arXiv subjects
Publications and source records attributed to Andrey Todorov.
We describe the proof that the period map from the Torelli space of Calabi-Yau manifolds to the classifying space of polarized Hodge structures is an embedding. The proof is based on the constructions of holomorphic affine structure on the Teichmüller space and Hodge metric completion of the Torelli space. A canonical global holomorphic section of the holomorphic $(n, 0)$ class on the Teichmüller space is constructed.
The result of this paper is proved in arXiv:1112.1163
We consider fibrations by abelian surfaces and K3 surfaces over a one dimensional base that are Calabi-Yau and we obtain dual fibrations that are derived equivalent to the original fibration. Finally, we relate the problem to mirror symmetry.
Given X a K3 surface, a mirror dual to X can be identified with a component of the moduli space of semistable sheaves on X. We consider fibrations by K3 surfaces over a one dimensional base that are Calabi-Yau and we obtain a dual fibration that turns to be derived equivalent to the original one and we relate the problem to mirror symmetry.
We give a simplified and more algebraic proof of the finiteness of the families of Calabi-Yau n-folds with non-vanishing of Yukawa-coupling over a fixed base curve and with fixed degeneration locus. We also give a generalization of this result. Our method is variation of Hodge structure and poly-stability of Higgs bundles.
In this paper we construct the analogue of Dedekind eta-function on the moduli space of polarized CY manifolds. We prove that the L-two norm of eta is the regularized determinants of the Laplacians of the CY metric on (0,1) forms. We construct the analogue of the Baily-Borel Compactification of the moduli space of polarized CY and prove that it has the same properties as the Baily-Borel compactification of the locally symmetric Hermitian spaces. We proved that the compactification constructed in the paper is the minimal.
In this paper it is proved that the volumes of the moduli spaces of polarized CY manifolds with respect to the Weil-Petersson metrics are finite and they are rational numbers.
In this paper we will use our results about the local deformation theory of Calabi-Yau manifolds to define a Higgs field on the tangent bundle of the moduli space of CY threefolds. Combining this Higgs field with the Levi-Chevitta connection of the Weil-Petersson metrics on the moduli space of three dimensional CY manifolds, we construct a new $Sp(2h^{2,1}% ,\mathbb{R)}$ connection, following the ideas of Cecotti and Vafa. We prove that this new connection is a flat connection. Using this flat connection, we apply the scheme of geometric quantization introduced by Axelrod, Della Pietra and Witten to the tangent bundle of the moduli space of three dimensional CY manifolds. By modifying the calculations of E. Witten done in 1993 to the tangent bundle of the moduli space of CY threefolds, we derive again the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vafa. We also introduced a Z structure on the tangent bundle of the moduli space of polarized CY threefolds by using the flat symplectic connection.
The Dedekind eta functions plays important role in different branches of Mathematics and Theoretical Physics. One way to construct Dedekind Eta function to use the explicit formula (Kroncker limit formula) for the regularized determinants of the Laplacian of the flat metric acting of (0,1) forms on elliptic curves. The holomorphic part of the regularized determinant is the Dedekind eta functions. In this paper we generalized the above approach to the case of K3 surfaces. We give an explicit formula of the regularized determinants of the Laplacians of Calabi Yau metrics on K3 Surfaces, following suggestions by R. Borcherds. The holomorphic part of the regularized determinants will be the higher dimensional analogue of Dedekind Eta function. We give explicit formulas for the number of non singular rational curves with a fixed volume with respect to a Hodge metric in the case of K3 surfaces with Picard group unimodular even lattice by using the holomorphic part exp_{3,19} of the regularized determinants det_{(0,1)}. We gave the combinatorial interpretation of the restriction of the automorphic form exp_{3,19} on the moduli of K3 surfaces with unimodular Picard lattices in the A and B models. The results obtained in this paper are related to some results of Bershadsky, Cecotti, Ouguri and Vafa. See BCOV.
