VARIETIES OF GENERAL TYPE WITH MANY VANISHING PLURIGENERA, AND OPTIMAL SINE AND SAWTOOTH INEQUALITIES LOUIS ESSER, TERENCE TAO, BURT TOTARO, AND CHENGXI WANG 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 equidis- tribution 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. Equiv- alently, we determine the optimal inequality of the form ∑ m k=1 a k sin kx ≤ 1, in the sense that ∑ m k=1 a k is maximal. The volume vol(X ) of an n-dimensional complex projective variety X measures the asymptotic growth of the plurigenera, and more precisely is defined by the formula vol(X ) := lim sup m→∞ h 0 (X, mK X )/(m n /n!) where K X is the canonical bundle. If K X is ample (or, more generally, nef), then vol(X ) is also equal to the intersection number K n X , by the asymptotic Riemann–Roch theorem. However, in general the volume need not be a natural number. A variety is said to be of general type if its volume is positive. It is a challenge to construct smooth projective varieties of general type with small volume, as they exhibit extreme behavior among all algebraic varieties. The known examples in high dimensions are: • Ballico–Pignatelli–Tasin gave examples of n-folds of general type with volume roughly 1/n n =1/e n log n [1, Theorem 1]. • Totaro and Wang obtained an improved family of examples of volume roughly 1/n n log n =1/e n(log n) 2 [17, Theorem 0.1]. In this paper, we go quite a bit further, giving examples of n-folds of volume roughly 1/e n 3/2 (Theorem 0.1). The examples will be resolutions of singularities of general hypersurfaces of degree d with canonical singularities in a weighted projective space P (a 0 ,...,a n+1 ). The volume can be calculated explicitly in terms of these numbers; the problem then becomes one of optimizing this volume over all admissible parameters of a certain form. In the class of examples we consider, we optimize the constant factor 2010 Mathematics Subject Classification. 14J40 (Primary) 11K06; 14E30; 26D05; 42A05 (Secondary). 1 arXiv:2107.11058v1 [math.AG] 23 Jul 2021