Considering a (co)homology theory $\mathbb{T}$ on a base category $\mathcal{C}$ as a fragment of a first-order logical theory we here construct an abelian category $\mathcal{A}[\mathbb{T}]$ which is universal with respect to models of $\mathbb{T}$ in abelian categories. Under mild conditions on the base category $\mathcal{C}$, e.g. for the category of algebraic schemes, we get a functor from $\mathcal{C}$ to ${\rm Ch}({\rm Ind}(\mathcal{A}[\mathbb{T}]))$ the category of chain complexes of ind-objects of $\mathcal{A}[\mathbb{T}]$. This functor lifts Nori's motivic functor for algebraic schemes defined over a subfield of the complex numbers. Furthermore, we construct a triangulated functor from $D({\rm Ind}(\mathcal{A}[\mathbb{T}]))$ to Voevodsky's motivic complexes.
We consider D-branes in string theory and address the issue of how to describe them mathematically as a fundamental object (as opposed to a solitonic object) of string theory in the realm in differential and symplectic geometry. The notion of continuous maps, $k$-times differentiable maps, and smooth maps from an Azumaya/matrix manifold with a fundamental module to a (commutative) real manifold $Y$ is developed. Such maps are meant to describe D-branes or matrix branes in string theory when these branes are light and soft with only small enough or even zero brane-tension. When $Y$ is a symplectic manifold (resp. a Calabi-Yau manifold; a $7$-manifold with $G_2$-holonomy; a manifold with an almost complex structure $J$), the corresponding notion of Lagrangian maps (resp. special Lagrangian maps; associative maps, coassociative maps; $J$-holomorphic maps) are introduced. Indicative examples linking to symplectic geometry and string theory are given. This provides us with a language and part of the foundation required to study themes, new or old, in symplectic geometry and string theory, including (1) $J$-holomorphic D-curves (with or without boundary), (2) quantization and dynamics of D-branes in string theory, (3) a definition of Fukaya category guided by Lagrangian maps from Azumaya manifolds with a fundamental module with a connection, (4) a theory of fundamental matrix strings or D-strings, and (5) the nature of Ramond-Ramond fields in a space-time. The current note D(11.1) is the symplectic/differential-geometric counterpart of the more algebraic-geometry-oriented first two notes D(1) ([L-Y1]) (arXiv:0709.1515 [math.AG]) and D(2) ([L-L-S-Y], with Si Li and Ruifang Song) (arXiv:0809.2121 [math.AG]) in this project.
In 1964, Golod and Shafarevich found that, provided that the number of relations of each degree satisfies some bounds, there exist infinitely dimensional algebras satisfying the relations. These algebras are called Golod–Shafarevich algebras. This paper provides bounds for the growth function on images of Golod–Shafarevich algebras based upon the number of defining relations. This extends results from Smoktunowicz and Bartholdi (Q J Math. doi:10.1093/qmath/hat0052013) and Smoktunowicz (J Algebra 381:116–130, 2013). Lower bounds of growth for constructed algebras are also obtained, permitting the construction of algebras with various growth functions of various entropies. In particular, the paper answers a question by Drensky (A private communication, 2013) by constructing algebras with subexponential growth satisfying given relations, under mild assumption on the number of generating relations of each degree. Examples of nil algebras with neither polynomial nor exponential growth over uncountable fields are also constructed, answering a question by Zelmanov (2013). Recently, several open questions concerning the commutativity of algebras satisfying a prescribed number of defining relations have arisen from the study of noncommutative singularities. Additionally, this paper solves one such question, posed by Donovan and Wemyss (Noncommutative deformations and flops, ArXiv:1309.0698v2 [math.AG]).
Nous donnons une condition nécessaire et suffisante pour que les coefficients de Taylor à l'origine de séries en plusieurs variables q_i(z)=z_iexp(G_i(z)/F(z)) soient entiers, avec z=(z_1,⋯,z_d) et où F(z) et G_i(z)+log(z_i)F(z), i=1,⋯,d, sont des solutions particulières de certains A-systèmes d'équations différentielles linéaires. Ce critère est basé sur les propriétés analytiques de l'application de Landau (classiquement associée aux suites de quotients de factorielles de formes linéaires). Pour démontrer ce critère, nous généralisons entre autres une version en plusieurs variables d'un théorème de Dwork concernant les congruences formelles entre séries formelles, démontrée par Krattenthaler et Rivoal dans og Multivariate p-adic formal congruences and integrality of Taylor coefficients of mirror maps fg [arXiv:0804.3049v3, math.NT]. Ce critère en plusieurs variables implique l'intégralité des coefficients de Taylor de nouvelles applications miroir d'une seule variable dans og Tables of Calabi--Yau equations fg [arXiv:math/0507430v2, math.AG] de Almkvist, van Enckevort, van Straten et Zudilin. Dans le cas particulier d'une variable, nous affinons notre critère et démontrons l'intégralité des coefficients de Taylor de racines d'applications miroir. Cela nous permet de démontrer une conjecture de Zhou énoncée dans og Integrality properties of variations of Mahler measures fg [arXiv:1006.2428v1 math.AG].