This is the first of a series of articles in which we are going to study the regularized determinants of the Laplacians of Calabi Yau metrics acting on (0,q) forms on the moduli space of CY manifolds with a fixed polarization. It is well known that in case of the elliptic curves the Kronecker limit formula gives an explicit formula for the regularized determinants of the flat metrics with fixed volume on the elliptic curves. The following formula holds in this case; the regularized determinant is the product of the imaginary part of the complex number in the Siegel upper half plane with the Dedekind eta function. It is well known fact that the Dedekind eta function in power 24 is a cusp automorphic form of weight 12 related to the discriminant of the elliptic curve. Thus we can view that the regularized determinant is the norm of a section of some power of the line bundle of the classes of cohomologies of (1,0) forms of the elliptic curves over its moduli space. Our purpose is to generalize this fact in the case of CY manifolds. In this paper we will establish the local analogue of the Kronecker limit formula for CY manifolds.
Based on the work of Borcherds we construct on the moduli space of K3 surfaces with B-field an automorphic form exp_{4,20} which vanishes on the totally geodesic subspaces orthogonal to -2 vectors of the even, unimodular lattice of signature (4,20). We give an explicit formula of the regularized determinants of the Laplacians of Calabi Yau metrics on K3 Surfaces, following suggestions by R. Borcherds. The holomorphic part of the regularized determinants will be the higher dimensional analogue of Dedekind Eta function. We give explicit formulas for the number of non singular rational curves with a fixed volume with respect to a Hodge metric in the case of K3 surfaces with Picard group unimodular even lattice. The counting of rational curves on special K3 surfaces using the regularized determinants of the Laplacian of CY metrics is related to some results of Bershadsky, Cecotti, Ouguri and Vafa.
In this paper we are begining to explore the complex counterpart of the Chern-Simon-Witten theory. We define the complex analogue of the Gauss linking number for complex curves embedded in a Calabi-Yau threefold using the formal path integral that leads to a rigorous mathematical expression. We give an analytic and geometric interpretation of our holomorphic linking following the parallel with the real case. We show in particular that the Green kernel that appears in the explicit integral for the Gauss linking number is replaced by the Bochner-Martinelli kernel. We also find canonical expressions of the holomorphic linking using the Grothendieck-Serre duality in local cohomology, the latter admits a generalization for an arbitrary field.
In this paper we first study the moduli spaces related to Calabi-Yau manifolds. We then apply the results to the following problem. Let $C$ be a fixed Riemann surface with fixed finite number of points on it. Given a CY manifold with fixed topological type, we consider the set of all families of CY manifolds of the fixed topological type over $C$ with degenerate fibres over the fixed points up to isomorphism. This set is called Shafarevich set. The analogue of Shafarevich conjecture for CY manifolds is for which topological types of CY the Shafarevich set is finite. It is well-known that the analogue of Shafarevich conjecture is closely related to the study of the moduli space of polarized CY manifolds and the moduli space of the maps of fixed Riemann surface to the coarse moduli space of the CY manifolds. We prove the existence of the Teichmüller space of CY manifolds together with a universal family of marked CY manifolds. From this result we derive the existence of a finite cover of the coarse moduli space which is a non-singular quasi-projective manifold. Over this cover we construct a family of polarized CY manifolds. We study the moduli space of maps of the fixed Riemann with fixed points on it to the moduli space of CY manifolds constructed in the paper such that the maps map the fixed points on the Riemann surface to the discriminant locus. If this moduli space of maps is finite then Shafarevich conjecture holds. We relate the analogue of Shafarevich problem to the non-vanishing of the Yukawa coupling. We give also a counter example of the Shafarevich problem for a class of CY manifolds.
