## The Lefschetz Theory: local theory

(4.1) On $ℂ$ the local Lefschetz results are the following. Let $D=\left\{z\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}|z|<1\right\}$ 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 $\mathrm{sp}:{H}^{i}\left({X}_{0},ℤ\right)\to {H}^{i}\left({X}_{t},ℤ\right)$: ${X}_{0}$ is a deformation retract of $X$, and $\mathrm{sp}$ is the composite arrow $Hi(X0,ℤ) ⟵∼ Hi(X,ℤ) ⟶ Hi(Xt,ℤ) ;$
2. the monodromy transformations $T:{H}^{i}\left({X}_{t},ℤ\right)\to {H}^{i}\left({X}_{t},ℤ\right)$, which describe the effect of the singular cycles of ${X}_{t}$ as "$t$ turns around $0$". This is also the action on ${H}^{i}\left({X}_{t},ℤ\right)$, the fibre in $t$ of the local system ${R}^{i}{f}_{*}\underset{_}{ℤ}|{D}^{*}$, of a positive generator of ${\pi }_{1}\left({D}^{*},t\right)$.

The Lefschetz theory writes α) and β) in terms of the vanishing cycle $\delta \in {H}^{n}\left({X}_{t},ℤ\right)$. This cycle is well defined up to sign as follows. For $i\ne n,n+1$, one has $Hi(X0,ℤ) ⟶∼ Hi(Xt,ℤ) (i≠n,n+1).$ For $i=n,n+1$, one has an exact sequence $0⟶ Hn(X0,ℤ) ⟶ Hn(Xt,ℤ) ⟶x↦ (x,δ) ℤ⟶ Hn+1 (X0,ℤ) ⟶ Hn+1 (Xt,ℤ) ⟶ 0 .$ For $i\ne n$ the monodromy $T$ is the identity. For $i=n$, one has $Tx=x±(x,δ) δ.$ The values of the $±$, of $T\delta$, and of $\left(\delta ,\delta \right)$ are the following: $nmod4 0 1 2 3 Tx=x± (x,δ)δ SPACE - - + + (δ,δ) 2 0 -2 0 Tδ -δ δ -δ δ$

The monodromy transformation $T$ respects the intersection form $\mathrm{Tr}\left(x\cup y\right)$ on ${H}^{n}\left({X}_{t},ℤ\right)$. 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 $\stackrel{‾}{\eta }$ (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 ${X}_{\stackrel{‾}{\eta }}$, we construct again a specialisation morphism

 $\mathrm{sp}:{H}^{i}\left({X}_{s},{ℚ}_{\ell }\right)\stackrel{\sim }{⟵}{H}^{i}\left(X,{ℚ}_{\ell }\right)⟶{H}^{i}\left({X}_{\stackrel{‾}{\eta }},{ℚ}_{\ell }\right).$ (4.2.1)
The role of $T$ is played by the action of the intertia group $I=\mathrm{Gal}\left(\stackrel{‾}{\eta }/\eta \right)$, acting on ${H}^{i}\left({X}_{\stackrel{‾}{\eta }},{ℚ}_{\ell }\right)$ by transport of structure (cf. (1.15)):
 $I=\mathrm{Gal}\left(\stackrel{‾}{\eta }/\eta \right)⟶\mathrm{GL}\left({H}^{i}\left({X}_{\stackrel{‾}{\eta }},{ℚ}_{\ell }\right)\right).$ (4.2.2)

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

 $\delta \in {H}^{n}\left({X}_{\stackrel{‾}{\eta }},{ℚ}_{\ell }\right)\left(m\right).$ (4.3.1)
This cycle is well defined up to sign as follows.

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

 ${H}^{i}\left({X}_{s},{ℚ}_{\ell }\right)\stackrel{\sim }{⟶}{H}^{i}\left({X}_{\stackrel{‾}{\eta }},{ℚ}_{\ell }\right)\phantom{\rule{2em}{0ex}}\left(i\ne n,n+1\right).$ (4.3.1)
For $i=n,n+1$, we have an exact sequence
 $0⟶{H}^{n}\left({X}_{s},{ℚ}_{\ell }\right)⟶{H}^{n}\left({X}_{\stackrel{‾}{\eta }},{ℚ}_{\ell }\right)\stackrel{x↦\mathrm{Tr}\left(x\cup \delta \right)}{⟶}{ℚ}_{\ell }\left(m-n\right)⟶{H}^{n+1}\left({X}_{s},{ℚ}_{\ell }\right)⟶{H}^{n+1}\left({X}_{\stackrel{‾}{\eta }},{ℚ}_{\ell }\right)⟶0.$ (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 $tℓ:I→ ℤℓ(1),$ and the action of $\sigma \in I$ is $x⟼x±tℓ (σ)(x,δ) δ.$

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 $ε:I→{±1} ,$ for which we have $σx=x, ifε(σ) =1, σx=x ±(x,δ)δ, ifε(σ) =-1.$

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

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