Showing papers in "Crelle's Journal in 2013"

The Witt group of non-degenerate braided fusion categories

Abstract: Abstract We give a characterization of Drinfeld centers of fusion categories as non-degenerate braided fusion categories containing a Lagrangian algebra. Further we study the quotient of the monoid of non-degenerate braided fusion categories modulo the submonoid of the Drinfeld centers and show that its formal properties are similar to those of the classical Witt group.

On Nichols algebras of diagonal type

Abstract: We give an explicit and essentially minimal list of defining relations of a Nichols algebra of diagonal type with finite root system. This list contains the well-known quantum Serre relations but also many new variations. A conjecture by Andruskiewitsch and Schneider states that any finite-dimensional pointed Hopf algebra over an algebraically closed field of characteristic zero is generated as an algebra by its group-like and skew-primitive elements. As an application of our main result, we prove the conjecture when the group of group-like elements is abelian.

Regularity of solutions to the parabolic fractional obstacle problem

Abstract: In recent years, there has been an increasing interest in studying constrained variational problems with a fractional diffusion. One of the motivations comes from mathematical finance: jumpdiffusion processes where incorporated by Merton [14] into the theory of option evaluation to introduce discontinuous paths in the dynamics of the stock’s prices, in contrast with the classical lognormal diffusion model of Black and Scholes [2]. These models allow to take into account large price changes, and they have become increasingly popular for modeling market fluctuations, both for risk management and option pricing purposes. Let us recall that an American option gives its holder the right to buy a stock at a given price prior (but not later) than a given time T > 0. If v(τ, x) represents the rational price of an American option with a payoff ψ at time T > 0, then v will solve (in the viscosity sense) the following obstacle problem: { min{Lv, v − ψ} = 0, v(T ) = ψ.

n-ANGULATED CATEGORIES

Abstract: We dene n-angulated categories by modifying the axioms of triangulated categories in a natural way. We show that Heller's parametrization of pre-triangulations extends to pre-n-angulations. We obtain a large class of examples of n-angulated cate- gories by considering (n 2)-cluster tilting subcategories of triangulated categories which are stable under the (n 2)nd power of the suspension functor. As an application, we show how n-angulated Calabi-Yau categories yield triangulated Calabi-Yau categories of higher Calabi-Yau dimension. Finally, we sketch a link to algebraic geometry and string theory.

Quantum Grothendieck rings and derived Hall algebras

Abstract: We obtain a presentation of the t-deformed Grothendieck ring of a quantum loop algebra of Dynkin type A, D, E. Specializing t at the the square root of the cardinality of a finite field F, we obtain an isomorphism with the derived Hall algebra of the derived category of a quiver Q of the same Dynkin type. Along the way, we study for each choice of orientation Q a tensor subcategory whose t-deformed Grothendieck ring is isomorphic to the positive part of a quantum enveloping algebra of the same Dynkin type, where the classes of simple objects correspond to Lusztig's dual canonical basis.

Finite Dimensional Representations of Khovanov-Lauda-Rouquier algebras I: Finite Type

Abstract: We classify simple representations of Khovanov-Lauda-Rouquier algebras in finite type. The classification is in terms of a standard family of representations that is shown to yield the dual PBW basis in the Grothendieck group. Finally, we describe the global dimension of these algebras.

A reduction theorem for the Alperin–McKay conjecture

Abstract: Abstract As a first step in a general verification of the Alperin–McKay conjecture, we prove a reduction of the conjecture to statements about simple groups. Furthermore we show in numerous cases, that simple groups satisfy the required conditions. The methods are also applied to obtain similar results for refinements due to Isaacs and Navarro.

Support theory via actions of tensor triangulated categories

Abstract: We give a definition of the action of a tensor triangulated category T on a triangulated category K. In the case that T is rigidly-compactly generated and K is compactly generated we show this gives rise to a notion of supports which categorifies work of Benson, Iyengar, and Krause and extends work of Balmer and Favi. We prove that a suitable version of the local-to-global principle holds very generally. A relative version of the telescope conjecture is formulated and we give a sufficient condition for it to hold.

