Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature.

*(English)*Zbl 0731.53063The author gives details of his very novel method of producing Kähler- Einstein metrics on certain compact complex manifolds with ample anti- canonical class (Fano manifolds). His method works, for instance, for Fermat hyper-surfaces of degree d in \(P^ n\), n/2\(\leq d\leq n\), certain complete intersection three-folds in \(P^ 5\) and \(P^ 6\), the blow up of \(P^{2n+1}\) along two skew n-planes, etc. For an account of the difficulty of the problem and earlier work, see Y.-T. Siu [Ann. Math., II. Ser. 127, 585-627 (1988; Zbl 0651.53035)]; some initial results of Siu’s paper form the point of departure for the present paper.

The author’s method is as follows. Let M be a Fano manifold, \(g_{\alpha {\bar \beta}}\) a Kähler metric on M whose Kähler form represents the anti-canonical class, \(R_{\alpha {\bar \beta}}\) the Ricci curvature of the metric, and F a smooth real function on M such that \(R_{\alpha {\bar \beta}}-g_{\alpha {\bar \beta}}=\partial_{\alpha}\partial_{\beta}F,\) normalized so that \(\int_{M}e^ F dV=volume\) of M. For each \(t\in [0,1]\), consider the so-called Monge-Ampère equation \(\det (g_{\alpha {\bar \beta}}+\partial_{\alpha}\partial_{{\bar \beta}}\phi)=\exp (-t\phi +F)\det g_{\alpha {\bar \beta}}\) for the unknown smooth function \(\phi =\phi_ t\). It is known that the set of t for which this equation can be solved contains \(t=0\) (Yau’s solution of the Calabi conjecture) and is open, and that, for \(t<1\), the solution \(\phi_ t\) is unique if it exists; \(\phi_ 1\) (if it exists) will provide a Kähler-Einstein metric (namely \(g_{\alpha {\bar \beta}}+\partial_{\alpha}\partial_{{\bar \beta}}\phi_ 1)\) on M. It is also known that a \(\phi_ 1\) exists unless there is an increasing sequence \((t_ k)\) in (0,1) for which the \(\phi_ k:=\phi_{t_ k}\) exist and \(\| \phi_ k\|_{\infty}\to \infty\) as \(k\to \infty.\)

Now the author uses such a seuence \(S:=(\phi_ k)\) to define a coherent sheaf of ideals I(S) on M (called a multiplier sheaf), which turns out to have many interesting properties: if \(V=V(S)\) is the (in general non- reduced) subscheme of M defined by I(S), then \(\emptyset \neq V\neq M\), and \(H^ i(M,I(S))=0\) for \(i\geq 1\) (in particular, V is connected; and \(V_{red}\) is a tree of smooth rational curves if dim V\(=1)\). This construction can be made invariant under any compact subgroup of Aut(M), because of the uniqueness of the \(\phi_ t\), \(t<1\). In all applications of the author’s method given in this paper, certain compact (usually finite) subgroups G of Aut(M) are utilized to show that there are no G- invariant subschemes of M with the properties enjoyed by the multiplier subschemes V(S), and it will follow that M is Kähler-Einstein.

To define the sheaf I(S), subtract constants from the \(\phi_ k\) and assume sup \(\phi\) \({}_ k=0\). For any \(L\in Pic(M)\), defined \(I_ S(L)=[s\in H^ 0(M,L):\) there exists a sequence \((s_ n)\) in \(H^ 0(M,L)\) and an \(r\in (m/(m+1),1)\) such that \((s_ n)\) converges uniformly to s and, for some sequence \((k_ n)\to \infty\), \((\int_{M}\| s_ n\|^ 2 \exp (-r\phi_{k_ n}) dV)\) is bounded as \(n\to \infty \}\); here \(m=\dim_{{\mathbb{C}}}M\), and \(\| \|\) is any metric on L. Now I(S) is the coherent sheaf of ideals on M determined by the graded ideal \(\oplus I_ S(H^{\mu})\) in the graded ring \(\oplus H^ 0(M,H^{\mu})\), for any ample \(H\in Pic(M).\)

Although a lot of hard nonlinear analysis is in the background of this paper, the results proved in this paper use only the linear analysis of the proof of Kodaira’s vanishing theorem. The paper is clearly written, and very readable.

