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Higher dimensional complex geometry. A summer seminar at the University of Utah, Salt Lake City, 1987. (English) Zbl 0689.14016
Centre National de la Recherche Scientifique. Astérisque, 166. Paris: Société Mathématique de France. 144 p. FF 100.00; $ 17.00 (1988).
The 24 chapter of this book consist of notes of a seminar held in 1987 at Salt Lake City.
An introduction to Mori’s minimal model program is presented in the 16 first chapters, which although of interest to specialists is understandable with a good background in algebraic geometry. This program aims at the classification of projective n-folds X according to the numerical positivity of their canonical bundle \(K_ X\). This (non- birational) invariant can be considered as a refinement of Kodaira dimension. Say that X is minimal if \(K_ X\cdot\) is nef (= numerically effective), i.e.: \(K_ X\cdot C\geq 0\) for every effective curve C of X.
When X is not minimal, S. Mori showed the existence of extremal rational curves C on X, for which: \(0<-K_ X\cdot C\leq n+1\), and which generate the extremal rays of the closed cone of effective curves of X. He showed when \(n=2, 3\) that each such extremal ray R determines an extremal contraction \(f: X\to Y\) with Y projective, which maps to points exactly the curves the numerical class of which is in R. This is the content of chapters 1 to 4, where proofs are given, which are very close to the geometric intuition.
Chapters 5 to 7: Let \(f: X\to Y\) be an extremal contraction of the n-fold X. If \(n\geq 3\) (respectively \(n\geq 4)\), then Y (respectively \(K_ Y)\) may not be smooth (respectively \({\mathbb{Q}}\)-Cartier). The process of contraction had then to stop. Chapter 5 explains how one is led to admit terminal singularities and “flips” in order to construct a minimal model program, that is a systematic procedure to produce minimal models (i.e.: with K nef) of projective terminal varieties by a succession of extremal contractions and flips.
Chapter 6 discusses terminal singularities, in particular in dimension 3.
Chapter 7 deals with extension of this program to some other usual situations.
Chapters 8 to 13: The first step of Mori’s program, that is the existence of extremal contractions of canonical projective varieties the canonical bundle of which is not nef, is proved. The proof consists of several fundamental results: Kawamata-Viehweg vanishing, Shokurov non-vanishing, base-point freeness and rationality theorems. The proofs given are the simplest known. Chapters 14 to 16: They deal with the existence of flips, which is known only in dimension 3. They provide an accessible introduction to the difficult paper of S. Mori where this result is proved.
Chapters 17 to 20: They treat some applications of harmonic maps to Kähler geometry, due to Carlson and Toledo: a continuous map \(f: M\to N\) from a compact Kähler manifold M to a compact Riemannian locally symmetric space is homotopic to a non-surjective harmonic map unless N is locally hermitian symmetric. - Chapter 20 gives an upper bound for the rank of variations of Hodge structures of weight 2.
Chapters 21 to 22 are devoted to the study of possibly singular curves of low genus g on generic hypersurfaces of degree d in \({\mathbb{P}}^ n:\) Chapter 21 deals with the cases \(g=0\), \(d\geq 2n-1\) (they don’t exist), and \((n+1)\leq d\leq (2n-2)\). - Chapter 22 deals with the case \(g=0, 1, 2\) and \(d=5\), \(n=4.\)
Chapters 23 to 24: The submanifolds Z of complete intersections X in Grassmannians \(Gr(r,n)\) are studied. An application is the following: Z is of general type if X is of type \((m_ 1,...,m_ k)\) with: \(m\geq \dim(X)+n+1\), and \(m:=m_ 1+...+m_ k.\)
In the following commented table of contents the authors’ names are listed only in case they are not the editors of this seminar notes.
Chapter 1: Finding rational curves when \(K_ X\) is negative (p. 9-15).
Chapter 2: Finding rational curves when \(K_ X\) is non-semi-positive (p. 16-18).
Chapter 3: Surface classification (p. 19-21).
Chapter 4: The cone of curves, smooth case (p. 22-27).
These chapters are devoted to the proof of the existence of extremal rational curves C (i.e.: such that \(0<K_ X\cdot C\leq n+1)\) which generate the extremal rays of the closed cone of (numerical equivalence of) effective curves on a projective n-fold X whose canonical bundle \(K_ X\) is not nef (chapters 1, 2, 4). - When X is a surface, such an extremal ray R determines an extremal contraction \(f: X\to Y\) mapping to points the curves the class of which lies in R. Such an extremal contraction is either: the constant map (iff \(X={\mathbb{P}}^ 2)\), or a ruling of X, or the contraction of a \((-1)\)-curve on X. - The proofs are very close to geometric intuition; the core of the passage through characteristic \(p>0\) is reduced to an elementary lemma in elimination theory. These lecture are thus a very accessible introduction to the techniques of S. Mori. (Notice that lemma 1.5 is false, although only its true part is used.)
