\documentclass[11pt]{article}

\title{On Lagrange and symmetric degeneracy loci}

\author{Maxim Kazarian\thanks{Supported by the grants RFBR 98-01-00612  and
INTAS 97-1644}
\\
Steklov Mathematical Institute\\
Independent Moscow University\\
\em kazarian@mccme.ru}
\date{}

\usepackage{array}
\setlength{\extrarowheight}{1ex}
\usepackage{amssymb}

%  papersize   
\newdimen\margn \margn=1in
%                        ~~~
\oddsidemargin=-1in                 \advance\oddsidemargin\margn
\topmargin=-1in                     \advance\topmargin\margn
\textwidth=210mm                    \advance\textwidth-2\margn
\textheight=297mm                   \advance\textheight-2\margn
\advance\textheight-\headheight     \advance\textheight-\headsep
\advance\textheight-\topskip        \evensidemargin=\oddsidemargin


%             s e p a r a t e
\def\spa#1{{\kern-.3ex\def\?##1{\kern.3ex##1\?}\?#1{\let\?=\relax}}}
%
%             sections, subsections, theorems
%
\def\mdskip{\ifnum\medskipamount>\lastskip\vskip-\lastskip
\vskip\medskipamount\fi}
\long\def\ex#1#2#3{\refstepcounter{subsection}\label{#1}\mdskip
{\thesubsection.\ \sc#2}.~#3\mdskip}
\long\def\th#1#2#3{\ex{#1}{#2}{{\em#3}}}
\def\QED{\ifmmode\eqno\square\else
{\parfillskip0pt\hfil$\square$\par}\mdskip\fi}

\long\def\ex#1.#2\par{\mdskip{\bf #1.} {#2}\par\mdskip}
\long\def\th#1.#2\par{\ex{#1}.{\em#2}\par}

\def\too#1{\stackrel{#1}\longrightarrow}
\def\({\ifmmode\left(\else{\rm(}\fi}
\def\){\ifmmode\right)\else{\rm)}\fi}

\def\Pf{\mathop{\rm Pf}}
\def\codim{\mathop{\rm codim}}
\def\Sing{\mathop{\rm Sing}}
\def\ind{\mathop{\rm ind}}
\def\rank{\mathop{\rm rk}}
\def\GL{\mathop{\rm GL}}
\def\Hom{\mathop{\rm Hom}}





\def\ot{\leftarrow}
\def\a{\alpha}
\def\b{\beta}
\def\g{\gamma}
\def\d{\partial}
\def\e{\varepsilon}
\def\s{\sigma}
\def\t{\tau}
\def\f{\varphi}
\def\l{\lambda}
\def\k{\kappa}
\def\o{\omega}
\def\<{\langle}
\def\>{\rangle}
\def\D{\partial}
\def\emptyset{\varnothing}
\def\const{{\rm const}}
\def\R{{\mathbb R}}
\def\Z{{\mathbb Z}}
\def\Q{{\mathbb Q}}
\def\C{{\mathbb C}}
\def\S{\Sigma}
\def\G{\Gamma}
\def\L{\Lambda}
\def\O{\Omega}
\def\Lc{{\mathcal L}}
\def\Fc{\mathcal F}
\def\F{{\mathbf F}}
\def\Db{{\mathbf D}}
\def\Lt{{\widetilde\Lambda}}
\def\Lct{{\widetilde\Lc}}
\def\pt{{\rm pt}}
\def\St{{\widetilde S}}


\def\Et{{\widetilde E}}
\def\Ft{{\widetilde F}}
\def\Lt{{\widetilde L}}
\def\s{Q}

\def\hL{{\widehat L}}
\def\hF{{\widehat F}}

\def\*{{\otimes}}

%
% temporary macros
%
%\def\cite#1{\begingroup\let\?\endgroup
%     \ifx[#1\def\?##1]##2{\endgroup\cite{##2,~##1}}\else[#1]\fi\?}
\def\?{\protect\marginpar[\hbox to \margn
    {\Large\hss$\surd$}]{\Large$\surd$}}
\long\def\comment#1{\relax}
%\def\ref#1{#1}

\sloppy

\begin{document}

\maketitle


The study of topology of the real Lagrange Grassmannians has various
important applications in the theory of Hamiltonian systems,
symplectic and contact geometry, and others fields of mathematics. 
The notion of the Lagrange Grassmannian can be complexified in
several ways (cf.~\cite{A,MSS}) and the corresponding objects of
complex geometry also have many important applications. In this paper
we study the most straightforward complex version of the isotrope and
the Lagrange Grassmannians, namely, the manifold of all complex
subspaces of a fixed dimension in $\C^{2n}$, which are isotrope with
respect to a fixed non-degenerate bilinear skew-symmetric form. 

The main result of the paper are various formulas for cohomology
classes dual to different kinds of Lagrange and orthogonal degeneracy
loci: Schubert cells in isotrope Grassmann and flag varieties,
intersections of isotrope subbundles of symplectic and orthogonal
bundles, degenerations of symmetric and skew-symmetric maps of
bundles etc. This subject was intensively studied last years,  here
is a non-complete list of references:
\cite{HT,JLP,Fu,P0,P1,P2,PR,FP,LP1,LP2}. A part of results presented
here can be found in these papers. Nevertheless, I decided to write
this note essentially for two reasons. First, to the best of my
knowledge, the main results of this paper, namely the formulas of
Theorems~1.1 and~2.1 are not known. Their particular case solves
J.~Harris' problem. I used these formulas in the computations of the
characteristic classes dual to Lagrange, Legendre, and critical point
function singularities~\cite{K3}. Two different proofs are presented
here. The analogues of these formulas for orthogonal case are
formulated in Appendix~D.  The second, and perhaps, more important
reason was an attempt to present a brief review of basic ideas and
methods of the theory which would be self-contained and clear for
non-specialists in intersection theory. I dare hope that the respect
to the theory will not be lost if it would be shown to be much
simpler then it is sometimes presented.  I am certainly sure that the
intension `to make things clear' will always be a strong tradition of
{\em Moscow Mathematical Journal}. 

%\footnote{This article is to appear in the first 
%issue of Moscow Mathematical Journal}



 I am extremely grateful to V.~Kryukov for careful reading the
initial version of the paper and many essential remarks. He pointed
my attention to the identities of Appendix~C.  The final form of the
statement of the main Theorem~1.1 is in much extent due to him. I
appreciate the hospitality of the I.~Newton Institute, Cambridge, where
the work on the paper was completed.



\section{Main results}\label{main}

Consider  vector bundles $L\subset E$ over some manifold $M$
and another flag $F$ of subbundles
 $$F_{k_1}\subset F_{k_2}\subset\dots\subset E,\quad\rank F_k=n+1-k,
 \qquad{\rm where}\quad n=\rank E-\rank L.$$
Different {\em degeneracy loci\/} are defined as the subsets of the 
base $M$ with the prescribed dimensions for the intersections of 
the corresponding fibers of the bundles $F_i$ and $L$. Namely, for a
given decreasing sequence of integers $\l_1>\l_2>\dots>\l_r>0$ we
consider the {\em degeneracy locus\/}
$S_{\l_1,\dots,\l_r}\subset M$ as the set of points $x\in M$ such
that
\begin{equation}  
  \dim F_{\l_ix}\cap L_x\ge  i\label{intdim}
\end{equation}
for all $i=1,\dots,r$ (assuming that the bundles 
$F_{\l_i}$ are present in the flag $F$.)

Assume that the fibers of $E$ are equipped with a linear symplectic
structure, i.e. we are given a nowhere degenerating section of the
bundle $\L^2E^*$. Assume also that the subbundles $F_i$ and $L$ are 
{\em isotrope\/} i.e.\ the fibers of $F_{k_i}$ and $L$ are isotrope
subspaces in the fibers of $E$. This assumption implies additional
restrictions on the degeneracy loci which change, in particular, their
expected codimensions. We present a formula for the cohomology 
classes dual to these loci in terms of the Chern classes of the bundles
$E,V,F_i$.




\ex Remark. Denote by $\rho(k)$ the largest number $k+j$ such that
$\l_k=\l_{k+1}+1=\dots=\l_{k+j}+j$.  The index $k$ is called {\em
redundant\/} if $\rho(k)\ne k$. Otherwise it is called {\em
essential}. The conditions~(\ref{intdim}) for redundant indices
follow from those for the essential ones. They can be
dropped in the definition of the degeneracy locus $S_{\l_1,\dots,\l_r}$.


For any collection of formal series
$c^{(i)}=1+c^{(i)}_1+c^{(i)}_2\dots$, $i=1,\dots,r$, and any sequence
(not necessary decreasing) of integers $\l_1,\dots,\l_r$ we define
the {\em generalized Schur $Q$-polynomials\/}
$\s_{\l_1,\dots,\l_r}(c^{(1)},\dots,c^{(r)})$ as follows: 
\begin{itemize}
\item
for $r=1$ we set $\s_k(c)=c_k$;
\item
for $r=2$ we set
$$
\s_{k,l}(c^{(1)},c^{(2)})=c^{(1)}_kc^{(2)}_l-2c^{(1)}_{k+1}c^{(2)}_{l-1}+
   2c^{(1)}_{k+2}c^{(2)}_{l-2}-2c^{(1)}_{k+3}c^{(2)}_{l-3}+\dots;
$$
\item 
for any even $r\ge4$ we set
\begin{eqnarray}
\s_{\l_1,\dots,\l_r}(c^{(1)},\dots,c^{(r)})&=
    &\Pf|\s_{\l_i,\l_j}(c^{(i)},c^{(j)})|_{1\le i,j\le r};
\label{seven}
\end{eqnarray}
\item
for any odd $r\ge3$ we set
\begin{eqnarray}
\s_{\l_1,\dots,\l_r}(c^{(1)},...,c^{(r)})=
\sum_{k=1}^r(-1)^{k-1}c^{(k)}_{\l_k}\;
\s_{\l_1,\dots,\widehat{\l_k},
\dots\l_r}
(c^{(1)},...,\widehat{ c^{(k)}},
...,c^{(r)}).
\label{sodd}
\end{eqnarray}
\end{itemize}

Here $\Pf$ is the Pfaffian of a skew-symmetric matrix (see
Appendix~C). This definition makes sense only if 
\begin{equation}
\s_{\l_i,\l_j}(c^{(i)},c^{(j)})+\s_{\l_j,\l_i}(c^{(j)},c^{(i)})=0
\quad\hbox{for all}\quad 1\le i,j\le r.
\label{correctly}
\end{equation}
If this condition holds, then the polynomial
$\s_{\l_1,\dots,\l_r}(c^{(1)},...,c^{(r)})$ depends skew-symmetrically 
with respect to the permutations of indices $\l_i$ and simultaneous
permutations of $c^{(i)}$,
 $$\s_{\l_{s(1)},\dots,\l_{s(r)}}(c^{(s(1))},...,c^{(s(r))})=
 (-1)^{|s|}\s_{\l_1,\dots,\l_r}(c^{(1)},...,c^{(r)}),$$
 where $s$ is a permutation and $|s|$ is its sign. This follows from
the fact that the Pfaffian is skew-symmetric with respect to
simultaneous permutations of rows and columns of the matrix. In particular,
$\s_{\l_1,\dots,\l_r}(c^{(1)},...,c^{(r)})$ vanishes if for some
$i\ne j$ one has $\l_i=\l_j$ and $c^{(i)}=c^{(j)}$.