Deformation of the O'Grady moduli spaces

Abstract: In this paper we study moduli spaces of sheaves on an abelian or projective K3 surface. If S is a K3, v = 2w is a Mukai vector on S, where w is primitive and w(2) = 2, and H is a v-generic polarization on S, then the moduli space M-v of H-semistable sheaves on S whose Mukai vector is v admits a symplectic resolution (M) over tilde (v). A particular case is the 10-dimensional O'Grady example (M) over tilde (10) of an irreducible symplectic manifold. We show that (M) over tilde (v) is an irreducible symplectic manifold which is deformation equivalent to (M) over tilde (10) and that H-2 (M-v, Z) is Hodge isometric to the sublattice v(perpendicular to) of the Mukai lattice of S. Similar results are shown when S is an abelian surface.

Isoparametric functions and exotic spheres

Abstract: The first part of the paper is to improve the fundamental theory of isoparametric functions on general Riemannian manifolds. Next we focus our attention on exotic spheres, especially on "exotic" 4-spheres (if exist) and the Gromoll-Meyer sphere. In particular, as one of main results we prove: there exists no properly transnormal function on any exotic 4-sphere if it exists. Furthermore, by projecting an $S^3$-invariant isoparametric function on $Sp(2)$, we construct a properly transnormal but not an isoparametric function on the Gromoll-Meyer sphere with two points as the focal varieties.

Upper motives of algebraic groups and incompressibility of Severi–Brauer varieties

Abstract: Let G be a semisimple affine algebraic group of inner type over a field F . We write XG for the class of all finite direct products of projective G-homogeneous F varieties. We determine the structure of the Chow motives with coefficients in a finite field of the varieties in XG. More precisely, it is known that the motive of any variety in XG decomposes (in a unique way) into a sum of indecomposable motives, and we describe the indecomposable summands which appear in the decompositions. In the case where G is the group PGLA of automorphisms of a given central simple F algebra A, for any variety in the class XG (which includes the generalized Severi-Brauer varieties of the algebra A) we determine its canonical dimension at any prime p. In particular, we find out which varieties in XG are p-incompressible. If A is a division algebra of degree p for some n ≥ 0, then the list of p-incompressible varieties includes the generalized Severi-Brauer variety X(p;A) of ideals of reduced dimension p for m = 0, 1, . . . , n.

