Pin^- structures関連論文

R. Benedetti, R. Silhol,
Spin and Pin^- structures, immersed and embedded surfaces and a result of Segre on real cubic surfaces,
Topology 34--3 (1995), 651--678.

pin and spin structures

Pin structures and spin structures on surfaces

For a closed orientable (resp. non-orientable) surface,

we can always give finite numbers of spin (resp. pin^±) structures.
There are some algebraic or geometric counterparts for them.

spin geometry

（この記事は単なるメモです．　後日，訂正などをいたします）

4次元向き付け可能閉多様体 X は，第2スティーフェルホイットニー類 w_2(X)=0のとき，スピン構造が入る．

概複素多様体 X に対しては，w_2(X)はc_1(X)のmod2 reduction

H. Blaine Lawsson, Jr. and Marie-Louise Michelsohn

"Spin Geomtry", Princeton University Press, 1989.

スピン幾何入門 　 　（資料ありがとうございます）

スピン幾何入門１

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閉曲面 S にはいつもスピン構造がはいり，

閉曲面上のスピン構造全体と，

H_1(S, Z_2)上の２次形式の全体は，１対１に対応する・・・

そこで，例えば，４次元球面の中の曲面に対して，Rokhlin's quatratic form (Rokhlin形式)という標準的な２次形式を考える．そして，それに対応するスピン構造というのがある．

hyperkaehler構造に関連する文献

A.L. Besse,
Einstein Manifolds,
Ergeb. Math. Grenzgeb. (3), {\bf 10}, Springer-Verlag, 1987.

S.K. Donaldson,
Yang--Mills invariants of smooth four-manifolds, (Donaldson's trick)
Geometry of low-dimensional manifolds (S.K. Donaldson, C.B. Thomas, eds.), Vol. 1.
Cambridge University Press, Cambridge (1990), 5--40.

A. Degtyarev, I. Itenberg, V. Kharlamov,
Real Enriques Surfaces,
Lect. Notes in Math. 1746 (2000).

Daniel Huybrechts,
Moduli spaces of hyperk\"ahler manifolds and mirror symmetry,
math.AG/0210219.

Alex Degtyarev, Ilia Itenberg, and Viatcheslav Kharlamov,

Finiteness and quasi-simplicity for symmetric K3-surfaces,

Duke Math. J. 122, Number 1 (2004), 1-49.

Degtyarevの論文

Degtyarevの論文：

A.I. Degtyarev,
Classification of surfaces of degree four having a nonsimple singular point,
Math. USSR-Izv., {\bf 35--3} (1990), 607--627.

A.I. Degtyarev,
Isotopic classification of complex plane projective curves of degree 5,
Leningrad Math. J., {\bf 1} (1990), ???

A.I. Degtyarev,
Stiefel orientations of a real algebraic variety,
Real Algebraic Geometry, Proceedings, Rennes 1991,
Lecture Notes in Math., {\bf 1524} (1992), 205--220.

A.I. Degtyarev,
Classification of quartics having a nonsimple singular point. II,
Advances in Soviet Math., {\bf 18} (1994), 23--54.

A.I. Degtyarev,
Vanishing of characteristic classes of a real algebraic variety,
Real Algebraic Geometry and Topology, Proceedings, Michigan 1993,
Contemp. Math., {\bf 182} (1995), 1--9.

A.I. Degtyarev,
Quadratic transformations {$\R P^2 \rightarrow \R P^2$}, }
Topology of Real Algebraic Varieties and Related Topics,
Advances in the Mathematical Sciences 29,
A. M. S. Translations Series 2, {\bf 173} (1996), 61--71.

A. Degtyarev, V. Kharlamov,
Topological classification of real Enriques surface,
Topology, {\bf 35--3} (1996), 711--729. %%この仕事の方が{DK96b}より先

A. Degtyarev, V. Kharlamov,
Halves of a real Enriques surface,
Comment. Math. Helv., {\bf 71--4} (1996),
628--663.

A. Degtyarev, V. Kharlamov,
Around real Enriques surfaces,
in Real algebraic and analytic geometry (Segovia, 1995),
Rev.-Mat.-Univ.-Complut.-Madrid {\bf 10} (1997), Special Issue, suppl., pp. 93--109.

Nikulinの格子の理論

●Nikulinによる格子（整数上対称双１次形式）の理論に関する文献：

I. Dolgachev and V. Nikulin

Exceptional singularities of V.I. .Arnold and K3-surfaces,

Proc.Minsk Topology Conf. Minsk. 1977 ( abstract ) .

V. Nikulin

Finite groups of automorphisms of Kahlerian surfaces of type K3 ,

Proc.Moscow Math.Soc., 38(1979),75-137.

V.V. Nikulin,
Integral symmetric bilinear forms and some of their applications,
Izv. Akad. Nauk SSSR Ser Mat., 43-1 (1979), 111--177.
= Math. USSR Izv., 14-1 (1980), 103--167.

keywords:

non-degenerate lattice,

unimodular, even lattice, genus,

primitive embedding,

finite quadratic form,

discriminant group,

discriminant form,

2-elementary lattice,

hyperbolic (=Lorenzian)

V. Nikulin

On arithmetic groups generated by reflections in Lobachevsky spaces, 44 (1980), 637-668 .

