Showing papers in "Compositio Mathematica in 2021"
TL;DR: In this article, an arithmetic count of the lines on a smooth cubic surface over an arbitrary field was given and generalized to complex and real algebraic geometry can be obtained with similar methods.
Abstract: We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field ,
\[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \]
where -homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.
30 citations
TL;DR: In this paper, it was shown that there exists a unique two-parameter -algebras, and the existence of such a two-dimensional graph is proven. But
Abstract: We prove the longstanding physics conjecture that there exists a unique two-parameter -algebras
21 citations
TL;DR: In this article, the authors proposed a method to combine the NSF and the Korean National Research Foundation (NRF) to solve the problem of artificial neural networks in the field of artificial intelligence.
Abstract: NSFNational Science Foundation (NSF) [DMS-1464693, DMS-1600653]; National Research Foundation of Korea (NRF) - Korean government (MSIT) [2019R1C1C1003473]
18 citations
TL;DR: For quantum symmetric pairs, this paper avoided a case-by-case rank-one analysis and removed the strong constraints on the parameters in a previous work, by avoiding the strong constraint on the symmetric parameters.
Abstract: For quantum symmetric pairs -canonical bases, by avoiding a case-by-case rank-one analysis and removing the strong constraints on the parameters in a previous work.
15 citations
TL;DR: In this paper, the analog of the Morel-Voevodsky localization theorem for framed motivic spaces was proved for arbitrary schemes, and a new construction of the motivic cohomology of arbitrary schemes was given.
Abstract: We prove the analog of the Morel–Voevodsky localization theorem for framed motivic spaces. We deduce that framed motivic spectra are equivalent to motivic spectra over arbitrary schemes, and we give a new construction of the motivic cohomology of arbitrary schemes.
13 citations
TL;DR: In this paper, a constructive existence theorem for abelian envelopes of non-abelian monoidal categories is proved for universal tensor categories with positive characteristic via tilting modules.
Abstract: We prove a constructive existence theorem for abelian envelopes of non-abelian monoidal categories. This establishes a new tool for the construction of tensor categories. As an example we obtain new proofs for the existence of several universal tensor categories as conjectured by Deligne. Another example constructs interesting tensor categories in positive characteristic via tilting modules for .
13 citations
TL;DR: In this article, it was shown that the algebraic $K-theory of valuation rings behave as though such rings were regular Noetherian, in particular an analogue of the Geisser-Levine theorem.
Abstract: We prove several results showing that the algebraic $K$-theory of valuation rings behave as though such rings were regular Noetherian, in particular an analogue of the Geisser--Levine theorem. We also give some new proofs of known results concerning cdh descent of algebraic $K$-theory.
10 citations
TL;DR: In this article, the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves is studied, and it is shown that all Hilbert schemes are encoded in the fine compactified Jacobians of connected subcurves via the perverse Leray filtration.
Abstract: We study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration. We also prove, along the way, a result of independent interest, giving sufficient conditions for smoothness of the total space of the relative compactified Jacobian of a family of locally planar curves.
10 citations
TL;DR: In this paper, the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor classes was studied, and it was shown that the 3-category of rigid tensor tensors with enough compact projectives is 2-dualizable.
Abstract: We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of .
8 citations
TL;DR: In this paper, a bimodule description of the Elias-Williamson category and re-prove the categorization theorem are given. But they do not give a description of a Coxeter system and a representation.
Abstract: For a Coxeter system and a representation . In this paper, we give a bimodule description of the Elias–Williamson category and re-prove the categorification theorem.
7 citations
TL;DR: In this article, it was shown that there are finitely many families of Calabi-Yau manifolds numerically trivial and not of product type, in dimension at most four.
Abstract: We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi–Yau manifolds numerically trivial and not of product type, in dimension at most four.
TL;DR: In this paper, the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds was shown.
Abstract: We show a rigidity theorem for the Seiberg–Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. These non-smoothable topological families provide new examples of -manifolds.
TL;DR: In this article, the additive higher Chow groups of regular schemes over a field induce a Zariski sheaf of pro-differential graded algebras, the Milnor range of which is isomorphic to the big de Rham-Witt complexes.
Abstract: We show that the additive higher Chow groups of regular schemes over a field induce a Zariski sheaf of pro-differential graded algebras, the Milnor range of which is isomorphic to the Zariski sheaf of big de Rham–Witt complexes. This provides an explicit cycle-theoretic description of the big de Rham–Witt sheaves. Several applications are derived.
TL;DR: In this paper, a necessary and sufficient condition for the graded algebra of automorphic forms on a symmetric domain of type IV being free was proved. But the necessary condition was not defined.
Abstract: We prove a necessary and sufficient condition for the graded algebra of automorphic forms on a symmetric domain of type IV being free. From the necessary condition, we derive a classification result. Let is a free algebra. Using the sufficient condition, we recover some well-known results.