\ex Remark. The distinction between the cases of even and odd $r$ is
apparent. For instance, the following reduction formula holds for any 
$r>1$ with positive $\l_1,\dots,\l_r$ 
 $$\s_{\l_1,\dots,\l_r,0}(c^{(1)},...,c^{(r+1)})=
 \s_{\l_1,\dots,\l_r}(c^{(1)},...,c^{(r)})$$
(whenever these classes are defined). 
 For $r=2$ this evidently follows from the definition. 
For greater $r$ it can be easily derived
from~(\ref{seven}) and~(\ref{sodd}) by induction in $r$.

Let $\l_1>\dots>\l_r>0$. If the flag $F$ contains the bundles
$F_{\l_i}$ for all essential indices $i$ then the degeneracy locus
$S_{\l_1,\dots,\l_r}\subset M$ is well defined.

\th Theorem {1.1}. Generically the cohomology class dual to
$S_{\l_1,\dots,\l_r}$ is given by
$$[S_{\l_1,\dots,\l_r}]=\s_{\l_1,\dots,\l_r},$$
where
$$\s_{\l_1,\dots,\l_r}=\s_{\l_1,\dots,\l_r}
(E{-}L{-}F_{\l_{\rho(1)}},\dots,E{-}L{-}F_{\l_{\rho(r)}}).$$
If the flag $F$ contains the bundles $F_{\l_i}$ for all redundant
indices $i$ then we have also
$$\s_{\l_1,\dots,\l_r}=\s_{\l_1,\dots,\l_r}
(E{-}L{-}F_{\l_1},\dots,E{-}L{-}F_{\l_r}).$$

First verify that {\em the classes entering these formulas are
defined}, that 
is the condition~(\ref{correctly}) is satisfied. Indeed, denoting
$X_k=E-L-F_k$, we get
\begin{eqnarray*}
\s_{k,l}(c(X_{k'}),c(X_{l'}))+
\s_{l,k}(c(X_{l'}),c(X_{k'}))=
2\sum_{i=-\infty}^\infty(-1)^i 
c_{k+i}(X_{k'})\;c_{l-i}(X_{l'})\\[.5ex]
=\pm2\;c_{k+l}(X_{k'}^*-X_{l'})
=\pm2\;c_{k+l}((L^\bot/L)^*\oplus F_{k'}^\bot/F_{l'}).
\end{eqnarray*}
Here we used the isomorphisms $E/L^\bot\cong L^*$,
$E/F_{k'}^\bot\cong F_{k'}^*$ provided by the symplectic
structure (the orthogonal complement is considered with respect to
the symplectic form). The bundle in the brackets has the rank
 $$(\rank E-2\rank L)+(\rank E-\rank F_{k'}-\rank
 F_{l'})=k'+l'-2.$$
The $(k+l)$th Chern class of this bundle vanishes if, for example,
$k'\le k$, $l'\le l$. Therefore all classes of the form
$\s_{\l_1,\dots,\l_r}(c(E{-}L{-}F_{\l_1'}),\dots,c(E{-}L{-}F_{\l_r'}))$
with $\l_i'\le\l_i$ are always well defined.\QED

\ex Remark. The {\em genericity condition\/} of Theorem is formulated
as follows.  Consider the locally trivial bundle $Y\to M$ of
`geometrical configurations' whose fibers are formed by products
$\F_x\times \L_x$ where $\F_x$ and $\L_x$ are manifolds of isotrope
flags and isotrope planes in $E_x$ of the dimensions corresponding to
the ranks of the bundles $F_k$, $L$.
The canonical bundles over $Y$ define degeneracy loci on $Y$ and the
genericity condition in this case is, by definition, satisfied. The
given flag of bundles $F$ and  the bundle $L$ define a section
$s:M\to Y$. The genericity condition means that this section is
transversal to every singularity locus on $Y$. In this case the
equality for $M$ is induced by $s^*$ from the corresponding equality
for $Y$. Remark that the class $s^*[S_{\l_1,\dots,\l_r}]$ is well
defined on $M$ and the equality of Theorem holds for this class even
if the section $s$ is not transversal. Similar trick can be applied
for other situations in order to avoid problems with
non-transversality. In particular it is implied in the definition of
the manifolds $Z_k$ in the next section.\QED 

\ex Remark.
Similar trick is used also to show that {\em it is sufficient to
prove  Theorem~{\rm 1.1} only for the case when the flag $F$ contains
subbundles $F_{\l_i}$ for all redundant indices $i$}.  Indeed,
consider the flag bundle $Y\to M$ the fibers of which are formed by
complete isotrope flags $F_{nx}'\subset
F_{(n-1)x}'\subset\dots\subset E_x$ such that
$F_{\l_ix}'=F_{\l_ix}$ for all essential indices $i$. Then the
validity of the assertion of Theorem~1.1 for $Y$  implies its
validity for $M$ since the induced homomorphism of the cohomology
$H^*(M)\to H^*(Y)$ is injective, see Appendix~A. \QED


\spa{Proof~of~the~second~equality~of~Theorem~{1.1}},
 \begin{equation}
 \s_{\l_1,\dots,\l_r}(c(X_1),...,c(X_r))=
 \s_{\l_1,\dots,\l_r}(c(X_{\rho(1)}),...,c(X_{\rho(r)})),
\quad X_k=E{-}L{-}F_{\l_k}.
 \label{reduct}
 \end{equation}
By definition, the classes 
$\s_{\l_1,\dots,\l_r}(c(X_1),...,c(X_r))$ are polilinear with 
respect to the total Chern classes $c(X_k)$. Suppose that
the index $k$ is redundant. Then from the equality
$$c_*(X_k)=c_*(F_{\l_{\rho(k)}}/F_{\l_k}+X_{\rho(k)})=
\sum_{j=0}^{\rho(k)-k}
c_j(F_{\l_{\rho(k)}}/F_{\l_k})\;c_{*-j}(X_{\rho(k)})
$$
we get
 $$\s_{\dots,\l_k,\dots}(\dots,X_k,\dots)=
 \sum_{j=0}^{\rho(k)-k} c_j(F_{\l_{\rho(k)}}/F_{\l_k})\;
\s_{\dots,\l_k-j,\dots}(\dots,X_{\rho(k)},\dots).$$
When we apply similar expansion to other redundant indices we
obtain a linear combination of different classes of the form 
$\s_{\l'}(c(X_{\rho(1)}),\dots,c(X_{\rho(r)}))$. In this
combination all terms except one with $\l'=\l$ 
will have repeating indices and so they vanish.\QED


Two different proofs of the first equality of Theorem~1.1 are
presented in two subsequent sections. In this Section we discuss some
applications of Theorem.

In particular case when the flag $F$ consists of only one plane
$L'=F_k$ the formula of Theorem~1.1 answers the problem of J.~Harris
about the class dual to the locus given by the intersection ranks for the
fibers of two isotrope bundles $L$ and $L'$ (see~\cite{PR}, where the
answer for the case when both $L$ and $L'$ are Lagrangian is given in
much more complicated form).  

\th Proposition. Let $L,L'$ be isotrope subbundles in a symplectic
bundle $E$. Then the cohomology class dual to the locus 
$\O_r=\{x\in M\bigm|\dim
L_x\cap\dim L'_x\ge r\}$ is given by
$$[\O_r]=\s_{m+r,m+r-1,\dots,m+1}(c(E{-}L{-}L'),\dots,c(E{-}L{-}L')),$$
where $m=\rank E-\rank L-\rank L'$.\QED

In particular, the last formula for the case $r=\rank L=\rank L'$
describes the cohomology class dual to the diagonal in the product 
of two isotrope Grassmannians (or, more general, to the diagonal
bundle in the fiber product of two isotrope Grassmannian
bundles associated with the given symplectic vector bundle $E\to M$
(cf.~\cite{P1} where this class for the case of Lagrange Grassmannians
is presented in much more complicated  form),
 \begin{equation}
[\Delta]=\s_{m+r,m+r-1,\dots,m+1}(c(E{-}L{-}L'),\dots,c(E{-}L{-}L')),
  \label{diagonal}
  \end{equation}
where $r=\rank L=\rank L'$, $m=\rank E-2r$, and
$L_1,L_2$ are the tautological bundles over the Grassmannians.


\mdskip
The simplest examples of degeneracy loci are provided by Schubert
varieties on Grassmannians and flag manifolds of isotrope subspaces
in $\C^{2n}$. In particular, this gives the following formula  (for
the case of Lagrange Grassmannians it was proved in~\cite{P1}).

\th Proposition. On any Grassmannian of isotrope subspaces in
symplectic vector space the cohomology class dual to the Schubert
cycle $S_\l$ defined by dimensions of intersections with a fixed
isotrope flag is given by 
\begin{eqnarray}
\label{grassm} [S_{\l_1,\dots,\l_r}]&=&
\s_{\l_1,\dots,\l_r}(c(-L),\dots,c(-L)). \end{eqnarray}


More examples are provided by symmetric degeneracy loci. Consider some
vector bundle $V\to M$ and a symmetric bundle map $f:V\to V^*$ (it
can be thought as the section of the bundle ${\rm Sym}^2 V^*$). Denote
by $\O_r\subset M$ the locus of points $x\in M$ for which the kernel
$\ker f_x$ has dimension at least $r$. Then we
have~(cf.~\cite{HT,JLP,P0})
  $$[\O_r]=\s_{r,r-1,\dots,1}(c(V^*-V),\dots,c(V^*-V)).$$
Indeed, the sum $V\oplus V^*$ carries the natural symplectic
structure (due to the natural isomorphism $V_x\oplus V_x^*\cong
T^*V_x$). The condition that the map $f_x$ is symmetric is equivalent
to the condition that the graph $L_x\subset V_x\oplus V^*_x$ of this
map is Lagrangian (in fact, the graphs of symmetric maps $V_x\to
V_x^*$ form an open cell in the Grassmannian of Lagrange subspaces in
$V_x\oplus V_x^*$). Hence the locus $\O_r$ is the degeneracy locus
$S_{r,r-1,\dots,1}$ defined with respect to the Lagrange subbundles
$L$ and $V\oplus0$ in $V\oplus V^*$ and we can apply the formula of
Theorem~1.1.\QED

In fact, all assertions about Lagrange degeneracy loci can be
reformulated in terms of the corresponding symmetric degeneracies.
(The inverse is also true, see Remark at the end of Appendix~A.)
The following assertions are the direct reformulations of Theorem~1.1 
for the case of symmetric degeneracy  loci in the same way as
explained above. 