V. Nikulin

On the quotient groups of automorphism groups of hyperbolic forms by the subgroups generated by 2-reflections.

Algebraic-geometrical applications. Modern Problems in Math., vol.18. Moscow. 1981, 1-114

(to be translated in J.Sov.Math.).

V. Nikulin

On classification of arithmetic groups generated by reflections in Lobachevsky spaces,

Izv.Akad.Nauk SSSR, Ser. Math., 45 (1981),113-142 .

Dolgachev, Igor
Integral quadratic forms :

applications to algebraic geometry.

Séminaire Bourbaki, 25 (1982-1983 ), Exposé No. 611, 28 p.

V.V. Nikulin,
Involutions of integral quadratic forms and their applications to real algebraic geometry,
Izv.-Akad.-Nauk-SSSR-Ser.-Mat. 47-1 (1983), 109--188.
English transl., Math.USSR Izv., 22 (1984), 99--172.

V.V. Nikulin,
Filterings of 2-elementary forms and involutions of integral bilinear symmetric
and skew-symmetric forms,

Izv.-Akad.-Nauk-SSSR-Ser.-Mat., 49-4 (1985), 847--873.

対称双１次形式に関する文献

向き付けられた4n次元 閉(=compact, 境界なし)多様体 X の

整係数2n次元コホモロジー/torsion 上の

cup product form は，

有限生成 Z-free module上の整数値対称双１次形式である．

これは，Poincare duality により，unimodularである．

有限生成 Z-free module上の整数値対称双１次形式については，

M. Eichler

Quadratische Formen und Orthogonalen Gruppen . Berlin. 1952.

和訳：　「2次形式と直交群」　(シュプリンガー東京)

J.-P. Serre,
A Course in Arithmetic,
Springer-Verlag, 1970.

J. Milnor, D. Husemoller,
Symmetric Bilinear Forms,

Springer-Verlag, 1973.

O'meara著　「・・・・・」

●コンパクト（複素）偶数次元 概複素多様体上の反正則対合は向きを保つので，

（上記の）cup product form を保つ．

つまり，formを保つhomomorphism, 特に，同型写像である．

(formを保つ同型は，isometryとも呼ばれる)

（編集中）

Huisman の論文

\bibitem{Huisman92a}
J. Huisman,
{\it The underlying real algebraic structure of complex elliptic curves,}
Math. Ann. {\bf 294} (1992), 19--35.

\bibitem{Huisman92}
J. Huisman,
{\it
Real abelian varieties with complex multiplication,}
Ph.D. thesis, Vrije Universiteit, Amsterdam, 1992.

\bibitem{Huisman99}
J. Huisman,
{\it
Real quotient singularities and nonsingular real algebraic curves
in the boundary of moduli space, }
Compositio Mathematica, {\bf 118} (1999), 43--60.

Krasnov の論文

V.A. Krasnov,
{\it Harnack-Thom inequalities for mappings of real algebraic varieties,}
Izv. Akad. Nauk SSSR Ser. math. {\bf 47} (1983), 268--297,
English transl. Math. USSR Izv. {\bf 22} (1984), 247--275.

\bibitem{Krasnov91}
V.A. Krasnov,
{\it Characteristic classes of vector bundles on a real algebraic variety,}
Izv. Akad. Nauk SSSR Ser. math. {\bf 55} (1991), 716--746,
English transl. Math. USSR Izv. {\bf 39} (1992), 703--730.

\bibitem{Krasnov93}
V.A. Krasnov,
{\it
Algebraic cycles on a real algebraic $GM$-manifold,}
Izv. Ross. Akad. Nauk Ser. Mat. {\bf 57} (1993), 153--173.
English transl. Russian Acad. Sci. Izv. Math. {\bf 43} (1994), 141--160.

\bibitem{Krasnov94}
V.A. Krasnov,
{\it
On equivariant Grothendiek cohomology of a real algebraic variety,
and its applications,}
Izv. Ross. Akad. Nauk Ser. Mat. {\bf 58} (1994), 36--52,
English transl. Russian Acad. Sci. Izv. Math. {\bf 44} (1995), 461--477.

\bibitem{Krasnov98}
V.A. Krasnov,
{\it
Real algebraic $GM{\Z}$-surfaces, }
Izv. Ross. Akad. Nauk Ser. Mat. {\bf 62--4} (1998), 51--80.
English transl., Izvestiya Mathematics {\bf 62--4} (1998), 695--721.

J. van Hamel の論文

J. van Hamel,
Real algebraic cycles on complex projective varieties,
Math. Z. 225 (1997), 177--198.

J. van Hamel,
Algebraic cycles and topology of real algebraic varieties,
Ph.D. thesis, Vrije Universiteit, Amsterdam, 1997.