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Search: All articles in the CJM digital archive with keyword variety

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1. CJM Online first

Zhang, Chao
Ekedahl-Oort strata for good reductions of Shimura varieties of Hodge type
For a Shimura variety of Hodge type with hyperspecial level structure at a prime~$p$, Vasiu and Kisin constructed a smooth integral model (namely the integral canonical model) uniquely determined by a certain extension property. We define and study the Ekedahl-Oort stratifications on the special fibers of those integral canonical models when $p\gt 2$. This generalizes Ekedahl-Oort stratifications defined and studied by Oort on moduli spaces of principally polarized abelian varieties and those defined and studied by Moonen, Wedhorn and Viehmann on good reductions of Shimura varieties of PEL type. We show that the Ekedahl-Oort strata are parameterized by certain elements $w$ in the Weyl group of the reductive group in the Shimura datum. We prove that the stratum corresponding to $w$ is smooth of dimension $l(w)$ (i.e. the length of $w$) if it is non-empty. We also determine the closure of each stratum.

Keywords:Shimura variety, F-zip
Categories:14G35, 11G18

2. CJM Online first

Zhang, Chao
Ekedahl-Oort strata for good reductions of Shimura varieties of Hodge type
For a Shimura variety of Hodge type with hyperspecial level structure at a prime~$p$, Vasiu and Kisin constructed a smooth integral model (namely the integral canonical model) uniquely determined by a certain extension property. We define and study the Ekedahl-Oort stratifications on the special fibers of those integral canonical models when $p\gt 2$. This generalizes Ekedahl-Oort stratifications defined and studied by Oort on moduli spaces of principally polarized abelian varieties and those defined and studied by Moonen, Wedhorn and Viehmann on good reductions of Shimura varieties of PEL type. We show that the Ekedahl-Oort strata are parameterized by certain elements $w$ in the Weyl group of the reductive group in the Shimura datum. We prove that the stratum corresponding to $w$ is smooth of dimension $l(w)$ (i.e. the length of $w$) if it is non-empty. We also determine the closure of each stratum.

Keywords:Shimura variety, F-zip
Categories:14G35, 11G18

3. CJM Online first

Xia, Eugene Z.
The algebraic de Rham cohomology of representation varieties
The $\operatorname{SL}(2,\mathbb C)$-representation varieties of punctured surfaces form natural families parameterized by monodromies at the punctures. In this paper, we compute the loci where these varieties are singular for the cases of one-holed and two-holed tori and the four-holed sphere. We then compute the de Rham cohomologies of these varieties of the one-holed torus and the four-holed sphere when the varieties are smooth via the Grothendieck theorem. Furthermore, we produce the explicit Gauss-Manin connection on the natural family of the smooth $\operatorname{SL}(2,\mathbb C)$-representation varieties of the one-holed torus.

Keywords:surface, algebraic group, representation variety, de Rham cohomology
Categories:14H10, 13D03, 14F40, 14H24, 14Q10, 14R20

4. CJM Online first

Manon, Christopher
Toric geometry of $SL_2(\mathbb{C})$ free group character varieties from outer space
Culler and Vogtmann defined a simplicial space $O(g)$ called outer space to study the outer automorphism group of the free group $F_g$. Using representation theoretic methods, we give an embedding of $O(g)$ into the analytification of $\mathcal{X}(F_g, SL_2(\mathbb{C})),$ the $SL_2(\mathbb{C})$ character variety of $F_g,$ reproving a result of Morgan and Shalen. Then we show that every point $v$ contained in a maximal cell of $O(g)$ defines a flat degeneration of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ to a toric variety $X(P_{\Gamma})$. We relate $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ and $X(v)$ topologically by showing that there is a surjective, continuous, proper map $\Xi_v: \mathcal{X}(F_g, SL_2(\mathbb{C})) \to X(v)$. We then show that this map is a symplectomorphism on a dense, open subset of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ with respect to natural symplectic structures on $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ and $X(v)$. In this way, we construct an integrable Hamiltonian system in $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ for each point in a maximal cell of $O(g)$, and we show that each $v$ defines a topological decomposition of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ derived from the decomposition of $X(P_{\Gamma})$ by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell in $O(g)$ all arise as divisorial valuations built from an associated projective compactification of $\mathcal{X}(F_g, SL_2(\mathbb{C})).$

Keywords:character variety, outer space, analytification, compactification, integrable system
Categories:14M25, 14T05, 14D20

5. CJM 2016 (vol 69 pp. 767)

Choi, Suyoung; Park, Hanchul
Wedge Operations and Torus Symmetries II
A fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. The authors in their previous paper provided a new way to find all characteristic maps on a simplicial complex $K(J)$ obtainable by a sequence of wedgings from $K$. The main idea was that characteristic maps on $K$ theoretically determine all possible characteristic maps on a wedge of $K$. In this work, we further develop our previous work for classification of toric spaces. For a star-shaped simplicial sphere $K$ of dimension $n-1$ with $m$ vertices, the Picard number $\operatorname{Pic}(K)$ of $K$ is $m-n$. We refer to $K$ as a seed if $K$ cannot be obtained by wedgings. First, we show that, for a fixed positive integer $\ell$, there are at most finitely many seeds of Picard number $\ell$ supporting characteristic maps. As a corollary, the conjecture proposed by V.V. Batyrev in 1991 is solved affirmatively. Second, we investigate a systematic method to find all characteristic maps on $K(J)$ using combinatorial objects called (realizable) puzzles that only depend on a seed $K$. These two facts lead to a practical way to classify the toric spaces of fixed Picard number.

