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\title[\ \kern-190pt\rlap{\blank{J.-P.$\,$Demailly, Seshadri Memorial Lectures, July$\,$31,$\,$2020}}\kern183pt\rlap{\blank{A brief survey on Seshadri constants}}\kern178pt\llap{\blank{\framenumbering~}}\kern-10pt]
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{A brief survey on Seshadri constants}

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{\vskip-3mm Jean-Pierre Demailly\vskip-3mm}

\institute[]{Institut Fourier, Université Grenoble Alpes\ \ \&\ \ Académie des Sciences de Paris}
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\date[]% (optional)
{Memorial Lectures for\\
Professor Conjeevaram Srirangachari Seshadri\\
organized by the TIFR School of Mathematics\\
July 31, 2020, 4 PM IST}

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\begin{document}
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%%    \frametitle{Outline}
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%%\begin{frame}
%%  \frametitle{Outline}
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% Since this a solution template for a generic talk, very little can
% be said about how it should be structured. However, the talk length
% of between 15min and 45min and the theme suggest that you stick to
% the following rules:  

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

\begin{frame}
\frametitle{The starting point of Seshadri constant theory}
\vskip-7pt
\claim{The paper that started the whole theory:}\vskip 5pt
\alert{C.S. Seshadri, {\it Annals of Mathematics}, Vol.\ 95
(May 1972) \hbox{511--556\kern-10pt}}
\vskip5pt
\pgfdeclareimage[height=6cm]{p1}{p1}
\pgfuseimage{p1}
\end{frame}

\begin{frame}
\frametitle{Seshadri's ampleness criterion}
\vskip-3pt
\pgfdeclareimage[height=8cm]{p39}{p39}
\pgfuseimage{p39}
\end{frame}

\begin{frame}
\frametitle{Definition of the Seshadri constants}
\vskip-5pt
\begin{block}{Definition (D, 1990)}
Let $X$ be a projective nonsingular variety and
$L$ a nef (or pseudo-ample) line bundle over~$X$.\pause\ Given a
point $x\in X$, one defines the \alert{Seshadri
constant $\varepsilon(L,x)$} of $L$ at $x$ to be\vskip5pt
\alert{\centerline{$\displaystyle
 \varepsilon(L,x)=\inf_{{\rm all~alg.~curves}\,C\ni x}~~
{L\cdot C\over \mult_x(C)}.$}}
\end{block}\pause
This is a very interesting numerical invariant that measures in a
deep manner the ``local positivity'' of the line bundle $L$ at point $x$.\pause

\begin{block}{Equivalent definition (already observed in Seshadri's paper~!)}
Let $\pi:\tilde X\to X$ be the blow-up of $X$ at $x\in X$,
and $E$ the exceptional divisor in $\tilde X$.\pause\ Then, for
$L\in\Pic(X)$ assumed to be nef,
one has\vskip5pt
\alert{\centerline{$\displaystyle
 \varepsilon(L,x)=\sup\{\gamma\ge 0\,/\;
\pi^*L-\gamma E~\hbox{is nef on $\tilde X$}\}.$}}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Reformulation of Seshadri's ampleness criterion}
\vskip-7pt
\begin{block}{Reformulation of Seshadri's ampleness criterion}
A nef line bundle $L\in\Pic(X)$ is ample if and only if one has
\alert{$\varepsilon(L):=\inf_{x\in X}\varepsilon(L,x)>0$}.  
\end{block}\pause
\vskip-3pt
A direct consequence of the fact $(\pi^*L-\gamma E)^n=L^n-\gamma^n\ge 0$ 
is that\vskip4pt
\alert{\centerline{$\displaystyle
\varepsilon(L,x)\le (L^n)^{1/n},\qquad\forall x\in X$.}}\pause\vskip2pt
A curve $C$ is said to be \alert{submaximal} if
\alert{$\displaystyle{L\cdot C\over \mult_x(C)}< (L^n)^{1/n}$}.\pause\vskip2pt
A large part of the investigations on Seshadri constants, especially
in the case of surfaces, rests upon the study of submaximal curves.\pause
\vskip5pt
\claim{Remark.} In [D, 1990], over $\bK=\bC$, the Seshadri constant is
related to more analytic invariants.\pause\ For instance, if $L$ is ample,
it can be shown that $\varepsilon(L,x)$ is the supremum of $\gamma\ge 0$ for
which $L$ possesses a singular Hermitian metric $h$ with
\alert{$\Theta_{L,h}\ge 0$}, that is smooth on $X\smallsetminus\{x\}$ with
a \alert{logarithmic pole of Lelong number $\gamma$ at $x$}.
\end{frame}

