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1. CJM Online first
Moduli Spaces of Vector Bundles over a Real Curve: $\mathbb Z/2$-Betti Numbers Moduli spaces of real bundles over a real curve arise naturally
as Lagrangian submanifolds of the moduli space of semi-stable
bundles over a complex curve. In this paper, we adapt the methods
of Atiyah-Bott's ``Yang-Mills over a Riemann Surface'' to compute
$\mathbb Z/2$-Betti numbers of these spaces.
Keywords:cohomology of moduli spaces, holomorphic vector bundles Categories:32L05, 14P25 |
2. CJM 2012 (vol 65 pp. 120)
Universal Families of Rational Tropical Curves We introduce the notion of families of $n$-marked
smooth rational tropical curves over smooth tropical varieties and
establish a one-to-one correspondence between (equivalence classes of)
these families and morphisms
from smooth tropical varieties into the moduli space of $n$-marked
abstract rational tropical curves $\mathcal{M}_{n}$.
Keywords:tropical geometry, universal family, rational curves, moduli space Categories:14T05, 14D22 |
3. CJM 2011 (vol 63 pp. 755)
On the Geometry of the Moduli Space of Real Binary Octics The moduli space of smooth real binary octics has five connected
components. They parametrize the real binary octics whose defining
equations have $0,\dots,4$ complex-conjugate pairs of roots
respectively. We show that each of these five components has a real
hyperbolic structure in the sense that each is isomorphic as a
real-analytic manifold to the quotient of an open dense subset of
$5$-dimensional real hyperbolic space $\mathbb{RH}^5$ by the action of an
arithmetic subgroup of $\operatorname{Isom}(\mathbb{RH}^5)$. These subgroups are
commensurable to discrete hyperbolic reflection groups, and the
Vinberg diagrams of the latter are computed.
Keywords:real binary octics, moduli space, complex hyperbolic geometry, Vinberg algorithm Categories:32G13, 32G20, 14D05, 14D20 |
4. CJM 2010 (vol 62 pp. 1131)
Moduli Spaces of Reflexive Sheaves of Rank 2
Let $\mathcal{F}$ be a coherent rank $2$ sheaf on a scheme $Y \subset \mathbb{P}^{n}$ of
dimension at least two and let $X \subset Y$ be the zero set of a section
$\sigma \in H^0(\mathcal{F})$. In this paper, we study the relationship between the
functor that deforms the pair $(\mathcal{F},\sigma)$ and the two functors that deform
$\mathcal{F}$ on $Y$, and $X$ in $Y$, respectively. By imposing some conditions on two
forgetful maps between the functors, we prove that the scheme structure of
\emph{e.g.,} the moduli scheme ${\rm M_Y}(P)$ of stable sheaves on a threefold $Y$
at $(\mathcal{F})$, and the scheme structure at $(X)$ of the Hilbert scheme of curves
on $Y$ become closely related. Using this relationship, we get criteria for the
dimension and smoothness of $ {\rm M_{Y}}(P)$ at $(\mathcal{F})$, without assuming $
{\textrm{Ext}^2}(\mathcal{F} ,\mathcal{F} ) = 0$. For reflexive sheaves on $Y=\mathbb{P}^{3}$ whose
deficiency module $M = H_{*}^1(\mathcal{F})$ satisfies $ {_{0}\! \textrm{Ext}^2}(M ,M ) = 0 $
(\emph{e.g.,} of diameter at most 2),
we get necessary and sufficient conditions of unobstructedness that coincide
in the diameter one case. The conditions are further equivalent to the
vanishing of certain graded Betti numbers of the free graded minimal
resolution of $H_{*}^0(\mathcal{F})$. Moreover, we show that every irreducible
component of ${\rm M}_{\mathbb{P}^{3}}(P)$ containing a reflexive sheaf of diameter
one is reduced (generically smooth) and we compute its dimension. We also
determine a good lower bound for the dimension of any component of ${\rm
M}_{\mathbb{P}^{3}}(P)$ that contains a reflexive stable sheaf with ``small''
deficiency module $M$.
Keywords:moduli space, reflexive sheaf, Hilbert scheme, space curve, Buchsbaum sheaf, unobstructedness, cup product, graded Betti numbers.xdvi Categories:14C05, qqqqq14D22, 14F05, 14J10, 14H50, 14B10, 13D02, 13D07 |
5. CJM 2007 (vol 59 pp. 845)
Representations of the Fundamental Group of an $L$-Punctured Sphere Generated by Products of Lagrangian Involutions |
Representations of the Fundamental Group of an $L$-Punctured Sphere Generated by Products of Lagrangian Involutions In this paper, we characterize unitary representations of $\pi:=\piS$ whose
generators $u_1, \dots, u_l$ (lying in conjugacy classes fixed initially)
can be decomposed as products of two Lagrangian involutions
$u_j=\s_j\s_{j+1}$ with $\s_{l+1}=\s_1$. Our main result is that such
representations are exactly the elements of the fixed-point set of an
anti-symplectic involution defined on the moduli space
$\Mod:=\Hom_{\mathcal C}(\pi,U(n))/U(n)$. Consequently, as this fixed-point set is
non-empty, it is a Lagrangian submanifold of $\Mod$. To prove this, we use
the quasi-Hamiltonian description of the symplectic structure of $\Mod$ and
give conditions on an involution defined on a quasi-Hamiltonian $U$-space
$(M, \w, \mu\from M \to U)$ for it to induce an anti-symplectic involution on
the reduced space $M/\!/U := \mu^{-1}(\{1\})/U$.
Keywords:momentum maps, moduli spaces, Lagrangian submanifolds, anti-symplectic involutions, quasi-Hamiltonian Categories:53D20, 53D30 |
6. CJM 2004 (vol 56 pp. 1228)
On the Connectedness of Moduli Spaces of Flat Connections over Compact Surfaces We study the connectedness of the moduli space
of gauge equivalence classes of flat $G$-connections on a compact
orientable surface or a compact nonorientable surface for a class
of compact connected Lie groups. This class includes all the
compact, connected, simply connected Lie groups, and some
non-semisimple classical groups.
Keywords:moduli space of flat $G$ connections Category:53 |