\th Proposition. Consider a flag of vector bundles
$F_n\subset\dots\subset F_1=V\to M$, $\rank F_k=n+1-k$, and a
symmetric bundle map $f:V\to V^*$. Denote by
$\O_{\l_1,\dots,\l_r}\subset M$ the locus of points $x\in M$ such
that $\dim F_{\l_ix}\cap\ker f_x\ge i$. Then generically
  $$[\O_{\l_1,\dots,\l_r}]=\s_{\l_1,\dots,\l_r}
       (c(V^*-F_{\l_\rho(1)}),\dots,c(V^*-F_{\l_\rho(r)})).\QED$$

A particular case of the last Proposition is the following
simplification of the formulas from~\cite{LP2} for degeneracies of
symmetric maps. Let $F\subset E$ be vector bundles over some base
$M$. The bundle map $f:F\to E^*$ is called {\em symmetric\/} if the
bilinear form $(u,v)\mapsto \<f_x(u),v\>$ on $F_x\times E_x$ is
symmetric when restricted to $F_x\times F_x$ for all $x\in M$. One can
easily see that the linear map $f_x:F_x\to E_x^*$ is symmetric if and
only if its graph $L(f_x)=\{y{\oplus} (f_x(y))\in E_x\otimes
E_x^*\bigm|y\in F_x\}$ is an isotrope subspace in  $E_x\oplus
E_x^*$. The kernel of $f_x$ is identified with the intersection of
the isotrope subspaces $L(f_x)$ and $E\oplus 0$. Hence,

\th Proposition. Let $F\to E^*$ be a symmetric map. Then the
cohomology class
Poincar\'e dual to the locus $\O_r\subset M$ of points $x\in M$ such
that $\dim\ker f_x\ge r$ is given by
 $$[\O_r]=\s_{n+r,n+r-1,\dots,n+1}(c(E^*-F),\dots,c(E^*-F)),
 \qquad n=\rank E-\rank F.\QED$$ 





The following interpretation of Theorem~2.1 of the next Section was
used in~\cite{K3}.  For a given vector bundle $V\to M$ denote by
$\Db_r(V)\to M$ the associated locally trivial flag bundle whose
fibers are formed by all flags $D^1,\dots,D^r\subset V_x$, $\dim
D^i=i$. If $f:V\to V^*$ is a generic symmetric bundle map define the
submanifold $Z_r\subset\Db_r(V)$ by the condition $D^r\subset \ker
f_x$. Denote $t_i=-c_1(D^i/D^{i-1})$. Let $p:Z_r\to M$ be  the
natural projection.

\th Theorem. For any monomial $t_1^{s_1}\cdots t_r^{s_r}\in
H^*(Z_r)$, we have
 $$p_*(t_1^{s_1}\cdots
 t_r^{s_r})=\s_{s_1+1,\dots,s_r+1}(c(V^*-V),\dots,c(V^*-V)).\QED$$

   
The analogues for the orthogonal case of the formulas of this Section
are formulated in Appendix~D. 



\section{The push-forward formula}\label{gysin}

In this section we prove Theorem~1.1 by means of resolutions of
degeneracy loci.  Let $L\subset E$ and $F_{k_1}\subset
F_{k_2}\subset\cdots\subset E$, $\rank F_k=n+1-k$, $n=\rank E-\rank
L$, be as in  previous Section. For a given decreasing (not necessary
strictly) sequence of integers $\l_1\ge\dots\ge \l_r$ such that the
planes $F_{\l_i}$ are present in the flag $F$, we
consider the flag bundle space $\Db_r\to M$, whose fibers are  formed
by all isotrope flags

 $$D^1_x\subset D^2_x\subset\dots D^r_x\subset E_x,
           \qquad\dim D^i=i,$$
such that $D^i_x\subset F_{\l_ix}$ for $i=1,\dots,r$. 
Denote by $Z_r\subset \Db_r$ the submanifold given by
the condition $D^r_x\subset L_x$, where $D^r_x$ is the largest plane of the
flag. Generically $Z_r$ is a smooth manifold of (complex) dimension
$\dim Z_r=\dim M-\sum\l_i$. We study the push-forward homomorphism
  $$p_*: H^*(Z_r)\to H^*(M)$$
corresponding to the natural projection $p:Z_r\to M$. We shall use
the same notations $D^i,L$ for the corresponding tautological bundles
on $\Db_r$, and $Z_r$. Denote 
$t_i=-c_1(D^i/D^{i-1})=c_1((D^i/D^{i-1})^*)$.

\th Theorem~{2.1}. For any monomial $t_1^{s_1}\cdots t_r^{s_r}\in H^*(Z_r)$ 
we have
 $$p_*(t_1^{s_1}\cdots
 t_r^{s_r})=\s_{s_1+\l_1,\dots,s_r+\l_r}(c(E{-}L{-}F_{\l_1}),\dots,
 c(E{-}L{-}F_{\l_r}))
  \in H^*(M).$$
 
Theorem~1.1 is a corollary of Theorem~2.1. Indeed, suppose that
$\l_1>\l_2>\dots>\l_r>0$. Then the image $p(Z_r)$ coincides with
$S_{\l_1,\dots,\l_r}$. Moreover, the restriction of $p$ to $Z_r$ 
is one-to-one over an open dense set in
$S_{\l_1,\dots,\l_r}$. Therefore, applying Theorem~2.1 to the case
$s_1=\dots=s_r=0$ we get the formula of Theorem 1.1:
 $$[S_{\l_1,\dots,\l_r}]=p_{*}(1)=
\s_{\l_1,\dots,\l_r}(c(E{-}L{-}F_{\l_1}),\dots,
 c(E{-}L{-}F_{\l_r})).\QED$$ 

To prove Theorem~2.1 we represent the map $p$ as the composition of the
following chain of bundles and embeddings
\begin{equation}
\begin{array}{ccccccccccccc}
&&P(E_{r-1})&&&&&&P(E_1)&&P(E)\\
&\nearrow&\downarrow&&&&&\nearrow&\downarrow&\nearrow&\downarrow\\
Z_r&\too{p_r}&Z_{r-1}&\too{p_{r-1}}&\cdots&
\too{p_3}&Z_2&\too{p_2}&Z_1&\too{p_1}&M
\end{array}\label{diag}
\end{equation}
where $E_k$ is the restriction to $Z_k$ of the bundle $F_{\l_k}/D^k$. 
At the points $y\in Z_{k}$ we
should have $D^{k}_y\subset L_y\subset D^{k\bot}_y$.  The
projective planes of the bundle $P(E_k)$ are formed by all  possible
positions of the line $D^{k+1}_y/D^{k}_y$ in $F_{\l_ky}/D^{k}_y$. The
submanifold $Z_{k+1}\subset P(E_{k})$ is given by condition: the
plane $L_y/D^{k}_y\subset D^{k\bot}_y/D^{k}_y$  contains the line
$D^{k+1}_y/D^{k}_y$. This explains the diagram~(\ref{diag}).


For the computation of the homomorphism $p_{r*}:H^*(Z_r)\to H^*(Z_{r-1})$ 
we remark that the classes $t_1,\dots,t_{r-1}$ come from
$Z_{r-1}$ and so the multiplication by these classes commutes with
$p_{r*}$. Hence, it is sufficient to compute $p_{r*}(t_r^{s})$.

\th Lemma. The homomorphism  $p_{r*}:H^*(Z_r)\to H^*(Z_{r-1})$ is
given by
 $$p_{r*}(t_r^{s})=R^{(r)}_0a_{s+\l_r}^{(r)}+R^{(r)}_1a_{s+\l_r-1}^{(r)}+
             R^{(r)}_2a_{s+\l_r-2}^{(r)}+\dots,$$
where $a^{(r)}_i=c_i(E{-}L{-}F_{\l_r})$ and
$R^{(r)}_i=c_i(D^{r-1}{-}D^{(r-1)*})$ are polynomials of  degree $i$
in $t_1,\dots,t_{r-1}$ that are independent of $s,\l$ and  given by
the expansion
 $$
 R_0^{(r)}+R_1^{(r)}+R_2^{(r)}+\dots=
 \prod_{i=1}^{r-1}\frac{1-t_i}{1+t_i}=
      \prod_{i=1}^{r-1}\(1-2t_i+
        2t_i^2-2t_i^3+\dots\). 
 $$

\spa{Proof~of~Theorem~{2.1}}.
Iterating this Lemma we can compute the direct image $p_*=p_{1*}\dots
p_{r*}$ of any particular monomial. This solves, in principle, the
problem of finding the cohomology class dual to any degeneracy locus.
It follows without further computations that $p_*(t_1^{s_1}\cdots
t_r^{s_r})$ is expressed as a universal polynomial (depending only on
$\l_i$, $s_i$, $i=1,\dots,r$) in classes
$a_i^{(k)}=c_i(E{-}L{-}F_{\l_k})$. It is a matter of algebra to show
that the result has the nice form of Theorem~2.1. This algebraic
proof is given in Appendix~C.\QED

\ex Example. For $r=2$ applying Lemma twice we get
\begin{eqnarray*}
[S_{k,l}]&=&p_{1*}p_{2*}(1)=
p_{1*}\(a_l^{(2)}-2t_1a_{l-1}^{(2)}+2t_1^2a_{l-2}^{(2)}-\dots\pm2t_1^l\)\\
&=&p_{1*}(1)\;a_l^{(2)}-2p_{1*}(t_1)\;a_{l-1}^{(2)}+
       2p_{1*}(t_1^2)\;a_{l-2}^{(2)}-\dots\pm2p_{1*}(t_1^l)\\
&=& a_k^{(1)}a_l^{(2)}-2a_{k+1}^{(1)}a_{l-1}^{(2)}
      +2a_{k+2}^{(1)}a_{l-2}^{(2)}-\dots\pm2a_{k+l}^{(1)}
      =\s_{k,l}(a^{(1)},a^{(2)}),
\end{eqnarray*}
where $a^{(1)}=c(E-L-F_k)$, $a^{(2)}=c(E-L-F_l)$.\QED


