On Chern numbers of algebraic varieties with arbitrary singularities

In 1965 the author introduced the notion of Chern classes for an algebraic variety with arbitrary singularities. Based on this definition the well-known Miyaoka-Yao inequalities have been proved and extended by quite simple direct computations.     

Singularities of generic projection hypersurfaces

Linearly projecting smooth projective varieties provide a method of obtaining hypersurfaces birational to the original varieties. We show that in low dimension, the resulting hypersurfaces only have Du Bois singularities. Moreover, we conclude that these Du Bois singularities are in fact semi log canonical. However, we demonstrate the existence of counterexamples in high dimension - the generic linear projection of certain varieties of dimension 30 or higher is neither semi log canonical nor Du Bois.

     

Existence of dicritical singularities of Levi-flat hypersurfaces and holomorphic foliations

We study holomorphic foliations tangent to singular real-analytic Levi-flat hypersurfaces in compact complex manifolds of complex dimension two. We give some hypotheses to guarantee the existence of dicritical singularities of these objects. As consequence, we give some applications to holomorphic foliations tangent to real-analytic Levi-flat hypersurfaces with singularities in \(\mathbb {P}^2\).     

Hilbert Series and Lefschetz Properties of Dimension One Almost Complete Intersections

We generalize some results about the graded Milnor algebras of projective hypersurfaces with isolated singularities to the more general case of an almost complete intersection ideal J of dimension one. When the saturation I of J is a complete intersection, we get formulas for some invariants. Examples of hypersurfaces V: f = 0 in ?ⁿ whose Jacobian ideals J satisfy this property and with nontrivial Alexander polynomials are given in any dimension. A Lefschetz property for the graded quotient I/J is proved for n = 2 and a counterexample due to A. Conca is given for such a property when n = 3.     

On the approximation of the elastica functional in radial symmetry

We prove: (1) a classification theorem for certain singularities of weak limits of immersed stable minimal hypersurfaces of arbitrary dimension and (2) a Bernstein theorem for complete immersed stable minimal hypersurfaces of dimension .Received: 1 September 2003, Accepted: 29 January 2004, Published online: 12 May 2004     

Rellich type inequalities in domains of the Euclidean space

For smooth functions supported in a domain of the Euclidean space we investigate two Rellich type inequalities with weights which are powers of the distance function. We prove that for an arbitrary plane domain there exist positive Rellich constants in these inequalities if and only if the boundary of the domain is a uniformly perfect set. Moreover, we obtain explicit estimates of constants in function of geometric domain characteristics. Also, we find sharp constants in these Rellich type inequalities for all non-convex domains of dimension d ≥ 2 provided that the domains satisfy the exterior sphere condition with certain restriction on the radius of spheres.     

Hochster’s theta pairing and algebraic equivalence

We define a variant of Hochster’s θ pairing and prove that it is constant in flat families of modules over hypersurfaces with isolated singularities. As a consequence, we show that the θ pairing factors through the Grothendieck group modulo algebraic equivalence. Moreover, our result allows us, in certain situations, to translate the properties of the θ pairing in characteristic zero [established in Moore et al. (Adv Math, 226(2):1692–1714, 2011) and Polishchuk and Vaintrob (Chern characters and Hirzebruch–Riemann-Roch formula for matrix factorizations. Preprint, arXiv:1002, 2010)] to the characteristic p setting. We also give an application of our result to the rigidity of Tor over hypersurfaces.     

Cheeger–Chern–Simons Theory and Differential String Classes

We construct new concrete examples of relative differential characters, which we call Cheeger–Chern–Simons characters. They combine the well-known Cheeger–Simons characters with Chern–Simons forms. In the same way as Cheeger–Simons characters generalize Chern–Simons invariants of oriented closed manifolds, Cheeger–Chern–Simons characters generalize Chern–Simons invariants of oriented manifolds with boundary. We study the differential cohomology of compact Lie groups G and their classifying spaces BG. We show that the even degree differential cohomology of BG canonically splits into Cheeger–Simons characters and topologically trivial characters. We discuss the transgression in principal G-bundles and in the universal bundle. We introduce two methods to lift the universal transgression to a differential cohomology valued map. They generalize the Dijkgraaf–Witten correspondence between 3-dimensional Chern–Simons theories and Wess–Zumino–Witten terms to fully extended higher-order Chern–Simons theories. Using these lifts, we also prove two versions of a differential Hopf theorem. Using Cheeger–Chern–Simons characters and transgression, we introduce the notion of differential trivializations of universal characteristic classes. It generalizes well-established notions of differential String classes to arbitrary degree. Specializing to the class \({\frac{1}{2} p_1 \in H^4(B{\rm Spin}_n;\mathbb{Z})}\), we recover isomorphism classes of geometric string structures on Spin _n -bundles with connection and the corresponding spin structures on the free loop space. The Cheeger–Chern–Simons character associated with the class \({\frac{1}{2} p_1}\) together with its transgressions to loop space and higher mapping spaces defines a Chern–Simons theory, extended down to points. Differential String classes provide trivializations of this extended Chern–Simons theory. This setting immediately generalizes to arbitrary degree: for any universal characteristic class of principal G-bundles, we have an associated Cheeger–Chern–Simons character and extended Chern–Simons theory. Differential trivialization classes yield trivializations of this extended Chern–Simons theory.     

