Logarithmic transformations of rigid analytic elliptic surfaces
Mitsui introduces a non-archimedean analogue of logarithmic transformations for elliptic fibrations over the field of complex numbers, working within the framework of rigid-analytic geometry. His surgery constructions apply to Tate elliptic fibrations over algebraically closed non-archimedean fields, and they can be used to construct interesting new examples of elliptic surfaces in positive characteristic: on the one hand, Mitsui is able to construct new types of algebraic elliptic surfaces, and on the other hand, he finds rigid-analytic elliptic surfaces whose combinations of invariants appear neither in the algebraic nor in the complex-analytic case. Let us be more precise about the content of the paper. The first section is introductory: the author summarizes the structure of his manuscript, and he outlines his main results. Moreover, he briefly mentions various standard techniques which can be used to construct elliptic fibrations with multiple fibers over fields of positive characteristic. In Section 2, the author sets up some notation, and he recalls basic facts from rigid geometry. Furthermore, he quotes results from \textit{E. Bombieri} and \textit{D. Mumford} [Complex Anal. Algebr. Geom., Collect. Pap. dedic. K. Kodaira, 23--42 (1977; Zbl 0348.14021), p.\ 27] regarding algebraic elliptic surfaces, most notably the canonical bundle formula. He shows (cf.\ his Proposition 2) that the total space of an elliptic fibration has trivial Euler characteristic if and only if every reduced fiber is an elliptic curve. The proof of this statement is reduced to the case where the fibration has a section; this reduction is achieved by passing to the relative Jacobian of the fibration and by using results from \textit{Q. Liu, D. Lorenzini} and \textit{M. Raynaud} [Invent. Math. 157, No. 3, 455--518 (2004; Zbl 1060.14037)]. In the third section, which is the heart of the paper, the author defines his non-archimedean logarithmic transformations. Let \(K\) be an algebraically closed non-archimedean field of arbitrary characteristic. By Proposition 10 of the appendix, any smooth elliptic fibration is, locally near a point of the base whose fiber has non-integral \(j\)-invariant, analytically isomorphic to a relative Tate elliptic curve. To define logarithmic transformations, it thus suffices to consider relative Tate curves over the closed unit disc \(C:=\text{Sp} K\langle t\rangle\). Let \(q\in C\) denote the point where the coordinate \(t\) vanishes. Given an element \(j\in K\langle t\rangle\) such that \(j(x)>0\) for all \(x\in C\), the author considers the associated relative Tate elliptic curve \(\pi_Z:Z\rightarrow C\) as well as its relative Tate uniformization \(\mathcal{X}\rightarrow Z\), where \(\mathcal{X}=\mathbb G_{m,C}\). Given coprime integers \(a\) and \(m\) with \(m>0\), he defines the associated logarithmic transformation \(L_q(m,a)(\pi_Z)\) of \(\pi_Z\) as follows: he first chooses integers \(b\) and \(n\) such that \(ab-mn=1\), and he regards \(\mathcal{Y}:=\{(y,u)\in\mathbb A^2_K\,|\,y\neq 0\,\text{and}\,|y^bu^m|\leq 1\}\) as a space over \(C\) via the identification \(t=y^bu^m\). He then identifies the complement \(\mathcal{X}^*\) of the special fiber in \(\mathcal{X}\) with the complement \(\mathcal{Y}^*\) of the special fiber of \(\mathcal{Y}\) by equating the vertical coordinate \(x\) on \(\mathcal{X}^*\) with \(y^nu^a\). He then naturally extends the induced \(Z\)-action from \(\mathcal{Y}^*\) to a well-behaved action of \(\mathbb Z\) on \(\mathcal{Y}\). Finally, he forms the rigid-analytic quotient of \(\mathcal{Y}\) modulo that action; the result is a relatively minimal elliptic fibration \(\pi_Y=L_q(m,a)(\pi_Z)\) which is isomorphic to \(\pi_Z\) away from \(q\) and which has the property that if \(\pi_Z^{-1}(q)\cong\mathbb G_{m,K}/\langle\alpha\rangle\), then the special fiber \(\pi_Y^{-1}(q)\) is tamely ramified with multiplicity \(m\), and its underlying reduced subscheme is isomorphic to \(\mathbb G_{m,K}/\langle\alpha^m\rangle\). The fibration \(\pi_Y\) does not depend on the choice of \(b\) or \(n\) up to canonical isomorphism. The author describes \(\pi_Y\) as the analytification of a GIT-quotient of an equivariant action of \(\mu_{m,K}\) on a smooth elliptic fibration which is defined over a degree \(m\) covering \(B\) of \(C\) ramified exactly at \(q\). Let us point out that while the fibration \(\pi_Y\) can be constructed directly in the category of schemes as a GIT quotient (in the case where the base is the spectrum of \(K\langle t\rangle\)), the complement of its special fiber is merely analytically isomorphic to the complement of the special fiber of \(\pi_Z\). In this sense, the logarithmic transformation is of purely analytical nature; cf.