\spa{Proof~of~Lemma}. Represent $p_r$ as the composition 
$Z_r\too{i}P(E_{r-1})\too q Z_{r-1}$ according to the 
diagram~(\ref{diag}). The homomorphism $i_*$ is given by the
multiplication by the fundamental cycle of $Z_r\subset P(E_{r-1})$.
The submanifold $Z_r\subset P(E_{r-1})$ is given, as explained above,
by the condition that the line $D^r_y/D^{r-1}_y\subset
D^{(r-1)\bot}_y/D^{r-1}_y$ is contained in the plane $L_y/D^{r-1}_y$.
This may be reformulated as vanishing of the section for the bundle
$\Hom(D^r/D^{r-1},D^{(r-1)\bot}/L)$. Therefore
 \begin{eqnarray*}
 i_*(t_r^s)&=&t_r^s\;c_{n-r+1}(\Hom(D^r/D^{r-1},D^{(r-1)\bot}/L))\\
   &=&t_r^s\;\sum_{i+j=n-r+1}c_i(D^{(r-1)\bot}/L)\;t_r^j\\
   &=&\sum_{i+j=n+s-r+1}c_i(D^{(r-1)\bot}/L)\;t_r^j\\ 
   &=&c_{n+s-r+1}(D^{(r-1)\bot}/L-D^r/D^{r-1})\\
   &=&c_{n+s-r+1}(D^{r-1}{-}D^{(r-1)*}{+}E{-}L{-}F_{\l_r}+F_{\l_r}/D^r)\\
   &=&\sum_{i+j=n+s-r+1} c_j(D^{r-1}{-}D^{(r-1)*}{+}E{-}L{-}F_{\l_r})\;
             c_{i}(F_{\l_r}/D^r)
   \end{eqnarray*}

Now we apply $q_*$. The classes
$c_j(D^{r-1}{-}D^{(r-1)*}+E{-}L{-}F_{\l_r})$ commute with this
homomorphism as they come from $Z_{k-1}$. The bundle $F_{\l_r}/D^r$
is the canonical quotient bundle over $P(E_{r-1})$. Therefore
$q_*c_{d}(F_{\l_r}/D^r)=1$, where $d=\rank (F_{\l_r}/D^r)=n+1-r-\l_r$
and  $q_*c_i(F_{\l_r}/D^r)=0$ for $i\ne d$. Therefore,
$$
\begin{array}[b]{rcl}
p_{r*}(t_r^s)=q_*i_*(t^s)&=&
c_{s+\l_r}(D^{r-1}{-}D^{(r-1)*}+E{-}L{-}F_{\l_r})\\[1ex]&=&
\displaystyle
\sum_{i=0}^\infty c_i(D^{r-1}{-}D^{(r-1)*})\;a_{s+\l_r-i}^{(r)}.
\end{array}
\QED$$





\section{Inverse induction}

In this section we present an alternative proof of Theorem~1.1.
Namely, we first establish the equality of Theorem in one particular
case when $M=\F_N$ is the flag manifold of complete isotrope subspaces in
$\C^{2N}$, the bundle $E=\C^{2N}$ is trivial, $F$ is the tautological
flag of bundles and $L\subset \C^{2N}$ is a fixed Lagrange subspace.
This space $\F_N$ for sufficiently large $N$ together with the induced
Schubert partition on it is considered as the classifying space for
our problem. It means that under condition of Theorem~1.1 there is a
classifying map $\k:M\to \F_N$ which induces both the partition of $M$ 
into the degeneracy loci and the characteristic classes. So the
equality of Theorem~1.1 for $M$ is induced from the corresponding 
equality for $\F_N$. Remark that in general we can not avoid the
consideration of $C^\infty$-manifolds and maps. The reader who does
not like this kind of arguments may consider the proof as a motivation
for finding new formulas (which can be applied as well in other
problems).

Let $\F=\F_n$ be the space of complete isotrope flags
$F_n\subset\dots\subset F_1\subset\C^{2n}$, $\dim F_k=n+1-k$. For a
strictly decreasing sequence of integers $n\ge\l_1>\dots>\l_r>0$ we
define the Schubert cycle $S_{\l_1,\dots,\l_r}\subset \F$ by
conditions $\dim(F_{\l_1}\cap L)\ge i$, $i=1,\dots,r$, where
$L\subset\C^{2n}$ is a fixed Lagrange subspace.

\th Theorem {3.1}. The cohomology class dual to the Schubert cycle in
$\F$ is given by
 $$[S_{\l_1,\dots,\l_r}]=\s_{\l_1,\dots,\l_r}
       (c(-F_{\l_1}),\dots,c(-F_{\l_r})).$$


In the proof we use the method of `divided differences',
see~\cite{BGG,D,Fu1,Fu,P2}. Namely, we introduce operations
$\d_k:H^*(\F_n)\to H^{*-2}(\F_n)$, $k=1,\dots,n$, that decrease by $1$
the complex codimension of the cycles representing the cohomology
classes. Then we compute these operations in terms of degeneracy loci
and in terms of the Chern classes. Then from the validity of Theorem
for the `deepest' locus we can establish the validity of Theorem for
other loci by reverse induction over the indices $\l$.

The operations $\d_k$ are defined as follows.  Denote 
by $\F^{(k)}$ the manifold of isotrope flags $F_n\subset\cdots
F_{k+1}\subset F_{k-1}\subset\cdots\subset F_1$ (with the $k$th
subspace of rank $n+1-k$ omitted). The natural projection
 $$p_k:\F \to \F^{(k)}$$
 is a smooth locally trivial bundle with the fibers isomorphic to $\C
P^1$. The total space $\F$ of this bundle can be identified with the
projectivization on the bundle $F_{k-1}/F_{k+1}$ of rank $2$ over
$\F^{(k)}$. This description is valid for all $k=1,2,\dots,n$ if
we set $F_{n+1}=0$, $F_0=F_2^\bot$. Indeed, since the plane $F_1$ is
Lagrangian, the condition $F_{2}\subset F_1$ implies $F_1\subset
F_2^\bot$.

\ex Definition. $\d_k=p_k^{*}\,p_{k*}:H^*(\F)\to H^{*-2}(\F)$. 

\th Lemma S. {Let $k\ge2$. Then 
$$\d_k[S_{\l_1,\dots,k,\dots,\l_r}]=
[S_{\l_1,\dots,k-1,\dots,\l_r}]$$
if $k\in\l$ and $k{-}1\not\in\l$; otherwise $\d_k[S_{\l_1,\dots,\l_r}]=0$.

For $k=1$ we have
 $$\d_1[S_{\l_1,\dots,\l_{r-1},1}]=[S_{\l_1,\dots,\l_{r-1}}]$$
and $\d_1[S_{\l_1,\dots,\l_r}]=0$ if $\l_r>1$.} 

\th Lemma C. In order to obtain the action of $\d_k$ in terms of Chern
classes one should to replace the classes of $[S_{\l_1,\dots}]$ 
in the statement of previous Lemma by the corresponding polynomials
$\s_{\l_1,\dots}(c(-F_{\l_1}),\dots)$.


\spa{Proof~of~Theorem~{3.1}}. Every Schubert cycle in $\F$ can be
obtained  from the cycle $S_{n,n-1,\dots,1}$ by a sequence of
operations $\d_k$ for different $k$. If we would prove the equality
of Theorem~3.1 for this cycle $S_{n,n-1,\dots,1}$ then by Lemmas S
and C we get that similar equality holds for the other Schubert
cycles. In the multi-index $(n,n-1,\dots,1)$ all entries except the
last one are redundant. Therefore,
$\s_{n,n-1,\dots,1}(c(-F_n),\dots,c(-F_1))=
\s_{n,n-1,\dots,1}(c(-F_1),\dots,c(-F_1))$. It follows that it is
sufficient to verify the required equality on the Lagrange
Grassmannian $\L_n$.  But in this
case the degree of the class
$\s_{n,n-1,\dots,1}(c(-F_1),\dots,c(-F_1))$ is equal  to the
dimension of $\L_n$ and $S_{n,n-1,\dots,1}$ is a point.
Therefore,
 $$\s_{n,n-1,\dots,1}(c(-F_1),\dots,c(-F_1))=b\;[\pt]
 =b\;[S_{n,n-1,\dots,1}]\in 
 H^{2\frac{n(n+1)}2}(\L_n)\cong H_0(\L_n),$$
where the constant $b\in\Z$ is equal to the value of the characteristic
class $\s_{n,n-1,\dots,1}(c(-F_1),\dots,c(-F_1))$ on the fundamental 
cycle of $\L_n$. Assume that $b\ne 1$. Then again by inverse induction 
we get that for all Schubert cycles on $\F_n$ we should have 
$\s_\l(c(-F_{\l_1}),\dots,c(-F_{\l_r}))=b\;[S_\l]$ and to
find the constant $b$ it is sufficient to compute $[S_\l]$ for any
particular $\l$. 

Set $\l=(n)$. Then $S_n\subset\F_n$ is given by the condition
$F_1\subset L$. This is equivalent to the vanishing of the section of
the bundle $\Hom(F_1,\C^{2n}/L)$. Therefore,
 $$[S_n]=c_n(\Hom(F_1,\C^{2n}/L))=c_n(-F_1)=\s_n(c(-F_1)).$$
 This proves that, in fact, $b=1$. Theorem is proved.\QED









\spa{Proof~of~Lemma~S}. If $k\not\in\l$ then the planes of the bundle 
$F_k$ are not used in the definition of the cycle $S_\l$. Therefore, 
this cycle is the inverse image of the corresponding cycle $S_\l'\subset
\F^{(k)}$. Hence
 $$p_{k*}[S_\l]=p_{k*}p^{k*}[S_\l']=
                   [S_\l']\;p_{k*}p^{k*}(1)=0,$$
and so $\d_k[S_\l]=0$. If $k=\l_i$ for some number $i$ but
$\l_{i+1}=k-1$ then the index $i$ is redundant and we can apply the
same arguments. 