TL;DR: In this article, the authors apply the results of Gross-Zagier, Gross-Kohnen and Zagier and their generalizations to give a short proof that the differences of singular moduli are not units.
Abstract: In this note, we will apply the results of Gross–Zagier, Gross–Kohnen–Zagier and their generalizations to give a short proof that the differences of singular moduli are not units. As a consequence, we obtain a result on isogenies between reductions of CM elliptic curves.
TL;DR: In this article, the strong Suslin reciprocity law conjectured by A. Goncharov was shown to be a generalization of Weil reciprocity to higher Milnor, and a hyperbolic polytope was constructed up to scissors congruence.
Abstract: We prove the strong Suslin reciprocity law conjectured by A. Goncharov. The Suslin reciprocity law is a generalization of the Weil reciprocity law to higher Milnor we construct a hyperbolic polytope (defined up to scissors congruence). The hyperbolic volume and the Dehn invariant of this polytope can be computed directly from the triple of rational functions on the curve.
TL;DR: In this article, the authors consider generic diffeomorphisms in a small neighborhood of the diffeomorphic structure under consideration and show that the super-exponential growth of number of periodic points can be achieved.
Abstract: We consider -generic diffeomorphisms in a small neighborhood of the diffeomorphism under consideration exhibit super-exponential growth of number of periodic points. We also give examples which show the necessity of the conditions we assume.
TL;DR: In this article, a framework was proposed to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle conjecture and good uniformity estimates.
Abstract: We propose a framework to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle's conjecture and good uniformity estimates. Using this method, we prove Malle's conjecture for quintic extensions over arbitrary number fields by adapting Bhargava's geometric sieve and averaging over fundamental domains of the parametrization space.
TL;DR: In this paper, a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety of Picard number 1 to its derived category of coherent sheaves was proposed.
Abstract: In our previous paper we suggested a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety of Picard number 1 to the structure of its derived category of coherent sheaves. Here we generalize this conjecture, make it more precise, and support it by the examples of (co)adjoint homogeneous varieties of simple algebraic groups of Dynkin types this is the first exceptional collection proved to be full.
TL;DR: In this article, a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli, was constructed using an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar.
Abstract: We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each . The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.
TL;DR: In this article, the trace of the n-framed surgery on a knot in S3 is a 4-manifold homotopy equivalent to the 2-sphere.
Abstract: The trace of the n-framed surgery on a knot in S3 is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded 2-sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable 3-dimensional knot invariants. For each n, this provides conditions that imply a knot is topologically n-shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.
TL;DR: In this article, a variant of the Vorst's conjecture has been shown in positive characteristic, which relates the regularity of a ring with the -theory to its regularity.
Abstract: Vorst's conjecture relates the regularity of a ring with the -theory. We show a variant of this conjecture in positive characteristic.
TL;DR: For DG Lie algebras equipped with a cyclic structure of degree 2 which is non-degenerate in cohomology, and does not rely on previous results on the same subject, see as mentioned in this paper.
Abstract: Let minimal models of DG Lie algebras equipped with a cyclic structure of degree 2 which is non-degenerate in cohomology, and does not rely (even for K3 surfaces) on previous results on the same subject.
TL;DR: In this article, an overlooked hypothesis in the definition of generically stable subset of the space of arcs X ∞ of a variety X defined over a perfect field k was revealed.
Abstract: The purpose of this note is to correct a mistake in the article “A curve selection lemma in spaces of arcs and the image of the Nash map” Compositio Math. 142 (2006), 119–130. It is due to an overlooked hypothesis in the definition of generically stable subset of the space of arcs X∞ of a variety X defined over a perfect field k.
TL;DR: In this article, the authors study the family of elliptic curves and prove upper bounds on the number of ellipses with bounded height, whose discriminants are divisible by high powers of primes.
Abstract: In this paper we study the family of elliptic curves . The key new ingredients necessary for the proofs are ‘uniformity estimates’, namely upper bounds on the number of elliptic curves with bounded height, whose discriminants are divisible by high powers of primes.
TL;DR: In this article, irregular constructible sheaves, which are -modules, are introduced and the algebraic version of the theory is also developed; see Section 2.2.1.
Abstract: We introduce irregular constructible sheaves, which are -modules. We also develop the algebraic version of the theory.
TL;DR: In this paper, a class of two-variable polynomials called exact polynomial which contains polynomial and gives a topological interpretation of its Mahler measure is studied.
Abstract: We study a class of two-variable polynomials called exact polynomials which contains -polynomial and give a topological interpretation of its Mahler measure.
TL;DR: In this article, the essential dimension of an unramified, non-abelian covering of a proper algebraic variety is shown to be bounded by a nontrivial lower bound.
Abstract: Consider the algebraic function is proper. As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety.