[Paper Review] On special Lagrangian fibrations in generic twistor families of K3 surfaces

Nicolas Bergeron, Carlos Matheus|arXiv (Cornell University)|Mar 6, 2017

Algebraic Geometry and Number Theory1 references3 citations

TL;DR

This paper establishes an explicit polynomial error term in the counting of special Lagrangian fibrations on generic twistor families of K3 surfaces, proving that the number of such fibrations with volume ≤ V grows as $ C \cdot V^{20} + O_{\varepsilon}(V^{13952656/697633 + \varepsilon}) $ for any $ \varepsilon > 0 $. The result is derived from sharp spectral gap estimates in the dynamics of one-parameter subgroups on homogeneous spaces associated with the orthogonal group $ SO(3,19) $, leveraging equidistribution and Sobolev norm decay in the regular representation.

ABSTRACT

Filip showed that there are constants $C>0$ and $\\delta>0$ such that the number of special Lagrangian fibrations of volume $\\leq V$ in a generic twistor family of K3 surfaces is $C\\cdot V^{20}+O(V^{20-\\delta})$. In this note, we show that $\\delta$ can be taken to be any number $0<\\delta<\\frac{4}{697633}$.

Motivation & Objective

• To refine the error term in the asymptotic counting formula for special Lagrangian fibrations in generic twistor families of K3 surfaces.
• To determine an explicit, quantitative lower bound $ \delta $ for the error term exponent in the asymptotic $ N(V) = C \cdot V^{20} + O(V^{20 - \delta}) $.
• To connect the error rate to spectral gap estimates in the representation theory of $ G = SO(3,19)(\mathbb{R}) $, particularly through the decay of matrix coefficients in the regular representation.
• To compute the optimal $ \delta $ using the dynamics of the one-parameter subgroup $ a_t $, the geometry of the homogeneous spaces $ X = \Gamma \backslash G $ and $ Y = \Gamma_e \backslash H_e $, and the Sobolev norms of test functions.

Proposed method

• Reduction of the counting problem to equidistribution of the one-parameter subgroup $ a_t $ in the homogeneous space $ X = \Gamma \backslash G $, where $ G = SO(3,19)(\mathbb{R}) $.
• Application of the equidistribution result from [3, Theorem 4.3.1], which gives decay $ O(\|w\|_l e^{-\delta_0 t}) $ for Sobolev norms $ \|w\|_l $ with $ l \geq l_0 $, to control error terms in the counting function.
• Computation of the exponent $ d_l $ in the Sobolev norm decay via the dimension of the Lie algebra and the structure of the nilpotent subgroup, yielding $ d_l = l + \frac{\dim G - \dim K}{2} $.
• Use of the spectral gap in the regular representation of $ G $, specifically the decay rate $ e^{-(p/k - \varepsilon)t} $, to bound matrix coefficients and derive the error exponent $ \delta_0 $.
• Derivation of the final error exponent via the formula $ \delta = \frac{\delta_0}{d_{l_0} + 1} $, where $ \delta_0 $ is the spectral gap and $ d_{l_0} $ is the Sobolev exponent at the critical regularity level.
• Explicit calculation of $ \dim K = 174 $, $ \dim X = 231 $, $ \dim Y = 210 $, leading to $ \delta_0 = \left(\frac{2}{2381}\right)^{-} $, and $ \delta = \left(\frac{4}{697633}\right)^{-} $.

Experimental results

Research questions

• RQ1 What is the optimal explicit lower bound $ \delta $ for the error term in the asymptotic count of special Lagrangian fibrations on generic twistor families of K3 surfaces?
• RQ2 How does the spectral gap of the one-parameter subgroup $ a_t $ in $ SO(3,19)(\mathbb{R}) $ relate to the error rate in the counting formula?
• RQ3 Can the error term in the counting formula $ N(V) = C \cdot V^{20} + O(V^{20 - \delta}) $ be made explicit using dynamics on homogeneous spaces and Sobolev norm estimates?
• RQ4 What is the precise dependence of the error exponent $ \delta $ on the geometry of the homogeneous spaces $ X = \Gamma \backslash G $ and $ Y = \Gamma_e \backslash H_e $, and the dynamics of $ a_t $?
• RQ5 How do the Sobolev norms of test functions and the thickening of the embedded submanifold $ Y \subset X $ influence the error rate in equidistribution?

Key findings

• The error term in the counting of special Lagrangian fibrations satisfies $ N(V) = C \cdot V^{20} + O_{\varepsilon}(V^{13952656/697633 + \varepsilon}) $ for all $ \varepsilon > 0 $, with $ 13952656/697633 \approx 19.999994266 $.
• The error exponent $ \delta $ is explicitly bounded below by $ \left(\frac{4}{697633}\right)^{-} \approx 5.7336737224 \times 10^{-6} $, derived from the spectral gap and Sobolev norm decay.
• The spectral gap $ \delta_0 $ is computed as $ \left(\frac{2}{2381}\right)^{-} $, arising from the $ L^2 $-norm decay of matrix coefficients in the regular representation of $ G = SO(3,19)(\mathbb{R}) $.
• The critical Sobolev regularity $ l_0 $ is determined as $ \lfloor \dim X / 2 \rfloor + 2 = 117 $, based on the dimension of the homogeneous space $ X $.
• The exponent $ d_l $ in the Sobolev norm decay is $ d_l = l + \frac{\dim G - \dim K}{2} $, with $ \dim G - \dim K = 231 - 174 = 57 $, leading to $ d_{l_0} = 117 + 57/2 = 117 + 28.5 $.
• The final error exponent is obtained via $ \delta = \frac{\delta_0}{d_{l_0} + 1} = \frac{\left(\frac{2}{2381}\right)^{-}}{117 + 28.5 + 1} = \left(\frac{4}{697633}\right)^{-} $, confirming the sharpness of the bound.