Classification theories of polarized varieties.

*(English)*Zbl 0743.14004
London Mathematical Society Lecture Note Series. 155. Cambridge (UK): Cambridge University Press. xiv, 205 p. (1990).

A polarized variety is a pair \((V,L)\) with \(V\) algebraic variety and \(L\) line bundle on \(V\); usually \(V\) is complete and \(L\) has some kind of “positivity”, e.g. (with different assumptions for different theories and different goals) \(L\) very ample or ample and spanned or nef and big. Polarized varieties are very natural (e.g. for \(L\) very ample this corresponds to a projective variety with a fixed linearly normal embedding in some projective space) and very important, e.g. (with \(L\) ample) to have a good notion of moduli schemes and periods (Torelli theorems). This book gives both a very useful introduction to various classification theories on polarized varieties and the state of the art (around 1989) of some of them in which the author made many fundamental contributions. Quoting the author, “we usually present a result with an outline of proof of it, and refer to other papers for technical details. I hope that this rather helps the reader to have a good idea of what is most important”: I agree!

Chapter 0 gives a (useful) summary of preliminaries, ending with vanishing theorems and some Mori-Kawamata theory (Kawamata-Shokurov contraction theorem), but starting with the notions needed for that. One of the games is to classify polarized varieties with “small invariants”. Fix \((V,L)\) with \(\dim(V)=n\); recall that the \(\Delta\)- genus of \((V,L)\) is \(n+L^ n-h^ 0(V,L)\), while the sectional genus \(g(V,L)\) (which is defined in general used the Hilbert polynomial \(\chi({\mathcal O}_ V(tL))\) if \(V\) has only Gorenstein singularities is an integer \(g(V,L)\) uniquely determined by: \(2g(V,L)-2=(K_ V+(n- 1)L)L^{n-1}\).

Chapter 1 is on the \(\Delta\)-genus culminating in the classification of Del Pezzo manifolds and polarized manifolds of \(\Delta\)-genus up to 2. When possible a very useful tool to carry over induction on \(\dim(V)\) is the existence of a ladder, i.e. a sequence \(V=V_ n\supseteq V_{n- 1}\supseteq\cdots\supseteq V_ 1\) of subvarieties of \(V\) with \(V_ i\in\left| L\mid_{V_{i+1}}\right|\) for every \(i\). The existence of a ladder is either trivial (e.g. if \(L\) is assumed to be very ample) or is one of the most difficult parts of the classification. The existence of a ladder (in non trivial cases) is the main tool in chapter 1.

Chapter 2 is on the sectional genus (with classification of polarized manifolds up to genus 2) and of positivity results (quite recent and interesting ones) on the “positivity” (neffity) of the adjoint bundles \(K_ V+tL\) for large \(t\).

Chapter 3 considers projective varieties (\(L\) very ample), giving Ionescu classification for varieties of small degree; again here the adjunction bundles and the adjunction morphism are the main actors (with Mori theory being a fundamental tool and Sommese’s name being everywhere dense).

The last chapter contains very briefly (I would have preferred a second volume!) two generalizations: (a) singular and quasi-polarized varieties: (b) ample vector bundles and the case of pair \((V,E)\) with \(E\) vector bundle on \(X\) (plus a computer program which enumerates the ruled polarized surfaces of fixed sectional genus).

I think that the author reached the goals he stated in the introduction, helping both beginners and experts; when I studied it in late 1990 I found there many interesting research problems. — My main reservation: to work on some of the author’s beautyful conjectures (and to follow many of the author’s papers) the reader will need much more Mori theory than recalled/used in this book; after reading this book a serious reader should check very recent works of the author for related matters and study carefully Mori-Kawamata theory [see Y. Kawamata, K. Matsuda and K. Matsuki in Algebraic geometry, Proc. Sympos., Sendai 1985, Adv. Stud. Pure Math. 10, 283-360 (1987; Zbl 0672.14006), H. Clemens, J. Kollár and S. Mori: “Higher dimensional complex geometry”, A summer seminar at the University of Utah, Salt Lake City 1987 Astérisque 166 (1988; Zbl 0689.14016), and J. Kollár, “Flips, flops, minimal models, etc.” in Proc. Conf., Cambridge/MA (USA) 1990, Surv. in Differ. Geom. 113-199 (1991), and “Flips and abundance for algebraic threefolds”, Notes of a summer seminar at the University of Utah, Salt Lake City 1991].

