ICTP - The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

School on Vanishing Theorems and Effective Results in Algebraic Geometry

Supported by the European Commission, Research DG, Human Potential Programme,
High Level Scientific Conferences HPCF-CT-1999-00140

25 April - 12 May 2000

Miramare - Trieste, Italy

DETAILED SCIENTIFIC PROGRAMME (Latex version retrievable) .

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\paragraph{Adjoint Linear Series.}
Let $C$ be a compact Riemann surface of genus $g$, and let $L$ be a holomorphic line bundle on $C$. A very basic classical theorem states that if $\deg L \ge 2g$ then $L$ is generated by its global sections, and that if $\deg L \ge 2g + 1$ then $L$ is very ample. Surprisingly enough, a clear picture of how this should generalize to higher dimensions has only recently come into focus. The key is to note that $\deg L \ge 2g$ if and only if $L$ is of the form $L = K_C + A $ for some ample divisor $A$ on $C$ with $\deg A \ge 2$, where as usual $K_C$ denotes a canonical divisor on $C$; similarly, $\deg L \ge 2g + 1$ if and only if $L = K_C + A$ with $\deg A \ge 3$.
This suggests that the correct higher-dimensional analogue of divisors of large degree on a curve are the so-called \textit{adjoint bundles} of the form $K + A$ for $K$ canonical and $A$ suitably positive.
Suppose then that $X$ is a smooth complex projective variety of dimension $n$, and denote by $K_X$ the canonical divisor of $X$. A conjecture of Fujita has sparked a great deal of activity: \begin{conjecture} If $B$ is an ample divisor on $X$,then:
\begin{enumerate}
\item [(i).] $\OO_X\big(K_X + (n+1)B \big)$ is globally generated \item[(ii).] $\OO_X\big(K_X + (n+2)B \big)$ is very ample. \end{enumerate} \end{conjecture}
\noi One actually expects more general statements dealing with linear series of the form $|K_X + L|$ provided that the $L$-degree of every subvariety of $X$ is sufficiently large. When $n = 2$ the conjecture follows from results of Reider. In higher dimensions, the first significant result is perhaps one obtained in the early 90's by Demailly, according to which $2K_X + 12 n^n B$ is always very ample. The proof uses deep analytic tools. Among other things, this paper pioneered the use of multiplier ideals and vanishing theorems for $\QQ$-divisors in connection with these questions. Ein and Lazarsfeld then observed that one could use algebro-geometric techniques developed by Kawamata, Shokurov and others to obtain effective results, and proved (i) when $n = 3$. Koll\'ar obtained non-vanishing results in all dimensions, and in a significant breakthrough Angehrn and Siu proved the freeness of $|K_X + \binom{n+1}{2} B|$ in all dimensions. While still based on the Kawamata-Shokurov method, the latter work introduced some interesting new ideas involving multiplier ideals. The theorem of Angehrn-Siu was was extended and improved by Koll\'ar, Helmke and Kawamata, and (i) is now known in dimensions $\le 5$. Interestingly enough, from the point of view of current techniques, very ampleness seems significantly harder to treat than global generation.
\vskip 10pt
\paragraph{Matsusaka's Theorem.}
A fundamental theorem of Matsusaka from the 1970's states that if $X$ is a smooth projective variety, and $L$ is an ample line bundle on $X$, then there is a number $m_0$ depending only on the Hilbert polynomial of $L$ such that $\OO_X(m_0L)$ is very ample. Koll\'ar and Matsusaka later showed that in fact $m_0$ depends only on the intersection numbers $(L^n)$ and $(K_X \cdot L^{n-1})$. It was long considered something of a mystery that no truely cohomological proof was known. However shortly after Demailly's paper appeared, Siu realized that one could use vanishing theorems for multiplier ideals to approach the problem cohomologically, and to obtain an effective computation of $m_0$. Siu's results were simplified and extended by Demailly. Given any new line bundle $B$ on $X$ (see below), Siu obtains in fact an effective computation of an sinteger $m_B$ such that $mL - B$ is nef for $m \ge m_B$.
\vskip 10 pt
\paragraph{Deformation Invariance of Plurigenera.}
One of the most striking applications of vanishing theorems for multiplier ideals is the proof by Siu of a long-standing conjecture concerning the deformation invariance of plurigenera. Suppose that $f : X \lra T$ is a one-paramater family of smooth varieties of general type. Siu proves that all the plurigenera \[ P_m(X_t) =_{\text{def}} h^0 \big( X_t,\OO_{X_t}(mK_{X_t}) \big) \] are constant as functions of $t$. Siu's approach, which is remarkably simple, involves comparing multiplier ideals constructed from pluricanonical series on the family with analogous ideals constructed on a particular fibre $X_0$. His argument was adapted by Kawamata to show that canonical singularities are invariant under deformation.
\vskip 10pt
\paragraph{Local Positivity of Line Bundles.}
Motivated by his work on Fujita's conjecture, Demailly introduced an interesting invariant -- the so-called Seshadri constant $\epsilon(L, x)$ -- which measures in effect how much of the positivity of an ample line bundle $L$ can be concentrated in the neighborhood of a point $x \in X$ on a smooth projective variety $X$. An elementary computation on surfaces by Ein and Lazarsfeld led to the surprising possibility that there might be universal lower bounds on $\epsilon(L,x)$ that hold at a very general point $x$ of any variety. Ein-K\"uchle-Lazarsfeld obtained such a bound depending on the dimension of $X$, but the natural conjecture is that in fact $\epsilon(L, x) \ge 1$ for very general $x$. These invariants are related to questions of symplectic packing and to a long-standing conjecture of Nagata on plane curves. When $X = (A, \Theta)$ is a principally polarized abelian variety, they are related to some work of Buser and Sarnak on lengths of periods.
\vskip 10pt
\paragraph{Effective Questions in Algebra.}
Applications of analytic techniques to questions in algebra go back at least to the classical work of Skoda and Brian\c con -- Skoda. On the algebraic side, the theory of tight closure developed by Hochster and Huneke has led to simple and purely algebraic approaches to some of these results. As work of Karen Smith and Huneke makes clear, there are suggestive points of contact with the more analytic techniques surrounding vanishing theorems and multiplier ideals. On the geometric front, Ein-Lazarsfeld remarked that elementary arguments with vanishing theorems lead to extensions and new perspectives on the effective Nullstellensatz of Brownawell, Koll\'ar and others.
\vskip 10 pt
\paragraph{Numerically Effective and Pseudo-Effective Vector Bundles.}
A numerically effective (nef) line bundle is by definition a line bundle of non-negative degree on every closed curve, whilst a pseudo-effective line bundle is a line bundle which is linear equivalent to a weak limit of effective ${\bf Q}$-divisors. These concepts have meaningful extensions to vector bundles and are central in the study of the structure of algebraic and compact K\"ahler mani\-folds. Much work has been done to understand the structure of compact K\"ahler manifolds with nef tangent or anticanonical bundles (Campana,Demailly, Peternell, Schneider, Qi Zheng). There is still a lot of activity in this area,e.g.\ around Lefschetz type theorems for cohomology groups with values in pseudo-effective line bundles. Viehweg has obtained various important results on positivity of direct image sheaves.
\vskip 10pt
\paragraph{Hyperbolicity.}
Hyperbolic varieties are tied to number theory and algebraic geometry in a number of interesting and intricate ways. It is expected (Green-Griffiths, Lang, Vojta), as a generalization of Falting's beautiful result on the Mordell conjecture, that a hyperbolic variety defined over ${\bf Q}$ only contains finitely many rational points. On the geometric side, the basic question is to understand the structure of entire transcendental curves drawn on an algebraic variety. There are only very partial answers known, mostly in the cases of subvarieties of abelian varieties or of algebraic surfaces (Ochiai, Green-Griffiths, Kawamata, Noguchi, Siu, Yeung, Demailly~$\ldots$). The most recent tools include a study of algebraic jet differential and jet bundles, in relation with deep techniques of value distribution theory.
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PARTICIPATION

