Deligne I Section 4

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 11 November 2011

The Lefschetz Theory: local theory

(4.1) On $ℂ$ the local Lefschetz results are the following. Let $D=\left\{z\right|\left|z\right|<1\}$ the unit disc, $D^{*}=D-\left\{0\right\}$ and $f:X\to D$ a morphism of analytic spaces. Suppose that

1. $X$ is nonsingular, and of pure dimension $n+1$;
2. $f$ is proper;
3. $f$ is smooth outside of a point $x$ of the special fiber $X_0=f^{-1}\left(0\right)$;
4. in $x$, $f$ presents a nondegenerate quadratic point.

Let $t\ne 0$ in $D$ and $X_t=f^{-1}\left(t\right)$; "the" general fiber. To the above data associate:

1. the specialiation morphisms $\mathrm{sp}:H^i(X_0,\mathbb{Z})\to H^i(X_t,\mathbb{Z})$: $X_0$ is a deformation retract of $X$, and $\mathrm{sp}$ is the composite arrow $$H^i(X_0,\mathbb{Z})\stackrel{\sim }{⟵}{H}^i(X,\mathbb{Z})⟶H^i(X_t,\mathbb{Z});$$
2. the monodromy transformations $T:H^i(X_t,\mathbb{Z})\to H^i(X_t,\mathbb{Z})$, which describe the effect of the singular cycles of $X_t$ as "$t$ turns around $0$". This is also the action on $H^i(X_t,\mathbb{Z})$, the fibre in $t$ of the local system $R^i{f}_{*}\underset{\_}{\mathbb{Z}}|D^{*}$, of a positive generator of ${\pi }_1(D^{*},t)$.

The Lefschetz theory writes α) and β) in terms of the vanishing cycle $\delta \in H^n(X_t,\mathbb{Z})$. This cycle is well defined up to sign as follows. For $i\ne n,n+1$, one has $$H^i(X_0,\mathbb{Z})\stackrel{\sim }{⟶}{H}^i(X_t,\mathbb{Z})(i\ne n,n+1).$$ For $i=n,n+1$, one has an exact sequence $$0⟶H^n(X_0,\mathbb{Z})⟶H^n(X_t,\mathbb{Z})\stackrel{x\mapsto (x,\delta )}{⟶}\mathbb{Z}⟶H^{n+1}(X_0,\mathbb{Z})⟶H^{n+1}(X_t,\mathbb{Z})⟶0.$$ For $i\ne n$ the monodromy $T$ is the identity. For $i=n$, one has $$Tx=x\pm (x,\delta )\delta .$$ The values of the $\pm$, of $T\delta$, and of $(\delta ,\delta )$ are the following: $$\begin{array}{lcccc}nmod4& 0& 1& 2& 3\\ Tx=x\pm (x,\delta )\delta \phantom{\mathrm{SPACE}}& -& -& +& +\\ (\delta ,\delta )& 2& 0& -2& 0\\ T\delta & -\delta & \delta & -\delta & \delta \end{array}$$

The monodromy transformation $T$ respects the intersection form $\mathrm{Tr}(x\cup y)$ on $H^n(X_t,\mathbb{Z})$. For $n$ odd, this is a symplectic transvection. For $n$ even, this is an orthogonal symmetry.

(4.2) Here is the analogue of (4.1) in abstract algebraic geometry. The disc $D$ is replaced by the spectrum of a discrete henselien valuation ring $A$ with algebraically closed residue field. Let $S$ be this spectrum, $\eta$ its generic point (spectrum of the field of fractions of $A$), $s$ its closed point (spectrum of the residue field). The role of $t$ is played by a generic geometric point $\bar{\eta }$ (spectrum of the algebraic closure of the field of fractions of $A$).

Let $F:X\to S$ be a proper morphism, with $X$ purely regular of dimension $n+1$. We assume that $f$ is smooth, except at an ordinary quadratic point $x$ in the special fiber $X_s$. Let $\ell$ be a pime number different from the characteristic $p$ of the residue field of $S$. Denoting the generic geometric fiber by $X_{\bar{\eta }}$, we construct again a specialisation morphism

$\mathrm{sp}:H^i(X_s,{\mathbb{Q}}_{\ell })\stackrel{\sim }{⟵}{H}^i(X,{\mathbb{Q}}_{\ell })⟶H^i(X_{\bar{\eta }},{\mathbb{Q}}_{\ell }).$

(4.2.1)

The role of $T$ is played by the action of the intertia group $I=\mathrm{Gal}\left(\bar{\eta }/\eta \right)$, acting on $H^i(X_{\bar{\eta }},{\mathbb{Q}}_{\ell })$ by transport of structure (cf. (1.15)):

$I=\mathrm{Gal}\left(\bar{\eta }/\eta \right)⟶\mathrm{GL}\left(H^i\right(X_{\bar{\eta }},{\mathbb{Q}}_{\ell }\left)\right).$

(4.2.2)

The maps in (4.2.1) and (4.2.2) completly describe the sheaves $R^i{f}_{*}{\mathbb{Q}}_{\ell }$ on $S$.

