Lectures on Vanishing Theorems

By H. Esnault, E. Viehweg. 2004

234 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f5b614e88ea73ca669d4a2f3d02f6be8aa676b06

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Citations in the corpus

GV-sheaves, Fourier-Mukai transform, and generic vanishing (2006, methodology): “When the nef part is trivial, one recovers the above results of Green-Lazarsfeld [GL1] and Hacon [Hac] — in this case, although the proof is just a variant of Hacon’s, it has the following extra feature: Kollár’s theorems on higher direct images of dualizing sheaves are not invoked, but rather just ”
p −1-Linear Maps in Algebra and Geometry (2012, background): “iﬀerentials are not OX-linear). However, the complex F∗Ω q X is a complex of OX-modules (notice that d(xp) = 0). We now state the Cartier isomorphism. We take this presentation from [Car57], [Kat70], [EV92], and [BK05]. Deﬁnition-Proposition 3.1.1. There is a natural isomorphism (of OX-modules): C−1”
Families of canonically polarized varieties over surfaces (2005, background): “Proof. To prove (2.14.1), consider the following diagram with exact rows [ EV92 , 2.3a]:”
A derived category approach to generic vanishing (2003, background): “[ EV ] §11), one can recover some similar statements for X a proper smooth scheme over a perfect field k admitting a lifting ˜”
A TRANSCENDENTAL APPROACH TO KOLL AR'S INJECTIVITY THEOREM II (2007, unknown): “ version of Koll´ar’s theorem (see [En, Theorem 0.2] and [F1, Corollary 1.4]). We note that Koll´ar’s proofs and the approach of Esnault– Viehweg are geometric (cf. [K1], [K2, Chapters 9 and 10], and [EV]) and are not related to curvature conditions. Therefore, we do not know the true relationship b”
Hodge metrics and the curvature of higher direct images (2007, background): “a [Ka1], Viehweg [Vi1], Koll´ar [Ko1], and so on, in more algebro-geometric setting. There are also positivity results on higher direct images by Moriwaki [Mw], Fujino [Fn] and so on. We refer to [Mr][EV][Vi2][Ko2] for further remarks on related results. On the analytic side, it can be understood as”
Orbifold generic semi-positivity: an application to families of canonically polarized manifolds (2013, methodology): “ocal description above over U, and extended to Y by applying (iU)∗ also. Remark 1.1. If (X,∆) is smooth, and if the support of ∆ is of simple normal crossings, we can obtain from [KMM87] (see equally [EV] and the references therein) a (non-cyclic) ﬁnite cover π : Y → X with Y smooth, and a branching”
On Hodge spectrum and multiplier ideals (2002, background): “ of D get ”worse” if J(D) is smaller. The multiplier ideals have been deﬁned ﬁrst analytically by A. M. Nadel in [Na90]. They appeared implicitly in algebraic geometry through vanishing theorems, see [EV92]. Since then, the multiplier ideals found numerous applications in algebraic geometry (see [De”
Fundamental Theorems for the Log Minimal Model Program (2009, methodology): “composition morphism. For the details, see [ EV , 3.5.”
Numerical Godeaux surfaces with an involution (2005, background): “ 2 Pk i=1 Ci is eﬀective, with zero integral part, and its support has normal crossings. Thus hi(W,OW (2KW +L)) = 0 for i>0 by Kawamata–Viehweg vanishing theorem (see, e.g., Corollary 5.12, c), of [EV]). For surfaces of general type with pg = 0 having an involution, one can be more speciﬁc. As sh”
The Grothendieck-Lefschetz theorem for normal projective varieties (2005, methodology): “ES 5 Theorem 3. Let W be a non-singular projective variety over k, and M a big and base-point free line bundle on W. Then Hi(W,M−1) = 0 for i<dimW. Proof. A proof of this statement can be found in [6], Cor. 5.6(b). 3. Proofof Theorem 2 3.1. Some preliminary reductions. From now on, we ﬁx the foll”
Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces (2009, methodology): “ it here for the convenience of the reader. We use vanishing results of Esnault-Viehweg [10, Theorem 5.1], based on logarithmic de Rham complexes. Recall that the characteristic is zero. Theorem 3.1 ([10], Theorem 3.2). Let X be a nonsingular projective variety and T a Q-divisor such that T ∼Q 0 and”
A canonical linear system associated to adjoint divisors in characteristic $p > 0$ (2011, methodology): “ng theorems and multiplier-ideal-like constructions in [EV83], cf. [Wal79]. Another proof of these bounds explicitly using the language of multiplier ideals can be found in [Laz04, Section 10.1], cf. [EV92]. This latter proof goes through without change and so we obtain a proof of the following char”
Cherednik algebras and Hilbert schemes in characteristic p (2003, background): “thus, the ﬁrst claim of part (i) of the Lemma is proved. Recall further that the variety M is symplectic, hence it has trivial canonical bundle. Therefore, by the Grauert-Riemenschneider theorem (see [EV] for chark >0 case), the higher direct image sheaves Rqf∗OM vanish for all q>0. Therefore,”