Assume now that $k=\l_i$ and that the index $i$ is not redundant.
Geometrically the homomorphism $\d_k$ can be described as follows.
Suppose that a cohomology class in the total space of the bundle
$p_k$ is represented by a cycle $C$ which meets every fiber  at at
most one point. Then the class $\d_k[C]$ is represented by the union
of all fibers through the points of $C$. It follows from this
description that the cycle representing the class $\d_k[S_\l]$  is
defined by  conditions $\dim F_{\l_j}\cap L\ge  j$ for $j\ne i$
and the corresponding condition for the index $i$ is replaced by the
following one: {\em there exist a plane $F'_{k}$ of dimension
$n+1-k$ such that $F_{(k+1)}\subset F_{k}'\subset F_{(k-1)}$ and
$\dim F_{k}'\cap L\ge  i$}. Clearly, this condition is equivalent
to the condition $\dim F_{(k-1)}\cap L\ge  i$, that is
$\d_k[S_\l]$ is represented by the cycle
$S_{\l_1,\dots,\l_{i-1},k-1,\l_{i+1},\dots,\l_r}$.\QED








\spa{Proof~of~Lemma~C}. 
Observe that $\d_k$ commutes with the multiplication by
characteristic classes of bundles that are defined on
$\F^{(k)}$. In particular, if
$k\not\in\l$ then the class $\s_\l$ is the pull back of the
corresponding class on $\F^{(k)}$. Therefore,
 $$\d_k\s_\l=\s_\l \d_k(1)=0.$$
Assume now that $k=\l_i>1$ for some (unique)
$i$. Then the equality
 $$c(-F_k)=c(F_{k-1}/F_k-F_{k-1})=
            (1+t)\;c(-F_{k-1}),$$
where $t=c_1(F_{k-1}/F_k)$ implies
 $$\s_{\dots,k,\dots}(\dots,c(-F_k),\dots)=A+t\;B,$$
where $A=\s_{\dots,k,\dots}(\dots,c(-F_{k-1}),\dots)$ 
and $B=
\s_{\dots,k-1,\dots}(\dots,c(-F_{k-1}),\dots)$ are classes that are defined on
$\F^{(k)}$. Therefore,
 $$\d_k\s_\l(\dots)=\d_k(A+t\;B)=
        A\d_k(1)+B\d_k (t).$$
$\d_k(1)$ vanishes by dimensional reason and the class
$t=c_1(F_{k-1}/F_k)$ is the top Chern class of the canonical
quotient bundle for the fiber bundle $P(F_{k-1}/F_{k+1})$ 
over $\F^{(k)}$. Therefore,
$\d_k(t)=p^{*k}p_{*k}(t)=1$. We get
finally
 $$\d_k\s_{\dots,k,\dots}(\dots,c(-F_k),\dots)=B=
 \s_{\dots,k-1,\dots}(\dots,c(-F_{k-1}),\dots).$$

Observe that if $k{-}1\in\l=(\dots,k,\dots)$ then $\l'=
(\dots,k-1,\dots)$ has repeating indices and so $B=0$ in this case. 
This proves Lemma in case when $k>1$.


In the case $k=1=\l_r$ the computation above can be applied 
as well if we denote $F_0=F_2^\bot$ and gives
  $$\d_1\s_{\dots,\l_{r-1},1}(\dots,c(-F_{\l_{r-1}}),c(-F_{1}))=
       \s_{\dots,\l_{r-1},0}(\dots,c(-F_{\l_{r-1}}),c(-F_{0})).$$
By Remark before Theorem~1.1, the class on the right hand side is equal to
$\s_{\dots,\l_{r-1}}(\dots,c(-F_{\l_{r-1}}))$.
 Lemma is proved.\QED







\spa{Proof~of~Theorem~{2.1}}. Assume we are given vector bundles
$V\subset E\to M$, $F_{k_1}\subset\dots\subset E\to M$, $E$ is
symplectic, $L$, $F_i$ are isotrope. Without loss of generality we
can assume that $E$ splits into the sum $E=L\oplus L^*\oplus K$, where $K$
is symplectic and the symplectic structure $\o_E$ on $E$ is the sum
of the corresponding symplectic structures $\o_{L\oplus L^*}$ on
$L\oplus L^*$ and $\o_K$ on $K$. Indeed, the fiber $L'_x$ of the
bundle $L'\cong L^*$ is chosen among the isotrope subspaces in $E_x$
transversal to $L^\bot_x$ and having the complementary dimension. The
space of possible choices is contractible (it is  homeomorphic to a
cell). Hence such subbundle $L'$ exists (in general, it is a complex
$C^\infty$-bundle). Then we set $K_x=L^\bot_x\cap L'^\bot_x$.

Now choose the bundle $U\to M$ such that $L\oplus K\oplus U=\C^N$ is a
trivial bundle (again, in general, $U$ is a complex $C^\infty$-bundle).
Define the bundles $\Et,\Lt,\Ft_i$ and the
symplectic structure $\o_\Et$ on $\Et$
according to the following table 
$$
\def\+{~\oplus~}
\begin{array}{c@{{}={}}c@{\+}c@{\+}
c@{\+}c@{\+}c@{\+}c}
\Et&L&L^*&K&K&U&U^*\\
\o_\Et&\multicolumn2{c@{\+}}{\o_{L\oplus L^*}}&
\o_K&-\o_K&\multicolumn2{c}{\o_{U\oplus U^*}}\\
\Lt&L&0&\multicolumn2{@{}c@{\+}}{\Delta}&U&0\\
\Ft_k&\multicolumn3{c@{\+}}{F_k}&0&0&U^*
\end{array}
$$
Here $\Delta\in K\oplus K$ is the diagonal bundle. Recall that the natural
symplectic structure on the sum $K_x\oplus K_x$ of two symplectic
spaces is defined as the {\em difference} of the symplectic
structures induced from the two summands. If it is defined this way
then the diagonal $\Delta_x=\{z{\oplus}z\bigm|z\in K_x\}$ as well as
the anti-diagonal $\Delta'_x\cong\Delta^*_x=\{z{\oplus}(-z)\bigm|z\in
K_x\}$ are Lagrange subspaces. In particular, they define the
canonical isomorphism $K\oplus K=\Delta\oplus\Delta^*$.  

By construction, the degeneracy loci for this new problem are the
same. But now the bundles $\Lt\cong L\oplus K\oplus U\cong\C^N$ and
$\Et=\Lt\oplus\Lt^*\cong\C^{2N}$ are trivial and the fibers of the
bundles $\Ft_k$ belong to the same symplectic space $\C^{2N}$.
Therefore, they define the map 
$$\k:M\to\F_N',$$ 
where $\F_N'$ is the the manifold of (incomplete) flags of
isotrope subspaces in $\C^{2N}$ of dimensions $N-n+1,N-n+2,\dots,N$. 

The formula of Theorem~3.1 can be applied to the classes of Schubert
cycles on $\F_N'$ since the projection $\F_N\to\F_N'$ induces
an injective homomorphism of cohomology, see Appendix~A. The degeneracy
locus $S_\l(M)\subset M$ is the inverse image of the Schubert cycle
$S_\l(\F_N')\subset\F_N'$. The characteristic classes
induced by $\k$ are $\k^*c(\Ft_k)=c(F_k+U^*)=c(F_k+L-E)$. Therefore,
 \begin{eqnarray*}
 [S_\l(M)]&=&\k^*[S_\l(\F_N')]=
 \k^*\s_\l(c(-\Ft_{\l_1}),\dots,c(-\Ft_{\l_r})\\
 &=&\s_\l(c(E-L-F_{\l_1}),\dots,c(E-L-F_{\l_r})
\end{eqnarray*}
Theorem~1.1 is proved.\QED


\ex Remark. The map $\k$ is holomorphic in case when the  degeneracy
locus is the diagonal in the product of two isotrope Grassmannians or
it is a Schubert cell on the Grassmannian of isotrope subspaces in
$\C^{2N}$. Therefore the proof of formulas~(\ref{diagonal}) (for the
case $E=\C^{2N}$) and~(\ref{grassm}) of Section~1 by the method of
this Section does not require considerations of $C^\infty$-maps and
bundles.

\ex Remark. Lemmas~C and~S remain valid if $\F_N$ is replaced by the
bundle of isotrope flags associated with some symplectic vector
bundle,~sf.~\cite{Fu}. It follows that the formula of Theorem~1.1
can be derived from the formula~(\ref{diag}) of Section~1 for the
class of the diagonal in the fiber product of two Lagrange Grassmann
bundles. The  direct proof of~(\ref{diag}) would imply the proof of
Theorem~1.1 by divided differences method without using
$C^\infty$-maps and bundles.



\section*{Appendix A. Complex Lagrange Grassmannian}\label{cohom}

It was shown in previous Section that the Lagrange Grassmannian
$\L_N$, $N\to\infty$ plays the role of a classifying space for many
geometrical problems. The limit cohomology group
$\lim_{N\to\infty}H^*(\L_N)$ is called {\em the ring of Lagrange
characteristic classes}. In this section we describe the topology of
$\L_N$. All results of this section are proved in~\cite{P1} but our
presentation is more elementary.

Consider an even-dimensional vector space $\C^{2n}$ and a fixed
skew-symmetric bilinear form (the {\em symplectic form}) $\sum dp_i\wedge
dq_i$ on it, where $dp_i,dq_i$ are elements of some fixed basis on the
dual space $\C^{*2n}$.

\ex Definition. An $n$-dimensional subspace is called {\em
Lagrangian} if the restriction of the symplectic form to it vanishes.
The {\em complex Lagrange Grassmannian} $\L^\C_n$ is the manifold of
all Lagrange subspaces in $\C^{2n}$.

The topology of the real Lagrange Grassmannian $\L_n^\R\cong {\rm
U}(n)/{\rm O(n)}$ is well known. Its
$\Z_2$-cohomology is
$H^*(\L^\R,\Z_2)\cong\L_{\Z_2}(\a_1,\dots,\a_n)$. The (integer)
cohomology of complex Lagrange Grassmannian
$\L_n^\C\cong{\rm Sp}(2n)/{\rm U}(n)$ has a similar description.

\th Theorem {\rm(\cite{P1})}. The ring $H^*(\L^\C_n)$ is isomorphic 
to the quotient of the polynomial ring in variables $a_1,a_2,\dots,a_n$ 
of degrees $2,4,\dots,2n$ over the ideal generated by elements
\begin{equation}\label{rel}
a_i^2-2a_{i+1}a_{i-1}+2a_{i+2}a_{i-2}-2a_{i+3}a_{i-3}+\dots.
\end{equation}
The group $H^*(\L^\C_n)$ is torsion free and the monomials
$a_1^{i_1}\dots a_n^{i_n}$, $i_k\in\{0,1\}$ form a free additive
basis.


In the relations above we assume $a_0=1$ and $a_i=0$ for $i<0$
or for $i>n$. The classes $a_i\in H^{2i}(\L^\C_n)$ are Chern classes
of the tautological bundle $L\to\L^\C_n$ (or inverse images of Chern
classes of the usual Grassmannian $G_{n,2n}^\C$ under the embedding
$\L^\C_n\subset G_{n,2n}^\C$. 

The second assertion of Theorem follows from the first one. To
express an element of this ring in terms of this basis one should
apply repeatedly relation of Theorem to every monomial which contains
squares of generators. This will require finite number of steps since
every newly appeared monomial has degree strictly less than the
original one if one uses the `strange' filtration with the
degree of $a_i$ equal $i^2$.

The relations~(\ref{rel}) can be rewritten in the form
  $$(1+a_1+a_2+\dots+a_n)(1-a_1+a_2-\dots\pm a_n)=1.$$
In this form they follow from the fact that the symplectic form
induces an isomorphism $\C^{2n}/L\cong L^*$.

In the proof of Theorem we use the following well-known lemma. Let
$E\to M$ be a complex vector bundle of rank $d$ over some manifold $M$. Let
$P=P(E)$ be the projectivization of the bundle $E$, i.e.\ the bundle
space over $M$ whose fibers are projective spaces formed by lines in
fibers of $E$. Consider $H^*(P)$ as a module over the ring
$A=H^*(M)$. Let $L\to P$ be the tautological line bundle and
$t=c_1(L^*)=-c_1(L)$.