Keywords:puzzle, toric variety, simplicial wedge, characteristic map
Categories:57S25, 14M25, 52B11, 13F55, 18A10

6. CJM 2016 (vol 68 pp. 280)

da Silva, Genival; Kerr, Matt; Pearlstein, Gregory
Arithmetic of Degenerating Principal Variations of Hodge Structure: Examples Arising from Mirror Symmetry and Middle Convolution
We collect evidence in support of a conjecture of Griffiths, Green and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi-Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions $1\leq d\leq6$ arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (which is $G_{2}$) of the family of 6-folds, and the theory of boundary components of Mumford-Tate domains.

Keywords:variation of Hodge structure, limiting mixed Hodge structure, Calabi-Yau variety, middle convolution, Mumford-Tate group
Categories:14D07, 14M17, 17B45, 20G99, 32M10, 32G20

7. CJM 2015 (vol 68 pp. 3)

Boden, Hans Ulysses; Curtis, Cynthia L
The SL$(2, C)$ Casson Invariant for Knots and the $\hat{A}$-polynomial
In this paper, we extend the definition of the ${SL(2, {\mathbb C})}$ Casson invariant to arbitrary knots $K$ in integral homology 3-spheres and relate it to the $m$-degree of the $\widehat{A}$-polynomial of $K$. We prove a product formula for the $\widehat{A}$-polynomial of the connected sum $K_1 \# K_2$ of two knots in $S^3$ and deduce additivity of ${SL(2, {\mathbb C})}$ Casson knot invariant under connected sum for a large class of knots in $S^3$. We also present an example of a nontrivial knot $K$ in $S^3$ with trivial $\widehat{A}$-polynomial and trivial ${SL(2, {\mathbb C})}$ Casson knot invariant, showing that neither of these invariants detect the unknot.

Keywords:Knots, 3-manifolds, character variety, Casson invariant, $A$-polynomial
Categories:57M27, 57M25, 57M05

8. CJM 2015 (vol 67 pp. 654)

Lim, Meng Fai; Murty, V. Kumar
Growth of Selmer groups of CM Abelian varieties
Let $p$ be an odd prime. We study the variation of the $p$-rank of the Selmer group of Abelian varieties with complex multiplication in certain towers of number fields.

Keywords:Selmer group, Abelian variety with complex multiplication, $\mathbb{Z}_p$-extension, $p$-Hilbert class tower
Categories:11G15, 11G10, 11R23, 11R34

9. CJM 2015 (vol 67 pp. 696)

Zhang, Tong
Geography of Irregular Gorenstein 3-folds
In this paper, we study the explicit geography problem of irregular Gorenstein minimal 3-folds of general type. We generalize the classical Noether-Castelnuovo type inequalities for irregular surfaces to irregular 3-folds according to the Albanese dimension.

Keywords:3-fold, geography, irregular variety
Category:14J30

10. CJM 2012 (vol 64 pp. 1222)

Bobiński, Grzegorz
Normality of Maximal Orbit Closures for Euclidean Quivers
Let $\Delta$ be an Euclidean quiver. We prove that the closures of the maximal orbits in the varieties of representations of $\Delta$ are normal and Cohen--Macaulay (even complete intersections). Moreover, we give a generalization of this result for the tame concealed-canonical algebras.

Keywords:normal variety, complete intersection, Euclidean quiver, concealed-canonical algebra
Categories:16G20, 14L30

11. CJM 2011 (vol 65 pp. 3)

Barto, Libor
Finitely Related Algebras in Congruence Distributive Varieties Have Near Unanimity Terms
We show that every finite, finitely related algebra in a congruence distributive variety has a near unanimity term operation. As a consequence we solve the near unanimity problem for relational structures: it is decidable whether a given finite set of relations on a finite set admits a compatible near unanimity operation. This consequence also implies that it is decidable whether a given finite constraint language defines a constraint satisfaction problem of bounded strict width.

Keywords:congruence distributive variety, Jónsson operations, near unanimity operation, finitely related algebra, constraint satisfaction problem
Categories:08B05, 08B10

12. CJM 2011 (vol 63 pp. 616)

Lee, Edward
A Modular Quintic Calabi-Yau Threefold of Level 55
In this note we search the parameter space of Horrocks-Mumford quintic threefolds and locate a Calabi-Yau threefold that is modular, in the sense that the $L$-function of its middle-dimensional cohomology is associated with a classical modular form of weight 4 and level 55.