\begin{frame}
\frametitle{Relation to the Fujita conjecture}
\vskip-6pt
\begin{block}{Proposition (D, 1990 -- implied by the Kodaira vanishing theorem)}
For $L\in\Pic(X)$ define\vskip0pt
\alert{\centerline{$\displaystyle
 \sigma(L,x)=\limsup_{k\to+\infty}{s(kL,x)\over k}$}}
where\vskip3pt
\alert{\centerline{$\displaystyle
s(L,x)=\max\{m\,/\;H^0(X,L)~\hbox{generates $m$-jets at $x$}\}.
$}}\pause\vskip5pt
(1) For $L$ ample, one has\vskip3pt
\alert{\centerline{$\displaystyle
    \varepsilon(L,x)=\sigma(L,x)$.}}
\pause\vskip5pt
(2) For $L$ ample such that $p=\lceil\varepsilon(L,x)\rceil>n=\dim X$,\vskip5pt
\alert{\centerline{$\displaystyle
    H^0(X,K_X+L)~\hbox{generates $(p-n-1)$-jets at $x$}.$}}\pause\vskip5pt
(3) As a consequence, if $\varepsilon(L)\geq 1$, then \alert{$K_X+(n+1)L$ is
generated by sections, and $K_X+(2n+1)L$ is very ample}\pause~~
[the Fujita conj.\ states that $L$ ample should imply
\alert{$K_X+(n+2)L$ very ample}.]
\end{block}\pause
\end{frame}

\begin{frame}
\frametitle{Seshadri constants on surfaces}
Miranda constructed a sequence of examples of smooth surfaces $X_p$,
ample line bundles $L_p$ on $X_p$ and points $x_p\in X_p$
such that \alert{$\lim\varepsilon(L_p,x_p)= 0$}, but is is unknown whether
one can possibly have\vskip5pt
\alert{\centerline{$\displaystyle
\varepsilon(X):=\inf_{L\in\Pic(X)\,{\rm ample},\;x\in X}
\varepsilon(L,x)=0\qquad???$}}\pause\vskip5pt
However, many results are known for surfaces.
\begin{block}{Theorem (Ein-Lazarsfeld,1993)}If $L$ is ample on
a smooth surface $X$, then \alert{$\varepsilon(L,x)\ge 1$} except for
\alert{countably many} points $x\in X$, and for
\alert{finitely many if $L^2>1$}.
\end{block}\pause

\begin{block}{Improvement by Geng Xu (1995)}If $L$ satisfies
\alert{$L^2\ge {1\over 3}(4a^2-4a+5)$} and \alert{$L\cdot C\ge a$} for some
every curve $C$ and some integer $a>1$, then \alert{$\varepsilon(L,x)\ge a$}
for all $x\in X$ outside a finite union of curves.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Seshadri constants on surfaces (sequel)}
\vskip-7pt
\begin{block}{Theorem (A.\ Steffens, 1998)}If $L$ is ample on
a smooth surface $X$ with Picard number $\rho(X)=1$, then
the generic value \alert{$\varepsilon(L,x_{\hbox{\sevenrm very general}})\ge \lfloor
\sqrt{L^2}\rfloor$},\\ and if $L^2$ is an integer, there is
equality.
\end{block}

\begin{block}{Theorem (T.\ Szemberg, 2008)}
If $L$ is ample on a smooth surface $X$ with Picard number $\rho(X)=1$, then
\vskip3pt
(1) \alert{
$\forall x\in X$,~ $\varepsilon(L,x)\ge 1$~~ if $X$ is not of general type.}
\pause\vskip6pt
(2) \alert{
$\forall x\in X$,~ $\varepsilon(L,x)\ge {1\over
1+(K_X^2)^{1/4}}$~~ if $X$ is of general type.}\vskip3pt\pause
Both bounds can be attained (and thus are sharp).
\end{block}\pause
A. Broustet in his PhD thesis (Grenoble, 2006) studied the case of
the anticanonical line bundle \alert{$L=-K_x$ on Del Pezzo surfaces}
\end{frame}

