A derived category approach to generic vanishing.

*(English)*Zbl 1137.14012Let \(X\) be a smooth complex variety, \(A\) its Albanese variety, and let \(a_X: X\to A\) denote the Albanese morphism. The so-called “generic vanishing theorem” by M. Green and R. Lazarsfeld [Invent. Math. 90, 389–407 (1987; Zbl 0659.14007)] states that then every irreducible component of the subvariety
\[
V^i(\omega_X):= \{P\in \text{Pic}^0(X)\mid h^i(X,\omega_X\otimes P)\neq 0\}\text{ of }\text{Pic}^0(X)
\]
is a translate of a subtorus of \(\text{Pic}^0(X)\) of codimension at least \(i- (\dim(X)-\dim(a_X(X))\). Moreover, if \(\dim(X)= \dim(a_X(X))\) then there is a filtration
\[
\{{\mathcal O}_X\}= V^{\dim(X)}(\omega_X)\subset\cdots\subset V^1(\omega_X)\subset V^0(\omega_X).
\]
The proof of this crucial theorem (and of its further generalizations) made essential use of transcendental methods, including Hodge theory.

In the paper under review, the author investigates the natural question of to what extent this result is of purely algebraic nature, and what transcendental aspects of its so far existing proof can be replaced by an algebraic approach in the spirit of H. Esnault and E. Viehweg [Lectures on vanishing theorems. DMV Seminar. 20. Basel: Birkhäuser Verlag (1992; Zbl 0779.14003)].

In fact, using relatively simple methods from the theory of derived categories, amongst them being Grothendieck duality, the projection formula, base change techniques, and the Fourier-Mukai transform, the author proves a very general generic vanishing theorem for coherent sheaves on abelian varieties. His general generic vanishing theorem implies the following concrete geometric results:

(1) For any coherent sheaf \(F\) on an abelian variety \(A\), every irreducible component of the subvariety \[ V^i(A, F\otimes P):= \{P\in \text{Pic}^0(A)\mid h^i(A, F\otimes P)\neq 0\} \] has codimension at least \(i\) in the dual abelian variety \(\widehat A= \text{Pic}^0(A)\), \(0\leq i\leq \dim(A)\).

(2) If \(X\) is a smooth projective variety with Albanese variety \(A\), and if \(F\) is taken to be the higher direct image sheaf \(R^i(a_X)_*)\omega_X)\), then a generalization of the generic vanishing results of Green-Lazarsfeld is obtained in a straightforward manner.

Moreover, a more general version of a long-standing conjecture of Green-Lazarsfeld is proved in the following form:

(3) If \(X\) is a smooth projective variety with \(\dim(a_X(X))= \dim(X)\) for the Albanese map then for the universal family of topological trivial line bundles \({\mathcal P}\to X\times\text{Pic}^0(X)\) with Poincaré bundle \({\mathcal P}\), the vanishing property \(R^i(\pi_{\widehat A})_*({\mathcal P}_= 0\) holds for all \(i<\dim(X)\).

Finally, as a further application of his algebraic approach developed in the current paper, the author extends an earlier result of his and J. A. Chen [Duke Math. J. 111, No. 1, 159–175 (2002; Zbl 1055.14010)] as follows: Let \(a: X\to A\) be a surjective morphism with connected fibers from a smooth complex variety \(X\) with Kodaira dimension zero to an abelian variety \(A\). Then, for all \(N> 0\) there exists a unipotent vector bundle \(V_N\) on \(A\) and an inclusion \(V_N\hookrightarrow R^0 a_*(\omega^{\otimes N}_X)\) which is generically an isomorphism and induces an isomorphism on the spaces of global sections.

In the paper under review, the author investigates the natural question of to what extent this result is of purely algebraic nature, and what transcendental aspects of its so far existing proof can be replaced by an algebraic approach in the spirit of H. Esnault and E. Viehweg [Lectures on vanishing theorems. DMV Seminar. 20. Basel: Birkhäuser Verlag (1992; Zbl 0779.14003)].

In fact, using relatively simple methods from the theory of derived categories, amongst them being Grothendieck duality, the projection formula, base change techniques, and the Fourier-Mukai transform, the author proves a very general generic vanishing theorem for coherent sheaves on abelian varieties. His general generic vanishing theorem implies the following concrete geometric results:

(1) For any coherent sheaf \(F\) on an abelian variety \(A\), every irreducible component of the subvariety \[ V^i(A, F\otimes P):= \{P\in \text{Pic}^0(A)\mid h^i(A, F\otimes P)\neq 0\} \] has codimension at least \(i\) in the dual abelian variety \(\widehat A= \text{Pic}^0(A)\), \(0\leq i\leq \dim(A)\).

(2) If \(X\) is a smooth projective variety with Albanese variety \(A\), and if \(F\) is taken to be the higher direct image sheaf \(R^i(a_X)_*)\omega_X)\), then a generalization of the generic vanishing results of Green-Lazarsfeld is obtained in a straightforward manner.

Moreover, a more general version of a long-standing conjecture of Green-Lazarsfeld is proved in the following form:

(3) If \(X\) is a smooth projective variety with \(\dim(a_X(X))= \dim(X)\) for the Albanese map then for the universal family of topological trivial line bundles \({\mathcal P}\to X\times\text{Pic}^0(X)\) with Poincaré bundle \({\mathcal P}\), the vanishing property \(R^i(\pi_{\widehat A})_*({\mathcal P}_= 0\) holds for all \(i<\dim(X)\).

Finally, as a further application of his algebraic approach developed in the current paper, the author extends an earlier result of his and J. A. Chen [Duke Math. J. 111, No. 1, 159–175 (2002; Zbl 1055.14010)] as follows: Let \(a: X\to A\) be a surjective morphism with connected fibers from a smooth complex variety \(X\) with Kodaira dimension zero to an abelian variety \(A\). Then, for all \(N> 0\) there exists a unipotent vector bundle \(V_N\) on \(A\) and an inclusion \(V_N\hookrightarrow R^0 a_*(\omega^{\otimes N}_X)\) which is generically an isomorphism and induces an isomorphism on the spaces of global sections.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14K05 | Algebraic theory of abelian varieties |

18E30 | Derived categories, triangulated categories (MSC2010) |

14F17 | Vanishing theorems in algebraic geometry |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

##### References:

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