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Louis Esser, Burt Totaro, Chengxi Wang, and myself have just uploaded to the arXiv our preprint “Varieties of general type with many vanishing plurigenera, and optimal sine and sawtooth inequalities“. This is an interdisciplinary paper that arose because in order to optimize a certain algebraic geometry construction it became necessary to solve a purely analytic question which, while simple, did not seem to have been previously studied in the literature. We were able to solve the analytic question exactly and thus fully optimize the algebraic geometry construction, though the analytic question may have some independent interest.

Let us first discuss the algebraic geometry application. Given a smooth complex {n}-dimensional projective variety {X} there is a standard line bundle {K_X} attached to it, known as the canonical line bundle; {n}-forms on the variety become sections of this bundle. The bundle may not actually admit global sections; that is to say, the dimension {h^0(X, K_X)} of global sections may vanish. But as one raises the canonical line bundle {K_X} to higher and higher powers to form further line bundles {mK_X}, the number of global sections tends to increase; in particular, the dimension {h^0(X, mK_X)} of global sections (known as the {m^{th}} plurigenus) always obeys an asymptotic of the form

\displaystyle  h^0(X, mK_X) = \mathrm{vol}(X) \frac{m^n}{n!} + O( m^{n-1} )

as {m \rightarrow \infty} for some non-negative number {\mathrm{vol}(X)}, which is called the volume of the variety {X}, which is an invariant that reveals some information about the birational geometry of {X}. For instance, if the canonical line bundle is ample (or more generally, nef), this volume is equal to the intersection number {K_X^n} (roughly speaking, the number of common zeroes of {n} generic sections of the canonical line bundle); this is a special case of the asymptotic Riemann-Roch theorem. In particular, the volume {\mathrm{vol}(X)} is a natural number in this case. However, it is possible for the volume to also be fractional in nature. One can then ask: how small can the volume get {\mathrm{vol}(X)} without vanishing entirely? (By definition, varieties with non-vanishing volume are known as varieties of general type.)

It follows from a deep result obtained independently by Hacon–McKernan, Takayama and Tsuji that there is a uniform lower bound for the volume {\mathrm{vol}(X)} of all {n}-dimensional projective varieties of general type. However, the precise lower bound is not known, and the current paper is a contribution towards probing this bound by constructing varieties of particularly small volume in the high-dimensional limit {n \rightarrow \infty}. Prior to this paper, the best such constructions of {n}-dimensional varieties basically had exponentially small volume, with a construction of volume at most {e^{-(1+o(1))n \log n}} given by Ballico–Pignatelli–Tasin, and an improved construction with a volume bound of {e^{-\frac{1}{3} n \log^2 n}} given by Totaro and Wang. In this paper, we obtain a variant construction with the somewhat smaller volume bound of {e^{-(1-o(1)) n^{3/2} \log^{1/2} n}}; the method also gives comparable bounds for some other related algebraic geometry statistics, such as the largest {m} for which the pluricanonical map associated to the linear system {|mK_X|} is not a birational embedding into projective space.

The space {X} is constructed by taking a general hypersurface of a certain degree {d} in a weighted projective space {P(a_0,\dots,a_{n+1})} and resolving the singularities. These varieties are relatively tractable to work with, as one can use standard algebraic geometry tools (such as the ReidTai inequality) to provide sufficient conditions to guarantee that the hypersurface has only canonical singularities and that the canonical bundle is a reflexive sheaf, which allows one to calculate the volume exactly in terms of the degree {d} and weights {a_0,\dots,a_{n+1}}. The problem then reduces to optimizing the resulting volume given the constraints needed for the above-mentioned sufficient conditions to hold. After working with a particular choice of weights (which consist of products of mostly consecutive primes, with each product occuring with suitable multiplicities {c_0,\dots,c_{b-1}}), the problem eventually boils down to trying to minimize the total multiplicity {\sum_{j=0}^{b-1} c_j}, subject to certain congruence conditions and other bounds on the {c_j}. Using crude bounds on the {c_j} eventually leads to a construction with volume at most {e^{-0.8 n^{3/2} \log^{1/2} n}}, but by taking advantage of the ability to “dilate” the congruence conditions and optimizing over all dilations, we are able to improve the {0.8} constant to {1-o(1)}.

Now it is time to turn to the analytic side of the paper by describing the optimization problem that we solve. We consider the sawtooth function {g: {\bf R} \rightarrow (-1/2,1/2]}, with {g(x)} defined as the unique real number in {(-1/2,1/2]} that is equal to {x} mod {1}. We consider a (Borel) probability measure {\mu} on the real line, and then compute the average value of this sawtooth function

\displaystyle  \mathop{\bf E}_\mu g(x) := \int_{\bf R} g(x)\ d\mu(x)

as well as various dilates

\displaystyle  \mathop{\bf E}_\mu g(kx) := \int_{\bf R} g(kx)\ d\mu(x)

of this expectation. Since {g} is bounded above by {1/2}, we certainly have the trivial bound

\displaystyle  \min_{1 \leq k \leq m} \mathop{\bf E}_\mu g(kx) \leq \frac{1}{2}.

However, this bound is not very sharp. For instance, the only way in which {\mathop{\bf E}_\mu g(x)} could attain the value of {1/2} is if the probability measure {\mu} was supported on half-integers, but in that case {\mathop{\bf E}_\mu g(2x)} would vanish. For the algebraic geometry application discussed above one is then led to the following question: for a given choice of {m}, what is the best upper bound {c^{\mathrm{saw}}_m} on the quantity {\min_{1 \leq k \leq m} \mathop{\bf E}_\mu g(kx)} that holds for all probability measures {\mu}?