Extension of plurisubharmonic functions with growth control

Abstract: Suppose that X is an analytic subvariety of a Stein manifold M and that φ is a plurisubharmonic (psh) function on X which is dominated by a continuous psh exhaustion function u of M . Given any number c > 1, we show that φ admits a psh extension to M which is dominated by cu on M . We use this result to prove that any ω-psh function on a subvariety of the complex projective space is the restriction of a global ω-psh function, where ω is the Fubini-Study Kahler form. Introduction Let X ⊂ Cn be a (closed) analytic subvariety. In the case when X is smooth it is well known that a plurisubharmonic (psh) function on X extends to a psh function on Cn [Sa] (see also [BL, Theorem 3.2]). Using different methods, Coltoiu generalized this result to the case when X is singular [Co, Proposition 2]. In this article we follow Coltoiu’s approach and show that it is possible to obtain extensions with global growth control: Theorem A. Let X be an analytic subvariety of a Stein manifold M and let φ be a psh function on X. Assume that u is a continuous psh exhaustion function on M so that φ(z) 1 there exists a psh function ψ = ψc on M so that ψ |X = φ and ψ(z) R− ρ(z) for all z ∈ CN \D with ‖z‖ = R. (ii) χ(R) > u(r(z)) − ρ(z) for all z ∈ ∂D with ‖z‖ = R. Then ũ(z) = { max{u(r(z)), χ(‖z‖) + ρ(z)}, if z ∈ D, χ(‖z‖) + ρ(z), if z ∈ CN \D, is a continuous psh exhaustion function on CN and ũ = u on V . Employing the methods of Coltoiu [Co] we now construct psh extensions with growth control over bounded sets in Cn. Proposition 1.2. Let χ be a psh function on a subvariety X ⊂ Cn and let v be a continuous psh function on Cn with χ 0, there exists a psh function χ = χR on C n so that χ | X = χ and χ(z) 1 is an increasing sequence defined in terms of the mj’s. Theorem A will follow by showing that it is possible to choose {mj} rapidly increasing so that lim γj is arbitrarily close to 1. We fix next an increasing sequence {mj}j≥−1 so that m−1 = m0 = 0 γj−1 > 1 for all j > 1. Proposition 1.3. Let X, φ, u be as in Theorem A with M = Cn, and let {mj}, {γj} be as above. There exists a psh function ψ on Cn so that ψ |X = φ and for all z ∈ Cn we have ψ(z) φj(z) for z ∈ X , ∫ X∩Kj−1 (ψj − φj) < 2 . (4) ψj(z) ≥ ρj(z) for z ∈ Dj , ψj(z) = ρj(z) for z ∈ C \Dj. (5) ψj(z) < ψj−1(z) for z ∈ Kj−1, where ψ0 = ρ0 = max{u, 0}. (6) Here the integral in (4) is with respect to the area measure on each irreducible component, i.e. ∫

Curve counting theories via stable objects II : DT/ncDT flop formula

Abstract: The goal of the present paper is to show the transformation formula of Donaldson-Thomas invariants on smooth projective Calabi-Yau 3-folds under birational transformations via categorical method. We also generalize the non-commutative Donaldson-Thomas invariants, introduced by B. Szendr{\H o}i in a local $(-1, -1)$-curve example, to an arbitrary flopping contraction from a smooth projective Calabi-Yau 3-fold. The transformation formula between such invariants and the usual Donaldson-Thomas invariants are also established. These formulas will be deduced from the wall-crossing formula in the space of weak stability conditions on the derived category.

On the Dirichlet problem for variational integrals in BV

Abstract: Abstract We investigate the Dirichlet problem for multidimensional variational integrals with linear growth which is formulated in a generalized way in the space of functions of bounded variation. We prove uniqueness of minimizers up to additive constants and deduce additional assertions about these constants and the possible (non-)attainment of the boundary values. Moreover, we provide several related examples. In the case of the model integral our results extend classical results from the scalar case N = 1—where the problem coincides with the non-parametric least area problem—to the general vectorial setting N ∈ ℕ.

The prime geodesic theorem

Abstract: We study the distribution of closed geodesics for the modular surface. We improve the error term in the prime geodesic theorem, and obtain results on prime geodesics in very short intervals conditionally on the generalized Riemann Hypothesis for Dirichlet L-functions. We emphasize a connection between the closed geodesics and certain Dirichlet L-functions.

Inequities in the Shanks–Rényi prime number race: An asymptotic formula for the densities

