What are the sharp polynomial upper bounds for anticanonical volumes?

From boundedness to a rate in epsilon

Birkar's proof of the Borisov–Alexeev–Borisov conjecture shows that, for fixed dimension n and fixed ε > 0, Fano varieties with ε-lc singularities form a bounded family, so their anticanonical volumes are bounded. That statement says nothing about how the bound behaves as ε shrinks. The paper asks for the rate: how fast must vol(−K_X) grow as the singularity parameter tends to zero?

The distinction matters because a bound for each fixed ε does not specify its dependence on ε. The authors also note that weighted projective spaces give examples whose volumes grow rapidly as singularities worsen, which pins down a candidate for the smallest possible exponent in each dimension. The theorem turns that candidate into a matching upper bound.

The parameter belongs to a boundary, not the variety

For 0 < ε ≤ 1, a variety of ε-Fano type is a normal projective Q-Gorenstein variety X for which there is an effective Q-divisor Δ such that (X, Δ) is ε-lc and −(K_X + Δ) is ample. The singularity parameter is therefore attached to a log Fano boundary, not merely to the underlying variety. The main theorem's estimate consequently covers ε-lc Fano and weak Fano varieties as special cases.

The paper shows why this formulation is needed. A Hirzebruch surface F_m with m ≥ 3 is smooth, hence 1-lc, and is of Fano type with boundary Δ_m = (1 − 1/m)S. Yet vol(−K_X) = (m + 2)²/m, which tends to infinity. Being η-lc and of Fano type without a controlled boundary does not bound the anticanonical volume; the boundary parameter does the work.

Exponents 0, 1, 4, 11, 26, 57

Theorem 1.1 states that for every positive integer n there is a constant C_n, depending only on n, such that vol(−K_X) ≤ C_n ε^−a_n with a_n = 2^n − n − 1, for every n-dimensional variety of ε-Fano type. The first exponents are a_1 = 0, a_2 = 1, a_3 = 4, a_4 = 11, a_5 = 26 and a_6 = 57. In dimension one the constant can be taken as C_1 = 2.

The exponent is sharp; the leading constant is not optimized. The fourfold statement is recorded separately as Corollary 1.2: every ε-lc weak Fano fourfold satisfies vol(−K_X) ≤ C_4 ε^−11, and eleven is already optimal for Fano fourfolds of Picard number one. The authors describe this as the first polynomial upper bound with the optimal exponent for arbitrary-dimensional ε-lc Fano varieties, and in particular the first such bound in dimension four.

A second theorem controls mixed boundaries

The volume bound is derived from a separate statement about interpolation. Theorem 1.3 says that for every n there is c_n > 0 with the following property: if X is a projective Q-Gorenstein variety of Fano type and B, D ≥ 0 are Q-divisors with B ∼_Q D ∼_Q −K_X and (X, B) is ε-lc, then (X, (1 − t)B + tD) is lc whenever 0 ≤ t ≤ c_n ε^p_n, where p_n = 2^n − 1. For n ≥ 2 the exponent p_n cannot be replaced by a smaller exponent uniformly in this class.

This theorem compares an arbitrary effective anticanonical divisor with a specified anticanonical boundary of controlled singularities. It controls all divisorial valuations, including those contracted on later models, which the authors say is needed when applying the canonical bundle formula and when moving between Mori fibre spaces. The interpolation exponent 2^n − 1 is larger than the volume exponent 2^n − n − 1, and the paper keeps the two inductions separate precisely to preserve the sharper volume exponent.

Projection, finite morphisms, and a bounded base

The proof runs two dimension inductions. The first establishes the interpolation theorem. The local input builds on Jiao's projection method: for a signed sub-pair on projective space, the loss of discrepancy under projection is expressed through finitely many quotients of coefficient ideals, and the normalized generating degree of the positive ideal is bounded by the square of the positive boundary degree, independently of the negative degree. The authors carry this construction over geometric discrete valuation rings and retain all vertical fixed divisors; a surface calculation then gives a relative discrepancy bound of order ε^2r in relative dimension r.

For the absolute comparison they keep the quotient by the restriction index through the induction, producing the recurrence p_j = 1 + 2p_{j−1} with p_1 = 1, hence p_j = 2^j − 1. To apply projective-space estimates to terminal Fano varieties they use a finite morphism of bounded degree: characteristic polynomials recover the largest order among all valuation extensions, and therefore the smallest discrepancy quotient, whereas the norm alone would only recover their sum. Fixed-parameter boundedness supplies these finite morphisms, so their constants do not depend on the varying ε.