Keywords: Calabi-Yau threefold, non-rigid Calabi-Yau threefold, two-dimensional Galois representation, modular variety, Horrocks-Mumford vector bundle
Categories:14J15, 11F23, 14J32, 11G40

13. CJM 2008 (vol 60 pp. 961)

Abrescia, Silvia
About the Defectivity of Certain Segre--Veronese Varieties
We study the regularity of the higher secant varieties of $\PP^1\times \PP^n$, embedded with divisors of type $(d,2)$ and $(d,3)$. We produce, for the highest defective cases, a ``determinantal'' equation of the secant variety. As a corollary, we prove that the Veronese triple embedding of $\PP^n$ is not Grassmann defective.

Keywords:Waring problem, Segre--Veronese embedding, secant variety, Grassmann defectivity
Categories:14N15, 14N05, 14M12

14. CJM 2007 (vol 59 pp. 36)

Develin, Mike; Martin, Jeremy L.; Reiner, Victor
Classification of Ding's Schubert Varieties: Finer Rook Equivalence
K.~Ding studied a class of Schubert varieties $X_\lambda$ in type A partial flag manifolds, indexed by integer partitions $\lambda$ and in bijection with dominant permutations. He observed that the Schubert cell structure of $X_\lambda$ is indexed by maximal rook placements on the Ferrers board $B_\lambda$, and that the integral cohomology groups $H^*(X_\lambda;\:\Zz)$, $H^*(X_\mu;\:\Zz)$ are additively isomorphic exactly when the Ferrers boards $B_\lambda, B_\mu$ satisfy the combinatorial condition of \emph{rook-equivalence}. We classify the varieties $X_\lambda$ up to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.

Keywords:Schubert variety, rook placement, Ferrers board, flag manifold, cohomology ring, nilpotence
Categories:14M15, 05E05

15. CJM 2000 (vol 52 pp. 265)

Brion, Michel; Helminck, Aloysius G.
On Orbit Closures of Symmetric Subgroups in Flag Varieties
We study $K$-orbits in $G/P$ where $G$ is a complex connected reductive group, $P \subseteq G$ is a parabolic subgroup, and $K \subseteq G$ is the fixed point subgroup of an involutive automorphism $\theta$. Generalizing work of Springer, we parametrize the (finite) orbit set $K \setminus G \slash P$ and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of $\theta$-stable (resp. $\theta$-split) parabolic subgroups. We also describe the decomposition of any $(K,P)$-double coset in $G$ into $(K,B)$-double cosets, where $B \subseteq P$ is a Borel subgroup. Finally, for certain $K$-orbit closures $X \subseteq G/B$, and for any homogeneous line bundle $\mathcal{L}$ on $G/B$ having nonzero global sections, we show that the restriction map $\res_X \colon H^0 (G/B, \mathcal{L}) \to H^0 (X, \mathcal{L})$ is surjective and that $H^i (X, \mathcal{L}) = 0$ for $i \geq 1$. Moreover, we describe the $K$-module $H^0 (X, \mathcal{L})$. This gives information on the restriction to $K$ of the simple $G$-module $H^0 (G/B, \mathcal{L})$. Our construction is a geometric analogue of Vogan and Sepanski's approach to extremal $K$-types.

Keywords:flag variety, symmetric subgroup
Categories:14M15, 20G05

16. CJM 1998 (vol 50 pp. 1253)

López-Bautista, Pedro Ricardo; Villa-Salvador, Gabriel Daniel
Integral representation of $p$-class groups in ${\Bbb Z}_p$-extensions and the Jacobian variety
For an arbitrary finite Galois $p$-extension $L/K$ of $\zp$-cyclotomic number fields of $\CM$-type with Galois group $G = \Gal(L/K)$ such that the Iwasawa invariants $\mu_K^-$, $ \mu_L^-$ are zero, we obtain unconditionally and explicitly the Galois module structure of $\clases$, the minus part of the $p$-subgroup of the class group of $L$. For an arbitrary finite Galois $p$-extension $L/K$ of algebraic function fields of one variable over an algebraically closed field $k$ of characteristic $p$ as its exact field of constants with Galois group $G = \Gal(L/K)$ we obtain unconditionally and explicitly the Galois module structure of the $p$-torsion part of the Jacobian variety $J_L(p)$ associated to $L/k$.

Keywords:${\Bbb Z}_p$-extensions, Iwasawa's theory, class group, integral representation, fields of algebraic functions, Jacobian variety, Galois module structure
Categories:11R33, 11R23, 11R58, 14H40

17. CJM 1997 (vol 49 pp. 675)

de Cataldo, Mark Andrea A.
Some adjunction-theoretic properties of codimension two non-singular subvarities of quadrics
We make precise the structure of the first two reduction morphisms associated with codimension two non-singular subvarieties of non-singular quadrics $\Q^n$, $n\geq 5$. We give a coarse classification of the same class of subvarieties when they are assumed not to be of log-general-type.}

Keywords:Adjunction Theory, classification, codimension two, conic bundles,, low codimension, non log-general-type, quadric, reduction, special, variety.
Categories:14C05, 14E05, 14E25, 14E30, 14E35, 14J10

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