Valentino Tosatti
We survey some recent developments on the problem of understanding degenerations of Calabi-Yau manifolds equipped with their Ricci-flat Kahler metrics, with an emphasis on the case when the metrics are volume collapsing.
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This note is a report on the observation that some singular varieties admit Calabi--Yau coverings. As an application, we construct 18 new Calabi--Yau 3-folds with Picard number one that have some interesting properties.
Uniform Skoda integrability and Calabi-Yau degeneration
本文研究 Calabi-Yau 流形的极化代数退化，通过证明一致 Skoda 型估计和一致 $L^\infty$-估计作出贡献。
This is a survey article of the recent progresses on the metric behaviour of Ricci-flat Kähler-Einstein metrics along degenerations of Calabi-Yau manifolds.
We use the gluing method to give a refined description of the collapsing Calabi-Yau metrics on Calabi-Yau 3-folds admitting a Lefschetz K3 fibration.
Collapsing Calabi-Yau fibrations and uniform diameter bounds
本文研究沿黎曼曲面全纯纤维化的 Calabi-Yau 度量坍缩，在典范奇异纤维假设下证明一致直径界，并探讨奇异纤维附近的几何后果。
The concept of non-Gorenstein involutions on Calabi-Yau threefolds is a higher dimensional generalization of non-symplectic involutions on $K3$ surfaces. We present some elementary facts about Calabi-Yau threefolds with non-Gorenstein involutions. We give a classification of the Calabi-Yau threefolds of Picard rank one with non-Gorenstein involutions whose fixed locus is not zero-dimensional.
This note studies the equivalencies among convergences of Ricci-flat Kähler-Einstein metrics on Calabi-Yau manifolds, cohomology classes and potential functions.
We prove that the only Calabi--Yau projective manifolds which bear holomorphic Cartan geometries are precisely the abelian varieties. (Nous démontrons que les seules variétés projectives de Calabi--Yau qui possèdent des géométrie holomorphes de Cartan sont les variétés abéliennes.)
In this paper, the relationship between the existence of special lagrangian submanifolds and the collapsing of Calabi-Yau manifolds is studied. First, special lagrangian fibrations are constructed on some regions of bounded curvature and sufficiently collapsed in Ricci-flat Calabi-Yau manifolds. Then, in the opposite direction,it is shown that the existence of special lagrangian submanifolds with small volume implies the collapsing of some regions in the ambient Calabi-Yau manifolds.
This paper is a sequel to arXiv:1012.2940. We further investigate the Gromov-Hausdorff convergence of Ricci-flat Kähler metrics under degenerations of Calabi-Yau manifolds. We extend Theorem 1.1 in arXiv:1012.2940 by removing the condition on existence of crepant resolutions for Calabi-Yau varieties.
We construct a family of Calabi-Yau metrics on $\C^3$ with properties analogous to the Taub-NUT metric on $\C^2$, and construct a family of Calabi-Yau 3-fold metric models on the positive and negative vertices of SYZ fibrations with properties analogous to the Ooguri-Vafa metric.
We construct non-Kähler simply connected Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers by smoothing normal crossing varieties with trivial dualizing sheaves.
We investigate a method of construction of Calabi--Yau manifolds, that is, by smoothing normal crossing varieties. We develop some theories for calculating the Picard groups of the Calabi--Yau manifolds obtained in this method. Some applications are included, such as construction of new examples of Calabi--Yau 3-folds with Picard number one with some interesting properties.
Sheng and Zuo's characteristic forms are invariants of a variation of Hodge structure. We show that they characterize Gross's canonical variations of Hodge structure of Calabi-Yau type over (Hermitian symmetric) tube domains.
New examples of affine Calabi-Yau 3-folds with maximal volume growth
本文构造了完整的Calabi-Yau度量的例子，这些例子是三维Calabi-Yau锥的平滑，不是低维Calabi-Yau锥的产品，并且在顶点外有orbifold奇点。
In this paper, we construct simply connected symplectic Calabi-Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along $T^4$. Using our construction, we also produce symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group $\Z$. In this paper, we also produce the first examples of simply connected symplectic Calabi-Yau and non-Calabi-Yau 6-manifolds via coisotropic Luttinger surgery.
