A numerical condition for a deformation of a Gorenstein surface singularity to admit a simultaneous log-canonical model (Q2723476)

We work with normal complex varieties \(X\) of dimension \(\geq 2\). The canonical divisor will be denoted by \(K_X\). A \(\mathbb{Q}\)-Cartier divisor on \(X\) is called log-canonical if it is of the form \(K_X+B\), (with \(B\) a linear combination of prime divisors with coefficients between 0 and 1) and there is a resolution of singularities \(f:Y\to X\) such that (letting \(B'\) be the strict transform of \(B\) to \(Y\) and \(E\), with irreducible components \(E_1,\dots,E_m\), the exceptional divisor) \((B'+E)_{\text{red}}\) has normal crossings and \(K_Y+B'= f^*(K_X+B) +\sum a_iE_i\), with \(a_i\) rational \(\geq -1\) for all \(i\). A birational morphism \(f:Y\to X\) is a log-canonical model of \(X\) if \(K_Y+E\) is log-canonical and \(f\)-ample.NEWLINENEWLINENEWLINENow, if \(\pi:X\to T\) is a one-parameter family of varieties a birational morphism \(f:Y\to X\) is called a simultaneous log-canonical model if \(KY+E\) is \(\mathbb{Q}\)-Cartier, for all \(t\) in \(T\) the fiber \(E_t\) is \(\mathbb{Q}\)-Cartier, for all \(t\) in \(T\) the fiber \(E_t\) is reduced and the induced morphism \(f_t:Y_t\to X_t\) is a log-canonical model of \(X_t\). In this paper, in the case where \(X_0=\pi^{-1}(t_0)\) is a surface with a single Gorenstein singularity \(x\) (i.e., essentially, for a deformation of the germ \((X_0,x))\), the author finds necessary and sufficient conditions for the existence of a simultaneous log-canonical model, in terms of certain numerical invariants associated to the singularity \(x\). Namely, if \(F\) is a surface with an isolated singularity \(y\), the log-plurigenera are the integers \(\lambda_m (F)=\dim_\mathbb{C}({\mathcal O}_X (mK_X)/g_*({\mathcal O}_M(K_M+A))\), where \(g:M\to F\) is a resolution with exceptional divisor \(A\) (with normal crossings), \(m=1,2, ...\) Also associate to \(y\) the number \(-P.P\), where now \(g\) is a minimal resolution and \(K_M+A=P+N\) is a Zariski decomposition (with \(P\) an \(f\)-nef divisor). Okuma proves that the following statements are equivalent:NEWLINENEWLINENEWLINE(a) The family \(\pi\) above (restricted to a suitable neighborhood of \(t_0)\) admits a simultaneous log-canonical model,NEWLINENEWLINENEWLINE(b) the functions \(\lambda_m(X_t)\) are constant, for all \(m\), for \(t\) near \(t_0\),NEWLINENEWLINENEWLINE(c) the number \(-P_t.P_t\) remains constant, for \(t\) near \(t_0\).NEWLINENEWLINENEWLINEThis is a log-version of a previous result of H. Laufer, involving canonical models and the constancy of \(-K_t.K_t\). The proof of the author's theorem is rather technical. It involves cohomological techniques, using certain auxiliary sheaves and numerical invariants.