Abstract: Chebyshev was the first to observe a bias in the distribution of primes in residue classes. The general phenomenon is that if a is a nonsquare ðmod qÞ and b is a square ðmod qÞ, then there tend to be more primes congruent to a ðmod qÞ than b ðmod qÞ in initial intervals of the positive integers; more succinctly, there is a tendency for pðx; q; aÞ to exceed pðx; q; bÞ. Rubinstein and Sarnak defined dðq; a; bÞ to be the logarithmic density of the set of positive real numbers x for which this inequality holds; intuitively, dðq; a; bÞ is the ''probability'' that pðx; q; aÞ > pðx; q; bÞ when x is ''chosen randomly''. In this paper, we establish an asymptotic series for dðq; a; bÞ that can be instantiated with an error term smaller than any negative power of q. This asymptotic formula is written in terms of a variance V ðq; a; bÞ that is originally defined as an infinite sum over all nontrivial zeros of Dirichlet L-functions corresponding to characters ðmod qÞ; we show how V ðq; a; bÞ can be evaluated exactly as a finite expression. In addition to providing the exact rate at which dðq; a; bÞ converges to 1= 2a sq grows, these evaluations allow us to compare the various density values dðq; a; bÞ as a and b vary modulo q; by analyzing the resulting formulas, we can explain and predict which of these densities will be larger or smaller, based on arithme- tic properties of the residue classes a and b ðmod qÞ. For example, we show that if a is a prime power and a 0 is not, then dðq; a; 1Þ < dðq; a 0 ; 1Þ for all but finitely many moduli q for which both a and a 0 are nonsquares. Finally, we establish rigorous numerical bounds for these densities dðq; a; bÞ and report on extensive calculations of them, including for ex- ample the determination of all 117 density values that exceed 9=10.

A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities

Abstract: We consider a subset S of the complex Lie algebra so(n, C) and the cone C(S) of curvature operators which are nonnegative on S. We show that C(S) defines a Ricci flow invariant curvature condition if S is invariant under AdSO(n,C). The analogue for Kahler curvature operators holds as well. Although the proof is very simple and short it recovers all previously known invariant nonnegativity conditions. As an application we reprove that a compact Kahler manifold with positive orthogonal bisectional curvature evolves to a manifold with positive bisectional curvature and is thus biholomorphic to CP. Moreover, the methods can also be applied to prove Harnack inequalities. We consider a Lie algebra g endowed with a scalar product 〈·, ·〉 which is invariant under the adjoint representation of the Lie algebra. The reader should think of g either as the space so(n) of skew adjoint endomorphism of R with the scalar product 〈A, B〉 = − 1 2 tr(AB) or of the Lie subalgebra u(n) ⊂ so(2n) corresponding to the unitary group U(n) ⊂ SO(2n) endowed with the induced scalar product. We consider the space of selfadjoint endomorphisms of S(g). Every selfadjoint endomorphism R ∈ S(g) is determined by the corresponding bilinear form (x, y) 7→ 〈Rx, y〉. The extension of this form to a complex bilinear form R : g⊗R C× g⊗R C → C will be denoted with the same letter R. Notice that for any x ∈ g⊗R C the number R(x, x) is real, where x 7→ x is complex conjugation. Also recall that the space of algebraic curvature operators S B(so(n)) is a linear subspace of S(so(n)). Similarly the space of algebraic Kahler curvature operators S K(u(n)) is a linear subspace of S (u(n)). The subspaces are also invariant under the Ricci flow ODE on S(g) R′ = R + R where 〈Rx, y〉 = − 12 tr(adx R ady R) for x, y ∈ g. We have the following basic result Theorem 1. Let S be a subset of the complex Lie algebra g ⊗R C and let GC denote a Lie group with Lie algebra g ⊗R C . If S is invariant under the adjoint representation of GC, then for h ∈ R the set C(S, h) := {R ∈ S(g) | R(v, v) ≥ h for all v ∈ S} is invariant under the ODE R′ = R + R. In many cases S is scaling invariant and then h = 0 is the only meaningful choice. For h = 0 the set C(S) := C(S, 0) is a cone and the curvature condition C(S) can be thought of as a nonnegativity condition. We recall that for a O(n)-invariant subset C ⊂ S(so(n)) we say that a manifold satisfies C if the curvature operator

A vanishing theorem forcharacteristic classes of odd-dimensional manifold bundles

