Complex Projective Hypersurfaces Research Articles

Effective homology and periods of complex projective hypersurfaces

We introduce a new algorithm for computing the periods of a smooth complex projective hypersurface. The algorithm intertwines with a new method for computing an explicit basis of the singular homology of the hypersurface. It is based on Picard–Lefschetz theory and relies on the computation of the monodromy action induced by a one-parameter family of hyperplane sections on the homology of a given section. We provide a SageMath implementation. For example, on a laptop, it makes it possible to compute the periods of a smooth complex quartic surface with hundreds of digits of precision in typically an hour.

Vanishing cohomology and Betti bounds for complex projective hypersurfaces

We employ the formalism of vanishing cycles and perverse sheaves to introduce and study the vanishing cohomology of complex projective hypersurfaces. As a consequence, we give upper bounds for the Betti numbers of projective hypersurfaces, generalizing those obtained by different methods by Dimca in the isolated singularities case, and by Siersma-Tib\u{a}r in the case of hypersurfaces with a $1$-dimensional singular locus. We also prove a supplement to the Lefschetz hyperplane theorem for hypersurfaces, which takes the dimension of the singular locus into account, and we use it to give a new proof of a result of Kato.

Systoles and Lagrangians of random complex algebraic hypersurfaces

Let n\geq 1 be an integer, and \mathcal L \subset \mathbb{R}^n be a compact smooth affine real hypersurface, not necessarily connected. We prove that there exist c>0 and d_0\geq 1 such that for any d\geq d_0 , any smooth complex projective hypersurface Z in \mathbb{C} P^n of degree d contains at least c\dim H_{*}(Z, \mathbb{R}) disjoint Lagrangian submanifolds diffeomorphic to \mathcal L , where Z is equipped with the restriction of the Fubini–Study symplectic form (Theorem 1.1). If moreover all connected components of \mathcal L have non-vanishing Euler characteristic, which implies that n is odd, the latter Lagrangian submanifolds form an independent family in H_{n-1}(Z, \mathbb{R}) (Corollary 1.2). These deterministic results are consequences of a more precise probabilistic theorem (Theorem 1.23) inspired by a 2014 result by J.-Y. Welschinger and the author on random real algebraic geometry, together with quantitative Moser-type constructions (Theorem 3.4). For n=2 , the method provides a uniform positive lower bound for the probability that a projective complex curve in \mathbb{C} P^2 of given degree equipped with the restriction of the ambient metric has a systole of small size (Theorem 1.6), which is an analog of a similar bound for hyperbolic curves given by M. Mirzakhani (2013). In higher dimensions, we provide a similar result for the (n-1) -systole introduced by M. Berger (1972) (Corollary 1.14). Our results hold in the more general setting of vanishing loci of holomorphic sections of vector bundles of rank between 1 and n tensored by a large power of an ample line bundle over a projective complex n -manifold (Theorem 1.20).

A note on non-uniform points for projections of hypersurfaces

Let X be an irreducible, reduced complex projective hypersurface of degree d. A uniform point for X is a point P such that the projection of X from P has maximal monodromy. We extend and improve some results concerning the finiteness of the locus of non-uniform points for projections of hypersurfaces obtained by the authors and Cuzzucoli (Ann. Mat. Pura ed Appl. 1923, 1–18 (2021)) only for P not contained in X.

Monodromy of projections of hypersurfaces

Let X be an irreducible, reduced complex projective hypersurface of degree d. A point P not contained in X is called uniform if the monodromy group of the projection of X from P is isomorphic to the symmetric group S_d. We prove that the locus of non-uniform points is finite when X is smooth or a general projection of a smooth variety. In general, it is contained in a finite union of linear spaces of codimension at least 2, except possibly for a special class of hypersurfaces with singular locus linear in codimension 1. Moreover, we generalise a result of Fukasawa and Takahashi on the finiteness of Galois points.

On tangential deformations of homogeneous polynomials

AbstractThe 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.

