Étale endomorphisms of 3-folds. I.
Let \(X\) be a smooth projective variety of dimension 3 defined over \(\mathbb{C}\). The purpose of this paper is to apply minimal model program (MMP) to develop a strategy for classifications of threefolds admitting a non-isomorphic étale endomorphism (an endomorphism is a morphism from the variety to itself) in case of \(\kappa(X)=-\infty\). The infinite sequence \(Z_\cdot =(v_n:Z_n\rightarrow Z_{n+1})_{n\in Z}\) is an étale sequence of constant Picard number (ESP for short) if the following three conditions are satisfied (1) any \(Z_n\) is a smooth projective variety, (2) any \(v_n\) is a non-isomorphic finite étale covering, and (3) the Picard number \(\rho(Z_n)=\dim_{\mathbb{R}}N_1(Z_n)\) is a constant. An ESP \(Y_{\cdot}=(g_n: Y_n\rightarrow Y_{n+1})\) is an ESP of fiber type (FESP for short) if any \(Y_n\) \((n\in \mathbb{Z})\) is a smooth projective 3-fold with an extremal ray of fiber type. Recall the notations in birational geometry. \(N_1(X)\) is the \({\mathbb{R}}\)-vector space of 1-cycles with real coefficients modulo numerical equivalence. \(\mathrm{NE}(X)= \{ \sum a_i[C_i]: C_i\subset X, 0\leq a_i\in {\mathbb{R}} \}\subset N_1(X) \), \(\overline{\mathrm{NE}}(X)=\) the closure of \(\mathrm{NE}(X)\) in \(N_1(X)\), where the \(C_i\) are irreducible curves on \(X\). Let \(V\) be a vector space over \(\mathbb{R}\). A subset \(N\subset V\) is a cone if \(0\in N\) and \(N\) is closed under multiplication by positive scalars. A subcone \(M\subset N\) is extremal (or an extremal face of \(N\)) if \(x, y\in N\), \(x+y\in M\) imply \(x, y\in M\). A 1-dimensional extremal subcone is called an extremal ray. In this paper, an extremal ray \(R\) on a smooth projective variety \(X\) means a \(1-\)dimensional face of \(\overline{\mathrm{NE}}(X)\) with \(K_XR<0\). The motivation of this paper is to apply MMP to classify smooth projective 3-folds \(X\) with \(\kappa(X)=-\infty\) admitting a non-isomorphic étale endomorphism. The author introduces the notions of an ESP and FESP, verify the finiteness of extremal rays of \(\overline{\mathrm{NE}}(X)\) in some cases and prove that for any extremal ray \(R\) of divisorial type, the contraction morphism \(\pi_R: X\rightarrow X'\) associated to \(R\) is the blowing-up of a smooth 3-fold \(X'\) along an elliptic curve.
scientific article; zbMATH DE number 6870388
scientific article