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According to a theorem of Cantat [Ann. of Math. (2) 174 (2011), pp. 299–340] and Urech [J. Reine Angew. Math. 770 (2021), pp. 27–57], an analog of the classical Tits alternative holds for the group of birational automorphisms of a compact complex Kähler surface. We established in Arzhantsev and Zaidenberg [Int. Math. Res. Not. IMRN 2022 (2022), pp. 8162–8195] the following Tits-type alternative: if X is a toric affine variety and G\subset \operatorname {Aut}(X) is a subgroup generated by a finite set of unipotent subgroups normalized by the acting torus then either G contains a nonabelian free subgroup or G is a unipotent affine algebraic group. In the present paper we extend the latter result to any group G of automorphisms of a complex affine surface generated by a finite collection of unipotent algebraic subgroups. It occurs that either G contains a nonabelian free subgroup or G is a metabelian unipotent algebraic group. [ABSTRACT FROM AUTHOR]

In this paper we show that evolution algebras over any given field \Bbbk are universally finite. In other words, given any finite group G, there exist infinitely many regular evolution algebras X such that Aut(X)\cong G. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme \mathbf {G} over \Bbbk is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra. [ABSTRACT FROM AUTHOR]

We study a fourth order partial differential equation (*) uijDijw = 0, w ≡ [det D2u]-θ, θ ≠ 0 of affine maximal type, which has attracted much attention in recent years. These models include the affine maximal equation for θ = N+1/N+2 (see, for example, [Invent. Math. 140 (2000), pp. 399-422]) and the Abreu equation for θ = 1 (see, for example, [Int. J. Math. 9 (1998), pp. 641-651] or [Calc. Var. Partial Differential Equations 43 (2012), pp. 25-44]. In this paper, we will prove a Bernstein theorem of (*) for all dimension N ≥ 2 and θ ∈ (0, 1/2 − √N−2/2√N) ∪ (1/2 + √N−2/2√N,+ ∞) under decaying convexity. Our result covers the affine maximal equation (θ = N+1/N+2) for dimension N ≤ 3 or the Abreu equation (θ = 1) for all dimension N\geq 1, which largely improves both cases by Du in [Nonlinear Anal. 187 (2019), pp. 170-179] (N=2 for θ = N+1/N+2 and N ≤ 9 for θ = 1). [ABSTRACT FROM AUTHOR]

Given an affine algebraic variety X, we prove that if the neutral component \mathrm {Aut}^\circ (X) of the automorphism group consists of algebraic elements, then it is nested, i.e., is a direct limit of algebraic subgroups. This improves our earlier result (see Perepechko and Regeta [Transform. Groups 28 (2023), pp. 401–412]). To prove it, we obtain the following fact. If a connected ind-group G contains a closed connected nested ind-subgroup H\subset G, and for any g\in G some positive power of g belongs to H, then G=H. [ABSTRACT FROM AUTHOR]

We produce counterexamples to the birational Torelli theorem for Calabi–Yau manifolds in arbitrarily high dimension: this is done by exhibiting a series of non-birational pairs of Calabi–Yau (n^2-1)-folds which, for n \geq 2 even, admit an isometry between their middle cohomologies. These varieties also satisfy an \mathbb {L}-equivalence relation in the Grothendieck ring of varieties, i.e. the difference of their classes annihilates a power of the class of the affine line. We state this last property for a broader class of Calabi–Yau pairs, namely all those which are realized as pushforwards of a general (1,1)-section on a homogeneous roof in the sense of Kanemitsu, along its two extremal contractions. [ABSTRACT FROM AUTHOR]

We give an exact formula for the dimension of the variety of homomorphisms from Sg to any semisimple real algebraic group, where Sg is a surface group of genus g = 2. [ABSTRACT FROM AUTHOR]

In this paper, the relative p-affine capacities are introduced, developed, and subsequently applied to the trace theory of affine Sobolev spaces. In particular, we geometrically characterize such a nonnegative Radon measure μ given on an open set O ⊆ Rn that naturally induces an embedding of the p-affine Sobolev class W1,p0,d (O) into the Lebesgue space Lq(O, μ) (under 1 ⩽ p ⩽ q < ∝) and the exponentially-integrable Lebesgue space exp ((nω1/nn ∣f∣)n/(n-1)) ∊ L1(O, μ) (under p = n) as well as the Lebesgue space L∝(O, μ) (under n < p < ∝) with μ(O) < ∝. The results discovered here are new and nontrivial. [ABSTRACT FROM AUTHOR]

Let G x X → X be an action of the connected algebraic group G on the irreducible, affine variety X. We discuss the relationship between [k[X]G] and k(X)G, where [k[X]G] denotes the quotient field of k[X]G. We are particularly interested in the following three questions. (1) When is the inclusion [k[X]G] ⊆k(X)G a finite extension of fields? (2) What is the role of G-invariant divisors? (3) What is the exact characterization of "observable in codimension one"? [ABSTRACT FROM AUTHOR]

We investigate flexibility of affine varieties with an action of a linear algebraic group. Flexibility of a smooth affine variety with only constant invertible functions and a locally transitive action of a reductive group is proved. Also we show that a normal affine complexity-zero horospherical variety with only constant invertible functions is flexible. [ABSTRACT FROM AUTHOR]

We prove that the multiplicities of certain maximal weights of g(An(1) )-modules are counted by pattern avoidance on words. This proves and generalizes a conjecture of Jayne-Misra. We also prove similar phenomena in types A2n(2) and Dn+1(2). Both proofs are applications of Kashiwara’s crystal theory. [ABSTRACT FROM AUTHOR]

We study the Lq-dimensions of self-affine measures and the Käenmäki measure on a class of self-affine sets similar to the class considered by Hueter and Lalley. We give simple, checkable conditions under which the Lq-dimensions are equal to the value predicted by Falconer for a range of q. As a corollary this gives a wider class of self-affine sets for which the Hausdorff dimension can be explicitly calculated. Our proof combines the potential theoretic approach developed by Hunt and Kaloshin with recent advances in the dynamics of self-affine sets. [ABSTRACT FROM AUTHOR]

We construct new irreducible weight modules over quantum affine algebras of type I with all weight spaces infinite dimensional. These modules are obtained by parabolic induction from irreducible modules over the Heisenberg subalgebra. [ABSTRACT FROM AUTHOR]

The basic result of Oka theory, due to Gromov, states that every continuous map f from a Stein manifold S to an elliptic manifold X can be deformed to a holomorphic map. It is natural to ask whether this can be done for all f at once, in a way that depends continuously on f and leaves f fixed if it is holomorphic to begin with. In other words, is O(S, X) a deformation retract of C(S,X)? We prove that it is if S has a strictly plurisubharmonic Morse exhaustion with finitely many critical points, in particular, if S is affine algebraic. The only property of X used in the proof is the parametric Oka property with approximation with respect to finite polyhedra, so our theorem holds under the weaker assumption that X is an Oka manifold. Our main tool, apart from Oka theory itself, is the theory of absolute neighbourhood retracts. We also make use of the mixed model structure on the category of topological spaces. [ABSTRACT FROM AUTHOR]