Louis Esser, Terry Tao, Chengxi Wang and I posted a new paper on the arXiv. As the list of authors might suggest, the paper uses a surprising combination of techniques.
We construct smooth projective varieties of general type with the smallest known volumes in high dimensions. These are n-folds with volume roughly 1/en3/2. Among other examples, we construct varieties of general type with many vanishing plurigenera, more than any polynomial function of the dimension.
The examples are resolutions of singularities of general hypersurfaces of degree d with canonical singularities in a weighted projective space P(a0,…,an+1). The volume can be calculated explicitly in terms of the weights ai, and so the problem is to make a good choice of these numbers. In the class of examples we consider, we minimize the volume by reducing to a purely analytic problem about equidistribution on the unit circle.
Namely, consider the sawtooth (or signed fractional part) function, the periodic function on the real line which is the identity on the interval (-1/2,1/2]. For each positive integer m and every probability measure on the real line, at least one of the dilated sawtooth functions g(kx) for k from 1 to m must have small expected value, and we determine the optimal bound in terms of m. We also solve exactly the corresponding optimization problem for the sine function.
Equivalently, we find the optimal inequality of the form ∑k=1m ak sin kx≤ 1 for each positive integer m, in the sense that ∑k=1m am is maximal. The figures show examples of these inequalities, which show striking cancellation among dilated sine or sawtooth functions.