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 prove a case of the conjecture of Douglas, Reinbacher and Yau about the existence of stable vector bundles with prescribed Chern classes on a Calabi-Yau threefold. For this purpose we prove the existence of certain stable vector bundle extensions over elliptically fibered Calabi-Yau threefolds.
We improve Gross-Wilson's local estimates to global ones. As an application, we study the blow-up limits of the degenerating Calabi-Yau metrics on singular fibers.
Complete Calabi-Yau metrics from smoothing Calabi-Yau complete intersections
本文在非紧流形上构造完备Calabi-Yau度量，推广了Székelyhidi的工作，所得流形在无穷远处的切锥为$\mathbb{C} \times V_0$，且度量纤维具有变化复结构和可能的孤立奇点。
In this paper we present a construction of stable bundles on Calabi-Yau threefolds using the method of bundle extensions. This construction applies to any given Calabi-Yau threefold with h^{1,1}>1. We give examples of stable bundles of rank 2 and 4 constructed out of pure geometric data of the given Calabi-Yau space. As an application, we find that some of these bundles satisfy the physical constraint imposed by heterotic string anomaly cancellation.
We survey our recent new results on the geometry of Teichmuller and moduli spaces of Riemann surfaces and Calabi-Yau manifolds.
Calabi-Yau metrics of Calabi type with polynomial rate of convergence
本文构造了定义在光滑反典范除子补集上、以多项式速度逼近Calabi模型空间的完备Calabi-Yau度量，并证明了其在固定上同调类中的唯一性。
In this paper, we study the convergence of Calabi-Yau manifolds under Kähler degeneration to orbifold singularities and complex degeneration to canonical singularities (including the conifold singularities), and the collapsing of a family of Calabi-Yau manifolds.
We study Calabi-Yau 3-folds M_0 with a conical singularity x modelled on a Calabi-Yau cone V. We construct desingularizations of M_0, obtaining a 1-parameter family of compact, nonsingular Calabi-Yau 3-folds which has M_0 as the limit. The way we do is to choose an Asymptotically Conical Calabi-Yau 3-fold Y modelled on the same cone V, and then glue into M_0 at x after applying a homothety to Y. We then get a 1-parameter family of nearly Calabi-Yau 3-folds M_t depending on a small real variable t. For sufficiently small t, we show that the nearly Calabi-Yau structures on M_t can be deformed to…
We first proved a compactness theorem of the Kähler metrics, which confirms a prediction of Chen. Then we prove several eigenvalue estimates along the Calabi flow. Combining the compactness theorem and these eigenvalue estimates, we generalize the method developed by Chen-Li-Wang to prove the small energy theorems of the Calabi flow.
A Gluing Theorem For Collapsing Warped-QAC Calabi-Yau Manifolds
本文通过粘合构造研究 $\CC^{n+2}$ 中坍缩扭曲 QAC Calabi–Yau 流形，验证了 Yang Li 关于奇异纤维分离时度量行为的猜想，并刻画其气泡树结构。
In this paper, by applying Greene-Shapere-Vafa-Yau semi-flat metric, we give a new proof of closed formula of Weil-Petersson metric on moduli space of Calabi-Yau varieties.
A Cheng-Yau type estimate for the symplectic Calabi-Yau equation
本文在Donaldson猜想的辛4-流形Calabi-Yau方程框架下，证明了一个在Kähler情形中与Cheng-Yau经典估计相似的先验估计。
In this paper we introduce a new equation on the compact Kahler manifolds. Solution of this equation corresponds to the Calabi-Yau metric. New equation differs from the Monge--Ampere equation considered by Calabi and Yau.
One of the main results of the paper arXiv:1108.0967 by Gross-Tosatti-Zhang establishes estimates on the collapsing of Ricci-flat Kahler metrics on holomorphic torus fibrations. We remove a projectivity assumption from these estimates and simplify some of the underlying analysis.
In this paper we state an analog of Calabi's conjecture proved by Yau. The difference with the classical case is that we propose deformation of the complex structure, whereas the complex Monge--Ampère equation describes deformation of the Kähler (symplectic) structure.