If one considers the deterministic case in which {\mu} is a Dirac mass supported at some real number {x_0}, then the Dirichlet approximation theorem tells us that there is {1 \leq k \leq m} such that {x_0} is within {\frac{1}{m+1}} of an integer, so we have

\displaystyle  \min_{1 \leq k \leq m} \mathop{\bf E}_\mu g(kx) \leq \frac{1}{m+1}

in this case, and this bound is sharp for deterministic measures {\mu}. Thus we have

\displaystyle  \frac{1}{m+1} \leq c^{\mathrm{saw}}_m \leq \frac{1}{2}.

However, both of these bounds turn out to be far from the truth, and the optimal value of {c^{\mathrm{saw}}_m} is comparable to {\frac{\log 2}{\log m}}. In fact we were able to compute this quantity precisely:

Theorem 1 (Optimal bound for sawtooth inequality) Let {m \geq 1}.
  • (i) If {m = 2^r} for some natural number {r}, then {c^{\mathrm{saw}}_m = \frac{1}{r+2}}.
  • (ii) If {2^r < m \leq 2^{r+1}} for some natural number {r}, then {c^{\mathrm{saw}}_m = \frac{2^r}{2^r(r+1) + m}}.
In particular, we have {c^{\mathrm{saw}}_m = \frac{\log 2 + o(1)}{\log m}} as {m \rightarrow \infty}.

We establish this bound through duality. Indeed, suppose we could find non-negative coefficients {a_1,\dots,a_m} such that one had the pointwise bound

\displaystyle  \sum_{k=1}^m a_k g(kx) \leq 1 \ \ \ \ \ (1)

for all real numbers {x}. Integrating this against an arbitrary probability measure {\mu}, we would conclude

\displaystyle  (\sum_{k=1}^m a_k) \min_{1 \leq k \leq m} \mathop{\bf E}_\mu g(kx) \leq \sum_{k=1}^m a_k \mathop{\bf E}_\mu g(kx) \leq 1

and hence

\displaystyle  c^{\mathrm{saw}}_m \leq \frac{1}{\sum_{k=1}^m a_k}.

Conversely, one can find lower bounds on {c^{\mathrm{saw}}_m} by selecting suitable candidate measures {\mu} and computing the means {\mathop{\bf E}_\mu g(kx)}. The theory of linear programming duality tells us that this method must give us the optimal bound, but one has to locate the optimal measure {\mu} and optimal weights {a_1,\dots,a_m}. This we were able to do by first doing some extensive numerics to discover these weights and measures for small values of {m}, and then doing some educated guesswork to extrapolate these examples to the general case, and then to verify the required inequalities. In case (i) the situation is particularly simple, as one can take {\mu} to be the discrete measure that assigns a probability {\frac{1}{r+2}} to the numbers {\frac{1}{2}, \frac{1}{4}, \dots, \frac{1}{2^r}} and the remaining probability of {\frac{2}{r+2}} to {\frac{1}{2^{r+1}}}, while the optimal weighted inequality (1) turns out to be

\displaystyle  2g(x) + \sum_{j=1}^r g(2^j x) \leq 1

which is easily proven by telescoping series. However the general case turned out to be significantly tricker to work out, and the verification of the optimal inequality required a delicate case analysis (reflecting the fact that equality was attained in this inequality in a large number of places).

After solving the sawtooth problem, we became interested in the analogous question for the sine function, that is to say what is the best bound {c^{\sin}_m} for the inequality

\displaystyle  \min_{1 \leq k \leq m} \mathop{\bf E}_\mu \sin(kx) \leq c^{\sin}_m.

The left-hand side is the smallest imaginary part of the first {m} Fourier coefficients of {\mu}. To our knowledge this quantity has not previously been studied in the Fourier analysis literature. By adopting a similar approach as for the sawtooth problem, we were able to compute this quantity exactly also:

Theorem 2 For any {m \geq 1}, one has

\displaystyle  c^{\sin}_m = \frac{m+1}{2 \sum_{1 \leq j \leq m: j \hbox{ odd}} \cot \frac{\pi j}{2m+2}}.

In particular,

\displaystyle  c^{\sin}_m = \frac{\frac{\pi}{2} + o(1)}{\log m}.

Interestingly, a closely related cotangent sum recently appeared in this MathOverflow post. Verifying the lower bound on {c^{\sin}_m} boils down to choosing the right test measure {\mu}; it turns out that one should pick the probability measure supported the {\frac{\pi j}{2m+2}} with {1 \leq j \leq m} odd, with probability proportional to {\cot \frac{\pi j}{2m+2}}, and the lower bound verification eventually follows from a classical identity

\displaystyle  \frac{m+1}{2} = \sum_{1 \leq j \leq m; j \hbox{ odd}} \cot \frac{\pi j}{2m+2} \sin \frac{\pi jk}{m+1}

for {1 \leq k \leq m}, first posed by Eisenstein in 1844 and proved by Stern in 1861. The upper bound arises from establishing the trigonometric inequality

\displaystyle  \frac{2}{(m+1)^2} \sum_{1 \leq k \leq m; k \hbox{ odd}}

\displaystyle \cot \frac{\pi k}{2m+2} ( (m+1-k) \sin kx + k \sin(m+1-k)x ) \leq 1

for all real numbers {x}, which to our knowledge is new; the left-hand side has a Fourier-analytic intepretation as convolving the Fejér kernel with a certain discretized square wave function, and this interpretation is used heavily in our proof of the inequality.

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