Let π: E → B be a fiber bundle with fiber having the mod 2 cohomology algebra of a real or a complex projective space and let π’: E’ → B be a vector bundle such that ℤ₂ acts fiber preserving and freely on E and E’-0, where 0 stands for the zero section of the bundle π’: E’ → B. For a fiber preserving ℤ₂-equivariant map f: E → E’, we estimate the cohomological dimension of the zero set $Z_f=x\in E\left|f\right(x)=0$. As an application, we also estimate the cohomological dimension of the ℤ₂-coincidence set $A_f=x\in E\left|f\right(x)=f(T\left(x\right))$ of a...

Let $X$ be a smooth projective curve defined over an algebraically closed field $k$, and let $F_X$ denote the absolute Frobenius morphism of $X$ when the characteristic of $k$ is positive. A vector bundle over $X$ is called virtually globally generated if its pull back, by some finite morphism to $X$ from some smooth projective curve, is generated by its global sections. We prove the following. If the characteristic of $k$ is positive, a vector bundle $E$ over $X$ is virtually globally generated if and only...

Let $E$ be a globally generated ample vector bundle of rank $\ge 2$ on a complex projective smooth surface $X$. By extending a recent result by A. Noma, we classify pairs $\left(X,E\right)$ as above satisfying $c_2\left(E\right)=2$.

Let $G_{n,k}$ (${\tilde{G}}_{n,k}$) denote the Grassmann manifold of linear $k$-spaces (resp. oriented $k$-spaces) in ${ℝ}^n$, $d_{n,k}=k(n-k)=\text{dim}{G}_{n,k}$ and suppose $n\ge 2k$. As an easy consequence of the Steenrod obstruction theory, one sees that $(d_{n,k}+1)$-fold Whitney sum $(d_{n,k}+1){\xi }_{n,k}$ of the nontrivial line bundle ${\xi }_{n,k}$ over $G_{n,k}$ always has a nowhere vanishing section. The author deals with the following question: What is the least $s$ ($=s_{n,k}$) such that the vector bundle $s{\xi }_{n,k}$ admits a nowhere vanishing section ? Obviously, $s_{n,k}\le d_{n,k}+1$, and for the special case in which $k=1$, it is known that $s_{n,1}=d_{n,1}+1$....

Any $r$-dimensional subbundle of the cotangent bundle on an $n$-dimensional manifold $M$ partitions $M$ into subsets $M_0,...,M_m$ ($m$ being the minimum of $r$ and $C(n-r,2)$, the combinations of $n-r$ things taken 2 at a time). $M_i$ is the set on which the first derived systems of the subbundle has codimension $i$. In this paper we prove the following: Theorem. Let $s\ge 2$ and let $Q$ be a generic $C^s$ $r$-dimensional subbundle of the cotangent bundle of an $n$-dimensional manifold $M$. The codimension...