The local cohomology of the Jacobian ring

We study the 0-th local cohomology module H^0_{\mathbf{m}}(R(f)) of the jacobian ring R(f) of a singular reduced complex projective hypersurface X , by relating it to the sheaf of logarithmic vector fields along X . We investigate the analogies between H^0_{\mathbf{m}}(R(f)) and the well known properties of the jacobian ring of a nonsingular hypersurface. In particular we study self-duality, Hodge theoretic and Torelli type questions for H^0_{\mathbf{m}}(R(f)) .

Intersection spaces, perverse sheaves and type IIB string theory

The method of intersection spaces associates rational Poincar\'e complexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA theory. For complex projective hypersurfaces with an isolated singularity, we show that the cohomology of intersection spaces is the hypercohomology of a perverse sheaf, the intersection space complex, on the hypersurface. Moreover, the intersection space complex underlies a mixed Hodge module, so its hypercohomology groups carry canonical mixed Hodge structures. For a large class of singularities, e.g., weighted homogeneous ones, global Poincar\'e duality is induced by a more refined Verdier self-duality isomorphism for this perverse sheaf. For such singularities, we prove furthermore that the pushforward of the constant sheaf of a nearby smooth deformation under the specialization map to the singular space splits off the intersection space complex as a direct summand. The complementary summand is the contribution of the singularity. Thus, we obtain for such hypersurfaces a mirror statement of the Beilinson-Bernstein-Deligne decomposition of the pushforward of the constant sheaf under an algebraic resolution map into the intersection sheaf plus contributions from the singularities.

A category of kernels for equivariant factorizations and its implications for Hodge theory

We provide a factorization model for the continuous internal Hom, in the homotopy category of k-linear dg-categories, between dg-categories of equivariant factorizations. This motivates a notion, similar to that of Kuznetsov, which we call the extended Hochschild cohomology algebra of the category of equivariant factorizations. In some cases of geometric interest, extended Hochschild cohomology contains Hochschild cohomology as a subalgebra and Hochschild homology as a homogeneous component. We use our factorization model for the internal Hom to calculate the extended Hochschild cohomology for equivariant factorizations on affine space. Combining the computation of extended Hochschild cohomology with the Hochschild-Kostant-Rosenberg isomorphism and a theorem of Orlov recovers and extends Griffiths’ classical description of the primitive cohomology of a smooth, complex projective hypersurface in terms of homogeneous pieces of the Jacobian algebra. In the process, the primitive cohomology is identified with the fixed subspace of the cohomological endomorphism associated to an interesting endofunctor of the bounded derived category of coherent sheaves on the hypersurface. We also demonstrate how to understand the whole Jacobian algebra as morphisms between kernels of endofunctors of the derived category. Finally, we present a bootstrap method for producing algebraic cycles in categories of equivariant factorizations. As proof of concept, we show how this reproves the Hodge conjecture for all self-products of a particular K3 surface closely related to the Fermat cubic fourfold.

Existence of global invariant jet differentials on projective hypersurfaces of high degree

Let $${X\subset\mathbb P^{n+1}}$$ be a smooth complex projective hypersurface. In this paper we show that, if the degree of X is large enough, then there exist global sections of the bundle of invariant jet differentials of order n on X, vanishing on an ample divisor. We also prove a logarithmic version, effective in low dimension, for the log-pair $${(\mathbb P^n,D)}$$ , where D is a smooth irreducible divisor of high degree. Moreover, these result are sharp, i.e. one cannot have such jet differentials of order less than n.

Differential equations on complex projective hypersurfaces of low dimension

AbstractLet n=2,3,4,5 and let X be a smooth complex projective hypersurface of $\mathbb {P}^{n+1}$. In this paper we find an effective lower bound for the degree of X, such that every holomorphic entire curve in X must satisfy an algebraic differential equation of order k=n=dim X, and also similar bounds for order k>n. Moreover, for every integer n≥2, we show that there are no such algebraic differential equations of order k<n for a smooth hypersurface in $\mathbb {P}^{n+1}$.

Boundary manifolds of projective hypersurfaces

We study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface. We show that, under certain Hodge-theoretic conditions, the cohomology ring of the complement of the hypersurface functorially determines that of the boundary. When the hypersurface defines a hyperplane arrangement, the cohomology of the boundary is completely determined by the combinatorics of the underlying arrangement and the ambient dimension. We also study the LS category and topological complexity of the boundary manifold, as well as the resonance varieties of its cohomology ring.

