Abstract We give an explicit construction of the cusp eigenforms on an elliptic curve defined over a finite field, using the theory of Hall algebras and the Langlands correspondence for function fields and ${\mathrm{GL} }_{n} $. As a consequence we obtain a description of the Hall algebra of an elliptic curve as an infinite tensor product of simpler algebras. We prove that all these algebras are specializations of a universal spherical Hall algebra (as defined and studied by Burban and Schiffmann [On the Hall algebra of an elliptic curve I, Preprint (2005), arXiv:math/0505148 [math.AG]] and Schiffmann and Vasserot [The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, Compositio Math. 147 (2011), 188–234]).
In this sequel to [L-Y1], [L-L-S-Y], and [L-Y2] (respectively arXiv:0709.1515 [math.AG], arXiv:0809.2121 [math.AG], and arXiv:0901.0342 [math.AG]), we study a D-brane probe on a conifold from the viewpoint of the Azumaya structure on D-branes and toric geometry. The details of how deformations and resolutions of the standard toric conifold Y can be obtained via morphisms from Azumaya points are given. This should be compared with the quantum-field-theoretic/D-brany picture of deformations and resolutions of a conifold via a D-brane probe sitting at the conifold singularity in the work of Klebanov and Witten [K-W] (arXiv:hep-th/9807080) and Klebanov and Strasser [K-S] (arXiv:hep-th/0007191). A comparison with resolutions via noncommutative desingularizations is given in the end.
In this continuation of [L-Y1], [L-L-S-Y], [L-Y2], and [L-Y3] (arXiv:0709.1515 [math.AG], arXiv:0809.2121 [math.AG], arXiv:0901.0342 [math.AG], arXiv:0907.0268 [math.AG]), we study D-branes in a target-space with a fixed $B$-field background $(Y,\alpha_B)$ along the line of the Polchinski-Grothendieck Ansatz, explained in [L-Y1] and further extended in the current work. We focus first on the gauge-field-twist effect of $B$-field to the Chan-Paton module on D-branes. Basic properties of the moduli space of D-branes, as morphisms from Azumaya schemes with a twisted fundamental module to $(Y,\alpha_B)$, are given. For holomorphic D-strings, we prove a valuation-criterion property of this moduli space. The setting is then extended to take into account also the deformation-quantization-type noncommutative geometry effect of $B$-field to both the D-brane world-volume and the superstring target-space(-time) $Y$. This brings the notion of twisted ${\cal D}$-modules that are realizable as twisted locally-free coherent modules with a flat connection into the study. We use this to realize the notion of both the classical and the quantum spectral covers as morphisms from Azumaya schemes with a fundamental module (with a flat connection in the latter case) in a very special situation. The 3rd theme (subtitled "Sharp vs. Polchinski-Grothendieck") of Sec. 2.2 is to be read with the work [Sh3] (arXiv:hep-th/0102197) of Sharp while Sec. 5.2 (subtitled less appropriately "Dijkgraaf-Holland-Su{\l}kowski-Vafa vs. Polchinski-Grothendieck") is to be read with the related sections in [D-H-S-V] (arXiv:0709.4446 [hep-th]) and [D-H-S] (arXiv:0810.4157 [hep-th]) of Dijkgraaf, Hollands, Su{\l}kowski, and Vafa.
Abstract M. Kapranov introduced and studied in math.AG/9802041 the noncommutative formal structure of a smooth affine variety. In this note we show that his construction is a particular case of micro-localization and extend the construction functorially to representation schemes of affine algebras. We describe explicitly the formal completions in the case of path algebras of quivers and initiate the study of their finite dimensional representations.
This note is but a research announcement, summarizing and explaining results proven and detailed in forthcoming papers. When one studies families of objects over curves, and the objects are parametrized by a Deligne-Mumford stack M, then the families are equivalent to morphisms of curves into M. In order to have complete moduli for such families, one needs to compactify the stack of stable maps into M. It turns out that in the boundary, the curve must acquire extra structure and you'd better read the paper to see what that structure is. Applications to fibered surfaces (see math.AG/9804097), admissible covers, and level structures are discussed.
The aim of this note is to give simple proofs of some results of Reichstein and Youssin (math.AG/9903162) about the behaviour of fixed points of finite group actions under rational maps. Our proofs work in any characteristic. We also give a short proof of the Nishimura lemma.
Following an approach of Dolgachev, Pinkham and Demazure, we classified in math.AG/0210153 normal affine surfaces with hyperbolic C^{*}-actions in terms of pairs of Q-divisors (D+,D-) on a smooth affine curve. In the present paper we show how to obtain from this description a natural equivariant completion of these C^*-surfaces. Using elementary transformations we deduce also natural completions for which the boundary divisor is a standard graph in the sense of math.AG/0511063 and show in certain cases their uniqueness. This description is especially precise in the case of normal affine surfaces completable by a zigzag i.e., by a linear chain of smooth rational curves. As an application we classify all zigzags that appear as boundaries of smooth or normal C^*-surfaces.