Here we prove that for a smooth projective variety $X$ of arbitrary dimension and for a vector bundle $E$ over $X$, the Harder-Narasimhan filtration of a Frobenius pull back of $E$ is a refinement of the Frobenius pull-back of the Harder-Narasimhan filtration of $E$, provided there is a lower bound on the characteristic $p$ (in terms of rank of $E$ and the slope of the destabilising sheaf of the cotangent bundle of $X$). We also recall some examples, due to Raynaud and Monsky,to show that some lower bound on $p$ is necessary. We further prove an analogue of this result for principal $G$-bundles over $X$. We also give a bound on the instability degree of the Frobenius pull back of $E$ in terms of the instability degree of $E$ and well defined invariants ot $X$ and $E$.
We prove the integrality of the Taylor coefficients of roots of mirror maps at the origin. By mirror maps, we mean formal power series z.exp(G(z)/F(z)), where F(z) and G(z)+log(z)F(z) are particular solutions of certain generalized hypergeometric differential equations. This enables us to prove a conjecture stated by Zhou in "Integrality properties of variations of Mahler measures" [arXiv:1006.2428v1 math.AG]. The proof of these results is an adaptation of the techniques used in our article "Critère pour l'intégralité des coefficients de Taylor des applications miroir", [J. Reine Angew. Math. (to appear)].
We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees. This gives complete control of the B-model side of mirror symmetry in terms of tropical geometry. For example, we expect our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. We anticipate this will lead to a proof of mirror symmetry via tropical methods. This paper is the key step of the program we initiated in math.AG/0309070.
Abstract Grothendieck–Chow motives of quadric hypersurfaces have provided many insights into the theory of quadratic forms. Subsequently, the landscape of motives of more general projective homogeneous varieties has begun to emerge. In particular, there have been many results which relate the motive of a one homogeneous variety to motives of other simpler or smaller ones (see for example [N.A. Karpenko, Cohomology of relative cellular spaces and of isotropic flag varieties, Algebra i Analiz 12 (1) (2000) 3–69. [Kar00a] ; V. Chernousov, S. Gille, A. Merkurjev, Motivic decomposition of isotropic projective homogeneous varieties, Duke Math. J. 126 (1) (2005) 137–159. [CGM05] ; P. Brosnan, On motivic decompositions arising from the method of Bialynicki-Birula, Invent. Math. 161 (1) (2005) 91–111. [Bro05] ; S. Nikolenko, N. Semenov, K. Zainoulline, Motivic decomposition of anisotropic varieties of type F 4 into generalized Rost motives, preprint, Max-Planck-Institut fur Mathematik, 90, 2005. [NSZ05] ; K.V. Zaĭnullin, N.S. Semenov, On the classification of projective homogeneous varieties up to motivic isomorphism, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 330 (2006) 158–172, 273. [ZS06] ; B. Calmes, V. Petrov, N. Semenov, K. Zainoulline, Chow motives of twisted flag varieties, Compos. Math. 142 (4) (2006) 1063–1080. [CPSZ06] ; K. Zainoulline, Motivic decomposition of a generalized Severi–Brauer variety, arXiv: math.AG/0601666 . [Zai] ]). In this paper, we exhibit a relationship between motives of two homogeneous varieties by producing a natural rational map between them. As an application, we compute the Chow group of zero-dimensional cycles on a homogeneous variety associated to a Hermitian form.
This paper is originally designed as a part of revision of the author's preprint math.AG/9908174 "P-adic Schwarzian triangle groups of Mumford type". Recently, Yves Andr'e pointed out a flaw in that preprint; more precisely, Proposition II there was not correct, and consequently, as he pointed out, some triangle groups are missing (but only in p=2,3,5). I apologize to the readers, and will release the other part soon by the original title. In this paper, we concentrate on the constructive part, but more systematically and in more general situation. More precisely, this paper presents a complete criterion for an abstract tree of groups to be realized in the context of p-adic uniformization. This criterion provides a practical way of constructing finitely generated discrete subgroups in PGL(2,K) over a p-adic field K. In the end of this paper, we will exhibit concrete examples of such construction, which in particular shows that there exist infinitely many triangle groups.
We describe the spaces of minimal rank last syzygies for the Mukai Varieties of sectional genus 6,7 and 8. Based on this we show: 1. The first geometric syzygies of a general canonical curve of genus 6 form a non degenerate configuration of 5 lines in P^4. 2. The first geometric syzygies of a general canonical curve of genus 7 form a non degenerate, linearly normal, ruled surface of degree 84 on a spinor variety S in P^15. 3. The second geometric syzygies of a general canonical curve of genus 8 form a non degenerate configuration of 14 conics on a 2-uple embedded P^5 in P^20. This proves a natural generalization of Green's conjecture [1984], namely that the geometric syzygies should span the space of all syzygies, in these cases. We have generalized results 1 and 3 to general curves of even genus in math.AG/0108078. Result 2 is the main new result of this paper.
Generalizing math.AG/0003076 we express universal triple Massey products between line bundles on elliptic curves in terms of Hecke's indefinite theta series. We show that all Hecke's indefinite theta series arise in this way.