Using results from non-abelian Hodge theory for klt spaces developed by Greb, Kebekus, Peternell and Taji, we deduce necessary and sufficient conditions for projective varieties with klt singularities to be uniformized by bounded symmetric domains. This generalizes a well known result of Simpson to the singular setting. We apply this to obtain explicit Miyaoka-Yau-type equalities to characterize singular quotients of the four classical irreducible bounded symmetric domains, and the polydisk.
Various results about a divisor function with Diophantine properties are obtained, including a simple asymptotic formula for its sum and a Voronoï-type formula. The proofs rely on analytic properties of certain Dirichlet series that are expressed in terms of Hecke’s zeta functions with Grössencharaktere associated to a real quadratic number field. Also used are new estimates and asymptotics for the standard hypergeometric function that are uniform in parameters, which are of independent interest.
The aim of this paper is to establish multiple positive normalized solutions (u,v,λ_1,λ_2)∈ H^1(ℝ^N,ℝ^2)×ℝ^2 to the following coupled Schrödinger system involving Sobolev critical exponent: -Δu+λ_1 u=μ_1|u|^p-2u+να|u|^α-2u|v|^β, x∈ℝ^N, -Δv+λ_2 v=μ_2|v|^q-2v+νβ|v|^β-2v|u|^α, x∈ℝ^N, ∫_ℝ^N|u|^2dx=a, ∫_ℝ^N|v|^2dx=b, N≥ 3, where μ_1,μ_2, ν, a, b>0. We are particularly interested in the mass mixed case that 2 1, β>1, and α+β=2^*:=2N/N-2. For sufficiently small ν>0, we demonstrate that the above system admits two positive solutions, one of which serves as a local minimizer, and the other as a mountain pass solution. By developing some new technical lemmas on the interaction estimates, we are managed to resolves Soave's open problem [J. Funct. Anal., 2020, Remark 1.1] within the context of the system case. Notably, our existence result holds true for all dimensions N≥ 3. Our results also significantly extend the result of Gou and Jeanjean [Nonlinearity, 2018, Theorem 1.1] to the Sobolev critical coupled case and removing the hypothesis “either p,q≤ α+β-2/N or |p-q|≤2/N" for N≥ 5. Additionally, we also establish a sequence of properties for the local minimizer, including local uniqueness, continuity with respect to the small parameter ν, and the limiting profiles for ν→ 0^+.
We provide several sufficient symmetry conditions for an embedded free boundary minimal annulus in the unit 3-ball to be congruent to the critical catenoid. First, we show that an annulus with either two symmetry planes or a symmetry plane that does not intersect the boundary must be congruent to the critical catenoid. As a corollary, any embedded free boundary minimal annulus in the half-ball satisfying certain boundary conditions is also the critical catenoid. Building on a symmetry principle, we further show that if the boundary consists of two congruent components and is invariant under reflection through a plane, then the surface is congruent to the critical catenoid.
For every stable presentably symmetric monoidal ∞ -category 𝒞 and every non-unital ∞ -operad 𝒪 in 𝒞 , where , we construct a Koszul duality adjunction TQ_𝒪: Alg_𝒪(𝒞) ⇆Coalg_𝒪^∨(𝒞): Prim_𝒪 between 𝒪 -algebras in 𝒞 and coalgebras over the Koszul dual ∞ -cooperad of 𝒪 . We prove that if all norm maps in 𝒞 associated to symmetric groups are equivalences, the unit of Koszul duality X →Prim_𝒪(TQ_𝒪(X)) identifies with the canonical map X → X^∧ := lim _n ≥ 1τ _n(𝒪) ∘ _𝒪X to the limit of the TQ_𝒪 -completion tower X ≃𝒪∘ _𝒪X → ... →τ _n(𝒪) ∘ _𝒪X → ... →τ _1(𝒪) ∘ _𝒪X= TQ_𝒪(X). We apply this result to the Koszul duality between the shifted spectral Lie ∞ -operad and the cocommutative cooperad to construct a derived enveloping Hopf algebra functor Alg_Lie(𝒞) →Hopf(𝒞) from Lie algebras in 𝒞 to cocommutative Hopf algebras in 𝒞 and deduce a derived version of the Milnor-Moore theorem: for every rational stable presentably symmetric monoidal ∞ -category 𝒞 the derived enveloping Hopf algebra functor Alg_Lie(𝒞) →Hopf(𝒞) is fully faithful.
In this paper, first we revisit the formal integration of Lie algebras, which give rise to braces in some special cases. Then we establish the formal integration theory for complete Rota-Baxter Lie algebras, that is, we show that there is a Rota-Baxter group with the underlying group structure given by the Baker-Campbell-Hausdorff formula, associated to any complete Rota-Baxter Lie algebra. In particular, we use the post-Lie Magnus expansion to give the explicit formula of the Rota-Baxter operator. Finally we show that one can obtain a graded Rota-Baxter Lie ring from a filtered Rota-Baxter group.
We study the future stability of cosmological fluids, in spacetimes with an accelerated expansion, which exhibit extreme tilt behavior, i.e. their fluid velocity becoming asymptotically null at timelike infinity. It has been predicted in the article [17] that / the latter behavior is dominant for sound speeds beyond radiation cs = 1 root 3, hence, bifurcating off of the stable orthogonal fluid behavior modeled by the classical FLRW family of solutions, for c(s)(2)E [0, 31]. First, we construct homogeneous solutions to the Einstein-Euler system with the latter behavior, in S3 spatial topology, for sound speeds c(s)(2)E (3/1, 1). Then, we study their future dynamics and prove a global stability result in the restricted range c(s)(2)E (3/1, 3/ 7). In particular, we show that extreme tilt behavior persists to sufficiently small perturbations of the homogeneous backgrounds, without any symmetry assumptions or analyticity. Our method is based on a bootstrap argument, in weighted Sobolev spaces, capturing the exponential decay of suitable renormalized variables. Extreme tilt behavior is associated with a degeneracy in the top order energy estimates that we derive, which allows us to complete our bootstrap argument only in the aforementioned restricted range of sound speeds. Interestingly, this is a degeneracy that does not appear in the study of formal series expansions. Moreover, for the Euler equations on a fixed FLRW background, our estimates can be improved to treat the entire beyond radiation interval c(s)(2)E (3/1, 1), a result already obtained in [21]. The latter indicates that the former issue is related to the general inhomogeneous geometry of the perturbed metric in the coupled to Einstein case.
We study the future stability of cosmological fluids, in spacetimes with an accelerated expansion, which exhibit extreme tilt behavior, i.e. their fluid velocity becoming asymptotically null at timelike infinity. It has been predicted in the article [17] that the latter behavior is dominant for sound speeds beyond radiation c_s=1/√(3) , hence, bifurcating off of the stable orthogonal fluid behavior modeled by the classical FLRW family of solutions, for c_s^2∈ [0,1/3] . First, we construct homogeneous solutions to the Einstein-Euler system with the latter behavior, in 𝕊^3 spatial topology, for sound speeds c_s^2∈ (1/3,1) . Then, we study their future dynamics and prove a global stability result in the restricted range c_s^2∈ (1/3,3/7) . In particular, we show that extreme tilt behavior persists to sufficiently small perturbations of the homogeneous backgrounds, without any symmetry assumptions or analyticity. Our method is based on a bootstrap argument, in weighted Sobolev spaces, capturing the exponential decay of suitable renormalized variables. Extreme tilt behavior is associated with a degeneracy in the top order energy estimates that we derive, which allows us to complete our bootstrap argument only in the aforementioned restricted range of sound speeds. Interestingly, this is a degeneracy that does not appear in the study of formal series expansions. Moreover, for the Euler equations on a fixed FLRW background, our estimates can be improved to treat the entire beyond radiation interval c_s^2∈ (1/3,1) , a result already obtained in [21]. The latter indicates that the former issue is related to the general inhomogeneous geometry of the perturbed metric in the coupled to Einstein case.
We prove equivalence of two integral representations for the wave functions of hyperbolic Calogero–Sutherland system. For this we study two families of Baxter operators related to hyperbolic Calogero–Sutherland and rational Ruijsenaars models; the first one as a limit from hyperbolic Ruijsenaars system, while the second one independently. Besides, computing asymptotics of integral representations and also the value at zero point, we identify them with renormalized Heckman–Opdam 𝔤𝔩_n hypergeometric function.
We show that for any separably closed field $k$ of characteristic $p>0$, the canonical functor from nilpotent $p$-adic spaces to $\mathbb{E}_{\infty}$-coalgebras over $k$ (given by singular chains with coefficients in $k$) is fully faithful. We also identify the essential image of simply connected spaces inside coalgebras. This dualizes and removes finiteness assumptions from a theorem of Mandell.
For a real n× m matrix ξ , we consider its sequence of best Diophantine approximation vectors x_i ∈ℤ^m, i =1,2,3,... , the sequences of its norms X_i = ‖x_i‖ and the norms of remainders L_i = ‖ξx_i‖ . It is known that, in the cases m=1 , bad approximability of ξ is equivalent to the boundedness of ratios X_i+1/X_i , while for n=1 bad approximability of ξ is equivalent to the boundedness of ratios L_i/L_i+1 . Moreover, carefully constructed example show that in the cases m=1 and n=1 boundedness of ratios L_i/L_i+1 and X_i+1/X_i respectively (the order of ratios changed), does not imply bad approximability of ξ . In the present paper, we study the impact of the boundedness of ratios on Diophantine properties of ξ , in particular, what restrictions it gives for Diophantine exponents ω (ξ) and ω̂(ξ) . One of our particular results deals with the case m=n=2 . We prove that for 2× 2 matrices ξ boundedness of both ratios X_i+1/X_i, L_i/L_i+1 implies inequality ω̂(ξ)⩽4/3 and that this result is optimal. Our methods combine parametric geometry of numbers as well as more classical tools.
We revise the notion of the blobbed topological recursion by extending it to the setting of generalized topological recursion as well as allowing blobs which do not necessarily admit topological expansion. We show that the so-called non-perturbative differentials form a special case of this revisited version of blobbed topological recursion. Furthermore, we prove the KP integrability of the differentials of blobbed topological recursion for the input data that include KP-integrable blobs. This result generalizes, unifies, and gives a new proof of the KP integrability of nonperturbative differentials conjectured by Borot–Eynard and recently proved by the authors.
We establish formulas for the Hilbert series of the Chow ring of a polymatroid using arbitrary building sets. For braid matroids and minimal building sets, our results produce new formulas for the Poincaré polynomial of the moduli space ℳ_0,n+1 of pointed stable rational curves, and recover several previous results by Keel, Getzler, Manin, and Aluffi–Marcolli–Nascimento. We also use our methods to produce examples of matroids and building sets for which the corresponding Chow ring has Hilbert series with non-log-concave coefficients. This contrasts with the real-rootedness and log-concavity conjectures of Ferroni–Schröter for matroids with maximal building sets, and of Aluffi–Chen–Marcolli for braid matroids with minimal building sets.
We continue our work on the model theory of free lattices, solving two of the main open problems from our first paper on the subject. Our main result is that the universal (existential) theory of infinite free lattices is decidable. Our second main result is a proof that finitely generated free lattices are positively distinguishable, as for each n ⩾ 1 there is a positive ∃∀ -sentence true in F_n and false in F_n+1 . Finally, we show that free lattices are first-order rigid in the class of finitely generated projective lattices, and that a projective lattice has the same existential (universal) theory of an infinite free lattice if and only if it has breadth > 4 (i.e., a single existential sentence is sufficient).
In this paper, we prove a Beilinson-type formula for the V-filtration of Kashiwara and Malgrange on a complex mixed Hodge module, using Hodge filtrations on the localization. Our formula expresses the V-filtration as the filtered 𝒟 -module underlying a pro-mixed Hodge module. We apply this to the theory of higher multiplier and Hodge ideals. Our first result shows that higher multiplier ideals can be obtained directly from Hodge ideals by taking a suitable limit. As a corollary, we deduce that Hodge ideals are left semi-continuous if and only if they coincide with higher multiplier ideals, thereby improving results of Saito and Mustaţă–Popa and resolving a folklore question. We further prove a birational transformation formula for higher multiplier ideals, generalising the classical formula for multiplier ideals and answering a question of Schnell and the second author. Finally, we provide very quick proofs of the main vanishing theorems for Hodge ideals, and strengthen a result of B. Chen.
Mutations occur in multiple algebraic contexts, often enjoying good combinatorial properties. In this paper we study mutations of pure-injective cosilting objects in compactly generated triangulated categories from a topological point of view. We consider the topologies studied by Gabriel, Burke and Prest on the set of indecomposable injective objects in a Grothendieck abelian category, transfer them to associated cosilting subcategories, and show that, in that context, right mutation induces a homeomorphism on two complementary subspaces. We then improve this result in the context of the derived category of a commutative noetherian ring, showing that right mutation is an open bijection. We end the paper with a detailed analysis of a range of cosilting subcategories over commutative noetherian rings for which the topology is completely known. As a byproduct of this analysis, we obtain that the category of modules over a commutative noetherian ring is the unique locally noetherian Grothendieck category in its derived-equivalence class.
We study the K-theoretic enumerative geometry of cyclic Nakajima quiver varieties, with particular focus on Hilb^n([ℂ^2/ℤ_l]) , the equivariant Hilbert scheme of points on ℂ^2 . The direct sum over n of the equivariant K-theories of these varieties is known to be isomorphic to the ring symmetric functions in l colors, with structure sheaves of torus fixed points identified with wreath Macdonald polynomials. Using properties of wreath Macdonald polynomials and the recent identification of the Maulik-Okounkov quantum affine algebra for cyclic quivers with the quantum toroidal algebras of type A, we derive an explicit formula for the generating function of capped vertex functions of Hilb^n([ℂ^2/ℤ_l]) with descendants given by exterior powers of the 0th tautological bundle. We also sharpen the large framing vanishing results of Okounkov, providing a class of descendants and cyclic quiver varieties for which the capped vertex functions are purely classical. Our results also suggest certain integrality and wall-crossing conjectures for capped vertex functions.
This paper studies rational functions 𝔍_α(q), which depend on a positive element α of the root lattice of a root system. These functions arise as Shapovalov pairings of Whittaker vectors in Verma modules of highest weight -ρ for quantum groups and as Hilbert series of Zastava spaces, and are related to the Toda system. They are specializations of multivariate functions more commonly studied in the literature. We investigate the denominator of these rational functions and give an explicit combinatorial formula for the numerator in type A. We also propose a conjectural realization of the numerator as the Poincaré polynomial of a smooth variety in type A.
