of the radius. We show that both polynomial and irrational solutions can be found for any value of the angular momentum and that the singlet and triplet manifolds, which are degenerate, have distinct geometric phases. We also study the nodal structure associated with these two-electron states.     

Betti Numbers of a Class of Barely <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> Manifolds

We calculate explicitly the Betti numbers of a class of barely G ₂ manifolds - that is, G ₂ manifolds that are realised as a product of a Calabi-Yau manifold and a circle, modulo an involution. The particular class which we consider are those spaces where the Calabi-Yau manifolds are complete intersections of hypersurfaces in products of complex projective spaces from which they inherit all their (1, 1)-cohomology and the involutions are free acting.     

A Construction of Frobenius Manifolds with Logarithmic Poles and Applications

A construction theorem for Frobenius manifolds with logarithmic poles is established. This is a generalization of a theorem of Hertling and Manin. As an application we prove a partial generalization of the reconstruction theorem of Kontsevich and Manin for projective smooth varieties with convergent Gromov-Witten potential. A second application is a construction of Frobenius manifolds out of a variation of polarized Hodge structures which degenerates along a normal crossing divisor when certain generation conditions are fulfilled.     

Invariant Manifolds for Impulsive Equations and Nonuniform Polynomial Dichotomies

For impulsive differential equations in Banach spaces, we construct stable and unstable invariant manifolds for sufficiently small perturbations of a polynomial dichotomy. We also consider the general case of nonuniform polynomial dichotomies. Moreover, we introduce the notions of polynomial Lyapunov exponent and of regularity coefficient for a linear impulsive differential equation, and we show that when the Lyapunov exponent never vanishes the linear equation admits a nonuniform polynomial dichotomy.     

Polynomial modular Frobenius manifolds

The moduli space of Frobenius manifolds carries a natural involutive symmetry, and a distinguished class–so-called modular Frobenius manifolds–lie at the fixed points of this symmetry. In this paper a classification of semi-simple modular Frobenius manifolds which are polynomial in all but one of the variables is begun, and completed for three and four dimensional manifolds. The resulting examples may also be obtained from higher dimensional manifolds by a process of folding. The relationship of these results with orbifold quantum cohomology is also discussed.     

Toric complete intersections and weighted projective space

It has been shown by Batyrev and Borisov that nef partitions of reflexive polyhedra can be used to construct mirror pairs of complete intersection Calabi–Yau manifolds in toric ambient spaces. We construct a number of such spaces and compute their cohomological data. We also discuss the relation of our results to complete intersections in weighted projective spaces and try to recover them as special cases of the toric construction. As compared to hypersurfaces, codimension two more than doubles the number of spectra with h¹¹=1. Altogether we find 87 new (mirror pairs of) Hodge data, mainly with h¹¹≤4.     

Quantization in the presence of Wess-Zumino terms

We show that for large classes of manifolds in odd space-time dimensions Wess-Zumino terms are equivalent to gauge invariant mass terms. As a consequence for quantum mechanical systems the effect of these terms is to restrict the Hilbert space through Gauss' law. We apply our method to obtain the general quantization rules for skyrmions with various baryon numbers.