On the heat operators of normal singular algebraic surfaces

Nagase, Masayoshi

doi:10.4310/jdg/1214442159

Cited by 16 publications

(15 citation statements)

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“…Then we have Lemma 4.3. On U-n~\p) 9 (1) dV g <dV s , In this section, we will prove Theorem 1 for / < 1 and then discuss a little bit the so-called (L 2 -, usual, logarithmic) irregularities. Since the case i=Q is trivial (see §3), we will treat the case i=l in the following.…”

Section: Afa(%)e*hi 2} (?£)mentioning

confidence: 99%

“…we can find a sequence ^6=^(3?) satisfying ]imi/rj=T/r 9 Hmd^j=d^r and Iim5^y=^ in the L 2 -sense. This is a quite nice property.…”

Section: Afa(%)e*hi 2} (?£)mentioning

confidence: 99%

“…And, because of (2.6) for i=Q and Lemma 2.1(1) for f=0, it suffices to prove the first two equalities at (1) In order to do so near S 9 we will make a very good resolution W N /W Q ) around the point. Then, for any point x^n~\p) 9 there exists a local coordinate neighborhood (17, (u 9 v)) around x and a permutation cr such that the n can be expressed on £7 as follows;…”

Section: Afa(%)e*hi 2} (?£)mentioning

confidence: 99%

“…On the neighborhood (17, (u 9 v)) around the point x&.n~\p) 9 we set IP -1.1. r,-i,i. ( Next, let us investigate the relationship between the square-integrabilities on 3C(g)=(3£, g) and 3E(ds*)=(3e, ds 2 ).…”

Section: Afa(%)e*hi 2} (?£)mentioning

confidence: 99%

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Pure Hodge Structure of the Harmonic $L^2$-Forms on Singular Algebraic Surfaces

Nagase¹

1988

Publ. Res. Inst. Math. Sci.

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. IntroductionLet X be an algebraic surface (over C) embedded in the projective space P N (C) and S be its singularity set. If S is empty, then the harmonic spaces M l (X) = {smooth /-form a on X\da = da=0} (or, equivalently, its de Rham cohomology groups) have the pure Hodge structure;rally changed into the Dolbeault-type harmonic space M P^q (X) in this case, but in the case we are going to discuss in this paper, that is, in the case where S is not empty, such a change has a subtle problem ( § 3) and it seems to be one of the key points not to try to do so.) Also there exists the hard Lefschetz structure compatible with (1.1); The author believes that the case i=2 omitted in Theorem 1(2) and in the above remark must hold.Conjecture A. Theorem 1(2) and the hard Lefschetz decomposition of the L 2 -cohomology groups hold also in the case i=p+q=2.Obviously this conjecture is equivalent to a part of it, i.e., M Moreover it suffices to prove a little bit stronger assertion; Ker ^= which is equivalent to Ker rf 2 =Ker rf^2. Thus Conjecture A can be deduced from the conjecture "d^i=d i for i=l 9 2" announced in [10].Acknowledgement. The author would like to thank Professors J. Noguchi, M. Oka and Y. Miyaoka for offering him valuable suggestions during the preparation of this paper. Also he would like to thank the referee for valuable comments.

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Section: Afa(%)e*hi 2} (?£)mentioning

confidence: 99%

“…we can find a sequence ^6=^(3?) satisfying ]imi/rj=T/r 9 Hmd^j=d^r and Iim5^y=^ in the L 2 -sense. This is a quite nice property.…”

Section: Afa(%)e*hi 2} (?£)mentioning

confidence: 99%

Section: Afa(%)e*hi 2} (?£)mentioning

confidence: 99%

Section: Afa(%)e*hi 2} (?£)mentioning

confidence: 99%

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Pure Hodge Structure of the Harmonic $L^2$-Forms on Singular Algebraic Surfaces

Nagase¹

1988

Publ. Res. Inst. Math. Sci.

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“…Constant functions are in the domain of minimal exterior differentiation on X-E [9], and thus they are in domain (dc). Nothing else is in kernel (dc).…”

Section: Proposition 22 {A°/(x)0} -» {A°c'*(x)dc} Induces H0ß(x) mentioning

confidence: 99%

𝐿²-Dolbeault complexes on singular curves and surfaces

Haskell¹

1989

Proc. Amer. Math. Soc.

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Abstract.Give the smooth part of a singular curve or normal surface the metric induced from the ambient projective space. On this incomplete manifold the minimal L2 ô-complex of (0, ?)-forms has finite-dimensional cohomology groups. The Euler characteristic of this cohomology equals the Todd genus of any desingularization of the singular variety.R. MacPherson ([4], [8]) has conjectured that the smooth part of a singular algebraic variety (when given one of certain natural metrics) carries an L dcomplex of (0, a)-forms whose Euler characteristic is equal to the Todd genus of any desingularization of the variety. Much progress on this conjecture has been made using complete metrics on the smooth parts of the varieties. The present paper focuses on the incomplete metrics induced from the ambient projective spaces. W. Pardon [ 10] has written an important paper showing that when X is a curve or normal surface whose smooth part is given the metric induced from the ambient projective space, then the complex of smooth (0, f?)-forms 2 -2 w satisfying w e L and dw e L has an Euler characteristic that is not necessarily equal to the Todd genus of a desingularization of X. However, an earlier paper [6] exhibited a d -complex on the smooth part of a curve X (with metric induced from the ambient projective space) that has the Euler characteristic conjectured by MacPherson. In the search for the complex whose existence is conjectured by MacPherson, one must choose carefully a closed extension of the d -operator. Let A' be a complex projective curve or normal surface, and let U be the set of smooth points of X. Give U the metric induced from the ambient projective space. The present paper shows that the minimal d-complex (defined in Definition 1.5) of (0,#)-forms on U has finite-dimensional cohomology groups and that these groups are isomorphic to the corresponding ¿9-cohomology groups on any desingularization of X . Thus the Euler characteristic of these

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Grieser

Lesch

2002

Math. Nachr.

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On the heat operators of normal singular algebraic surfaces

Cited by 16 publications

References 8 publications

Pure Hodge Structure of the Harmonic $L^2$-Forms on Singular Algebraic Surfaces

Pure Hodge Structure of the Harmonic $L^2$-Forms on Singular Algebraic Surfaces

𝐿²-Dolbeault complexes on singular curves and surfaces

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