(4.3)We put $n=2m$ for $n$ even, and $n=2m+1$ for $n$ odd. (4.2.1) and (4.2.2) can again be described in terms of a vanishing cycle

$\delta \in H^n(X_{\bar{\eta }},{\mathbb{Q}}_{\ell })\left(m\right).$

(4.3.1)

This cycle is well defined up to sign as follows.

For $i\ne n,n+1$ we have

$H^i(X_s,{\mathbb{Q}}_{\ell })\stackrel{\sim }{⟶}{H}^i(X_{\bar{\eta }},{\mathbb{Q}}_{\ell })(i\ne n,n+1).$

(4.3.1)

For $i=n,n+1$, we have an exact sequence

$0⟶H^n(X_s,{\mathbb{Q}}_{\ell })⟶H^n(X_{\bar{\eta }},{\mathbb{Q}}_{\ell })\stackrel{x\mapsto \mathrm{Tr}(x\cup \delta )}{⟶}{\mathbb{Q}}_{\ell }(m-n)⟶H^{n+1}(X_s,{\mathbb{Q}}_{\ell })⟶H^{n+1}(X_{\bar{\eta }},{\mathbb{Q}}_{\ell })⟶0.$

(4.3.3)

The action (4.2.2) of $I$ (the local monodromy) is trivial if $i\ne n$. For $i=n$, it is described as follows.

A) $n$ odd. -- We use the canonical homomorphism $$t_{\ell }:I\to {\mathbb{Z}}_{\ell }\left(1\right),$$ and the action of $\sigma \in I$ is $$x⟼x\pm t_{\ell }\left(\sigma \right)(x,\delta )\delta .$$

B) $n$ even. -- We will not use this case. We will note only that, if $p\ne 2$, there exists a unique character of order two $$\epsilon :I\to \{\pm 1\},$$ for which we have $$\begin{array}{ll}\sigma x=x,& \text{if}\epsilon \left(\sigma \right)=1,\\ \sigma x=x\pm (x,\delta )\delta ,& \text{if}\epsilon \left(\sigma \right)=-1.\end{array}$$

(4.4) These results carry the following information on the $R^i{f}_{*}{\mathbb{Q}}_{\ell }$.

1. If $\delta \ne 0$:
1. For $i\ne n$, the sheaf $R^i{f}_{*}{\mathbb{Q}}_{\ell }$ is constant.
2. Let $j$ be the inclusion of $\eta$ in $S$. We have $$R^n{f}_{*}{\mathbb{Q}}_{\ell }=j_{*}{j}^{*}{R}^n{f}_{*}{\mathbb{Q}}_{\ell }.$$
2. If $\delta =0$: (This is an exceptional case. Since $(\delta ,\delta )=\pm 2$ for $n$ even, it cannot occur except when $n$ is odd.)
1. For $i\ne n,n+1$, the sheaf $R^i{f}_{*}{\mathbb{Q}}_{\ell }$ is constant.
2. Let ${{\mathbb{Q}}_{\ell }(m-n)}_s$ be the sheaf ${\mathbb{Q}}_{\ell }(m-n)$ on $\left\{s\right\}$, extended by zero on $S$. We have an exact sequence $$0⟶{{\mathbb{Q}}_{\ell }(m-n)}_s⟶R^{n+1}{f}_{*}{\mathbb{Q}}_{\ell }⟶j_{*}{j}^{*}{R}^{n+1}{f}_{*}{\mathbb{Q}}_{\ell }⟶0,$$ where $j_{*}{j}^{*}{R}^{n+1}{f}_{*}{\mathbb{Q}}_{\ell }$ is a constant sheaf.

Notes and References

This is an attempt to translate Section 4 of [DeI].

References

[DeI] P. Deligne, La Conjecture de Weil I, Publ. Math. IHÉS (1974) 273-307. MR0340258.

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