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\begin{document}

\begin{center}
{\Large{\underline{MATH 658, Spring 2000 - The topology algebraic varieties}}}
\end{center}
\begin{center}
Liviu I. Nicolaescu
\end{center}

 Norman Steenrod once said that a topologist is `` {\em a person who sees no difference between a donut  and a mug}''. For this course, the fact that the surface of a donut  happens to be an algebraic variety (elliptic curve to be precise) will be particularly relevant.


In general, the algebraic varieties  are sets  of solutions of {\em polynomial}  equations and  their structure  has many special features.   I will  be interested   mainly  in their topological properties.    The class of  algebraic varieties may seem small, special and thus uninteresting. By the end of this course I hope to convince you of the contrary, and  present a few   interesting features of these  special topological spaces.\footnote{Even if you  don't plan to  attend this class you ought to read the captivating brief  historical  survey ``{\em Singularities}'' by Alan Durfee, xxx-Archive {\sf math.AG/9801123},  to   learn about some exciting piece of mathematics.}   The course will consist of three parts.

\medskip

\noindent {\bf 1. Picard-Lefschetz theory}

\noindent {\bf 2. Singularities of curves and surfaces and their resolutions.}

\noindent {\bf 3.  The Milnor fiber,  the link  and the vanishing cohomology of a singularity. }

\medskip

The {\em  Picard-Lefschetz theory} is  the complex analytic  counterpart of  the versatile Morse theory.  Historically, Picard-Lefschetz theory came first, at the dawn of the 20th century,   and had a dramatic impact on the  development of modern algebraic topology.    The recent years  have witnessed an increased   the interest  in this old theory due  mainly to the surprising     results  of S. Donaldson on the topology of  symplectic manifolds.

Roughly speaking, Morse theory   describes a procedure of reconstructing  a smooth manifold $M$   assuming we know a smooth function $f: M\ra {\bR}$  and its behavior near its critical points.     Picard-Lefschetz theory proposes  a general  reconstruction process  for a {\em complex analytic} manifold $M$  knowing a {\em holomorphic} function $f: M\ra {\bC}$ and its behavior near its critical points.  Picard-Lefschetz theory requires that all critical points of $f: M\ra {\bC}$ are nondegenerate, just as  Morse theory does.   For this part of the course I plan to follow  closely the modern approach in \cite{Lamot}.

Naturally, one can ask what happens if we relax   the above requirement and allow {\em degenerate} critical points as well.  In this case one is   lead to the study of singularities of   hypersurfaces
\[
X= P^{-1}(0)\subset {\bC}^n
\]
where $P=P(z_1, \cdots, z_n)$ is a polynomial in $n$-complex variables such that $0\in X$ is a singular point.           I will discuss two  methods for studying the singularities of   such hypersurfaces. Both   are based on the same    principle: investigate the singular object $X$ by ``approximating'' it with a smooth one.

The first ``approximation'' technique  is the so called {\em resolution of singularities} and I will discuss it  only in low dimensions $n=2,3$.   The case $n=2$ leads to beautiful connections with {\em knot theory}, while the case $n=3$ leads to  interesting and important examples of $3$-manifolds.    My presentation will   be inspired from \cite{BK, Laufer}.

Another,  simpler way of approximating $X$ is by deforming the  defining equation $P=0$ to $P=\ve$.  This process is called  {\em smoothing} and the resulting hypersurface
\[
X_\ve:=\{\vec{z}\in {\bC}^n;\;\;P(\vec{z})=\ve\}
\]
is called the {\em Milnor fiber} of the singularity.   The  odd-dimensional manifold $N_\ve:= X \{|\vec{z}|=\ve$ is called the {\em link} of the singularity.       When $n=2$, so that $X$ is a complex curve, the link is a one-dimensional submanifold  of  the sphere $S^3$, a.k.a. {\em knot}.   The knots obtained as links of singularities are called {\em algebraic}.