Mathematicians from all countries that are members of the UN, UNESCO or IAEA can attend the School. The main purpose of the Centre is to help research workers from developing countries through a programme of training activities within a framework of international cooperation. However, students and post-doctoral scientists from developed countries are also welcome to attend. As the School will be conducted in English, participants should have an adequate working knowledge of that language. Participants should preferably have completed some years of study and research after a first degree.

There is no registration fee for participation in the School.

As a rule, all expenses of the participants should be borne by the home institution. However, a limited number of financial grants are available for participants from developing countries. As scarcity of funds allows travel to be granted only in few exceptional cases, every effort should be made by candidates to secure support for their fares (or at least half of their fares) from their home country.

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* Member States of the European Union
Austria, Belgium, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Luxemburg, Netherlands, Portugal, Spain, Sweden, United Kingdom.

** Associated States
Bulgaria, Cyprus, Czech Republic, Estonia, Hungary, Iceland, Israel, Latvia, Liechtenstein, Lithuania, Norway, Poland, Romania, Slovakia, Slovenia.

	
  	School on Vanishing Theorems and Effective Results in Algebraic Geometry
	(c/o Ms. A. Bergamo)
	Strada Costiera 11
	I-34014 Trieste
	Italy
	Tel.: +39 040 2240201           
	Fax:  +39 040 2240490

	
                                                Trieste, August 1999

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