18 research outputs found

    Strong Stability of Cotangent Bundles of Cyclic Covers

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    Let XX be a smooth projective variety over an algebraically closed field kk of characteristic p>0p>0 of dimX4\dim X\geq 4 and Picard number ρ(X)=1\rho(X)=1. Suppose that XX satisfies H^i(X,F^{m*}_X(\Omg^j_X)\otimes\Ls^{-1})=0 for any ample line bundle \Ls on XX, and any nonnegative integers m,i,jm,i,j with 0i+j<dimX0\leq i+j<\dim X, where FX:XXF_X:X\rightarrow X is the absolute Frobenius morphism. We prove that by procedures combining taking smooth hypersurfaces of dimension 3\geq 3 and cyclic covers along smooth divisors, if the resulting smooth projective variety YY has ample (resp. nef) canonical bundle ωY\omega_Y, then \Omg_Y is strongly stable ((resp. strongly semistable)) with respect to any polarization.Comment: To appear in Comptes Rendus Math\'ematiqu

    L2L^2-Dolbeault resolution of the lowest Hodge piece of a Hodge module

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    Let XX be a complex space and MM a pure Hodge module with strict support XX. The purpose of this paper is to introduce a coherent subsheaf S(M,φ)S(M,\varphi) of M. Saito's S(M)S(M) which is a combination of S(M)S(M) and the multiplier ideal sheaf I(φ)\mathscr{I}(\varphi) while constructing a resolution of S(M,φ)S(M,\varphi) by differential forms with certain L2L^2-boundary conditions. This could be viewed as a wide generalization of MacPherson's conjecture on the L2L^2-representation of the Grauert-Riemenschneider sheaf. As applications, various vanishing theorems for S(M)S(M) (Saito's vanishing, Kawamata-Viehweg vanishing and some new ones like Nadel vanishing, partial vanishing) are proved via standard differential geometrical arguments.Comment: 25 pages. Comments are welcome
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