Chapter 5: Introduction to Mori’s program (p. 28-37).
Let \(f: X\to Y\) be an extremal contraction of the n-fold X. If \(n\geq 3\) (resp. \(n\geq 4)\), then Y (resp. \(K_ Y)\) may not be smooth (resp. \({\mathbb{Q}}\)-Cartier). The process of contraction has then to stop. Chapter 5 explains how one is led to admit terminal singularities and “flips” in order to construct a minimal model program, that is a systematic procedure to produce minimal models (i.e.: with K nef) of projective terminal varieties by a succession of extremal contractions and flips.
Chapter 6: Singularities in the minimal model program (p. 38-46).
This chapter deals with terminal singularities. The notion of discrepancy for \({\mathbb{Q}}\)-Cartier singularities is introduced, as well as the notion of canonical and terminal singularities. Terminal singularities are shown to be rational, at least in dimension 3. Canonical singularities are discussed in dimensions 2 and 3. In partigular: a 3-fold singularity is terminal Gorenstein iff it is a hypersurface double point.
Chapter 7: Extensions of the minimal model program (p. 47-49).
This chapter is concerned with extensions of Mori’s program to the relative case and to the case of varieties on which a finite group acts.
Chapter 8: Vanishing theorems (p. 50-56).
A simple proof of the Kawamata-Viehweg vanishing theorem and of its corollaries is given. The proof rests on Hodge theory and the fact that complex n-dimensional affine varieties have the homotopy type of a real n-dimensional CW-complex.
Chapter 9: Introduction to the proof of the cone theorem (p. 57-59). Chapter 10: Basepoint-free theorem (p. 60-62).
Chapter 11: K.Matsuki: The cone theorem (p. 63-66).
Chapter 12: Rationality theorem (p. 67-73).
Chapter 13: K. Matsuki: Non-vanishing theorem (p. 74-76).
The cone theorem for canonical projective varieties is proven. The proof consists of several fundamental intermediate results: Shokurov non- vanishing, base-point free and rationality theorems. The proofs given here are the simplest known.
The first part of Mori’s program is thus established. The second part, namely the existence of flips is discussed in chapter 14 to chapter 16 in the 3-dimensional case.
Chapter 14: Introduction to flips (p. 77-82).
The termination of sequences of flips is established, as a consequence of the decreasing character of Shokurov’s notion of difficulty under flips. General properties of “extremal neighborhoods” X (of an irreducible curve C contracted to a point under a small extremal contraction in dimension 3) are proved: C is rational smooth, and does not contain more than two points at which X is not Gorenstein. Moreover:
(1) \((K_ X\cdot C)\in [-1,0);\)
(2) \(((\omega_ X/I\cdot \omega_ X)/Tors)\cong {\mathcal O}_ C(-1);\)
(3) \(((I/I^ 2)/Tors)\cong {\mathcal O}(a)\oplus {\mathcal O}(b)\) with a,b\(\geq - 1\), where I is the ideal of C in X.
It is also shown that the construction of flips can be reduced to that of (easier) flops, provided \(| -2K_ X|\) contains an element E such that the associated double cover has only canonical singularities. This double cover exists if \(| -K_ X|\) contains an element D which has only Du Val singularities. Such a D exists when X has only cyclic quotient terminal singularities.
Chapter 15: Singularities on an extremal neighborhood (p. 83-91).
This chapter introduces to the paper of S. Mori where the existence of terminal flips in dimension 3 is proved. It is thus intended to classify the triples \((X,C,p)\) where X is an extremal neighborhood of the flipped curve C, and \(p\in C\). Mori’s invariants \(i_ p\) and \(w_ p\) are introduced, and it is shown that the relations (2) and (3) of chapter 14 translate into the following inequalities, which are global near \(C\): \(((\sum w_ p <1))\) and ((\(\sum i_ p )\leq 3)\), the sum being taken over all points of C. From this it is deduced that, after passing to the index one cover of X, either C becomes planar, or \(mult_ p(C)\leq 3.\)
Chapter 16: Small resolutions of terminal singularities (p. 92-96).
Chapter 16 discusses small resolutions f: \(X\to Y\) of 3-fold terminal singularities Y, and explains how to construct the flop of f from a local equation \([x^ 2+q(y,z,t)]=0\) of the index one cover of Y using te involution which maps x to \((-x).\)
Chapter 17: D. Toledo: Kähler structures on locally symmetric spaces (p. 97-100).
