In this paper, we study the cohomology of semisimple local systems in the spirit of classical Hodge theory. On the one hand, we establish a generalization of Hodge-Riemann bilinear relations. For a semisimple local system on a smooth projective variety, we define a canonical isomorphism from the complex conjugate of its cohomology to the cohomology of the dual local system, which is a generalization of the classical Weil operator for pure Hodge structures. This isomorphism establishes a relation between the twisted Poincaré pairing, a purely topological object, and a positive definite Hermitian pairing. On the other hand, we prove a global invariant cycle theorem for semisimple local systems.As an application, we give a new and geometric proof of Sabbah's Decomposition Theorem for the direct images of semisimple local systems under proper algebraic maps, without using the category of polarizable twistor D-modules.
Let X be the projective plane, a Hirzebruch surface, or a general K3 surface. In this paper, we study the birational geometry of various nested Hilbert schemes of points parameterizing pairs of zero-dimensional subschemes on X. We calculate the nef cone for two types of nested Hilbert schemes. As an application, we recover a theorem of Butler on syzygies on Hirzebruch surfaces. 1 NEF CONES OF NESTED HILBERT SCHEMES OF POINTS ON SURFACES 2 product of projective lines, on Hirzebruch surfaces, and on del Pezzo surfaces of degree at least two [BC13]. Recently, there has been tremendous progress using Bridgeland stability to compute the nef cones of Hilbert schemes of points on K3 surfaces [BM14b], Abelian surfaces ([MM13], [YY14]), Enriques surfaces [Nue16], and all surfaces with Picard number one and irregularity zero [BHL + 16].To calculate the nef cone for nested Hilbert schemes, we must first understand the Picard group and the Néron-Severi group.Proposition A. Let X be a smooth projective surface of irregularity zero and fix n ≥ 2. ThenIn particular, the Néron-Severi group N 1 (X [n+1,n] ) has rank 2(ρ(X) + 1), where ρ(X) is the picard number of X.Knowing the Picard groups, we can describe the nef cones. To easily state our theorem, let us first recall the nef cone of the Hilbert schemes of points on P 2 . The nef cone of P 2[n] is spanned by the two divisorswhere H[n] is the class of the pull-back of the ample generator via the Hilbert-Chow morphism and B[n] is the exceptional locus.Theorem B. The nef cone of P 2[n+1,n] , n > 1, is spanned by the four divisors, and pr * a (D n [n + 1]). Similar results hold for the Hirzebruch surfaces F i , i ≥ 0, and general K3 surfaces S as well as for the universal families on all of these surfaces.Knowing the nef cone allows us to recover a theorem of Butler about projective normality of line bundles on Hirzebruch surfaces [But94].Proposition C. Let X be a Hirzebruch surface, and A be an ample line bundles on X, then L = K X + nA is projectively normal for n ≥ 4.
The purpose of this note is to study the (total) geometric genus of the compactified Severi curve V L,g−1 consisting of nodal elliptic curves on a general primitively polarized K3 surface (X, L) of genus g.We prove that the genus is bounded below by O(e C √ g ) and conjecture an asymptotic of O(e Cg ).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.