We develop some techniques to study the adiabatic limiting behaviour of Calabi-Yau metrics on the total space of a fibration, and obtain strong control near the singular fibres by imposing restrictions on the singularity types. We prove a uniform lower bound on the metric up to the singular fibre, under fairly general hypotheses. Assuming a result in pluripotential theory, we prove a uniform fibre diameter bound for a Lefschetz K3 fibred Calabi-Yau 3-fold, which reduces the study of the collapsing metric to a locally non-collapsed situation, and we identify the Gromov-Hausdorff limit of the re…
Complex structure degenerations and collapsing of Calabi-Yau metrics
研究极化Calabi-Yau流形退化族的度量坍缩问题，针对射影空间中Calabi-Yau超曲面退化为两Fano超曲面横截并的情形，用粘合与奇异摄动法构造环面对称Kähler度量，建立度量坍缩与复结构退化的精确关系，全维结果含复二维情形均为新发现，并探讨推广可能。
Gromov-Hausdorff limits of collapsing Calabi-Yau fibrations
本研究探讨射影流形上收敛于半充足类的Calabi-Yau度量的Gromov-Hausdorff极限，证明其与纤维化基空间同胚且判别轨迹余维数至少为2，解决了Tosatti的相关猜想。
This is a short expository note about Calabi-Yau manifolds and degenerations of their Ricci-flat metrics.
We prove that the only complex parabolic geometries on Calabi-Yau manifolds are the homogeneous geometries on complex tori. We also classify the complex parabolic geometries on homogeneous compact Kähler manifolds.
In this paper we construct all smooth torus fibres of the generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties near the large complex limit.
We develop some consequences of the connection between Calabi-Yau structures and torsion-free $G_2$ structures on compact and asymptotically cylindrical six- and seven-dimensional manifolds. Firstly, we improve the known proof that matching asymptotically cylindrical Calabi-Yau threefolds can be glued. Secondly, we give an alternative proof that the moduli space of Calabi-Yau structures on a six-dimensional real manifold is smooth, and extend it to the asymptotically cylindrical case. Finally, we prove that the gluing map of Calabi-Yau threefolds, extended between these moduli spaces, is a loc…
In this paper we prove the existence and uniqueness of the form-type Calabi-Yau equation on Kähler manifolds of nonnegative orthogonal bisectional curvature.
Dealing with the generalized Calabi-Yau equation proposed by Gromov on closed almost-Kähler manifolds, we extend to arbitrary dimension a non-existence result proved in complex dimension 2.
We consider a generalization of Calabi-Yau structures in the context of $α$-Sasakian manifolds. We study deformations of a special class of Legendrian submanifolds and classify invariant contact Calabi-Yau structures on 5-dimensional nilmanifolds. Finally we generalize to codimension $r$.
A generalized Calabi-Yau structure is a geometrical structure on a manifold which generalizes both the concept of the Calabi-Yau structure and that of the symplectic one. In view of a result of Lin and Tolman in generalized complex cases, we introduce in this paper the notion of a generalized moment map for a compact Lie group action on a generalized Calabi-Yau manifold and construct a reduced generalized Calabi-Yau structure on the reduced space. As an application, we show some relationship between generalized moment maps and the Bergman kernels, and prove the Duistermaat-Heckman formula for …
The form-type Calabi-Yau equation on a class of complex manifolds
该研究针对一类复流形上的形式型Calabi-Yau方程，通过定义astheno-Ricci曲率，证明了该曲率非正时方程存在解，推进了复几何中相关方程的可解性研究。
We study and construct non-abelian hermitian Yang-Mills (HYM) instantons on Calabi-Yau cones. By means of a particular isometry preserving ansatz, the HYM equations are reduced to a novel Higgs-Yang-Mills flow on the Einstein-Kahler base. For any 2d-dimensional Calabi-Yau cone, we find explicit solutions of the flow equations that correspond to non-trivial SU(d) HYM instantons. These can be regarded as deformations of the spin connection of the Calabi-Yau cone.
We show that Calabi-Yau manifolds are emergent from the commutative limit of six-dimensional noncommutative Hermitian U(1) instantons. Therefore we argue that the noncommutative Hermitian U(1) instantons correspond to quantized Calabi-Yau manifolds.
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