For volumes, a further reduction produces a polarized base whose degree is bounded independently of ε, using Birkar's theorem on singularities of Fano fibrations together with a lifting result of Mauri–Moraga. A flag on this base and connectedness then give a volume bound proportional to M_r(ε/4) μ_r(ε/4)^{−(n−r)}, where M_r bounds fibre volumes and μ_r is the interpolation threshold. The largest exponent occurs for a one-dimensional base: a_n = a_{n−1} + p_{n−1} = 2^n − n − 1.

Weighted projective spaces attain the exponent

Sharpness comes from a family of well-formed weighted projective Fano varieties X_{n,h} of dimension n, with Q-factorial singularities and Picard number one. For every n ≥ 2 and integer h ≥ 3, X_{n,h} is 1/h-lc and vol(−K_{X_{n,h}}) ∼ h^{2^n − n − 1} as h → ∞, where ∼ means the ratio tends to one. The construction sets u_1 = h, u_{j+1} = u_j(1 + u_j), q_j = 1 + u_j, U = u_{n+1}, L = U/h, and takes X_{n,h} = P(1, a_1, …, a_n) with a_j = L/q_j.

The discrepancy check covers all valuations, not only the invariant divisors: the toric pair (X, (1 − 1/h)S) has log discrepancy function positive on every ray generator, and because the log canonical divisor has Cartier index dividing h, every log discrepancy lies in h^{−1}Z, so all are at least 1/h. The volume computation uses the Hilbert series to get S^n = 1/∏ a_j and ∏ a_j = L^{n−1}, yielding vol(−K_X) = (U/h^{n+1})(1 + (h² − h)/U)^n, and induction gives u_j ∼ h^{2^{j−1}}, hence U ∼ h^{2^n}.

The same examples also show that the interpolation exponent is optimal. There are effective anticanonical Q-divisors B_h, D_h on X_{n,h} with (X_{n,h}, B_h) 1/h-lc for which lc of (X_{n,h}, (1 − t)B_h + tD_h) implies t ≤ h/u_{n+1}, and h/u_{n+1} ∼ h^{−(2^n − 1)}.

Extra fibre structure lowers the exponent to seven

The estimate over a bounded base also improves when the fibre volume is already bounded independently of the singularity parameter. Corollary 9.2 states that for n ≥ 2, if (X, Δ) is an η-lc log Fano n-fold admitting a contraction to a smooth curve whose general fibre is terminal Fano, then vol(−K_X) ≤ E_n η^{−(2^{n−1} − 1)} for some E_n depending only on n. For fourfolds under this additional hypothesis the exponent is seven, not eleven.

The authors explain why this extra hypothesis cannot simply be imposed on the final morphism of the general reduction: lifted birational contractions need not preserve terminality, and composing base contractions need not preserve the Fano condition on the resulting generic fibre. That is why the general argument still uses the variable-parameter volume induction.

Two further limits are stated explicitly. The optimal numerical constant in the main bound remains open: formula (9.5) has leading coefficient one, while the upper-bound argument loses substantial constants in projection, finite transfer and the change of base, and a numerical value of C_4 would also require an effective version of the bounded polarization used in the reduction. Separately, Theorem 4.3 supplies an exponent 2r for the relative discrepancy estimate with bounded horizontal degree, and the authors do not claim its optimality in every relative dimension.

The exponent, not the constant, is the result

The paper's decisive comparison is between two exponents that a reader could easily conflate. The interpolation theorem controls how far a boundary can be mixed toward an arbitrary effective anticanonical divisor, with exponent 2^n − 1. The volume theorem controls the growth of vol(−K_X) itself, with exponent 2^n − n − 1. The second is smaller, and the proof structure is arranged to keep it that way: the two inductions are separated, and the volume step adds a_{n−1} + p_{n−1} rather than reusing the interpolation exponent directly.

For fourfolds the practical reading is concrete. The general theorem gives ε^−11 for ε-lc weak Fano fourfolds, and the toric weighted projective examples show eleven cannot be lowered even under Picard number one. If a fourfold additionally contracts to a smooth curve with terminal Fano general fibre, the exponent drops to seven. The constant C_4, by contrast, is left dimension dependent and is not made numerical, so the result should be read as a statement about the sharp rate in ε rather than about an explicit threshold value.

Sources used in this answer