Abstract: We show how the Atiyah-Singer family index theorem for both, usual and self-adjoint elliptic operators fits naturally into the framework of the Madsen-Tillmann-Weiss spectra. Our main theorem concerns bundles of odd-dimensional manifolds. Using completely functional-analytic methods, we show that for any smooth proper oriented fibre bundle $E \\to X$ with odd-dimensional fibres, the family index $\\ind (B) \\in K^1 (X)$ of the odd signature operator is trivial. The Atiyah-Singer theorem allows us to draw a topological conclusion: the generalized Madsen-Tillmann-Weiss map $\\alpha: B \\Diff^+ (M^{2m-1}) \\to \\loopinf \\MTSO(2m-1)$ kills the Hirzebruch $\\cL$-class in rational cohomology. If $m=2$, this means that $\\alpha$ induces the zero map in rational cohomology. In particular, the three-dimensional analogue of the Madsen-Weiss theorem is wrong. For 3-manifolds $M$, we also prove the triviality of $\\alpha: B \\Diff^+ (M) \\to \\MTSO (3)$ in mod $p$ cohomology in many cases. We show an appropriate version of these results for manifold bundles with boundary.

Nilpotent operators and weighted projective lines

Abstract: We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link is established via weighted projective lines and (stable) categories of vector bundles on those. The setup yields a new approach to attack the subspace problem. In particular, we deduce the main results of Ringel and Schmidmeier for nilpotency degree p from properties of the category of vector bundles on the weighted projective line of weight type (2,3,p), obtained by Serre construction from the triangle singularity x^2+y^3+z^p. For p=6 the Ringel-Schmidmeier classification is thus covered by the classification of vector bundles for tubular type (2,3,6), and then is closely related to Atiyah's classification of vector bundles on a smooth elliptic curve. Returning to the general case, we establish that the stable categories associated to vector bundles or invariant subspaces of nilpotent operators may be naturally identified as triangulated categories. They satisfy Serre duality and also have tilting objects whose endomorphism rings play a role in singularity theory. In fact, we thus obtain a whole sequence of triangulated (fractional) Calabi-Yau categories, indexed by p, which naturally form an ADE-chain.

Derived equivalences for cotangent bundles of Grassmannians via categorical SL2 actions

Abstract: We construct an equivalence of categories from a strong categorical sl(2) action, following the work of Chuang-Rouquier. As an application, we give an explicit, natural equivalence between the derived categories of coherent sheaves on cotangent bundles to complementary Grassmannians.

Existence and uniqueness of constant mean curvature spheres in

Abstract: We study the classification of immersed constant mean curvature (CMC) spheres in the homogeneous Riemannian 3-manifold Sol_3, i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for every H>1/(\\sqrt{3}), there exists a unique (up to left translations) immersed CMC H sphere S_H in Sol_3 (Hopf-type theorem). Moreover, this sphere S_H is embedded, and is therefore the unique (up to left translations) compact embedded CMC H surface in Sol_3 (Alexandrov-type theorem). The uniqueness parts of these results are also obtained for all real numbers H such that there exists a solution of the isoperimetric problem with mean curvature H.

Convergence of the Kähler–Ricci flow on Fano manifolds

Abstract: Abstract In this article, we give an alternative proof for the convergence of the Kähler –Ricci flow on a Fano manifold (M, J). The proof differs from the one in our previous paper [J. Amer. Math. Sci. 17 (2006), 675 –699]. Moreover, we generalize the main theorem given there to the case that (M, J) may not admit any Kähler–Einstein metrics.

Cohomogeneity one actions on symmetric spaces of noncompact type

Abstract: An isometric action of a Lie group on a Riemannian manifold is of cohomogeneity one if the corresponding orbit space is one-dimensional. In this article we develop a conceptual approach to the classification of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type in terms of orbit equivalence. As a consequence, we find many new examples of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type. We apply our conceptual approach to derive explicit classifications of cohomogeneity one actions on some symmetric spaces.