\begin{frame}
\frametitle{Seshadri constants over a subscheme}
\vskip-6pt
\begin{block}{More general definition (Ein-Lazarsfeld)}
Let $X$ be a projective nonsingular variety and
$Z$ a (non necessarily reduced) subscheme, $\pi:\tilde X\to X$
the blow-up of $X$ with centre $Z$, and $E$ the exceptional divisor.\pause\
Then, for $L\in\Pic(X)$ nef, one defines\vskip5pt
\alert{\centerline{$\displaystyle
\varepsilon(L,Z)=\sup\{\gamma\ge 0\,/\;
\pi^*L-\gamma E~\hbox{is nef on $\tilde X$}\}.$}}
\end{block}\pause
\pause
\claim{\bf Remark 1.} The case where $Z=\{x_1,\ldots,x_p\}$ is a finite set
(or possibly a $0$-dimensional scheme) is already very interesting.\pause\\
One says that \alert{$\varepsilon(L,x_1,\ldots, x_p)$} is a
\alert{multipoint Seshadri constant}.\vskip6pt\pause
\claim{\bf Remark 2.} Ein, Lazarsfeld, Mustata, Nakamaye and Popa also
extended the concept to pseudoeffective non nef line bundles, by
introducing ``moving'' Seshadri numbers, based on a use of approximate
Zariski decomposition of $L$ as a $\bQ$-divisor (2006).\pause
\end{frame}

\begin{frame}
\frametitle{Relation to the Nagata conjecture}
\vskip-6pt
The concept of Seshadri constant is already highly non trivial on rational
surfaces. For instance, the famous \alert{Nagata conjecture}, has
attracted lot of work by Hirschowitz, Harbourne, Biran, Bauer, Szemberg,
Dumnicki and others. It can be reformulated~:\pause\vskip-3pt

\begin{block}{Nagata conjecture (1959), reformulated} Let $x_1,\ldots,x_p$ be
$p$ \alert{very general} points in $\bP^2$, $p\geq 9$. Then the multipoint
Seshadri constant of $\cO(1)$ on $\bP^2$ satisfies\vskip5pt
\alert{\centerline{$\displaystyle
\varepsilon(\cO(1),x_1,\ldots,x_p)={1\over\sqrt{p}}$.}}\pause\vskip5pt
\end{block}\pause\vskip-3pt
A simple counting argument implies that
$\varepsilon(\cO(1),x_1,\ldots,x_p)\leq{1\over\sqrt{p}}$, and the main
difficulty is to find good configurations of points to get lower bounds.\pause\
In case $p=q^2$ is a perfect square, a square grid works, hence equality.
For $4<p<9$, one
is in the Del Pezzo case, and the inequality turns out to be strict.
\end{frame}

\begin{frame}
\frametitle{Higher dimensional case}
\vskip-6pt
The geometry of curves makes the situation much more involved in the
higher dimensional case.\pause\ A result valid in arbitrary dimension is

\begin{block}{Theorem (Ein, Küchle, Lazarsfeld, 1995)}
Let $L$ be ample on a non singular $n$-dimensional projective variety $X$
over $\bC$. Then \alert{$\varepsilon(L,x)\ge 1/n$} at a
\alert{very general} point $x\in X$.
\end{block}\pause\vskip-2pt
The proof uses a ``differentiation argument'' of sections of $kL-pE_x$,
considered on the universal family, i.e., sections of $k({\rm pr}_1^*L)-pE$
on the blow up of $X\times X$ along the diagonal.\pause\\ By a more elaborate
use of the EKL argument, M.\ Nakamaye got in 2004 the improved
\alert{very generic} lower bound\vskip5pt
\alert{\centerline{$\displaystyle
 \varepsilon(L,x)\ge {3n+1\over  3n^2}\quad\Big(\hbox{resp.}~~{1\over 2}~~
\hbox{if $n=3\Big)$}.$}}\pause\vskip5pt
\claim{\bf Question} (even for $n=\dim X=2$~!).
Is there a lower bound for \alert{$\varepsilon(X)=\inf_{L\in\Pic(X)\,{\rm ample}}
\varepsilon(L)$} depending only on the geometry \hbox{of $X$~?\kern-10pt}
\end{frame}