In this paper the relations between the existence of Lagrangian fibration of Hyper-Kähler manifolds and the existence of the Large Radius Limit is established. It is proved that if the the rank of the second homology group of a Hyper-Kähler manifold N of complex dimension $2n\geq4$ is at least 5, then there exists an unipotent element T in the mapping class group $Γ$(N) such that its action on the second cohomology group satisfies $(T-id)^{2}\neq0$ and $(T-id)^{3}=0.$ A Theorem of Verbitsky implies that the symmetric power $S^{n}(T)$ acts on $H^{2n}$ and it satisfies $(S^{n}% (T)-id)^{2n}\neq0$ and $(S^{n}(T)-id)^{2n+1}=0.$ This fact established the existence of Large Radius Limit for Hyper-Kähler manifolds for polarized algebraic Hyper-Kähler manifolds. Using the theory of vanishing cycles it is proved that if a Hyper-Kähler manifold admits a Lagrangian fibration then the rank of the second homology group is greater than or equal to five. It is also proved that the fibre of any Lagrangian fibration of a Hyper-Kähler manifold is homological to a vanishing invariant $2n$ cycle of a maximal unipotent element acting on the middle homology. According to Clemens this vanishing invariant cycle can be realized as a torus. I conjecture that the SYZ conjecture implies finiteness of the topological types of Hyper-Kähler manifolds of fix dimension.
The computations that are suggested by String Theory in the B model requires the existence of degenerations of CY manifolds with maximum unipotent monodromy. In String Theory such a point in the moduli space is called a large radius limit (or large complex structure limit). In this paper we are going to construct one parameter families of $n$ dimensional Calabi-Yau manifolds, which are complete intersections in toric varieties and which have a monodromy operator $T$ such that (T$^{N}-id)^{n+1}=0$ but (T$^{N}-id)^{n}\neq0,$ i.e the monodromy operator is maximal unipotent.
In this article we give a general approach to the following analogue of Shafarevich's conjecture for some polarized algebraic varieties; suppose that we fix a type of an algebraic variety and look at families of such type of varieties over a fixed Riemann surface with fixed points over which we have singular varieties, then one can ask if the set of such families, up to isomorphism, is finite. In this paper we give a general approach to such types of problems. The main observation is the following; suppose that the moduli space of a fixed type of algebraic polarized variety exists and suppose that in some projective smooth compactification of the coarse moduli the discriminant divisor supports an ample one, then it is not difficult to see that this fact implies the analogue of Shafarevich's conjecture. In this article we apply this method to certain polarized algebraic K3 surfaces and also to Enriques surfaces.
In this paper we generalized the variational formulas for the determinants of the Laplacians on functions of CY metrics to forms of type (0,q) on CY manifolds. We also computed the Ray Singer Analytic torsion on CY manifolds we proved that it is bounded by a constant. In case of even dimensional CY manifolds the Ray Singer Analytic torsion is zero. The interesting case is the odd dimensional one.
In this paper we construct the analogue of Dedekind eta function for odd dimensional CY manifolds. We use the theory of determinant line bundles. We constructed a canonical holomorphic section $η^{N}$ of some power of the determinant line bundle on the moduli space of odd dimensional CY manifolds. According to Viehweg the moduli space of moduli space of polarized odd dimensional CY manifolds $\mathcal{M}(M)$ is quasi projective. According to a Theorem due to Hironaka we can find a projective smooth variety $\bar {\mathcal{M}(M)}$ such that $\bar{\mathcal{M}(M)}\backslash$ $\mathcal{M}(M)=\mathcal{D}_{\infty}$ is a divisor of normal crossings. We also showed by using Mumford's theory of metrics with logarithmic growths that the determinant line bundle can be canonically prolonged to $\bar {\mathcal{M}(M)}$ $.$ We also showed that there exists section $η$ of some power of the determinant line bundle which vanishes on $\mathcal{D}_{\infty}$ and has a Quillen norm the Ray Singer Analytic Torsion$.$ This section is that analogue of the Dedekind eta Function $η.$