We give some bounds on the anticanonical degrees of Fano varieties with Picard number 1 and mild singularities, extending results of Koll\'ar et al. from the early 90's and improving them even in the smooth case. The proof is based on a study of positivity properties of sheaves of differential operators on ample line bundles, and avoids the use of rational curves and bend-and-break. This note is a self-contained exposition of the main ideas of math.AG/9811022
We prove two results on the defining ideals of certain varieties of matrices. Let us fix two positive integers r, e. Let M(r) be the set of r x r matrices over a field K. We consider the closed subscheme of the nilpotent variety of M(r) over K defined by the conditions char_A(T)=T^r, A^e=0. We prove that when the characteristic of K is zero this scheme is reduced. Also for e=2 we prove that this scheme is reduced over a field K of arbitrary characteristic. These results were motivated by the questions of G. Pappas and M. Rapoport (compare their paper ''Local models in the ramified case I. The EL-case", math.AG/0006222) and give answers to some of their conjectures.
Abstract For a symmetrizable Kac–Moody algebra the category of admissible representations of the category O is an analogue of the category of finite-dimensional representations of a semisimple Lie algebra. The monoid which is associated to this category and the category of restricted duals by a Tannaka–Krein reconstruction contains the Kac–Moody group as open dense unit group. It has similar properties as a reductive algebraic monoid. In particular, there are Bruhat and Birkhoff decompositions, the Weyl group replaced by the Weyl monoid [C. Mokler, An analogue of a reductive algebraic monoid, whose unit group is a Kac–Moody group, arXiv: math.AG/0204246 , Mem. Amer. Math. Soc., in press]. We determine the closure relations of the Bruhat and Birkhoff cells, which give extensions of the Bruhat order from the Weyl group to the Weyl monoid. We show that the Bruhat and Birkhoff cells are irreducible and principal open in their closures. We give product decompositions of the Bruhat and Birkhoff cells. We define extended length functions, which are compatible with the extended Bruhat orders. We show a generalization of some of the Tits axioms for twin BN-pairs.
We show that crystals with the properties of crystalline cohomology of ordinary Calabi-Yau threefolds in characteristic p > 0, exhibit a remarkable similarity with the well known structure on the cohomology of complex Calabi-Yau threefolds near a boundary point of the moduli space with maximal unipotent local monodromy. In particular, there are canonical coordinates and an analogue of the prepotential of the Yukawa coupling. Moreover in Formulas (2.25) and (2.29) we show p-adic analogues of the integrality properties for the canonical coordinates and the prepotential of the Yukawa coupling, which have been observed in the examples of Mirror Symmetry. http://www.arxiv.org/abs/math.AG/0212061
In this paper we construct 206 examples of Calabi-Yau manifolds with different Euler numbers. All constructed examples are smooth models of double coverings of $P^3$ branched along an octic surface. We allow 11 types of (not necessary isolated) singularities in the branch locus. Thus we broaden the class of examples studied in math.AG/9902057. For every considered example we compute the Euler number and give a precise description of a resolution of singularities.
Abstract After discussing some general problems for heterotic compactifications involving fivebranes we construct bundles, built as extensions, over an elliptically fibered Calabi–Yau threefold. For these we show that it is possible to satisfy the anomaly cancellation topologically without any fivebranes. The search for a specific Standard Model or GUT gauge group motivates the choice of an Enriques surface or certain other surfaces as base manifold. The burden of this construction is to show the stability of these bundles. Here we give an outline of the construction and its physical relevance. The mathematical details, in particular the proof that the bundles are stable in a specific region of the Kahler cone, are given in the mathematical companion paper math.AG/0611762 .
Let $(X, \omega, c_X)$ be a real symplectic 4-manifold with real part $R X$. Let $L \subset R X$ be a smooth curve such that $[L] = 0 \in H_1 (R X ; Z / 2Z)$. We construct invariants under deformation of the quadruple $(X, \omega, c_X, L)$ by counting the number of real rational $J$-holomorphic curves which realize a given homology class $d$, pass through an appropriate number of points and are tangent to $L$. As an application, we prove a relation between the count of real rational $J$-holomorphic curves done in math.AG/0303145 and the count of reducible real rational curves done in math.SG/0502355. Finally, we show how these techniques also allow to extract an integer valued invariant from a classical problem of real enumerative geometry, namely about counting the number of real plane conics tangent to five given generic real conics.