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      Canonical metric on moduli spaces of log Calabi-Yau varieties

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          Abstract

          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.

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          Mirror symmetry, Langlands duality, and the Hitchin system

          We study the moduli spaces of flat SL(r)- and PGL(r)-connections, or equivalently, Higgs bundles, on an algebraic curve. These spaces are noncompact Calabi-Yau orbifolds; we show that they can be regarded as mirror partners in two different senses. First, they satisfy the requirements laid down by Strominger-Yau-Zaslow (SYZ), in a suitably general sense involving a B-field or flat unitary gerbe. To show this, we use their hyperkahler structures and Hitchin's integrable systems. Second, their Hodge numbers, again in a suitably general sense, are equal. These spaces provide significant evidence in support of SYZ. Moreover, they throw a bridge from mirror symmetry to the duality theory of Lie groups and, more broadly, to the geometric Langlands program.
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            Global generation of adjoint bundles

             Hajime Tsuji (1996)
            In 1988, I. Reider proved that for a smooth projective surface X and an ample line bundle L on X, K x + 3L is globally generated and K x + 4L is very ample ([12]). In fact his theorem is much stronger than this (see [12] for detail). Recently a lot of results have been obtained about effective base point freeness (cf. [1, 3, 8, 13, 14, 15]). In particular J. P. Demailly proved that 2K X + 12n n L is very ample for a smooth projective n -fold X and an ample line bundle L on X . [2] will give a good overview for these recent results. The motivation of these works is the following conjecture posed by T. Fujita.
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              Author and article information

              Journal
              16 September 2017
              Article
              1709.05471

              http://creativecommons.org/licenses/by/4.0/

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              41 pages
              math.DG

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