Chapter 18: D. Toledo: Proof of Sampson’s theorem (p. 101-104).
Chapter 19: D. Toledo: Abelian subalgebras of Lie algebras (p. 105-108).
It is shown that any continuous map \(f: M\to N\) from a compact Kähler manifold M to a compact Riemannian locally symmetric space is homotopic to a non-surjective harmonic map, unless N is locally hermitian symmetric. One can assume f to be harmonic, by a theorem of Eells-Sampson (chapter 17). Another theorem of Sampson, proved in chapter 18, shows that for any \(x\in M\), \(df: W:=TM_ x^{1,0}\to (TN_{f(x)})^{{\mathbb{C}}}:=V\) maps W to an abelian subspace of V, i.e.: \([df,df]=0\). The last step, proved in chapter 19, shows that if W is any abelian subspace of V, then \(\dim_{{\mathbb{C}}}(W)\leq \dim_{{\mathbb{R}}}(V)\), with equality iff N is locally hermitian symmetric.
Chapter 20: J. Carlson: Maximal variations of Hodge structures (p. 109- 116).
By arguments analogous to that of chapter 19, upper bounds, sometimes sharp, are obtained for the rank r of local variations of Hodge structures of weight 2. For example: \(r\leq (h^{2,0}\cdot h^{1,1})\) when \(h^{1,1}\) is even and \(h^{2,0}\geq 3.\)
Chapter 21: Subvarieties of generic hypersurfaces (p. 117-122).
A typical example of the results proved here is the following: Let f: \(C\to V\subseteq {\mathbb{P}}^ n\) be a finite map from a smooth rational curve C to a generic hypersurface V of degree \( m\) in \({\mathbb{P}}^ n\). Let \(N_{f,V}:=(f^*T_ V/T_ C)\) be the normal sheaf to f. Then: \[ \text{rank}[N_{f,V}/\text{image}(H^ 0(C,N_{f,V}\otimes {\mathcal O}_ C)]>(m-(n+1)). \] In particular: V does not contain any rational curve if \(m\geq (2n-1).\)
Chapter 22: Conjectures about curves on generic quintic threefolds (p. 123-128).
The following two conjectures are stated, where V is a generic quintic hypersurface of \({\mathbb{P}}^ 4:\)
(1) For every degree \(d\geq 1\), V contains only finitely many rational curves of degree d.
(2) V is not covered by elliptic curves.
It is further shown that (1) implies (2), and that:
(3) V is covered by curves of genus 2 (more precisely plane quintics with 4 nodes.)
Chapter 23: L. Ein: Submanifolds of generic complete intersections in Grassmannians (p. 129-133).
Let X be a generic complete intersection of type \((m_ 1,...,m_ k)\) in the Grassmannian \(G=Grass(r,n)\). Let \(m:=(m_ 1+...+m_ k)\). Let Z be a submanifold of X, and let \(m_ 0:=\inf \{\mu: h^ 0(Z,K_ Z\otimes {\mathcal O}_ Z(\mu))>0\}\). Let c be the codimension in X of the subvariety covered by deformations (in X) of Z. It is shown here, using the Koszul resolution of the ideal of the graph in \((Z\times G)\) of the natural inclusion, that \(c\geq (m+m_ 0-(n+1))\). Thus: Z is of general type if: \(m\geq (\dim(X)+n+1)\). - The Hilbert scheme of X is also shown to be smooth at Z if \(\dim(Z)=1\) and \(h^ 1(Z,N_{Z| G})=0\). This last result uses the theorem of Gruson-Lazarsfeld-Peskine shown in chapter 24.
Chapter 24: L. Ein: A theorem of Gruson-Lazarsfeld-Peskine and a lemma of Lazarsfeld (p. 134-139).
Let \(C\subseteq {\mathbb{P}}^ n\) be a smooth curve of degree d, which spans \({\mathbb{P}}^ n\). Then: \(H^ 0({\mathbb{P}}^ n,{\mathcal O}(a))\to H^ 0(C,{\mathcal O}(a))\) is surjective if \(a\geq (d-h+1)\). This result, due to Gruson- Lazarsfeld-Peskine is shown here.
Reviewer: F.Campana

MSC:
14J30 \(3\)-folds
14C20 Divisors, linear systems, invertible sheaves
00Bxx Conference proceedings and collections of articles
14-06 Proceedings, conferences, collections, etc. pertaining to algebraic geometry
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14J15 Moduli, classification: analytic theory; relations with modular forms
14J10 Families, moduli, classification: algebraic theory
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