Decomposition into pairs-of-pants for complex algebraic hypersurfaces

It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary dimension admits a similar decomposition. The n-dimensional pair-of-pants is diffeomorphic to CP n minus n+2 hyperplanes. Alternatively, these decompositions can be treated as certain fibrations on the hypersurfaces. We show that there exists a singular fibration on the hypersurface with an n-dimensional polyhedral complex as its base and a real n-torus as its fiber. The base accommodates the geometric genus of a hypersurface V. Its homotopy type is a wedge of h n, o ( V) spheres S n .

Homology of a Perturbation of a Complex Projective Hypersurface

A nonsingular hypersurface X in \(\mathbb{C}P^{n-1}\) with n≥3 is studied. The main result of the paper says that the homology coming from the affine part of a hypersurface of smaller degree forms a direct summand in the homology of X, which is independent over integers with the class of a multiple hyperplane section. The proof is outlined. Bibliography: 3 titles.

Endomorphisms of hypersurfaces and other manifolds

We prove in this note the following result: Theorem .− A smooth complex projective hypersurface of dimension ≥ 2 and degree ≥ 3 admits no endomorphism of degree > 1 . Since the case of quadrics is treated in [PS], this settles the question of endomorphisms of hypersurfaces. We prove the theorem in Section 1, using a simple but efficient trick devised by Amerik, Rovinsky and Van de Ven [ARV]. In Section 2 we collect some general results on endomorphisms of projective manifolds; we classify in particular the Del Pezzo surfaces which admit an endomorphism of degree > 1 .

The equivalence problem for higher-codimensional CR structures

The equivalence problem for CR structures can be viewed as a special case of the equivalence problem for G-structure. This paper uses Cartan’s methods (in modernized form) to show that a CR manifold of codimension 3 or greater with suitably generic Levi form admits a canonical connection on a reduced structure bundle whose group is isomorphic to the multiplicative group of complex numbers. As corollaries, it follows that the CR manifold admits a canonical ane connection, and consequently that the automorphisms of the CR manifold constitute a Lie group. The most dicult technical step is to construct a smooth moduli space for generic vector-valued hermitian forms, which is tied to the CR manifold via the Levi map. The techniques used to construct this space are drawn from the classical invariant theory of complex projective hypersurfaces.

Koszul complexes and hypersurface singularities

The behavior of Poincaré series and Betti numbers of complex projective hypersurfaces in terms of their singularities is compared.

Variation of complex structures and variation of Lie algebras

Given a family of complex projective hypersurfaces in CP", the Torelli problem studied by P. Griffiths and his school asks whether the period map is injective on that family, i.e., whether the family of complex hypersurfaces can be distinguished by means of their Hodge structures. A complex projective hypersurface in CP" can be viewed as a complex hypersurface with isolated singularity in C "+ t. Let V= {z~C "+1 : f(z)=O} be a complex hypersurface with isolated singularity at the origin. The moduli algebra of (V, 0) is A(V)

On complex projective hypersurfaces

In this paper we prove various results concerning monodromy groups associated with nonsingular complex projective hypersurfaces. Most of these results are already known but proofs are either unavailable or are algebraic and require a lot of machinery. The groups in question are those obtained from the second Lefschetz theorem (see (1)) applied to (a) the general Veronese variety, (b) a nonsingular projective hypersurface. By embedding the monodromy group of an extraordinary local isolated singularity (discovered by Libgober (8)) in these global monodromy groups we obtain necessary and sufficient conditions for the global groups to be finite. For case (a) we also obtain information on the structure of the dual to the Veronese variety which is of use when considering the monodromy group. The author gratefully acknowledges the financial support of the Stiftung Volkswagenwerk for a vist to the IHES during which this paper was written.

Homology of Complex Projective Hypersurfaces with Isolated Singularities

We use information concerning the homology of links and Milnor fibers at each singularity of a hypersurface $\tilde V$, of degree $d$ in $C{P^{n + 1}}$ with only isolated singularities, to determine $\tilde V$’s homology except for the question of torsion in ${H_n}(\tilde V;Z)$.