## 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)

Zbl 0598.14015
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### References:

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