\th Lemma. Additively the group $H^*(P)$ is isomorphic to
$H^*(M\times \C P^{d-1})$ independently of the bundle $E$. Moreover,
it is freely generated by elements $1,t,\dots,t^{d-1}$ as an $A$-module.
As a ring $H^*(P)$ is isomorphic to the quotient ring of $A[t]$ over
the ideal generated by 
  $$t^d+c_1(E)t^{d-1}+c_2(E) t^{d-2}+\dots+c_d(E)\in A[t].$$
The Gysin homomorphism  $\pi_*:H^*(P)\to H^*(M)$ maps the element 
$u_0+u_1t+\dots+u_{d-1}t^{d-1}\in H^*(P)$ to $u_{d-1}\in H^*(M)$.

Recall that the {\em Gysin} or {\em push-forward} homomorphism or
{\em transfer} $\pi_*$ associated with a proper map $\pi:P\to M$ of
smooth manifolds is the composition of Poincar\'e duality in $P$,
usual homomorphism of homology, and Poincar\'e duality in $M$. The
main property of Gysin  homomorphism is the identity
$\pi_*(\pi^*a\;b)=a\pi_*(b)$ for any elements $a\in H^*(M)$, $b\in
H^*(P)$. In other words, $\pi_*$ is the homomorphism of
$H^*(M)$-modules.

The proof of Lemma is simple. First note that $\pi_*t^i=0$ for $i<d-1$
(by dimensional reason) and $\pi_*t^{d-1}=1\in H^0(M)$ (this means
that $d-1$ hyperplanes in $\C P^{d-1}$ intersect at one point).
Therefore, 
  $$\pi_*(u_0+u_1t+\dots+u_{d-1}t^{d-1})=
      u_0\pi_*(1)+u_1\pi_*(t)+\dots+u_{d-1}\pi_*(t^{d-1})=u_{d-1}.$$

Therefore the elements $1,t,\dots,t^{d-1}$ are independent over $A$
in $H^*(P)$ since any relation of the form $u_0+\dots+u_k t^k=0$ with
$u_k\ne0$ would imply $0=\pi_*(t^{d-k-1}(u_0+\dots+u_kt^k))=u_k$. It
follows from the spectral sequence of the bundle that the elements of
the form $u_0+\dots+u_{d-1}t^{d-1}$ exhaust all cohomology of $P$.
This proves Lemma. Note that $t^d+c_1(E)t^{d-1}+c_2(E)
t^{d-2}+\dots+c_d(E)$ is equal to zero since it is the $d$th Chern
class of the $(d{-}1)$-dimensional quotient bundle $\pi^*E/L$.\QED  

The lemma is applied as follows. Denote by $\F _{i_1,\dots,i_r}$ the
space of flags consisting of {\em isotrope} subspaces
$U_{i_1}\subset\dots\subset U_{i_r}$, $\dim_\C U_l=l$. Consider the
following diagram of projections 
$$\C P^{2n-1}=\F _1\ot \F _{1,2}\ot\dots\ot \F _{1,2,\dots,n}
                  \to \F _{2,\dots,n}\to\cdots\to \F _n=\L^\C_n.$$
All arrows in this diagram are projectivizations of certain vector
bundles. This allows us to compute inductively the cohomology groups of
all spaces in this diagram. It follows without any calculation that
all these cohomology groups are torsion free. Minimal calculations give
the total rank of $H^*(\L^\C_n)$. Comparing this diagram with a
similar diagram for usual flags and Grassmannian we see that the Chern
classes $a_1,\dots,a_n$ generate all cohomology ring of $\L^\C_n$. We 
know already some set of relations, and, comparing
dimensions we see that there are no other relations.\QED

\ex Exercise. Find generators and relations for the cohomology ring
of any flag manifold of isotrope subspaces on $\C^{2n}$. Present a
free additive basis for each case. 

The correspondence between symmetric and Lagrange degeneracy loci is
formalized as follows. Consider the homomorphism
 $$ \psi:\Z[a_1,a_2,\dots]\to\Z[c_1,c_2\dots],\qquad
 1+a_1+a_2+\dots\mapsto \frac{1+c_1+c_2+\dots}
 {1-c_1+c_2-\dots}.$$
 
\th Proposition. The homomorphism $\psi$ induces an embedding
of the ring of Lagrange characteristic classes to the polynomial ring
$\Z[c_1,c_2,\dots]$.


\spa{Proof}.
Since both $\Z[c_1,c_2,\dots]$ and
the ring of Lagrange characteristic classes are torsion free it is
sufficient to prove this assertion over $\Q$. Introduce a new system
of generators $\tilde a_i$, $\tilde c_i$ in the polynomial rings
by setting 
$$\tilde c_1+\tilde c_2+\dots=\log(1+c_1+c_2+\dots)$$
and similarly for $\tilde a_i$ (the classes $\tilde c_k$, $\tilde a_k$
coincide up to $\pm1/(k-1)!$ with the homogeneous components of the
corresponding Chern characters). In
terms of these generators the homomorphism
 $$\psi:\Q[\tilde a_1,\tilde a_2,\dots]\too{}
 \Q[\tilde c_1,\tilde c_2,\dots]$$
is given by $\tilde a_{2k-1}\mapsto 2 \tilde c_{2k-1}$, 
$\tilde a_{2k}\mapsto 0$. The ideal~(\ref{rel}) of defining relations 
in the ring of Lagrange characteristic classes is generated by 
$2\tilde a_2,2\tilde a_4,\dots$.
Proposition follows.\QED

\ex Remark.
This proposition implies that all identities valid in the case of
symmetric degeneracies are satisfied also in the ring of Lagrange
characteristic classes. Therefore, Theorems~1.1 and~2.1 are
equivalent to the corresponding statements about symmetric
degeneracies formulated at the end of Section~\ref{main}. Remark that
in particular, the solution to J.~Harris' problem (see Section~1) is
contained implicitly already in~\cite{HT}! (That is the polynomial
$\s_{m+r,\dots,m+1}(c(E{-}L{-}L'),\dots,c(E{-}L{-}L'))$ in classes
$a_i=c_i(E{-}L{-}L')$ is uniquely determined by the condition that the
image of this polynomial under $\psi$ coincides with the class found
in~\cite{HT}.) I would claim even more, that some papers on
symmetric (or skew-symmetric) degeneracy loci and those on isotrope
degeneracy loci are in much extent duplicates of each others. 



\section*{Appendix~B. Twisted degeneracy loci}

Assume that the symplectic form on the fibers of vector bundle $E\to
M$ takes values not in $\C$ but in fibers of some line bundle $I\to
M$. In other words, this twisted symplectic structure on the fibers
is given by nowhere degenerating section of $\L^2E^*\otimes I$. The
degeneracy loci for isotrope subbundles $L\subset E$ and
$F_{k_1}\subset F_{k_2}\subset\dots\subset E$, $k_i=\rank E-\rank
L-\rank F_{k_i}+1$, are defined in the same way as in Section~1. The
cohomology classes dual to these loci can be obtained in the
following way. 

Assume first that the bundle $I$ is a tensor square of another line
bundle: $I=J^{\otimes 2}$. Then $\L^2E^*\* I=\L^2(E\* J^*)^*$ and we
can apply Theorem~1.1 to the bundles $\Et=E\* J^*$, $\Lt=L\*
J^*\subset \Et$, $\Ft_k=F_{k}\* J^*\subset\Et$:
\begin{equation}
[S_{\l_1,\dots,\l_r}]=\s_{\l_1,\dots,\l_r}
(c(\Et{-}\Lt{-}\Ft_{\l_1}),\dots,c(\Et{-}\Lt{-}\Ft_{\l_1})).
\label{tw1}
\end{equation}
Then we substitute
\begin{equation}
c(\Et{-}\Lt{-}\Ft_{k})=(1-u/2)^{k-1}\(
1+\frac{c_1(E{-}L{-}F_k)}{1-u/2}+\frac{c_2(E{-}L{-}F_k)}{(1-u/2)^2}+
%\frac{c_3(E{-}L{-}F_k)}{(1-u/2)^3}+
\dots\),
\label{tw2}
\end{equation}
where $u=c_1(I)=2c_1(J)$. The expression for $[S_\l]$ in terms of
$c_i(E{-}L{-}F_k)$ and $u=c_1(I)$ obtained in this way can be applied
for general case since it is universal (the existence of universal
polynomial expressing $[S_\l]$ in terms of $c_1(I),c_i(E{-}L{-}F_k)$
follows from either of the two methods of the proof of Theorem~1.1
presented in this paper, cf.\ also~\cite{HT}). Moreover,
though we used division by $2$ to obtain it, {\em this polynomial
expression for $[S_\l]$ has integer coefficients and so it can be
applied for any coefficient ring}. This (algebraic) assertion has the
following topological proof.

Fix some integer $N\gg0$ and define the manifold $Y=Y_N$
as the total space of the bundle over $\C P^{N}$ with the fibers
formed by all isotrope flags in the twisted symplectic spaces
$\C^{N}\oplus(\C^{N}\*I_x)$, $x\in \C P^{N}$, where $I\to\C
P^{N}$ is the tautological line bundle (the twisted symplectic form
on $\C^{N}\oplus(\C^{N}\*I_x)=(T^*\C^{N})\*I_x$ with values in 
$I_x$ is the obvious generalization of the standard symplectic form
on  $\C^{2N}=T^*\C^{N}$). It is not difficult to compute
explicitly the cohomology ring of $Y$ (using the same arguments as in
Appendix~A). This ring is generated by classes $u=c_1(I)$ and
by the Chern (or Segre) classes $a_i^{(k)}=c_i(-F_k')$ 
of the tautological bundles $F_k'\to Y$.

Now, we {\em define} the polynomial $\s_{\l_1,\dots,\l_r}\in H^*(Y)$
as the cohomology class (expressed in terms of the generators
$u,a_i^{(k)}$) dual to the twisted Schubert cycle
$S_{\l_1,\dots,\l_r}\subset Y$ defined with respect to the Lagrange
subbundle $\C^{N}\oplus0\subset \C^{N}\oplus(\C^{N}\*I)$.  By
definition, this is a polynomial with integer coefficients. Since
$H^*(Y)$ is torsion free, this polynomial coincides with that
defined by~(\ref{tw1})--(\ref{tw2}) (with $a_i^{(k)}$ in place of
$c_i(E{-}L{-}F_k)$).

