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【期刊论文】Shafarevich's Conjecture for CY Manifolds Ⅰ (Moduli of CY Manifolds)
刘克峰, Kefeng Liu, Andrey Todorov, Shing-Tung Yau, Kang Zuo
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-1年11月30日
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 Teichmuller 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.
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【期刊论文】Heat Kernels, Symplectic Geometry, Moduli Spaces and Finite Groups
刘克峰, Kefeng Liu
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-1年11月30日
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【期刊论文】GEOMETRIC ASPECTS OF THE MODULI SPACE OF RIEMANN SURFACES
刘克峰, KEFENG LIU, XIAOFENG SUN, AND SHING-TUNG YAU
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-1年11月30日
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【期刊论文】Adiabatic Limits and Foliations
刘克峰, Kefeng Liu and Weiping Zhang
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-1年11月30日
We use adiabatic limits to study foliated manifolds. The Bott connection naturally shows up as the adiabatic limit of Levi-Civita connections. As an application, we then construct certain natural elliptic operators associated to the foliation and present a direct geometric proof of a vanshing theorem of Connes[Co], which extends the Lichnerowicz vanishing theorem [L] to foliated manifolds with spin leaves, for what we call almost Riemannian foliations. Several new vanishing theorems are also proved by using our method.
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