Abstracts
Numerically reﬂective involutions of Enriques surfaces
Shigeru Mukai
A (holomorphic) automorphism of an Enriques surface S is numerically reﬂective(resp. numerically trivial) if it acts on the Q-cohomology group H 2 (S, Q)(≅ Q10 )by reﬂection (resp. trivially). For K3 surfaces we have• a numerically trivial automorphism is trivial, and• no automorphisms are numerically reﬂective.
But these are no more true for Enriques surfaces. In my talk I summarized theclassiﬁcation and gave a very rough sketch of the proof. The details will be published elsewhere.
1. Numerically trivial involutions
Let XBP be the minimal model of the function ﬁeld√()11(1)
C x, y, a(x + ) + b(y + ) + 2cx
yof two variables, where a, b ∈ C× and c ∈ C are constants. XBP is the mimimal resolution of the double P1 × P1 with branch the union of the coordinate
quadrilateral and the curveΓ0 : a(x2 + 1)y + bx(y 2 + 1) + 2cxy = 0(2)of bidegree (2, 2).

y=0
Γ0
y=∞

x=∞

x=0

√
Assume further that a ± b ± c ̸= 0. Then the involution ε : (x, y, ) 7→√(1/x, 1/y, − ) has no ﬁxed points on XBP . Hence the quotient SBP = XBP /ε isan Enriques surface. Let σBP be the involution of SBP induced from the covering√√
7→ −of XBP . Then σBP is homologically trivial, that is, its acts
involution
on the Z-homology group H2 (SBP , Z) trivially ([1, (4.8)], [4, Exmaple 2]).
Theorem 1 Every homologically trivial automorphism of an Enriques surface
is either trivial or the above involution σBP y SBP .
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Oberwolfach Report 44/2009

Theorem 2 ([2]) Let σ be a numerically trivial involution of an Enriquessurface, and assume that σ is neither trivial nor σBP . Then the universal coveris a Kummer surface Km(E1 × E2 ) of product type and σ is either of Liberman
type ([4, Exmaple 1]) or Kondo-Mukai type ([4, Exmaple 2], see also §2).
2. Numerically reflective involutions
Let XGBP be the minimal model of the ﬁeld√()
1
1x y(3)C x, y, a(x + ) + b(y + ) + c( + ) + 2d ,xyyx
where a, b, c ∈ C× and d ∈ C are constants. XGBP is the mimimal resolution ofthe double P2 with branch the union of the coordinate triangle and the cubic curve(4)Γ1 : a(x2 + 1)y + bx(y 2 + 1) + c(x2 + y 2 ) + 2dxyz = 0.

Assume further that(a + b + c + d)(a + b − c − d)(a − b + c − d)(a − b − c + d) ̸= 0.
√√Then the involution ε : (x, y, ) 7→ (1/x, 1/y, − ) has no ﬁxed points and we
obtain the Enriques quotient SGBP := XGBP /ε. The involution σGBP of SGBP
induced from the covering involution is numerically reﬂective if (4) is irreducible
and numerically trivial otherwise. By [2, Remark 9], σGBP is equivalent to [2,Example 2] in the latter case.
Let C be a curve of genus 2 and G a Göpel subgroup of the 2-torsion group ofits Jacobian J(C). For a non-bielliptic pair (C, G), we constructed an Enriquessurface Km(C)/εG and a numerically reﬂective involution σG y Km(C)/εG in[3], where Km(C) is the Kummer surface of J(C).
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Theorem 3 Let σ be a numerically reﬂective involution of an Enriques surface
S. Then either(1) σ is isomorphic to the involution σGBP , or(2) the universal cover of S is isomorphic to the Jacobian Kummer surface
Km(C) and (S, σ) is isomorphic to (Km(C)/εG , σG ) for a curve C ofgenus 2 and a Göpel subgroup G.
References[1] Barth, W. and Peters, C.: Automorphism of Enriques surfaces, Invent. math., 73(1983),
383–411.
[2] Mukai, S.: Numerically trivial involutions of Enriques surfaces, RIMS preprint #1544, May
2006.
[3] Mukai, S.:
Kummer’s quartics and numerically reﬂective involutions of Enriques surfaces, RIMS preprint #1633, June 2008 (http://www.kurims.kyotou.ac.jp/preprint/ﬁle/RIMS1633.pdf).
[4] Mukai, S. and Namikawa, Y.: Automorphisms of Enriques surfaces which act trivially on
the cohomology groups, Invent. math., 77(1984), 383–397.

Reporter: Fabio Perroni