\begin{frame}
\frametitle{More is known for special classes of varieties~$\ldots$}
\vskip-7pt  
\begin{block}{Case of Abelian varieties (Nakamaye, 1996)}
Let $X=\bC^n/\Lambda$ be an Abelian variety and $L\in\Pic(X)$ be ample.
Then \alert{$\varepsilon(L,x)=\varepsilon(L)\ge 1$}, and the equality occurs
if and only if \alert{$X\simeq E\times Y$} where $E={}$elliptic curve and
$Y={}$Abelian variety of dimension $n-1$, with
$L\equiv{\rm pr}_1^*\cO_E([p_0])+{\rm pr_2}^*A$ and $C=E\times\{y_0\}$.
\end{block}\pause\vskip-3pt
The subject is still very much alive~!\pause\
Since 2010, there have been \alert{45 arXiv submissions} dealing with
Seshadri constants.\pause\ The last one
at this date is from June 2020, by I.~Biswas, J.~Dasgupta, K.~Hanumanthu
and B.~Khan. It gives an estimate of Seshadri constants of
nef line bundles on Bott towers, namely a particular class of
projective non singular toric varieties of the form\vskip3pt
\alert{\centerline{$\displaystyle
 X_n\to X_{n-1}\to\cdots\to X_2\to X_1\to X_0=\{{\rm point}\}$,}}\vskip3pt
so that each $X_k=\bP(\cO_{X_{k-1}}\oplus\cL)$ is a $\bP^1$-bundle
over~$X_{k-1}$.\pause\vskip3pt
One of the main points is to \alert{identify the nef cone of
$\Pic(X_n)\simeq\bZ^n$}.\vskip3pt
\end{frame}

\begin{frame}

\centerline{\bf References}
\vskip15pt

\Bibitem[1]&C.S.\ Seshadri&Quotient spaces modulo reductive algebraic
groups&Annals of Mathematics, Vol.~95 (May 1972) 511–556&
\vskip3pt

\Bibitem[2]&J.-P.\ Demailly&Singular Hermitian metrics on positive line
bundles&Complex algebraic varieties (Bayreuth, 1990), Lect.\ Notes Math.\
1507, Springer-Verlag, 1992, pp. 87–104&
\vskip3pt

\Bibitem[3]&T.\ Fujita&On polarized manifolds whose adjoint bundles are
not semipositive&Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math.,
10, North-Holland, Amsterdam, 167–178&
\vskip3pt

\Bibitem[4]&L.\ Ein \&\ R.\ Lazarsfeld&Seshadri constants on smooth
surfaces&In Journées de Géométrie Algébrique d’Orsay (Orsay, 1992);
Astérisque No.~218 (1993), 177–186&
\end{frame}

\begin{frame}
\Bibitem[5]&Geng Xu&Curves in $\bP^2$ and symplectic packings&Math.\ Ann.\ 299
(1994), 609–613&
\vskip3pt

\Bibitem[6]&A.\ Steffens&Remarks on Seshadri constants&Math.\ Z.~227 (1998),
505-510&
\vskip3pt

\Bibitem[7]&T.\ Szemberg&An effective and sharp lower bound on Seshadri
constants on surfaces with Picard number~$1$&J.~Algebra 319 (2008) 3112–3119&
\vskip3pt

\Bibitem[8]&L.~Ein, R.~Lazarsfeld, M.~Mustata, M.~Nakamaye,
M.~Popa&Asymptotic invariants of base loci&Ann.\ Inst.\ Fourier (Grenoble)
56 (2006), 1701–1734&
\vskip3pt

\Bibitem[9]&M.~Nagata, Masayoshi&On the $14$-th problem of Hilbert&American
Journal of Mathematics, 81 (3): 766–772&
\vskip3pt

\Bibitem[10]&L.~Ein, O.~Küchle, R.~Lazarsfeld&Local positivity of ample
line bundles&J.~Differential Geom.\ 42 (1995), 193–219&
\end{frame}
    
\begin{frame}
\Bibitem[11]&M.\ Nakamaye&Seshadri constants on abelian varieties&American
Journal of Math. 118 (1996), 621–635&
\vskip3pt

\Bibitem[12]&I.~Biswas, J.~Dasgupta, K.~Hanumanthu, B.~Khan&Seshadri
constants on Bott towers&math.AG, arXiv:2006.12723&
\vskip5cm
\strut
\end{frame}
    
\begin{frame}
\frametitle{The end}
\centerline{\huge\bf Thank you for your attention}
\vskip5pt
\pgfdeclareimage[height=5.5cm]{Seshadri}{Seshadri}
\strut\kern1.75cm\pgfuseimage{Seshadri}
\vskip3pt
\centerline{Professor C.S.\ Seshadri in Bangalore (2010)}

\end{frame}

\end{document}

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