A FILTRATION ON EQUIVARIANT BOREL–MOORE HOMOLOGY

Let $G/H$ be a homogeneous variety and let $X$ be a $G$ -equivariant embedding of $G/H$ such that the number of $G$ -orbits in $X$ is finite. We show that the equivariant Borel–Moore homology of $X$ has a filtration with associated graded module the direct sum of the equivariant Borel–Moore homologies of the $G$ -orbits. If $T$ is a maximal torus of $G$ such that each $G$ -orbit has a $T$ -fixed point, then the equivariant filtration descends to give a filtration on the ordinary Borel–Moore homology of $X$ . We apply our findings to certain wonderful compactifications as well as to double flag varieties.

Baba Malay

This article sheds light on the systematic differences between the variety of Baba Malay spoken in Malacca and that spoken in Singapore. In the literature, the creole is usually discussed as if it is a single homogeneous variety. Language documentation work conducted in both Malacca and Singapore shows that there are crucial differences between how Baba Malay is spoken in both places. These differences are systematic, and they pervade areas of morphology and syntax. All differences lead to the same conclusions. Evidence shows that the variety of Baba Malay in Malacca is much more heavily influenced by its lexifier, Malay, than the variety that is spoken in Singapore. Singapore Baba Malay is also more influenced by its substrate, Hokkien, than Malacca Baba Malay. This divergence between the two varieties is attributed to their specific ecologies. Crucially, the impetus for establishing two varieties of Baba Malay becomes even more vital than ever, given the fact that both varieties are highly endangered, and that researchers engaged in research on Baba Malay should be aware of both varieties and their social environments in order to fully represent the creole.

BALANCED METRICS ON HOMOGENEOUS VECTOR BUNDLES

Let E → M be a holomorphic vector bundle over a compact Kähler manifold (M, ω) and let E = E1 ⊕ ⋯ ⊕ Em → M be its decomposition into irreducible factors. Suppose that each Ej admits a ω-balanced metric in Donaldson–Wang terminology. In this paper we prove that E admits a unique ω-balanced metric if and only if [Formula: see text] for all j, k = 1,…, m, where rj denotes the rank of Ej and Nj = dim H0(M, Ej). We apply our result to the case of homogeneous vector bundles over a rational homogeneous variety (M, ω) and we show the existence and rigidity of balanced Kähler embedding from (M, ω) into Grassmannians.

Motivic decomposition of anisotropic varieties of type F4 into generalized Rost motives

We prove that the Chow motive of an anisotropic projective homogeneous variety of type F4 is isomorphic to the direct sum of twisted copies of a generalized Rost motive. In particular, we provide an explicit construction of a generalized Rost motive for a generically splitting variety for a symbol in . We also establish a motivic isomorphism between two anisotropic non-isomorphic projective homogeneous varieties of type F4. All our results hold for Chow motives with integral coefficients.

Zero Cycles on a Twisted Cayley Plane

Let k be a field of characteristic not 2, 3. Let G be an exceptional simple algebraic group over k of type F4, 1E6 or E7 with trivial Tits algebras. Let X be a projective G-homogeneous variety. If G is of type E7, we assume in addition that the respective parabolic subgroup is of type P7. The main result of the paper says that the degree map on the group of zero cycles of X is injective.

Topological triviality of families of functions on analytic varieties

We present in this paper sufficient conditions for the topological triviality of families of germs of functions defined on an analytic variety V. The main result is an infinitesimal criterion based on a convenient weighted inequality, similar to that introduced by T. Fukui and L. Paunescu in [8]. When V is a weighted homogeneous variety, we obtain as a corollary, the topological triviality of deformations by terms of non negative weights of a weighted homogeneous germ consistent with V. Application of the results to deformations of Newton non-degenerate germs with respect to a given variety is also given.

Tangency quantum cohomology and characteristic numbers

This work establishes a connection between gravitational quantum cohomology and enumerative geometry of rational curves (in a projective homogeneous variety) subject to conditions of infinitesimal nature like, for example, tangency. The key concept is that of modified psi classes, which are well suited for enumerative purposes and substitute the tautological psi classes of 2D gravity. The main results are two systems of differential equations for the generating function of certain top products of such classes. One is topological recursion while the other is Witten-Dijkgraaf-Verlinde-Verlinde. In both cases, however, the background metric is not the usual Poincaré metric but a certain deformation of it, which surprisingly encodes all the combinatorics of the peculiar way modified psi classes restrict to the boundary. This machinery is applied to various enumerative problems, among which characteristic numbers in any projective homogeneous variety, characteristic numbers for curves with cusp, prescribed triple contact, or double points.

The heterozygous form of 4.1(-) hereditary elliptocytosis [the 4.1(-) trait]

Using clinical, morphological, genetic, and biochemical criteria, we studied ten white and North African families with hereditary elliptocytosis (HE). In four families, elliptocytic individuals displayed a highly significant reduction of band 4.1, which was recorded using two electrophoretic procedures. The 4.1a/4.1b ratio was also significantly reduced, as is usually observed in suspensions enriched in young red cells. This form of HE was invariably associated with the following characteristics: absence of clinical signs; numerous, smooth and well-elongated elliptocytes; dominant transmission; and, when investigated, normal osmotic fragility. Its frequency, among all forms of HE, is about one third as a first estimate, at least in whites and North Africans. In the other six families studied, elliptocytic subjects presented normal 4.1 bands. Again, the 4.1a/4.1b ratio was decreased, reflecting the red cell age- dependent changes in these two components. In three of these families, elliptocytosis was accompanied by clinical signs of variable intensity, and the mode of inheritance could not be unequivocally determined. Therefore, HE with a partially reduced band 4.1 defines a homogeneous variety of HE that can be isolated from other forms of HE. We suggest that it be termed the 4.1 (-) trait, so as to correspond with a previously proposed terminology.

The heterozygous form of 4.1(-) hereditary elliptocytosis [the 4.1(-) trait]

Using clinical, morphological, genetic, and biochemical criteria, we studied ten white and North African families with hereditary elliptocytosis (HE). In four families, elliptocytic individuals displayed a highly significant reduction of band 4.1, which was recorded using two electrophoretic procedures. The 4.1a/4.1b ratio was also significantly reduced, as is usually observed in suspensions enriched in young red cells. This form of HE was invariably associated with the following characteristics: absence of clinical signs; numerous, smooth and well-elongated elliptocytes; dominant transmission; and, when investigated, normal osmotic fragility. Its frequency, among all forms of HE, is about one third as a first estimate, at least in whites and North Africans. In the other six families studied, elliptocytic subjects presented normal 4.1 bands. Again, the 4.1a/4.1b ratio was decreased, reflecting the red cell age- dependent changes in these two components. In three of these families, elliptocytosis was accompanied by clinical signs of variable intensity, and the mode of inheritance could not be unequivocally determined. Therefore, HE with a partially reduced band 4.1 defines a homogeneous variety of HE that can be isolated from other forms of HE. We suggest that it be termed the 4.1 (-) trait, so as to correspond with a previously proposed terminology.