For general case it is not difficult to verify that the classes
$c_i(E{-}L{-}F_k)$ satisfy all relations for the classes $a_i^{(k)}$,
provided that $N$ is sufficiently large. In other words the 
characteristic homomorphism 
 $$\k^*:H^*(Y)\to H^*(M),\quad  a_i^{(k)}\mapsto c_i(E{-}L{-}F_k)$$
is well defined though it is not necessary induced by a map
$\k:M\to Y$. This proves that the class defined
by~(\ref{tw1})--(\ref{tw2}) is integer.\QED






\section*{Appendix C. Identities in Pfaffians}

Recall that the Pfaffian of a skew-symmetric matrix $\o=\|\o_{i,j}\|$ of even
order $2n$ is, by definition,
 $$\Pf\|\o_{i,j}\|=\sum\pm\o_{i_1,i_2}\cdots\o_{i_{2n-1},i_{2n}},$$
 where the sum is over all $(2n-1)!!$ ways to represent
$\{1,2,\dots,2n\}$ as a union of $n$ pairs
$\{i_1,i_2\}\cup\dots\cup\{i_{2n-1},i_{2n}\}$ and $\pm$ is
the sign of the permutation $(1,2,\dots,2n)\mapsto(i_1,i_2,\dots,i_{2n})$.

Consider a skew-symmetric function $f[k,l]=-f[l,k]$ with values in
some commutative ring and defined on some discrete set $I$. For any ordered 
collection $(\l_1,\dots,\l_r)$, $\l_i\in I$, we set
\begin{itemize}
\item $f[\l_1,\dots,\l_r]=\Pf\|f[\l_i,\l_j]\|$,  $r$  even;
\item $f[\l_1,\dots,\l_r]=f[\l_1,\dots,\l_r,0]=\sum_{i=1}^r(-1)^{i-1}
  f[\l_1,\dots,\widehat{\l_i},\dots,\l_r]$, $r$  odd, \\where for
  $0\not\in I$ we put formally $f[k,0]=-f[0,k]=1$.
\end{itemize}



\th Proposition. The function $f$ satisfies the identity
 $$f[\l_1,\dots,\l_r]=\prod_{1\le i<j\le r}f[\l_i,\l_j]$$
if and only if it satisfies this identity for $r=3$.

\spa{Proof}. This proposition generalizes similar statement
from~\cite{Kn}, where the identity for even $r$ was proved under
assumption that it is satisfied for $r=4$. Observe that adding $0$
does not spoil the identity for $r=3$. Then, it is easily verified
for $r=4$, and so the result follows form Knuth's theorem. Remark
that the direct proof of Proposition is a little simpler than the
original proof of Knuth's theorem.\QED

\ex Example. The identity is satisfied for the function 
$\displaystyle g[x,y]=\frac{y-x}{x+y}$. 

\ex Example {\rm(V. Kryukov)}. Denote by $\Lc$ the completion of the ring of Laurent
polynomials $\Z[t_1,t_1^{-1},\dots,t_r,t_r^{-1}]$ with help of 
infinite formal series in
$t_1/t_2,t_2/t_3,\dots,t_{r-1}/t_r$ (the variables $t_i$ can enter in
monomials with arbitrary large positive or negative exponents but every
monomial of the product $R_1 R_2$ of two such series is determined by only
finite number of monomials in $R_1$ and $R_2$). For $i<j$ set
$$f[i,j]=-f[j,i]=\frac{1-t_i/t_j}{1+t_i/t_j}=1-2\frac{t_i}{t_j}
+2\frac{t_i^2}{t_j^2}-2\frac{t_i^3}{t_j^3}+\dots\in\Lc$$
This function also satisfies the condition of Proposition. Indeed, if
for $i<j<k$ we set $a=t_i/t_j$, $b=t_j/t_k$, then the identity we
should verify reduces to the correct identity
$$\frac{1-a}{1+a}-\frac{1-ab}{1+ab}+\frac{1-b}{1+b}=
  \frac{(1-a)(1-ab)(1-b)}{(1+a)(1+ab)(1+b)}.\QED$$
  
  
  
\spa{Completion~of~the~proof~of~Theorem~{2.1}}. Denote by $A$ the
polynomial ring in variables $a_k^{(i)}$. Consider the homomorphism
$$p_{k*}:A[t_1,\dots,t_k]\to  A[t_1,\dots,t_{k-1}],$$
that commutes with $t_1,\dots,t_{k-1}$ and
$$p_{k*}t_k^s=\sum_{i=0}^\infty R^{(k)}_ia_{s+\l_r-i}^{(k)},
\quad{\rm where}\quad R_0^{(k)}+R_1^{(k)}+R_2^{(k)}+\dots=
 \prod_{i=1}^{k-1}\frac{1-t_i}{1+t_i}.$$
 Our goal is to compute $p_{1*}\dots
 p_{r*}t_1^{s_1}{\cdots}t_r^{s_r}$. Let $\Lc_k\subset\Lc$ denote the
subring of series that do not contain negative exponents of variables
$t_1,\dots,t_k$. Consider the $\Z[t_1,\dots,t_k]$-linear homomorphism
 $$\Lc_k\too{\f}A[t_1,\dots,t_k],\qquad 
 t_1^{m_1}\cdots t_r^{m_r}\mapsto t_1^{m_1}\cdots 
 t_k^{m_k}\; a_{m_{k+1}}^{(k+1)}\cdots a_{m_{r}}^{(r)}.$$
In terms of this homomorphism the image $p_{k*}t_k^s$ is given by 
 $$p_{k*}t_k^s=\f\;\sum_{i=0}^\infty
 R_i^{(k)}t_k^{s+\l_k-i}=\f\;t_k^{\l_k+s}\sum_{i=0}^\infty
 R_i^{(k)}t_k^{-s}=\f\;t_k^{\l_k+s}
 \prod_{i=1}^{k-1}\frac{1-t_i/t_k}{1+t_i/t_k}.$$
In other words, in terms of $\f$ the homomorphism $p_{k*}$ is just the
multiplication by $t_k^{\l_k} \prod_{i=1}^{k-1}\frac{1-t_i/t_k}{1+t_i/t_k}$.
After $r$ steps we obtain
 $$p_{1*}\dots p_{r*}t_1^{s_1}\cdots t_r^{s_r}=\f\;
 t_1^{s_1+\l_1}\cdots t_r^{s_r+\l_r}
 \prod_{1\le i<j\le r}\frac{1-t_i/t_j}{1+t_i/t_j}.$$
 Now we use the identity for the function $f$ of Example above. 
Assume $r$ is even. Denoting $\tilde
f[i,j]=t_i^{s_i+\l_i}t_j^{s_j+\l_j}f[i,j]$, we have
 \begin{eqnarray*}
 p_{1*}... p_{r*}t_1^{s_1}... t_r^{s_r}&=&
 \f\; t_1^{s_1+\l_1}\cdots t_r^{s_r+\l_r}f[1,2,\dots,r]\\
 &=&\f\;\sum \pm \tilde f[i_1, i_2]\;\tilde f[i_3,i_4]\cdots\\
 &=&\sum \pm \f(\tilde f[i_1, i_2])\;\f(\tilde f[i_3,i_4])\cdots\\
 &=&\sum \pm \s_{\l_{i_1}+s_{i_1},\l_{i_2}+s_{i_2}}(a^{(i_1)},a^{(i_2)})
    \; \s_{\l_{i_3}+s_{i_3},\l_{i_4}+s_{i_4}}(a^{(i_3)},a^{(i_4)})\cdots\\
 &=&\s_{\l_1+s_1,\dots,\l_n+s_n}(a^{(1)},\dots,a^{(r)}).
 \end{eqnarray*}

The case of odd $r$ is considered in a similar way or we can simply
reduce the problem to the previous case by setting $\l_{r+1}=0$.\QED






\section*{Appendix D. Orthogonal degeneracy loci}

In this section we state the analogue of Theorem~1.1 for the case when
the bundle $E$ is {\em orthogonal\/} i.e.\ it is equipped with a
nondegenerate {\em symmetric\/} bilinear form given as a nowhere
degenerating section of the bundle ${\rm Sym}^2E^*$. The proofs
will appear elsewhere. 
 
The main difficulty arising in the orthogonal case is that the
isotrope bundles have characteristic classes that are not expressed
in terms of their Chern classes. In other words, the cohomology rings
of the isotrope Grassmann and flag manifolds are not generated by the
Chern classes of the tautological bundles. Almost all such classes
still can be expressed via the Chern classes if the division by $2$ is
allowed. Therefore in this Appendix we assume that the coefficient
ring of all cohomology groups contains $1/2$.  

There are still extra characteristic classes in the case when  $\rank
E=2n$ is even. For instance the Grassmannian of {\em maximal\/} (i.e.\
$n$-dimensional) isotrope subspaces in $\C^{2n}$ has two components.
Namely, the dimension of the intersection $\dim(L_1\cap L_2)$ of two
maximal isotrope subspaces $L_1,L_2\subset \C^{2n}$ may jump only by
even numbers when $L_1$ and $L_2$ are changing continuously. These
two planes belong to the same component iff $\dim(L_1\cap L_2)\equiv
n\pmod2$. For an isotrope subbundle $L\subset E$ the
missed characteristic class is defined as follows. Assume 
that the bundle $L$ is a subbundle of some
maximal isotrope bundle $\hL$. Then {\em the top Chern
class $c_m(\hL/L)$, $m=n-\rank L$, is independent on $\hL$ up to a
sign}. More precisely,

\th Lemma. If $\hL,\hL'\subset E$, $\rank E=2n$, are two maximal
isotrope subbundles containing $L$, then $c_m(\hL/L)=\pm
c_m(\hL'/L)$, where the sign $\pm$ is positive \(negative\) if
$\dim(\hL_x\cap\hL'_x)\equiv n\pmod 2$ \(resp.\
$\dim(\hL_x\cap\hL'_x)\equiv n-1\pmod 2$\) for any point $x\in M$.


\ex Definition. {Let $F,L\subset E$ be two isotrope subbundles. If
the rank $\rank E=2n$ is even we set
 $$e(L,F)=(-1)^{\dim(\hL_x\cap\hF_x)}c_k(\hL/L\oplus\hF/F),
 \quad k=2n-\rank L-\rank F,$$
where $\hL,\hF$ are some maximal isotrope subbundles containing $L$
and $F$ respectively.

If $\rank E=2n+1$ is odd we set $e(L,F)=0$.}

\ex Remark.  The Lemma implies that the class $e(L,F)$
depends neither on the bundles $\hL,\hF$ nor on the choice of the
point $x\in M$. Moreover this class is well defined even if the maximal
isotrope bundles $\hL,\hF$ do not exist. Indeed, we can
pass from $M$ to the total space of the bundle $G\to M$ whose fibers
are formed by pairs of maximal isotrope subspaces in $E_x$ containing
$L_x$ and $F_x$ respectively, $x\in M$; it is sufficient to define 
the corresponding characteristic class on $G$ since the induced 
homomorphism of the cohomology $H^*(M)\to H^*(G)$ is injective (over
$\Z[\frac12]$).