On classical analogues of free entropy dimension

We define a classical probability analogue of Voiculescu's free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on Rn. We show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension. First, it can be viewed as a kind of fractal dimension. Second, if one extends Bochner's inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical Bochner's inequalities, then we show that the classical probability entropy dimension controls the rate of increase of optimal constants in Bochner's inequality for a measure regularized by convolution with the Gaussian law as the regularization is removed. We introduce a free analogue of the Bochner inequality and study the related free entropy dimension quantity. We show that it is greater or equal to the non-microstates free entropy dimension.     

On closed minimal hypersurfaces with constant scalar curvature in \mathbb{S}^7

We consider minimal closed hypersurfaces ${M \subset \mathbb{S}^7(1)}$ with constant scalar curvature. We prove that if M fulfills particular additional assumptions, then it is isoparametric. This gives a partial answer to the question made by S.-S. Chern about the pinching of the scalar curvature for closed minimal hypersurfaces in dimension 6.     

Geometry of Affine Warped Product Hypersurfaces

The purpose of this article is the study of warped product manifolds which can be realized either as centroaffine or graph hypersurfaces in some affine space. First, we show that there exist many such realizations. Then we establish general optimal inequalities in terms of the warping function and the Tchebychev vector field for such affine hypersurfaces. We also investigate warped product affine hypersurfaces which verify the equality case of the inequalities. Several applications are also presented.     

Integral Grothendieck–Riemann–Roch theorem

We show that, in characteristic zero, the obvious integral version of the Grothendieck–Riemann–Roch formula obtained by clearing the denominators of the Todd and Chern characters is true (without having to divide the Chow groups by their torsion subgroups). The proof introduces an alternative to Grothendieck’s strategy: we use resolution of singularities and the weak factorization theorem for birational maps.     

Cobordism of algebraic knots

Summary The isolated singularities of complex hypersurfaces are studied by considering the topology of the highly connected submanifolds of spheres determined by the singularity. By introducing the notion of the link of a perturbation of the singularity and using techniques of surgery theory, we are able to describe which invariants associated to a singularity can be used to determine the cobordism type of the singularity.It is shown that the cobordism type is determined by the set of weakly distinguished bases. This result is used to draw a distinction between the classical case of two variables and the higher dimensional problem. That is, we show that the result of Le which states that the cobordism and topological classifications of singularities coincide in the classical dimension does not hold for singularities of functions of more than three variables. Examples of topologically distinct but cobordant singularities are obtained using results of Ebeling.     

Singular set of some Kähler orbifolds

We consider some examples of orbifolds with positive first Chern class. Applying a result of Ding and Tian, we show that the singularities must be very mild if the orbifold admits a Kähler-Einstein metric.

     

Semidefinite Representation of Convex Hulls of Rational Varieties

Using elementary duality properties of positive semidefinite moment matrices and polynomial sum-of-squares decompositions, we prove that the convex hull of rationally parameterized algebraic varieties is semidefinite representable (that is, it can be represented as a projection of an affine section of the cone of positive semidefinite matrices) in the case of (a) curves; (b) hypersurfaces parameterized by quadratics; and (c) hypersurfaces parameterized by bivariate quartics; all in an ambient space of arbitrary dimension.     

On tangential deformations of homogeneous polynomials

The Jacobian ideal provides the set of infinitesimally trivial deformations for a homogeneous polynomial, or for the corresponding complex projective hypersurface. In this article, we investigate whether the associated linear deformation is indeed trivial, and show that the answer is no in a general situation. We also give a characterization of tangentially smoothable hypersurfaces with isolated singularities. Our results have applications in the local study of variations of projective hypersurfaces, complementing the global versions given by J. Carlson and P. Griffiths, R. Donagi and the author, and in the study of isotrivial linear systems on the projective space, showing that a general divisor does not belong to an isotrivial linear system of positive dimension.     

Projective Hypersurfaces with many Singularities of Prescribed Types

Patchworking of singular hypersurfaces is used to constructprojective hypersurfaces with prescribed singularities. Forall n 2, an asymptotically proper existence result is deducedfor hypersurfaces in Pⁿ with singularities of corank at most2 prescribed up to analytical or topological equivalence. Inthe case of T-smooth hypersurfaces with only simple singularities,the result is even asymptotically optimal, that is, the leadingcoefficient in the sufficient existence condition cannot beimproved, which is new even in the case of plane curves. Furthermore,an asymptotically proper existence result is proved for hypersurfacesin Pⁿ with quasihomogeneous singularities. The estimates substantiallyimprove all known (general) existence results for hypersurfaceswith these singularities.