The author’s method is as follows. Let M be a Fano manifold, \(g_{\alpha {\bar \beta}}\) a Kähler metric on M whose Kähler form represents the anti-canonical class, \(R_{\alpha {\bar \beta}}\) the Ricci curvature of the metric, and F a smooth real function on M such that \(R_{\alpha {\bar \beta}}-g_{\alpha {\bar \beta}}=\partial_{\alpha}\partial_{\beta}F,\) normalized so that \(\int_{M}e^ F dV=volume\) of M. For each \(t\in [0,1]\), consider the so-called Monge-Ampère equation \(\det (g_{\alpha {\bar \beta}}+\partial_{\alpha}\partial_{{\bar \beta}}\phi)=\exp (-t\phi +F)\det g_{\alpha {\bar \beta}}\) for the unknown smooth function \(\phi =\phi_ t\). It is known that the set of t for which this equation can be solved contains \(t=0\) (Yau’s solution of the Calabi conjecture) and is open, and that, for \(t<1\), the solution \(\phi_ t\) is unique if it exists; \(\phi_ 1\) (if it exists) will provide a Kähler-Einstein metric (namely \(g_{\alpha {\bar \beta}}+\partial_{\alpha}\partial_{{\bar \beta}}\phi_ 1)\) on M. It is also known that a \(\phi_ 1\) exists unless there is an increasing sequence \((t_ k)\) in (0,1) for which the \(\phi_ k:=\phi_{t_ k}\) exist and \(\| \phi_ k\|_{\infty}\to \infty\) as \(k\to \infty.\)

Now the author uses such a seuence \(S:=(\phi_ k)\) to define a coherent sheaf of ideals I(S) on M (called a multiplier sheaf), which turns out to have many interesting properties: if \(V=V(S)\) is the (in general non- reduced) subscheme of M defined by I(S), then \(\emptyset \neq V\neq M\), and \(H^ i(M,I(S))=0\) for \(i\geq 1\) (in particular, V is connected; and \(V_{red}\) is a tree of smooth rational curves if dim V\(=1)\). This construction can be made invariant under any compact subgroup of Aut(M), because of the uniqueness of the \(\phi_ t\), \(t<1\). In all applications of the author’s method given in this paper, certain compact (usually finite) subgroups G of Aut(M) are utilized to show that there are no G- invariant subschemes of M with the properties enjoyed by the multiplier subschemes V(S), and it will follow that M is Kähler-Einstein.

To define the sheaf I(S), subtract constants from the \(\phi_ k\) and assume sup \(\phi\) \({}_ k=0\). For any \(L\in Pic(M)\), defined \(I_ S(L)=[s\in H^ 0(M,L):\) there exists a sequence \((s_ n)\) in \(H^ 0(M,L)\) and an \(r\in (m/(m+1),1)\) such that \((s_ n)\) converges uniformly to s and, for some sequence \((k_ n)\to \infty\), \((\int_{M}\| s_ n\|^ 2 \exp (-r\phi_{k_ n}) dV)\) is bounded as \(n\to \infty \}\); here \(m=\dim_{{\mathbb{C}}}M\), and \(\| \|\) is any metric on L. Now I(S) is the coherent sheaf of ideals on M determined by the graded ideal \(\oplus I_ S(H^{\mu})\) in the graded ring \(\oplus H^ 0(M,H^{\mu})\), for any ample \(H\in Pic(M).\)

Although a lot of hard nonlinear analysis is in the background of this paper, the results proved in this paper use only the linear analysis of the proof of Kodaira’s vanishing theorem. The paper is clearly written, and very readable.

Reviewer: R.R.Simha (Bombay)