Consider an isotrope 
subbundle $L\subset E$ and a flag of isotrope subbundles
$$F_{k_1}\subset F_{k_2}\subset\dots\subset E,\quad
\rank F_{k}=\rank E-\rank V-k.$$
(Remark the shift by $1$ in the numeration of the planes $F_k$
comparing with the symplectic case.) The degeneracy loci
$S_{\l_1,\dots,\l_r}\subset M$, $\l_1>\dots>\l_r$ are defined 
similarly to the symplectic case by conditions
 $$\dim F_{\l_ix}\cap L_x\ge i,\quad i=1,\dots,r.$$ 
We may always assume that $\l_r>0$. Indeed, if $\rank E$ is even and 
both $F_0$ and $L$ are maximal then the fact that $\dim(F_{0x}\cap
L_x)=\const\pmod2$ implies that the condition on the dimension of the
intersection $F_{0x}\cap L_x$ is equivalent to the corresponding
condition on the dimension of the intersection $F_{1x}\cap L_x$.

\th Theorem.  Generically the cohomology class dual to the locus
$S_{\l_1,\dots,\l_r}$ is given by
 $$[S_{\l_1,\dots,\l_r}]=P_{\l_1,\dots,\l_r},$$
where the characteristic class $P_{\l_1,\dots,\l_r}\in
H^{2\sum\l_i}(M)$ is defined as follows:
\begin{itemize}
\item if $r=1$ then $2P_k=a_k^{(k)}-e(L,F_k)$, where
$a_i^{(k)}=c_i(E{-}L{-}F_k)$;
\item if $r=2$ then $\displaystyle 
4P_{k,l}=\(a_k^{(k)}{-}e(L,F_k)\)\(a_l^{(l)}{+}e(L,F_l)\)+
2\sum_{i=1}^l(-1)^ia_{k+i}^{(k)}a_{l-i}^{(l)}$;
\item if $r>3$ is even then
$P_{\l_1,\dots,\l_r}=\Pf|P_{\l_i,\l_j}|_{1\le i,j\le r}$;
\item if $r>2$ is odd then $\displaystyle 
P_{\l_1,\dots,\l_r}=\sum_{i=1}^r(-1)^{i-1}P_{\l_i}
      P_{\l_1,\dots,\widehat{\l_i},\dots,\l_r}.$
\end{itemize}

The genericity condition is formulated in the same way as in the
remark after
Theorem~1.1. All corollaries of Theorem~1.1 listed in Section~1 and
Appendix~B have the corresponding reformulations for the case of
orthogonal degeneracy loci (getting rid of redundant indicies,
Schubert classes on isotrope flag and Grassmann manifolds, degeneracy
of skew-symmetric maps of bundles, classes of twisted degeneracy loci
etc.). We present just two examples;  the interested  reader can
easily formulate the others. The formulas of the statements below
simplify the corresponding formulas from~\cite{PR,LP2}.

Let $L,L'\subset E$ be two isotrope subbundles in
an orthogonal bundle $E$. For any integer $r>0$ consider the locus
$\O_r=\{x\in M\bigm| \dim(L_x\cap L_x')\ge r\}$. Denote $a=c(E-L-L')$, 
$m=\rank E-\rank L-\rank L'$. Then $m$ is strictly positive unless 
$\rank E$ is even and both $L,L'$ are maximal.

\th Theorem {\rm(cf.~\cite{PR})}. Assume that $m>0$. Then generically
$$[\O_r]=\frac 1{2^r}\(
\s_{m+r-1,\dots,m+1,m}(a,\dots,a)+(-1)^r
e(L,L')\,\s_{m+r-1,\dots,m+1}(a,\dots,a)\).$$
If $m=0$ \(i.e.\ $\rank E=2n$ is even and the isotrope sbbundles 
$L,L'$ are maximal\) we assume that
$\dim(L_x\cap L'_x)\equiv r\pmod 2$. Then generically
$$[\O_r]=\frac1{2^{r-1}}\s_{r-1,\dots,1}(a,\dots,a).$$

Let $F\subset E$ be vector bundles over some base
$M$. The bundle map $f:F\to E^*$ is called {\em skew-symmetric\/} if the
bilinear form $\<f_x(u),v\>$ on $F_x\times E_x$ is
skew-symmetric when restricted to $F_x\times F_x$ for all $x\in M$.
Denote $m=\rank E-\rank F$, $a=c(E^*-F)$.

\th Corollary {\rm(cf.~\cite{LP2})}. Let $F\to E^*$ be a
skew-symmetric map of vector bundles $F\subset E$. 
If $F\ne E$ \(i.e.\ $m>0$\) then the
Poincar\'e dual to the locus $\O_r\subset M$ of points $x\in M$ such
that $\dim\ker f_x\ge r$ is given by
 $$[\O_r]=\frac1{2^r}\(\s_{m+r-1,\dots,m}(a,\dots,a)+ \strut
 (-1)^rc_m(E/F)\,\s_{m+r-1,\dots,m+1}(a,\dots,a)\).$$
If $F=E$ then generically
 $$[\O_r]=\frac1{2^{r-1}}\s_{r-1,\dots,1}(a,\dots,a).$$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{BGG}

\bibitem[A]{A} \spa{V.I.~Arnold}, {Complex Lagrange Grassmannian},
preprint Paris-IX (2000).

\bibitem[AG]{AG} \spa{V.I.~Arnold,~A.B.~Givental}, {\em Symplectic
Geometry}, Dynamical Systems. IV. Symplectic geometry and its
applications, Encycl. Math. Sci. 4, 1-136 (1990);

\bibitem[Au]{Au} \spa{M.~Audin}, {\em Classes caract\'eristique d'immersions
lagrangiennes d\'efinies par des vari\'et\'es de caustiques (d'apr\'es
V.A. Vassiliev)}, S\'eminare Sud-Rhodanien de G\'eom\'etry, travaux en
cours, {\bf1}, Paris: Hermann, 1984.

\bibitem[BGG]{BGG} \spa{I.N.~Bernstein,~I.M.~Gelfand,~S.I.~Gelfand},
{\em  Schubert cells, and the cohomology of the spaces $G/P$}. (Russian) 
Uspehi Mat. Nauk {\bf 28} (1973), no. 3(171), 3--26.

\bibitem[D]{D} \spa{M.~Demazure}, {\em D\'esingularisation des
vari\'et\'es de Schubert generalis\'ees},
Ann. Sci. \'Ecole Norm. Sup.  {\bf 7} (1974), 53--88.

\bibitem[F1]{Fu1} \spa{W.~Fulton}, {\em Flags, Schubert polynomials, 
degeneracy loci, and determinantal formulas}, Duke Math. J. {\bf65} 
(1992), 381--420.


\bibitem[F2]{Fu} \spa{W.~Fulton}, {\em Determinantl formulas for
orthogonal and symplectic degeneracy loci,} J. Diff. Geom., {\bf43}
(1996), 276--290. 

\bibitem[FP]{FP} \spa{W.~Fulton,~P.~Pragacz}, {\em Schubert Varieties and
Degeneracy loci}, Lect. Notes in Math., Vol. {\bf1689} (1998). 

\bibitem[Fuks]{F} \spa{D.B.~Fuks}, {\em Maslov-Arnold Characteristic
classes} (Russian), Soviet. Math. Docl. {\bf9} (1968), 96--99. 

\bibitem[JLP]{JLP} \spa{T.~J{\'o}zefiak,~A.~Lascoux,~P.~Pragacz}, {\em
Classes of determinantal varieties associated with symmetric and
antisymmetric matricies} (Russian), Izvestija AN SSSR, {\bf45}
(1981), 662--673.

\bibitem[HT]{HT} \spa{J.~Harris, L.~W.~Tu} {\em On symmetric and
skew-symmetric determinantal varieties,} Topology, {\bf23} (1984),
71--84.

\bibitem[K1]{K1} \spa{M.~Kazarian}, {\em Characteristic classes of
Singularity theory}, in: Arnold-Gelfand Mathematical Seminars,
Birkh\"auser, 1977, pp. 325--340.

\bibitem[K2]{K2} \spa{M.~Kazarian}, {\em Characteristic classes of
Lagrange and Legendre singularities}
(Russian), Uspekhi Mat. Nauk {\bf50} (304), 1995, 45--70.

\bibitem[K3]{K3} \spa{M.~Kazarian}, {\em Thom polynomials for
Lagrange, Legendre,  and critical point function singularities},
preprint (2000)

\bibitem[Kn]{Kn} \spa{D.~Knuth}, {\em Overlapping Pfaffians}, Electronic
J. Combinatorics, {\bf 3}, (1996), 151-163.


\bibitem[LP1]{LP1} \spa{A.~Lascoux,~P.~Pragacz}, {\em Operator calculus
for $\widetilde Q$-polynomials and Schubert polynomials}. Adv. Math.
140 (1998), no. 1, 1--43.


\bibitem[LP2]{LP2} \spa{A.~Lascoux,~P.~Pragacz}, {\em Schur $Q$-functions
and degeneracy locus formulas for morphisms with symmetries},
to appear in the Proceedings of Intersection Theory Conference in Bologna, 
"Progress in Mathematics", Birkhauser.


\bibitem[MSS]{MSS}
\spa{A.S.~Mishchenko,~B.Yu.~Sternin,~V.E.~Shatalov,} {\em Geometry of
Lagrangian manifolds and the canonical Maslov operator  in complex
phase space}, J. Sov. Math. {\bf 13}, 1-23 (1980).

\bibitem[MN]{MN} \spa{J.-M.~Morvan,~L.~Niglio}, {\em Classes
caract\'eristiques des couples de susfibr\'es Lagrangiens}, Ann.
Inst. Fourier, Grenoble, {\bf36} (1986), 193-209.

\bibitem[P1]{P0} \spa{P.~Pragacz}, {\em Cycles of isotropic subspaces 
and formulas for symmetric degeneracy loci}, Topics in algebra, Part
2 (Warsaw, 1988), 189--199, Banach Center  Publ., 26, Part 2, 

\bibitem[P2]{P1} \spa{P.~Pragacz}, {\em Algebro-geometric applications of
Schur $S$- and $Q$-polynomials}, Springer Lect. Notes in Math., Vol.
{\bf1478} (1991), 130--191. 

\bibitem[P3]{P2} \spa{P.~Pragacz}, {\em Symmetric polynomials and divided
differences in formulas of intersection theory}, Parameter spaces (P.
Pragacz ed.), Vol. {\bf36}, Banach Center Publications, 1996,
125--177.

\bibitem[PR]{PR} \spa{P.~Pragacz,~J.~Ratajski}, {\em Formulas for
Lagrangian and orthogonal degeneracy loci; $\widetilde Q$-polynomial
approach}, Composito Math., {\bf107} (1997), 11--87.

\bibitem[V]{V1} \spa{V.A.~Vassiliev}, {\em Lagrange and Legendre
Characteristic classes}, 2nd edition, Gordon and Breach, 1993.
(Extended Russian translation: MCCME, 2000.)

\end{thebibliography}

\end{document}





