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A transcendental approach to Kollár’s injectivity theorem. (English) Zbl 1270.32004

The author proves an analytic generalization of J. Kollár’s injectivity theorem [Ann. Math. (2) 123, 11–42 (1986; Zbl 0598.14015)]. Let \(X\) be an \(n\)-dimensional compact Kähler manifold and \((E,h_E)\) and \((L,h_L)\) be a holomorphic vector bundle and a holomorphic line bundle on \(X\) with smooth hermitian metrics; suppose that there exists a holomorphic line bundle \(F\) on \(X\) with a singular hermitian metric \(h_F\) such that:
(i) there exists \(Z\subset X\) such that \(h_F\) is smooth on \(X\setminus Z\);
(ii) \(\sqrt{-1}\Theta(F)\geq -\gamma\) for \(\gamma\) a smooth \((1,1)\)-form on \(X\);
(iii) \(\sqrt{-1}(\Theta(E)+Id_E\otimes\Theta(F))\geq_{Nak} 0\) on \(X\setminus Z\);
(iv) \(\sqrt{-1}(\Theta(E)+Id_E\otimes\Theta(F)-\epsilon Id_E\otimes\Theta(L))\geq_{Nak} 0\) on \(X\setminus Z\) for some \(\epsilon>0\).
Then, for \(s\) a nonzero holomorphic section of \(L\), the multiplication homomorphism
\[ H^q(X,K_X\otimes E\otimes F\otimes\mathcal{I}(h_F))\rightarrow H^q(X,K_X\otimes E\otimes F\otimes \mathcal{I}(h_F)\otimes L) \] is injective for \(q\geq 0\), where \(\mathcal{I}(h_F)\) is the multiplier ideal associated to \(h_F\).
The proof, illustrated in Section 3, is based on interpreting the cohomology groups as spaces of harmonic forms on a Zariski open subset with a suitable complete Kähler metric; an important tool for it are \(L^2\)-estimates for the \(\bar{\partial}\)-equation on complete Kähler manifolds. Section 4 then contains applications of the main results to injectivity and vanishing theorems in algebraic geometry.

MSC:

32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)

Citations:

Zbl 0598.14015
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Full Text: arXiv Euclid

References:

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