The Lawson-Yau formula and its generalization

AbstractThe Euler characteristic of Chow varieties of algebraic cycles of a given degree in complex projective spaces was computed by Blaine Lawson and Stephen Yau by using holomorphic symmetries of cycles spaces. In this paper we compute this in a direct and elementary way and generalize this formula to the l-adic Euler–Poincaré characteristic for Chow varieties over any algebraically closed field. Moreover, the Euler characteristic for Chow varieties with certain group action is calculated. In particular, we calculate the Euler characteristic of the space of right quaternionic cycles of a given dimension and degree in complex projective spaces

Exactly nn-resolvable Topological Expansions

For $\kappa$ a cardinal, a space X=(X,\sT) is $\kappa$-{\it resolvable} if $X$ admits $\kappa$-many pairwise disjoint \sT-dense subsets; (X,\sT) is {\it exactly} $\kappa$-{\it resolvable} if it is $\kappa$-resolvable but not $\kappa^+$-resolvable. The present paper complements and supplements the authors' earlier work, which showed for suitably restricted spaces (X,\sT) and cardinals $\kappa\geq\lambda\geq\omega$ that (X,\sT), if $\kappa$-resolvable, admits an expansion \sU\supseteq\sT, with (X,\sU) Tychonoff if (X,\sT) is Tychonoff, such that (X,\sU) is $\mu$-resolvable for all $\mu<\lambda$ but is not $\lambda$-resolvable (cf. Theorem~3.3 of \cite{comfhu10}). Here the "finite case" is addressed. The authors show in ZFC for $1<n<\omega$: (a) every $n$-resolvable space (X,\sT) admits an exactly $n$-resolvable expansion \sU\supseteq\sT; (b) in some cases, even with (X,\sT) Tychonoff, no choice of \sU is available such that (X,\sU) is quasi-regular; (c) if $n$-resolvable, (X,\sT) admits an exactly $n$-resolvable quasi-regular expansion \sU if and only if either (X,\sT) is itself exactly $n$-resolvable and quasi-regular or (X,\sT) has a subspace which is either $n$-resolvable and nowhere dense or is $(2n)$-resolvable. In particular, every $\omega$-resolvable quasi-regular space admits an exactly $n$-resolvable quasi-regular expansion. Further, for many familiar topological properties \PP, one may choose \sU so that (X,\sU)\in\PP if (X,\sT)\in\PP

Tychonoff Expansions with Prescribed Resolvability Properties

The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) and to Comfort and Garc\'ia-Ferreira (2001): (1) Is every $\omega$-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of ${\mathcal{KID}}$ expansion, the authors show that {\it every} suitably restricted Tychonoff topological space (X,\sT) admits a larger Tychonoff topology (that is, an "expansion") witnessing such failure. Specifically the authors show in ZFC that if (X,\sT) is a maximally resolvable Tychonoff space with S(X,\sT)\leq\Delta(X,\sT)=\kappa, then (X,\sT) has Tychonoff expansions \sU=\sU_i ($1\leq i\leq5$), with \Delta(X,\sU_i)=\Delta(X,\sT) and S(X,\sU_i)\leq\Delta(X,\sU_i), such that (X,\sU_i) is: ($i=1$) $\omega$-resolvable but not maximally resolvable; ($i=2$) [if $\kappa'$ is regular, with S(X,\sT)\leq\kappa'\leq\kappa] $\tau$-resolvable for all $\tau<\kappa'$, but not $\kappa'$-resolvable; ($i=3$) maximally resolvable, but not extraresolvable; ($i=4$) extraresolvable, but not maximally resolvable; ($i=5$) maximally resolvable and extraresolvable, but not strongly extraresolvable.Comment: 25 pages, 0 figure

An Isocurvature Mechanism for Structure Formation

We examine a novel mechanism for structure formation involving initial number density fluctuations between relativistic species, one of which then undergoes a temporary downward variation in its equation of state and generates superhorizon-scale density fluctuations. Isocurvature decaying dark matter models (iDDM) provide concrete examples. This mechanism solves the phenomenological problems of traditional isocurvature models, allowing iDDM models to fit the current CMB and large-scale structure data, while still providing novel behavior. We characterize the decaying dark matter and its decay products as a single component of ``generalized dark matter''. This simplifies calculations in decaying dark matter models and others that utilize this mechanism for structure formation.Comment: 4 pages, 3 figures, submitted to PRD (rapid communications

Hiding dark energy transitions at low redshift

We show that it is both observationally allowable and theoretically possible to have large fluctuations in the dark energy equation of state as long as they occur at ultra-low redshifts z<0.02. These fluctuations would masquerade as a local transition in the Hubble rate of a few percent or less and escape even future, high precision, high redshift measurements of the expansion history and structure. Scalar field models that exhibit this behavior have a sharp feature in the potential that the field traverses within a fraction of an e-fold of the present. The equation of state parameter can become arbitrarily large if a sharp dip or bump in the potential causes the kinetic and potential energy of the field to both be large and have opposite sign. While canonical scalar field models can decrease the expansion rate at low redshift, increasing the local expansion rate requires a non-canonical kinetic term for the scalar field.Comment: 4 pages, 2 figures; submitted to Phys. Rev. D (Brief Report