Eckart Viehweg

Lectures on vanishing theorems

1 Kodairas vanishing theorem, a general discussion.- 2 Logarithmic de Rham complexes.- 3 Integral parts of Q-divisors and coverings.- 4 Vanishing theorems, the formal set-up.- 5 Vanishing theorems for invertible sheaves.- 6 Differential forms and higher direct images.- 7 Some applications of vanishing theorems.- 8 Characteristic p methods: Lifting of schemes.- 9 The Frobenius and its liftings.- 10 The proof of Deligne and Illusie [12].- 11 Vanishing theorems in characteristic p.- 12 Deformation theory for cohomology groups.- 13 Generic vanishing theorems [26], [14].- Appendix: Hypercohomology and spectral sequences.- References.

The Arason invariant and mod 2 algebraic cycles

Introduction 73 1. Review of the Arason invariant 75 2. The special Clifford group 76 3. K-cohomology of split reductive algebraic groups 78 4. K-cohomology of BG 86 5. GL(N) and Cliff(n, n) 92 6. Two invariants for Clifford bundles 95 7. Snaking a Bloch-Ogus differential 100 8. Proof of Theorem 1 101 9. Application to quadratic forms 102 Appendix A. Toral descent 104 Appendix B. The Rost invariant 108 Appendix C. An amusing example 113 Acknowledgements 116 References 116

Base Spaces of Non-Isotrivial Families of Smooth Minimal Models

Given a polynomial h of degree n let M h be the moduli functor of canonically polarized complex manifolds with Hilbert polynomial h. By [23] there exist a quasi-projective scheme M h together with a natural transformation

Chow groups of projective varieties of very small degree

\Psi :\mathcal{M}_h \to Hom(\_,M_h )

On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds

such that M h is a coarse moduli scheme for M h . For a complex quasi-projective manifold U we will say that a morphism ϕ U → M h factors through the moduli stack, or that ϕ is induced by a family, if ϕ lies in the image of Ψ(U), hence if ϕ = Ψ(ƒ: V → U).

Complex multiplication, Griffiths-Yukawa couplings, and rigidity for families of hypersurfaces

In this note we prove an effective version of the positivity theorems for certain direct image sheaves for fibre spaces over curves and apply it to obtain bounds for the height of points on curves of genus g ≥ 2 over complex function fields. Similar positivity theorems over higher dimensional basis and their applications to moduli spaces [13] were presented by the second author at the conference on algebraic geometry, Humboldt Universitat zu Berlin, 1988.

Numerical bounds for semi-stable families of curves or of certain higher-dimensional manifolds

(see [12], [5] and the references given there). These facts, together with various conjectures on the cohomology and Chow groups of algebraic varieties, suggest that the Chow groups of X might satisfy CHl(X)⊗Q = CHl(Pk)⊗Q = Q (∗) for l ≤ κ− 1 (compare with Remark 5.6 and Corollary 5.7). This is explicitly formulated by V. Srinivas and K. Paranjape in [16], Conjecture 1.8; the chain of reasoning goes roughly as follows. Suppose X is smooth. One expects a good filtration 0 = F j+1 ⊂ F j ⊂ . . . ⊂ F 0 = CH(X ×X)⊗Q, whose graded pieces F /F l+1 are controlled by H2j−l(X × X) (see [10]). According to Grothendieck’s generalized conjecture [8], the groups H (X) should be generated by the image under the Gysin morphism of the homology of a codimension κ subset, together with the classes coming from P. Applying this to the diagonal in X × X should then force the triviality of the Chow groups in the desired range. For zero-cycles, the conjecture (∗) follows from Roitman’s theorem (see [17] and [18]):

Effective Iitaka fibrations

For a non-isotrivial family of surfaces of general type over a complex projective curve, we give upper bounds for the degree of the direct images of powers of the relative dualizing sheaf. They imply that, fixing the curve and the possible degeneration locus, the induced morphisms to the moduli scheme of stable surfaces of general type are parameterized by a scheme of finite type. The method extends to families of canonically polarized manifolds, but the modular interpretation requires the existence of relative minimal models.