Expand all Collapse all | Results 1 - 8 of 8 |
1. CMB 2010 (vol 54 pp. 520)
Simple Helices on Fano Threefolds
Building on the work of Nogin,
we prove that the braid group $B_4$ acts transitively on full exceptional
collections of vector bundles on Fano threefolds with $b_2=1$ and
$b_3=0$. Equivalently,
this group acts transitively on the set of simple helices (considered
up to a shift in the derived category) on such a Fano threefold. We
also prove that on
threefolds with $b_2=1$ and very ample anticanonical class, every
exceptional coherent
sheaf is locally free.
Categories:14F05, 14J45 |
2. CMB 2009 (vol 53 pp. 218)
Restriction of the Tangent Bundle of $G/P$ to a Hypersurface Let P be a maximal proper parabolic subgroup of a connected simple linear algebraic group G, defined over $\mathbb C$, such that $n := \dim_{\mathbb C} G/P \geq 4$. Let $\iota \colon Z \hookrightarrow G/P$ be a reduced smooth hypersurface of degree at least $(n-1)\cdot \operatorname{degree}(T(G/P))/n$. We prove that the restriction of the tangent bundle $\iota^*TG/P$ is semistable.
Keywords:tangent bundle, homogeneous space, semistability, hypersurface Categories:14F05, 14J60, 14M15 |
3. CMB 2009 (vol 52 pp. 175)
Connections on a Parabolic Principal Bundle, II In \emph{Connections on a parabolic principal bundle over a curve, I}
we defined connections on a parabolic
principal bundle. While connections on usual principal bundles are
defined as splittings of the Atiyah exact sequence, it was noted in
the above article that the Atiyah exact sequence does not generalize to
the parabolic principal bundles.
Here we show that a twisted version
of the Atiyah exact sequence generalizes to the context of
parabolic principal bundles. For usual principal bundles, giving a
splitting of this twisted Atiyah exact sequence is equivalent
to giving a splitting of the Atiyah exact sequence. Connections on
a parabolic principal bundle can be defined using the
generalization of the twisted Atiyah exact sequence.
Keywords:Parabolic bundle, Atiyah exact sequence, connection Categories:32L05, 14F05 |
4. CMB 2007 (vol 50 pp. 427)
On the Image of Certain Extension Maps.~I Let $X$ be a smooth complex projective curve of genus $g\geq
1$. Let $\xi\in J^1(X)$ be a line bundle on $X$ of degree $1$. Let
$W=\Ext^1(\xi^n,\xi^{-1})$ be the space of extensions of $\xi^n$
by $\xi^{-1}$. There is a rational map
$D_{\xi}\colon G(n,W)\rightarrow SU_{X}(n+1)$,
where $G(n,W)$ is the Grassmannian variety of $n$-linear subspaces
of $W$ and $\SU_{X}(n+1)$ is the moduli space of rank $n+1$ semi-stable
vector
bundles on $X$ with trivial determinant. We prove that if $n=2$,
then $D_{\xi}$ is
everywhere defined and is injective.
Categories:14H60, 14F05, 14D20 |
5. CMB 2006 (vol 49 pp. 72)
Additive Riemann--Hilbert Problem in Line Bundles Over $\mathbb{CP}^1$ In this note we consider $\overline\partial$-problem in
line bundles over complex projective space $\mathbb{CP}^1$
and prove that the
equation can be solved for $(0,1)$ forms with compact support. As a
consequence, any Cauchy-Riemann function on a compact real hypersurface in
such line bundles is a jump of two holomorphic functions defined on the
sides of the hypersurface. In particular, the results can be applied to
$\mathbb{CP}^2$ since by removing a point from it we get a line bundle over
$\mathbb{CP}^1$.
Keywords:$\overline\partial$-problem, cohomology groups, line bundles Categories:32F20, 14F05, 32C16 |
6. CMB 2003 (vol 46 pp. 429)
The Grothendieck Trace and the de Rham Integral On a smooth $n$-dimensional complete variety $X$ over ${\mathbb C}$ we
show that the trace map ${\tilde\theta}_X \colon\break
H^n (X,\Omega_X^n)
\to {\mathbb C}$ arising from Lipman's version of Grothendieck duality
in \cite{ast-117} agrees with
$$
(2\pi i)^{-n} (-1)^{n(n-1)/2} \int_X \colon H^{2n}_{DR} (X,{\mathbb
C}) \to {\mathbb C}
$$
under the Dolbeault isomorphism.
Categories:14F10, 32A25, 14A15, 14F05, 18E30 |
7. CMB 2001 (vol 44 pp. 452)
Some Adjunction Properties of Ample Vector Bundles Let $\ce$ be an ample vector bundle of rank $r$ on a projective
variety $X$ with only log-terminal singularities. We consider the
nefness of adjoint divisors $K_X + (t-r) \det \ce$ when $t \ge \dim X$
and $t>r$. As an application, we classify pairs $(X,\ce)$ with
$c_r$-sectional genus zero.
Keywords:ample vector bundle, adjunction, sectional genus Categories:14J60, 14C20, 14F05, 14J40 |
8. CMB 1999 (vol 42 pp. 209)
Ample Vector Bundles of Curve Genus One We investigate the pairs $(X,\cE)$ consisting of a smooth complex
projective variety $X$ of dimension $n$ and an ample vector bundle
$\cE$ of rank $n-1$ on $X$ such that $\cE$ has a section whose
zero locus is a smooth elliptic curve.
Categories:14J60, 14F05, 14J40 |