Chapter 0 gives a (useful) summary of preliminaries, ending with vanishing theorems and some Mori-Kawamata theory (Kawamata-Shokurov contraction theorem), but starting with the notions needed for that. One of the games is to classify polarized varieties with “small invariants”. Fix \((V,L)\) with \(\dim(V)=n\); recall that the \(\Delta\)- genus of \((V,L)\) is \(n+L^ n-h^ 0(V,L)\), while the sectional genus \(g(V,L)\) (which is defined in general used the Hilbert polynomial \(\chi({\mathcal O}_ V(tL))\) if \(V\) has only Gorenstein singularities is an integer \(g(V,L)\) uniquely determined by: \(2g(V,L)-2=(K_ V+(n- 1)L)L^{n-1}\).

Chapter 1 is on the \(\Delta\)-genus culminating in the classification of Del Pezzo manifolds and polarized manifolds of \(\Delta\)-genus up to 2. When possible a very useful tool to carry over induction on \(\dim(V)\) is the existence of a ladder, i.e. a sequence \(V=V_ n\supseteq V_{n- 1}\supseteq\cdots\supseteq V_ 1\) of subvarieties of \(V\) with \(V_ i\in\left| L\mid_{V_{i+1}}\right|\) for every \(i\). The existence of a ladder is either trivial (e.g. if \(L\) is assumed to be very ample) or is one of the most difficult parts of the classification. The existence of a ladder (in non trivial cases) is the main tool in chapter 1.

Chapter 2 is on the sectional genus (with classification of polarized manifolds up to genus 2) and of positivity results (quite recent and interesting ones) on the “positivity” (neffity) of the adjoint bundles \(K_ V+tL\) for large \(t\).

Chapter 3 considers projective varieties (\(L\) very ample), giving Ionescu classification for varieties of small degree; again here the adjunction bundles and the adjunction morphism are the main actors (with Mori theory being a fundamental tool and Sommese’s name being everywhere dense).

The last chapter contains very briefly (I would have preferred a second volume!) two generalizations: (a) singular and quasi-polarized varieties: (b) ample vector bundles and the case of pair \((V,E)\) with \(E\) vector bundle on \(X\) (plus a computer program which enumerates the ruled polarized surfaces of fixed sectional genus).

I think that the author reached the goals he stated in the introduction, helping both beginners and experts; when I studied it in late 1990 I found there many interesting research problems. — My main reservation: to work on some of the author’s beautyful conjectures (and to follow many of the author’s papers) the reader will need much more Mori theory than recalled/used in this book; after reading this book a serious reader should check very recent works of the author for related matters and study carefully Mori-Kawamata theory [see Y. Kawamata, K. Matsuda and K. Matsuki in Algebraic geometry, Proc. Sympos., Sendai 1985, Adv. Stud. Pure Math. 10, 283-360 (1987; Zbl 0672.14006), H. Clemens, J. Kollár and S. Mori: “Higher dimensional complex geometry”, A summer seminar at the University of Utah, Salt Lake City 1987 Astérisque 166 (1988; Zbl 0689.14016), and J. Kollár, “Flips, flops, minimal models, etc.” in Proc. Conf., Cambridge/MA (USA) 1990, Surv. in Differ. Geom. 113-199 (1991), and “Flips and abundance for algebraic threefolds”, Notes of a summer seminar at the University of Utah, Salt Lake City 1991].

Reviewer: E.Ballico (Povo)

##### MSC:

14C20 | Divisors, linear systems, invertible sheaves |

14E30 | Minimal model program (Mori theory, extremal rays) |

14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |

14J30 | \(3\)-folds |

14N05 | Projective techniques in algebraic geometry |

14J10 | Families, moduli, classification: algebraic theory |