Variant Of Theorem Research Articles

Stone Pseudovarieties

We extend the theory of profinite algebras, as it is used in the algebraic theory of regular languages, to the more general setting of Stone topological algebras with the ultimate goal of going beyond classes of regular languages. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed topological signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. Looking at the dual Stone spaces of Boolean algebras of subsets of a given topological algebra, we find a simple characterization of when the dual space admits a natural structure of topological algebra; the characterization is given in terms of how the Boolean algebra behaves under the inverses of the operation evaluation mappings of each arity. This provides an alternative, which is presented in the form of an equivalence of categories, to the duality theory of M. Gehrke. As an application, a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is shown to be also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures, which was recently extended to topological signatures jointly by the authors and H. Goulet-Ouellet. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for Stone varieties is also established and it is shown how Reiterman’s theorem can be derived from it.

Filtered colimit elimination from Birkhoff's variety theorem

Birkhoff's variety theorem, a fundamental theorem of universal algebra, asserts that a subclass of a given algebra is definable by equations if and only if it satisfies specific closure properties. In a generalized version of this theorem, closure under filtered colimits is required. However, in some special cases, such as finite-sorted equational theories and ordered algebraic theories, the theorem holds without assuming closure under filtered colimits. We call this phenomenon “filtered colimit elimination,” and study a sufficient condition for it. We show that if a locally finitely presentable category A satisfies a noetherian-like condition, then filtered colimit elimination holds in the generalized Birkhoff's theorem for algebras relative to A.

A vanishing theorem for varieties with finitely many solvable group orbits

AbstractWe reprove and generalize a result proved by Yavin that the intersection cohomology groups of a toric variety with coefficient in a nontrivial rank one local system vanish. We prove a similar vanishing result for a certain class of varieties on which a connected linear solvable group acts, including all spherical varieties.

Analytic Ax–Schanuel for semi-abelian varieties and Nevanlinna theory

Let $A$ be a semi-abelian variety with an exponential map $\exp : \mathop{\mathrm{Lie}}(A) \to A$. The purpose of this paper is to explore Nevanlinna theory of the entire curve $\hskip1pt{\widehat{\mathrm{exp}}}\hskip2pt f := (\exp f, f) : \mathbf{C} \to A \times \mathop{\mathrm{Lie}}(A)$ associated with an entire curve $f : \mathbf{C} \to \mathop{\mathrm{Lie}}(A)$. Firstly we give a Nevanlinna theoretic proof to the analytic Ax–Schanuel Theorem for semi-abelian varieties, which was proved by J. Ax 1972 in the case of formal power series (Ax–Schanuel Theorem). We assume some non-degeneracy condition for $f$ such that the elements of the vector-valued function $f(z) - f(0) \in \mathop{\mathrm{Lie}}(A) \cong \mathbf{C}^n$ are $\mathbf{Q}$-linearly independent in the case of $A = (\mathbf{C}^{*})^{n}$. Our proof is based on the Log Bloch–Ochiai Theorem and a key estimate which we show. Our next aim is to establish a Second Main Theorem for $\hskip1pt{\widehat{\mathrm{exp}}}\hskip2pt f$ and its $k$-jet lifts with truncated counting functions at level one. We give some applications to a problem of a type raised by S. Lang and the unicity. The results clarify a relationship between the problems of Ax–Schanuel type and Nevanlinna theory.

The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus

The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus

On higher dimensional extremal varieties of general type

Relations among fundamental invariants play an important role in algebraic geometry. It is known that an $n$-dimensional variety of general type whose image of its canonical map is of maximal dimension, satisfies $\textrm{Vol} \geq 2 (p_g - n)$. In this article, we investigate the very interesting extremal situation of varieties with $\textrm{Vol} = 2(p_{g} - n)$, which we call Horikawa varieties for they are natural higher dimensional analogues of Horikawa surfaces. We obtain a structure theorem for Horikawa varieties and explore their pluriregularity. We use this to prove optimal results on projective normality of pluricanonical linear systems. We study the fundamental groups of Horikawa varieties, showing that they are simply connected. We prove results on deformations of Horikawa varieties, whose implications on the moduli space make them the higher dimensional analogue of curves of genus 2. Even though there are infinitely many families of Horikawa varieties in any given dimension $n$, we show that when the image of the canonical map is singular, the geometric genus of the Horikawa varieties is bounded by $n + 4$.

An effective decomposition theorem for Schubert varieties

Given a Schubert variety S contained in a Grassmannian Gk(Cl), we show how to obtain further information on the direct summands of the derived pushforward Rπ⁎QS˜ given by the application of the decomposition theorem to a suitable resolution of singularities π:S˜→S. As a by-product, Poincaré polynomial expressions are obtained along with an algorithm which computes the unknown terms in such expressions and which shows that the actual number of direct summands happens to be less than the number of supports of the decomposition.

Infinitesimal categorical Torelli theorems for Fano threefolds

Let X be a smooth Fano variety and Ku(X) its Kuznetsov component. A Torelli theorem for Ku(X) states that Ku(X) is uniquely determined by a certain polarized abelian variety associated to it. An infinitesimal Torelli theorem for X states that the differential of the period map is injective. A categorical variant of the infinitesimal Torelli theorem for X states that the morphism η:H1(X,TX)→HH2(Ku(X)) is injective. In the present article, we use the machinery of Hochschild (co)homology to relate the aforementioned three Torelli-type theorems for smooth Fano varieties via a commutative diagram. As an application, we prove infinitesimal categorical Torelli theorems for a class of prime Fano threefolds. We then prove, infinitesimally, a restatement of the Debarre–Iliev–Manivel conjecture regarding the general fibre of the period map for ordinary Gushel–Mukai threefolds.

Correction to: A Donaldson–Uhlenbeck–Yau theorem for normal varieties and semistable bundles on degenerating families

Correction to: A Donaldson–Uhlenbeck–Yau theorem for normal varieties and semistable bundles on degenerating families

A Donaldson–Uhlenbeck–Yau theorem for normal varieties and semistable bundles on degenerating families

In this paper, we first prove a Donaldson–Uhlenbeck–Yau theorem over projective normal varieties smooth in codimension two. As a consequence we deduce the polystability of (dual) tensor products of stable reflexive sheaves, and we give a new proof of the Bogomolov–Gieseker inequality, along with a precise characterization of the case of equality. This also improves several previously known algebro-geometric results on normalized tautological classes. We study the limiting behavior of semistable bundles over a degenerating family of projective normal varieties. In the case of a family of stable vector bundles, we study the degeneration of the corresponding HYM connections and these can be characterized from the algebro-geometric perspective. In particular, this proves another version of the singular Donaldson–Uhlenbeck–Yau theorem for the normal projective varieties in the central fiber. As an application, we apply the results to the degeneration of stable bundles through the deformation to projective cones, and we explain how our results are related to the Mehta–Ramanathan restriction theorem.

On the distribution of rational points on ramified covers of abelian varieties

We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields $k$ of characteristic zero. For example, given a ramified cover $\pi : X \to A$, where $A$ is an abelian variety over $k$ with a dense set of $k$-rational points, we prove that there is a finite-index coset $C \subset A(k)$ such that $\pi (X(k))$ is disjoint from $C$. Our results do not seem to be in the range of other methods available at present; they confirm predictions coming from Lang's conjectures on rational points, and also go in the direction of an issue raised by Serre regarding possible applications to the inverse Galois problem. Finally, the conclusions of our work may be seen as a sharp version of Hilbert's irreducibility theorem for abelian varieties.

Expectations, concave transforms, chow weights and Roth’s theorem for varieties

We present a new perspective, which is at the intersection of K-stability and Diophantine arithmetic geometry. It builds on work of Boucksom and Chen (Compos Math 147(4):1205–1229, 2011), Boucksom et al. (Math Ann 361(3–4):811–834, 2015), Boucksom et al. (Ann Inst Fourier (Grenoble) 67(2):743–841, 2017), Codogni and Patakfalvi (Invent Math 223(3):811–894, 2021), Fujita (Amer J Math 140(2):391–414, 2018), Fujita (J reine angew Math 751:309–338, 2019), Li (Duke Math J 166(16):3147–3218, 2017), Ferretti (Compos Math 121(3):247–262, 2000), Ferretti (Duke Math J 118(3): 493–522, 2003), McKinnon and Roth (Invent Math 200(2): 513–583, 2015), Ru and Vojta (Amer J Math 142(3):957–991, 2020), Ru and Wang (Algebra Number Theory 11(10): 2323–2337, 2017), Ru and Wang (Int J Number Theory 18(1):61–74, 2022), Heier and Levin (Amer J Math 143(1): 213–226, 2021) and our related works (including Grieve (Michigan Math J 67(2):371–404, 2018), Grieve (Houston J Math 44(4):1181–1202, 2018), Grieve (Asian J Math 24(6):995–1005, 2020), Grieve (Acta Arith 195(4):415–428, 2020), Grieve (Res Number Theory 7(1):1, 14, 2021) and Grieve (C R Math Acad Sci Soc R Can 44(1):16–32, 2022)). Our main results may be described via the expectations of the Duistermaat-Heckman measures. For example, we prove that the expected orders of vanishing for very ample linear series along subvarieties may be calculated via the theory of Chow weights and Okounkov bodies. This result provides a novel unified viewpoint to the existing works that surround these topics. It expands on work of Mumford (Enseignement Math (2) 23(1-2):39–110, 1977), Odaka (Ann of Math (2) 177(2):645–661, 2013) and Fujita (Amer J Math 140(2):391–414, 2018). Moreover, it has applications for Diophantine approximation and K-stability. As one result of this flavour, we define a concept of uniform arithmetic $$\mathrm {K}$$ -stability for Kawamata log terminal pairs. We then establish a form of K. F. Roth’s celebrated approximation theorem for certain collections of divisorial valuations which fail to be uniformly arithmetically $$\mathrm {K}$$ -stable. This result builds on the Roth type theorems of McKinnon and Roth (Invent Math 200(2):513–583, 2015). Moreover, it complements the concept of arithmetic canonical boundedness, in the sense of McKinnon and Satriano (Trans Amer Math Soc 374(5):3557–3577, 2021), in addition to our results that establish instances of Vojta’s Main Conjecture for $${\mathbb {Q}}$$ -Fano varieties Grieve (Asian J Math 24(6):995–1005, 2020).

An effective function field version of Schmidt's subspace theorem for projective varieties, with arbitrary families of homogeneous polynomials

In this paper, by giving a new lower bound estimate for the Chow weight of a projective variety, we will prove an effective Schmidt subspace theorem for projective varieties on the rational function field with arbitrary families of higher degree homogeneous polynomials. Our result is a generalization and also an improvement of the previous results.

Correction to “A finite basis theorem for difference-term varieties with a finite residual bound”

There is a gap in our proof [Trans. Amer. Math. Soc. 368 (2016), pp. 2115–2143, Lemma 6.2]. We direct readers to a paper that fills the gap.

Eilenberg's variety theorem without Boolean operations

Eilenberg's variety theorem marked a milestone in the algebraic theory of regular languages by establishing a formal correspondence between properties of regular languages and properties of finite monoids recognizing them. Motivated by classes of languages accepted by quantum finite automata, we introduce basic varieties of regular languages, a weakening of Eilenberg's original concept that does not require closure under any boolean operations, and prove a variety theorem for them. To do so, we investigate the algebraic recognition of languages by lattice bimodules, generalizing Klíma and Polák's lattice algebras, and we utilize the duality between algebraic completely distributive lattices and posets.

On the Bertini regularity theorem for arithmetic varieties

On the Bertini regularity theorem for arithmetic varieties

A global Torelli theorem for singular symplectic varieties

We systematically study the moduli theory of symplectic varieties (in the sense of Beauville) which admit a resolution by an irreducible symplectic manifold. In particular, we prove an analog of Verbitsky’s global Torelli theorem for the locally trivial deformations of such varieties. Verbitsky’s work on ergodic complex structures replaces twistor lines as the essential global input. In so doing we extend many of the local deformation-theoretic results known in the smooth case to such (not-necessarily-projective) symplectic varieties. We deduce a number of applications to the birational geometry of symplectic manifolds, including some results on the classification of birational contractions of K3^[n] -type varieties.

Mixed Ax–Schanuel for the universal abelian varieties and some applications

In this paper we prove the mixed Ax–Schanuel theorem for the universal abelian varieties (more generally any mixed Shimura variety of Kuga type), and give some simple applications. In particular, we present an application for studying the generic rank of the Betti map.

On weighted $$\ell ^p$$- convergence of Fourier series: a variant of theorems of Wiener and Lévy

Let $$1<p<\infty$$ , let q be the conjugate index of p, and let $$\omega$$ be an almost monotone algebra weight on $$\mathbb Z$$ . Let f be a complex valued continuous function on the unit circle such that Fourier series of f is p-th power $$\omega$$ -absolutely convergent, i.e., $$\sum _{n\in {\mathbb {Z}}}|{\widehat{f}}(n)|^p \omega (n)^p<\infty$$ . If f is nowhere vanishing, then there is an almost monotone algebra weight $$\nu$$ on $${\mathbb {Z}}$$ such that $$\nu \le K\omega$$ for some $$K>0$$ and the Fourier series of $$\frac{1}{f}$$ is p-th power $$\nu$$ -absolutely convergent. If $$\varphi$$ is holomorphic on the range of f, then there exists an almost monotone algebra weight $$\chi$$ on $${\mathbb {Z}}$$ such that $$\chi \le K\omega$$ for some constant $$K>0$$ and the Fourier series of $$\varphi \circ f$$ is p-th power $$\chi$$ -absolutely convergent. This gives p-th power analogue of a Theorem by Bhatt and Dedania; and rectifies Theorems of Kinani and Bouchikhi.

Line bundles on rigid varieties and Hodge symmetry

We prove several related results on the low-degree Hodge numbers of proper smooth rigid analytic varieties over non-archimedean fields. Our arguments rely on known structure theorems for the relevant Picard varieties, together with recent advances in p-adic Hodge theory. We also define a rigid analytic Albanese naturally associated with any smooth proper rigid space.