diff d5ba301..763cb89 singular_free_tangent_sheaf_v2.txt
diff --git a/singular_free_tangent_sheaf_v2.txt b/singular_free_tangent_sheaf_v2.txt index 9fa3c43..cf318b2 100644 --- a/singular_free_tangent_sheaf_v2.txt +++ b/singular_free_tangent_sheaf_v2.txt @@ -8,7 +8,7 @@ xy - z^p = 0 has tangent sheaf freely generated by x d_x - y d_y and d_z, while the local ring at the origin is not regular (Graf, J. Algebra 601, 2022, 115-128). -./graf-2201.06646-positive-characteristic.pdf +./pdf/graf-2201.06646-positive-characteristic.pdf Over the complex numbers the answer is also negative. There exist a finitely generated normal integral complex algebra A of Krull @@ -31,7 +31,7 @@ locally complete intersections in characteristic zero were covered by Kallstrom (J. Algebra 337, 2011, 169-180; corrected version arXiv:1003.4241v6, 2018). -./kallstrom-1003.4241-complete-intersections.pdf +./pdf/kallstrom-1003.4241-complete-intersections.pdf Extension of holomorphic differential forms gives the conjecture for isolated singularities in dimension at least three (van Straten and @@ -41,9 +41,9 @@ and Peternell (Publ. Math. IHES 114, 2011, 87-169), by Druel (Bull. London Math. Soc. 46, 2014, 827-835), and by Graf and Kovacs (Doc. Math. 19, 2014, 815-830). -./greb-kebekus-kovacs-peternell-2011-1003.2913-diff-forms-lc.pdf -./druel-1304.6000-log-canonical-spaces.pdf -./graf-kovacs-1301.7315-optimal-extension-1forms.pdf +./pdf/greb-kebekus-kovacs-peternell-2011-1003.2913-diff-forms-lc.pdf +./pdf/druel-1304.6000-log-canonical-spaces.pdf +./pdf/graf-kovacs-1301.7315-optimal-extension-1forms.pdf For normal complex surfaces, the geometric genus together with the configuration of the exceptional divisor gives sharp criteria. Graf @@ -58,9 +58,9 @@ surfaces whose tangent bundle on the smooth locus is trivial and whose smooth locus has logarithmic Kodaira dimension at most one (J. London Math. Soc. 90, 2014, 270-286). -./graf-1801.05753-low-genus.pdf -./bergner-graf-2020-lipman-zariski-genus-one-higher.pdf -./bgk-1403.5613-zariski-lipman-normal-surfaces.pdf +./pdf/graf-1801.05753-low-genus.pdf +./pdf/bergner-graf-2020-lipman-zariski-genus-one-higher.pdf +./pdf/bgk-1403.5613-zariski-lipman-normal-surfaces.pdf The surface to be constructed lies outside every one of these positive results. Its canonical discrepancy is -5, so it is not log @@ -73,7 +73,7 @@ iterated blowup at the module of derivations resolves a singularity: on a variety with free tangent sheaf every such blowup, with or without subsequent normalization, is already the identity. -./barajas-chavez-romano-2510.07607-blowup-derivations.pdf +./pdf/barajas-chavez-romano-2510.07607-blowup-derivations.pdf An additional intrinsic constraint comes from the bracket of any frame. Jorder proved that a normal complex space whose tangent sheaf @@ -81,7 +81,7 @@ locally admits a basis of pairwise commuting vector fields is smooth (Math. Z. 278, 2014, 893-899). Hence no frame for the tangent sheaf of the surface below can be made commuting near its singular point. -./jorder-1311.5141-weak-lipman-zariski.pdf +./pdf/jorder-1311.5141-weak-lipman-zariski.pdf Conventions. Line bundles on curves are written additively; aL + bH denotes the tensor product L^a tensor H^b over O_C. Two-dimensional @@ -101,8 +101,8 @@ Grauert and Remmert, Coherent Analytic Sheaves, Grundlehren 265 Geometry (2012), for coherent analytic sheaves and normal complex spaces. -./serre-1956-gaga.pdf -./demailly-2012-complex-analytic-differential-geometry.pdf +./pdf/serre-1956-gaga.pdf +./pdf/demailly-2012-complex-analytic-differential-geometry.pdf The construction begins on the quadric surface Sigma = PP^1 x PP^1 with ruling line bundles F = O(1,0) and S = O(0,1). Let Gamma be a @@ -1292,7 +1292,7 @@ zero completion implies that the stalk itself is zero, by Nakayama's lemma. Away from x the map pi is an isomorphism, so the same sheaf vanishes there. Thus R^1 pi_* E = 0. -./grauert-1960-theorem-garbentheorie.pdf +./pdf/grauert-1960-theorem-garbentheorie.pdf The sheaf pi_* E is also coherent. Cartan's theorem B on the Stein space X, plus the Leray spectral sequence for pi, gives @@ -1381,7 +1381,7 @@ curves and rational singularities" (2001; arXiv:math/0109221), and Serre, "Un theoreme de dualite", Comment. Math. Helv. 29 (1955), 9-26. -./flenner-zaidenberg-2001-math0109221-rational-curves-singularities.pdf +./pdf/flenner-zaidenberg-2001-math0109221-rational-curves-singularities.pdf By the ordinary determinant formula, the pullback of omega^o is @@ -1433,7 +1433,7 @@ local rings", Publ. Math. IHES 36 (1969), 23-58. The finite- determinacy statement is proved below, with fixed estimates and no complete-intersection hypothesis. -./artin-1969-algebraic-approximation.pdf +./pdf/artin-1969-algebraic-approximation.pdf Put P = C[[x_1, ..., x_e]], n = (x_1, ..., x_e), I = (f_1, ..., f_s), R = P / I, where f = (f_1, ..., f_s) in n P^s is a row.