\ also Remark 3. Section 4 is devoted to the computation of invariants of the elliptic fibration \(\pi_Y:Y\rightarrow C\) over a smooth proper \(K\)-curve \(C\) which is obtained from an elliptic curve \(E\) over \(K\) with non-integral \(j\)-invariant together with the induced trivial Tate elliptic fibration \(X=E\times C\rightarrow C\) via application of the logarithmic transformations \(L_{q}(m_q,a_q)\), where \(q\) varies in a finite set \(S\) of closed points in \(C\) and where the \(a_q\in\mathbb Z\) and \(m_q\in\mathbb Z_{>0}\), with \(q\in S\), are coprime integers as above. Let \(T\subseteq S\) denote the set of points such that \(m_q\) is divisible by the characteristic of \(K\). If \(T\) is nonempty, then \(h^0(\Omega^1_Y)=g(C)+\sharp T-1\) (Theorem 4), where \(g(C)\) denotes the genus of \(C\). If \(T\) is empty, then \(h^0(\Omega^1_Y)\) is \(g(C)\) or \(g(C)+1\), depending on whether a certain residue \(\eta=\sum a_q/m_q\in K\) is zero (cf.\ Theorem 5). The proofs of these statement are achieved by means of elementary considerations involving the explicit description of logarithmic transformations as it is given in Section 3, the Riemann-Roch theorem for curves and the short exact sequence attached to the first Cousin problem on \(C\). In Section 5, the author states a criterion which guarantees algebraicity of the rigid-analytic elliptic fibration \(\pi_Y:Y\rightarrow C\) that was studied in Section 4. Algebraicity of \(Y\) is implied by certain properties of a divisor \(D\) on \(C\) defined in terms of the integers \(a_q\) and \(m_q\): if a multiple of \(D\) is principal, then \(Y\) is algebraic (Theorem 6). The proof of this fact uses B.\ Conrad's theory of relative ampleness in rigid geometry and the GAGA theorems. Having established this criterion, the author computes (cf.\ Theorem 7) that whenever \(Y\) is algebraic, the equalities \(h^1(\mathcal O_Y)=g(C)+1\) and \(h^2(\mathcal O_Y)=g(C)\) hold. He then explains, using Theorems 4, 6 and 7, how to choose a set of points \(S\) and integers \(m_q\), \(a_q\) such that the resulting logarithmic transformations give examples of algebraic elliptic surfaces \(Y\) in characteristic \(p>0\) where \(h^0(\Omega^1_Y)\) is arbitrarily greater than \(h^1(\mathcal O_Y)\) (cf.\ Example 1). As it turns out, it is possible to find such examples where \(m_q=1\) for all \(q\). In Section 6, the author specializes to the case \(C=\mathbb P^1_K\), and he gives a number of explicit examples (cf.\ Examples 2--3) for which he states the respective types of the multiple fibers. Section 7 provides criteria for non-algebraicity of elliptic surfaces \(Y\) of the type studied in Section 4. If \(p=0\) and if the residue \(\eta\) mentioned above is nonzero, then \(Y\) is non-algebraic, because otherwise Hodge symmetry would be violated, cf.\ Proposition 6. For the same reason, elliptic surfaces \(Y\) in characteristic \(p>0\) with certain combinations of invariants cannot be lifted to characteristic zero (cf. Lemma 12 and Proposition 7). A non-algebraicity criterion that works in arbitrary characteristic is given in Example 4: by Theorem 3.3 of \textit{T. Katsura} and \textit{K. Ueno} [Math. Ann. 272, 291--330 (1985; Zbl 0553.14019)], algebraic elliptic fibrations over \(\mathbb P^1_K\) can only have certain multiplicities. In Example 5, the author discusses a situation, again over \(\mathbb P^1_K\), where the Katsura-Ueno criterion does not apply and where non-algebraicity follows because algebraicity would contradict non-liftability. Finally, in Example 6 the author reviews a construction by Katsura and Ueno [Zbl 0553.14019, Example 8.4] of an elliptic fibration via GIT quotient techniques with two multiple fibers, exactly one of which is wildly ramified. He resolves the tame multiple fiber via the inverse of a suitable logarithmic transformation, thus obtaining an elliptic fibration having a single multiple fiber that is wildly ramified; non-algebraicity follows, again, in [Zbl 0553.14019, Theorem 3.3]. In the appendix, the author proves that two smooth elliptic fibrations which admit sections and which have the same \(j\)-invariant are locally isomorphic in the rigid-analytic category. This result is required to extend the explicit constructions of Section 3 from Tate elliptic curves over the closed unit disc to general smooth elliptic fibrations with everywhere non-integral \(j\)-invariant.
scientific article; zbMATH DE number 6149474
scientific article