Band 2015, Heft 701

We extend the formalism and results on motivic integration from [Invent. Math. 173 (2008), 23–121] to mixed characteristic discretely valued Henselian fields with bounded ramification. We also generalize the equicharacteristic zero case of loc. cit. by giving, in all residue characteristics, an axiomatic approach (instead of only using Denef–Pas languages) and by using richer angular component maps. In this setting we prove a general change of variables formula and a general Fubini Theorem. Our set-up can be specialized to previously known versions of motivic integration by e.g. the second author and J. Sebag and to classical p -adic integrals.

Let F be a local field with finite residue field of characteristic p . Let G be a connected reductive group over F and B a minimal parabolic subgroup of G with Levi decomposition B=ZU$B=ZU$. Let K be a special parahoric subgroup of G , in good position relative to ( Z , U ). Fix an absolutely irreducible smooth representation of K on a vector space V over some field C of characteristic p . Writing ℋ(G,K,V)$\mathcal {H}(G,K,V)$ for the intertwining Hecke algebra of V in G , we define a natural algebra homomorphism from ℋ(G,K,V)$\mathcal {H}(G,K,V)$ to ℋ(Z,Z∩K,VU∩K)$\mathcal {H}(Z,Z\cap K,V^{U\cap K})$, we show it is injective and identify its image. We thus generalize work of F. Herzig, who assumed F of characteristic 0, G unramified and K hyperspecial, and took for C an algebraic closure of the prime field 𝔽 p . We show that in the general case ℋ(G,K,V)$\mathcal {H}(G,K,V)$ need not be commutative; that is in contrast with the cases Herzig considers and with the more classical situation where V is trivial and the field of coefficients is the field of complex numbers.

Let 𝔤 = 𝔫 ⊕ 𝔥 ⊕ 𝔫 - be a simple Lie algebra over ℂ of type A , D , E , and let U q ( L 𝔤) be the associated quantum loop algebra. Following Nakajima [Ann. of Math. (2) 160 (2004), 1057–1097], Varagnolo–Vasserot [Studies in memory of Issai Schur, Birkhäuser-Verlag, Basel (2002), 345–365], and the first author [Adv. Math. 187 (2004), 1–52], we study a t -deformation 𝒦 t of the Grothendieck ring of a tensor category 𝒞 ℤ of finite-dimensional U q ( L 𝔤)-modules. We obtain a presentation of 𝒦 t by generators and relations. Let Q be a Dynkin quiver of the same type as 𝔤. Let DH( Q ) be the derived Hall algebra of the bounded derived category D b (mod( F Q )) over a finite field F , introduced by Toën [Duke Math. J. 135 (2006), 587–615]. Our presentation shows that the specialization of 𝒦 t at t = | F | 1/2 is isomorphic to DH( Q ). Under this isomorphism, the classes of fundamental U q ( L 𝔤)-modules are mapped to scalar multiples of the classes of indecomposable objects in DH( Q ). Our presentation of 𝒦 t is deduced from the preliminary study of a tensor subcategory 𝒞 Q of 𝒞 ℤ analogous to the heart mod( F Q ) of the triangulated category D b (mod( F Q )). We show that the t -deformed Grothendieck ring 𝒦 t , Q of 𝒞 Q is isomorphic to the positive part of the quantum enveloping algebra of 𝔤, and that the basis of classes of simple objects of 𝒦 t , Q corresponds to the dual of Lusztig's canonical basis. The proof relies on the algebraic characterizations of these bases, but we also give a geometric approach in the last section. It follows that for every orientation Q of the Dynkin diagram, the category 𝒞 Q gives a new categorification of the coordinate ring ℂ[ N ] of a unipotent group N with Lie algebra 𝔫, together with its dual canonical basis.

Let Y → ℙ n be a flat family of integral Gorenstein curves, such that the compactified relative Jacobian X = J ̅ d ( Y /ℙ n ) is a Lagrangian fibration. We prove that the degree of the discriminant locus Δ ⊂ ℙ n is at least 4 n + 2, and we prove that X is a Beauville–Mukai integrable system if deg Δ > 4 n + 20.

Making use of noncommutative motives we relate exceptional collections (and more generally semi-orthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M (𝒳) ℚ of every smooth and proper Deligne–Mumford stack 𝒳, whose bounded derived category 𝒟 b (𝒳) of coherent schemes admits a full exceptional collection, decomposes into a direct sum of tensor powers of the Lefschetz motive. Examples include projective spaces, quadrics, toric varieties, homogeneous spaces, Fano threefolds, and moduli spaces. On the other hand we prove that if M (𝒳) ℚ decomposes into a direct sum of tensor powers of the Lefschetz motive and moreover 𝒟 b (𝒳) admits a semi-orthogonal decomposition, then the noncommutative motive of each one of the pieces of the semi-orthogonal decomposition is a direct sum of ⊗-units. As an application we obtain a simplification of Dubrovin's conjecture.

In this paper we classify branched Willmore spheres with at most three branch points (including multiplicity), showing that they may be obtained from complete minimal surfaces in ℝ 3 with ends of multiplicity at most three. This extends the classification result of Bryant. We then show that this may be applied to the analysis of singularities of the Willmore flow of non-Willmore spheres with Willmore energy 𝒲(f)≤16π$ \mathcal {W} (f) \le 16\pi $.

We prove that a Mori dream space over a field of characteristic zero is of Calabi–Yau type if and only if its Cox ring has at worst log canonical singularities. By slightly modifying the arguments we also reprove the characterization of varieties of Fano type by log terminality of their Cox rings.

Suppose that W is a finite, unitary reflection group acting on the complex vector space V . Let 𝒜 = 𝒜( W ) be the associated hyperplane arrangement of W . Terao [J. Fac. Sci. Univ. Tokyo 27 (1980), 293–320] has shown that each such reflection arrangement 𝒜 is free. There is the stronger notion of an inductively free arrangement. In 1992, Orlik and Terao [Arrangements of hyperplanes, Springer-Verlag, Berlin 1992, Conjecture 6.91] conjectured that each reflection arrangement is inductively free. It has been known for quite some time that the braid arrangement as well as the Coxeter arrangements of type B ℓ and type D ℓ are inductively free. Barakat and Cuntz [Adv. Math. 229 (2012), 691–709] completed this list only recently by showing that every Coxeter arrangement is inductively free. Nevertheless, Orlik and Terao's conjecture is false in general. In a paper which will appear in Tôhoku Math. J., we already gave two counterexamples to this conjecture among the exceptional complex reflection groups. In this paper we classify all inductively free reflection arrangements. In addition, we show that the notions of inductive freeness and that of hereditary inductive freeness coincide for reflection arrangements. As a consequence of our classification, we get an easy, purely combinatorial characterization of inductively free reflection arrangements 𝒜 in terms of exponents of the restrictions to any hyperplane of 𝒜.

In this paper, we can completely determine the conformal class of Willmore circle-foliated surfaces by a solution of elastic equation and a constant. Moreover, the circle-foliated minimal surfaces in three space forms are also classified.