# 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

## 1 Answer

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 $X$.

• What does "general complete intersections" mean in Voisin's paper "The generalized Hodge and Bloch conjectures are equivalent for general complete intersections". – mwZhang Oct 22 '13 at 3:15
• "General complete intersection" is what you think, but it is your version of the generalized Bloch conjecture which is different (and much weaker) from hers. Read the introduction of the paper, it is quite clear. – abx Oct 24 '13 at 16:44