# About generalized Bloch conjecture

Conjecture((generalized Bloch conjecture)): Suppose that $\text{H}^{k,0}(X)=0$ for all $k>0$. Then $\text{CH}_0(X)=\mathbb{Z}.$

Is generalized Bloch conjecture known for complete intersections of sufficiently large degrees? In which case generalized Bloch conjecture is known?

• Maybe you should say what you call the generalized Bloch conjecture ... – abx Oct 21 '13 at 14:10
• I gave the statement of conjecture. – mwZhang Oct 21 '13 at 15:05
• What do you mean by "complete intersections of sufficiently large degrees"? For a fixed ambient variety $Y$, for a Cartier divisor $X$, if $\mathcal{O}_Y(X)$ is "sufficiently ample", then $h^0(X,\omega_X)$ will be nonzero. It seems to me it is better to consider complete intersections of sufficiently small degree so that every $h^{k,0}(X)$ equals $0$. – Jason Starr Oct 21 '13 at 15:06

Are you considering complete intersections in projective space? In that case, for a complete intersection $X$ in $\mathbb{P}^n$ of type $(d_1,\dots,d_c)$, then $\omega_X$ is isomorphic to $$\mathcal{O}_{\mathbb{P}^n}(-n-1+d_1 + \dots + d_c)$$. Thus $h^0(X,\omega_X)$ is nonzero unless $d_1+\dots+d_c \leq n$. In this case, $X$ is a Fano manifold. Thus, by work of Kollár-Miyaoka-Mori and Campana, $X$ is rationally chain connected (I think you can usually prove this directly for complete intersections; though definitely not for general Fano manifolds). Thus $\text{CH}_0(X)$ is just $\mathbb{Z}$ since every pair of points is connected by a chain of rational curves in $X$.