A Satake isomorphism for representations modulo p of reductive groups over local fields

Abstract: 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 . Let K be a special parabolic 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 ) for the intertwining Hecke algebra of V in G, we define a natural algebra homomorphism from H(G,K, V ) to H(Z,Z ∩ K,V U∩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 Fp. We show that in the general case 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.

Descente galoisienne sur le groupe de Brauer

Abstract: Soit X une variété projective et lisse sur un corps k de caractéristique zéro. Le groupe de Brauer de X s’envoie dans les invariants, sous le groupe de Galois absolu de k, du groupe de Brauer de la même variété considérée sur une clôture algébrique de k. Nous montrons que le quotient est fini. Sous des hypothèses supplémentaires, par exemple sur un corps de nombres, nous donnons des estimations sur l’ordre de ce quotient. L’accouplement d’intersection entre les groupes de diviseurs et de 1-cycles modulo équivalence numérique joue ici un rôle important. For a smooth and projective variety X over a field k of characteristic zero we prove the finiteness of the cokernel of the natural map from the Brauer group ofX to the Galois-invariant subgroup of the Brauer group of the same variety over an algebraic closure of k. Under further conditions, e.g., over a number field, we give estimates for the order of this cokernel. We emphasise the rôle played by the exponent of the discriminant groups of the intersection pairing between the groups of divisors and curves modulo numerical equivalence.

The Witten–Reshetikhin–Turaev invariants of finite order mapping tori I

Abstract: We identify the leading order term of the asymptotic expansion of the Witten-Reshetikhin-Turaev invariants for finite order mapping tori with classical invariants for all simple and simply-connected compact Lie groups. The square root of the Reidemeister torsion is used as a density on the moduli space of flat connections and the leading order term is identified with the integral over this moduli space of this density weighted by a certain phase for each component of the moduli space. We also identify this phase in terms of classical invariants such as Chern-Simons invariants, eta invariants, spectral flow and the rho invariant. As a result, we show agreement with the semiclassical approximation as predicted by the method of stationary phase.

Hypertrees, projections, and moduli of stable rational curves

Abstract: We give a conjectural description for the cone of effective divisors of the Grothendieck-Knudsen moduli space M 0;n of stable ra- tional curves with n marked points. Namely, we introduce new combi- natorial structures called hypertrees and show that they give exceptional divisors onM 0;n with many remarkable properties. x1. INTRODUCTION A major open problem inspired by the pioneering work of Harris and Mumford (HM) on the Kodaira dimension of the moduli space of stable curves, is to understand geometry of its birational models, and in particular to describe its cone of effective divisors and a decomposition of this cone into Mori chambers (HK) encoding ample divisors on birational models. Here we study the genus zero case. The moduli spaces M 0;n parame- trize stable rational curves, i.e., nodal trees of P 1 's with n marked points and without automorphisms. For any subsetI of marked points, M 0;n has a natural boundary divisor I whose general element parametrizes stable rational curves with two irreducible components, one marked by points in I and another marked by points inI c . We will introduce new combinatorial objects called hypertrees with an eye towards the following 1.1. CONJECTURE. The effective cone of M 0;n is generated by boundary divisors and by divisorsD (defined below) parametrized by irreducible hypertrees.

Equivariant algebraic cobordism

Abstract: We construct an equivariant algebraic cobordism theory for schemes with an action by a linear algebraic group over a field of characteristic zero.

Essential dimension of algebraic tori

Abstract: The essential dimension is a numerical invariant of an algebraic group G which may be thought of as a measure of complexity of G-torsors over fields. A recent theorem of N. Karpenko and A. Merkurjev gives a simple formula for the essential dimension of a finite p-group. We obtain similar formulas for the essential p-dimension of a broad class of groups, which includes all algebraic tori.

The Thurston norm and twisted Alexander polynomials

Abstract: Using recent results of Agol, Przytycki-Wise and Wise we show that twisted Alexander polynomials detect the Thurston norm of any irreducible 3-manifold which is not a closed graph manifold.