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      Varieties of general type with many vanishing plurigenera, and optimal sine and sawtooth inequalities

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          Abstract

          We construct smooth projective varieties of general type with the smallest known volumes in high dimensions. Among other examples, we construct varieties of general type with many vanishing plurigenera, more than any polynomial function of the dimension. As part of the construction, we solve exactly an optimization problem about equidistribution on the unit circle in terms of the sawtooth (or signed fractional part) function. We also solve exactly the analogous optimization problem for the sine function. Equivalently, we determine the optimal inequality of the form \(\sum_{k=1}^m a_k\sin kx\leq 1\), in the sense that \(\sum_{k=1}^m a_k\) is maximal.

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          Journal
          23 July 2021
          Article
          2107.11058
          b85b178c-37b2-4778-abfe-93389f2f866f

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          14J40 (Primary) 11K06, 14E30, 26D05, 42A05 (Secondary)
          29 pages, 6 figures
          math.AG math.CA

          Geometry & Topology,Mathematics
          Geometry & Topology, Mathematics

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