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Search: MSC category 14F05 ( Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] )

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1. CMB Online first

Fiori, Andrew
A Riemann-Hurwitz Theorem for the Algebraic Euler Characteristic
We prove an analogue of the Riemann-Hurwitz theorem for computing Euler characteristics of pullbacks of coherent sheaves through finite maps of smooth projective varieties in arbitrary dimensions, subject only to the condition that the irreducible components of the branch and ramification locus have simple normal crossings.

Keywords:Riemann-Hurwitz, logarithmic-Chern class, Euler characteristic
Categories:14F05, 14C17

2. CMB 2010 (vol 54 pp. 520)

Polishchuk, A.
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

3. CMB 2009 (vol 53 pp. 218)

Biswas, Indranil
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

4. CMB 2009 (vol 52 pp. 175)

Biswas, Indranil
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

5. CMB 2007 (vol 50 pp. 427)

Mejía, Israel Moreno
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

6. CMB 2006 (vol 49 pp. 72)

Dwilewicz, Roman J.
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

7. CMB 2003 (vol 46 pp. 429)

Sastry, Pramathanath; Tong, Yue Lin L.
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

8. CMB 2001 (vol 44 pp. 452)

Ishihara, Hironobu
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

9. CMB 1999 (vol 42 pp. 209)

Lanteri, Antonio; Maeda, Hidetoshi
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

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