The topologies of the Milnor fiber and the link of a singularity are    nontrivially constrained by  the  {\em complex geometry} of  the singularity.   For example, J.  Milnor  has   shown that    amongst these  links one can detect  manifolds which   ``talk like a sphere, walk like a sphere'' but   they are not quite spheres: they  are {\em homeomorphic} but  {\em  not diffeomorphic} to  spheres.

As $\ve\ra 0$, some of the cycles of $X_\ve$ collapse.  These are called {\em vanishing cycles} and the  subgroup they generate is called  {\em the vanishing (co)homology} of the singularity. My presentation will be inspired from \cite{AGV, Dimca, Milnor}.  If time permits, I plan to say a few words about   the   connection between the asymptotics of certain oscillatory  integrals and  vanishing cohomology.


To  enjoy this class you need to be  familiar with basic   facts of algebraic topology, especially  singular (co)homology and {\em very little} analysis. I believe  the things I will talk about will be of interest to topologists, algebraic and/or differential geometers alike. The algebraic varieties represent an  excellent testing ground for many    topological questions and  ought to belong to the bag of examples of any  topologist or geometer.

The class will be example-oriented because,  as E.C. Zeeman once put it, ``a good example is worth ten theorems''.  This  is quite an ecclectic subject  and  there will be  many  special  interesting things  I won't be able to cover in class.    That is why, part of the requirement of this class will  be  to present a one hour  informal talk   on a subject of your choice from  a list I will submit  in class.  In  compiling this list I will try to  satisfy all the backgrounds in class. E.g. a student   pursuing a degree in algebraic geometry  will  be able to choose an elementary  topic in this field which will be relevant to all of us in class.


Notes for this course are available (as PDF  files) from my homepage

\verb+ http://www.nd.edu/~lnicolae/+



\begin{thebibliography}{XXXXX}

\bibitem{AGV} V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko: {\sl Singularities of Differentiable Maps. Volume II}, Monographs in Mathematics, vol. 83, Birkh\"{a}user, 1988.

\bibitem{BK} E. Brieskorn, H. Kn\"{o}rer: {\sl Plane Algebraic Curves}, Birkh\"{a}user, 1986.

\bibitem{Dimca} A. Dimca: {\sl Singularities and Topology of Hypersurfaces}, Universitext, Springer Verlag, 1992.

\bibitem{Lamot} K. Lamotke: {\sl The topology of complex projective varieties after S. Lefschetz}, Topology, {\bf 20}(1981), 15-51.

\bibitem{Laufer} H.B. Laufer: {\sl Normal Two-Dimensional Singularities}, Annals of Mathematics Studies, vol. {\bf 71}, Princeton University Press, 1971.

\bibitem{Milnor} J. Milnor: {\sl  Singular Points of Complex Hypersurfaces}, Annals of Math. Studies, vol. 61, Princeton Univ. Press, 1968.

\end{thebibliography}

\newpage

\thispagestyle{empty}

\begin{center}
{\bf TENTATIVE LIST OF TOPICS  FOR MATH 658}
\end{center}


\noindent {\bf I. Complex manifolds -Definitions and Examples}

1.The projective spaces.

2. Complex submanifolds: implicit function theorem, critical and regular points, critical values.

3. Algebraic manifolds.

4.  ``Baby'' Lefschetz theory:  Riemann-Hurwitz theorem and the  Pl\"{u}cker formula.

5.  The tautological line bundle and the  blow-up construction

\medskip

\noindent {\bf II.  Lefschetz theory}

1. Linear systems, pencils.

2. Lefschetz pencils, examples.

3. Vanishing cycles. Weak and Hard Lefschetz theorems.



4. Morse lemma and Picard-Lefschetz formul{\ae}

5. Deligne's  interpretation of the Hard Lefschetz theorem.


\medskip

\noindent {\bf III. Invariants of  critical points}


1. Mather-Tougeron sufficiency theorem.

2.  Milnor number

3.  Morsification and Milnor fibrations

4. Monodromy.

5. Resolutions of singularity  (curves and surfaces only)

















\end{document}
