Let 𝑘 ≥ 2 be a given integer. We study the set of 3-fold canonical thresholds ct(𝑋;𝑆) with \frac{1}{𝑘} < ct(𝑋;𝑆) < \frac{1}{𝑘−1} where 𝑆 is a ℚ-Cartier prime divisor of a projective 3-fold 𝑋. Express ct(𝑋;𝑆) as the rational number \frac{𝑎}{𝑚} where 𝑎 (resp. 𝑚) denotes the weighted discrepancy (resp. weighted multiplicity). We conclude that if 𝑎 ≥ 54𝑘⁴, then we may choose positive integers 𝑝 and 𝑞 satisfying ct(𝑋;𝑆) = \frac{𝑎}{𝑚} = \frac{1}{𝑘} + \frac{𝑞}{𝑝} and 𝑞 < 6𝑘³. As a consequence, the set of accumulation points of the set of 3-fold canonical thresholds consists of {0} ∪ { \frac{1}{𝑘} }_{𝑘∈ℤ _{≥ 2}}. Moreover, we generalize the ACC for the set of 3-fold canonical thresholds to pairs.

We study the integrability to second order of infinitesimal Einstein deformations on compact Riemannian and in particular on Kähler manifolds. We find a new way of expressing the necessary and sufficient condition for integrability to second order, which also gives a very clear and compact way of writing the Koiso obstruction. As an application we consider the Kähler case, where the condition can be further simplified and in complex dimension 3 turns out to be purely algebraic. One of our main results is the complete and explicit description of integrable to second order infinitesimal Einstein deformations on the complex 2-plane Grassmannian, which also has a quaternion Kähler structure. As a striking consequence we find that the symmetric Einstein metric on the Grassmannian Gr₂(ℂ^𝑛+2) for 𝑛 odd is rigid.

We develop a variational principle for mean dimension with potential of ℝ^𝑑-actions. We prove that mean dimension with potential is bounded from above by the supremum of the sum of rate distortion dimension and a potential term. A basic strategy of the proof is the same as the case of ℤ-actions. However measure theoretic details are more involved because ℝ^𝑑 is a continuous group. We also establish several basic properties of metric mean dimension with potential and mean Hausdorff dimension with potential for ℝ^𝑑-actions.

In this paper, we derive the generalized hypergeometric functions (period integrals) used in mirror computation of Calabi–Yau hypersurface in 𝐶𝑃^𝑁−1 as generating functions of intersection numbers on the moduli space of quasimaps from 𝐶𝑃¹ with 2 + 1 marked points to 𝐶𝑃^𝑁−1.

Let 𝑓 be a polynomial map from ℝ^𝑚 to ℝ^𝑛 with 𝑚 > 𝑛 > 0 and 𝑡₀ be a regular value of 𝑓. For a small open ball 𝐷_{𝑡₀} centered at 𝑡₀, we show that the map 𝑓 : 𝑓⁻¹(𝐷_{𝑡₀}) → 𝐷_{𝑡₀} is a Serre fibration if and only if 𝑓 is a Serre fibration over a finite number of certain simple arcs starting at 𝑡₀. We characterize the fibration 𝑓 : 𝑓⁻¹(𝐷_{𝑡₀}) → 𝐷_{𝑡₀} by relative homotopy groups defined for these arcs and use it to prove the assertion.

Briançon and Speder gave an example of a 𝜇-constant family of weighted homogeneous polynomials for which 𝜇^* is not constant. In this note we analyze this example. We study similar weighted homogeneous polynomials and determine the number of possible different topology of curves which are obtained as generic plane sections.

In this paper, we consider infinite length versions of multiple zeta-star values. We give several explicit formulas for the infinite length versions of multiple zeta-star values. We also discuss analytic properties of the map from indices to the infinite length versions of multiple zeta-star values.

For a flat proper morphism of finite presentation between schemes with almost coherent structural sheaves (in the sense of Faltings), we prove that the higher direct images of quasi-coherent and almost coherent modules are quasi-coherent and almost coherent. Our proof uses Noetherian approximation, inspired by Kiehl's proof of the pseudo-coherence of higher direct images. Our result allows us to extend Abbes–Gros' proof of Faltings' main 𝑝-adic comparison theorem in the relative case for projective log-smooth morphisms of schemes to proper ones, and thus also their construction of the relative Hodge–Tate spectral sequence.

Let 𝑅 be a commutative noetherian local ring with residue field 𝑘. Denote by 𝖣^𝖻(𝑅) the bounded derived category of finitely generated 𝑅-modules. In this paper, we study the structure of the Verdier quotient 𝖣^𝖻(𝑅)/𝗍𝗁𝗂𝖼𝗄(𝑅 ⊕ 𝑘). We give necessary and sufficient conditions for it to admit an additive generator.

Borel's stability and vanishing theorem gives the stable cohomology of GL(𝑛, ℤ) with coefficients in algebraic GL(𝑛, ℤ)-representations. By combining the Borel theorem with the Hochschild–Serre spectral sequence, we compute the twisted first cohomology of the automorphism group Aut(𝐹_𝑛) of the free group 𝐹_𝑛 of rank 𝑛. We also study the stable rational cohomology of the IA-automorphism group IA_𝑛 of 𝐹_𝑛. We propose a conjectural algebraic structure of the stable rational cohomology of IA_𝑛, and consider some relations to known results and conjectures. We also consider a conjectural structure of the stable rational cohomology of the Torelli groups of surfaces.

In this paper, for each 𝑑 > 0, we study the minimum integer ℎ_3,2𝑑 ∈ ℕ for which there exists a complex polarized K3 surface (𝑋, 𝐻) of degree 𝐻² = 2𝑑 and Picard number 𝜌(𝑋) := rank Pic 𝑋 = ℎ_3,2𝑑 admitting an automorphism of order 3. We show that ℎ_3,2 ∈ {4, 6} and ℎ_3,2𝑑 = 2 for 𝑑 > 1. Analogously, we study the minimum integer ℎ^*_3,2𝑑 ∈ ℕ for which there exists a complex polarized K3 surface (𝑋, 𝐻) as above plus the extra condition that the automorphism acts as the identity on the Picard lattice of 𝑋. We show that ℎ^*_3,2𝑑 is equal to 2 if 𝑑 > 1 and equal to 6 if 𝑑 = 1. We provide explicit examples of K3 surfaces defined over ℚ realizing these bounds.