Varieties of general type with doubly exponential asymptotics
Louis Esser (UCLA)
Abstract: By a theorem of Hacon–McKernan, Takayama, and Tsuji, for every $n$ there is a constant $r_n$ for which every smooth variety $X$ of dimension $n$ of general type has birational pluricanonical maps $|mK_X|$ for $m \geq r_n$. In joint work with Burt Totaro and Chengxi Wang (see arxiv.org/abs/2109.13383), we show that the constants $r_n$ grow at least doubly exponentially. Conjecturally, it's expected that the optimal bound is in fact doubly exponential. We do this by finding weighted projective hypersurfaces of general type with extreme behavior: this includes examples of very small volume and many vanishing plurigenera. We also consider the analogous questions for other classes of varieties and provide some conjecturally optimal examples. For instance, we conjecture the terminal Fano variety of minimal volume and the canonical Calabi-Yau variety of minimal volume in each dimension.
Audience: researchers in the topic
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