arXiv subjects
Publications and source records attributed to Mihnea Popa.
We prove that smooth projective varieties with equivalent derived categories have isogenous (and sometimes isomorphic) Picard varieties. In particular their irregularity and number of independent vector fields are the same. This is turn implies, in combination with the invariance of Hochschild homology, that all Hodge numbers are the same for arbitrary derived equivalent threefolds.
The cohomology algebra of the canonical bundle of a compact Kähler manifold is naturally viewed as a module over an exterior algebra. We use the Bernstein-Gel'fand-Gel'fand correspondence, together with Generic Vanishing theory, in order to understand the regularity properties of this module. We also relate it to the infinitesimal theory of the canonical linear series inside paracanonical space. Finally, we apply vector bundle methods on the polynomial ring side to obtain inequalities for the holomorphic Euler characteristic and the Hodge numbers of compact Kähler manifolds without irregular fibrations.
We use the language of multiplier ideals in order to relate the syzygies of an abelian variety in a suitable embedding with the local positivity of the line bundle inducing that embedding. This extends to higher syzygies a result of Hwang and To on projective normality.
This article is based on lecture notes prepared for the August 2006 Cologne Summer School. The first part contains background material and references for beginners. The second (and main) part is a survey of the current status in the theory of pluri-theta linear series and generalized theta divisors on moduli spaces of vector bundles on curves. It emphasizes relatively new techniques employed in the analysis of linear series on these moduli spaces, namely the use of moduli spaces of stable maps for understanding Quot schemes, and the Fourier-Mukai functor in the study of coherent sheaves on abelian varieties. In addition, it briefly describes recent important developments, most significant of which is the proof of the Strange Duality conjecture due to Belkale and Marian-Oprea.
The paper contains theorems on extending sections of line bundles from divisors to the ambient space, inspired by various results of Siu, Kawamata, and especially Hacon-McKernan and Takayama. Applications are given to basepoint-freeness, to the invariance of analogues of plurigenera and of part of the boundary of the pseudo-effective cone in families, and to a simplified approach towards part of the results used for proving the existence of flips by Hacon-McKernan.
This is a mostly expository paper, intended to explain a very natural relationship between two a priori distinct notions appearing in the literature: Generic Vanishing in the context of vanishing theorems and birational geometry, and Perverse Coherent Sheaves in the context of derived categories. I describe homological and commutative algebra criteria for checking either condition, and then recall applications to geometric situations. A few new results, and especially new proofs, are included as well.
We use homological methods to establish a formal criterion for Generic Vanishing, in the sense originated by Green and Lazarsfeld and pursued further by Hacon and the first author, but in the context of an arbitrary Fourier-Mukai correspondence. For smooth projective varieties we apply this to deduce a Kodaira-type generic vanishing theorem for adjoint bundles of the form $K_X + L$ with $L$ a nef line bundle, and in fact a more general generic Nadel-type vanishing theorem for multiplier ideal sheaves. Still in the context of the Picard variety, the same method generates various other generic vanishing results, by reduction to standard vanishing theorems. We further use the formal criterion in order to address examples related to generic vanishing on higher rank moduli spaces (on curves and on some threefold Calabi-Yau fiber spaces).
We extend to manifolds of arbitrary dimension the Castelnuovo-de Franchis inequality for surfaces. The proof is based on the theory of Generic Vanishing and higher regularity, and on the Evans-Griffith Syzygy Theorem in commutative algebra. Along the way we give a positive answer, in the setting of Kähler manifolds, to a question of Green-Lazarsfeld on the vanishing of higher direct images of Poincaré bundles. We indicate generalizations to arbitrary Fourier-Mukai transforms.
We note that the vanishing and injectivity theorems of Kollár and Esnault-Viehweg can be used to give a quick algebraic proof of a strengthening of the Ein-Lazarsfeld Skoda-type division theorem for global sections of adjoint line bundles vanishing along suitable multiplier ideal sheaves, and to extend it to higher cohomology classes as well. For global sections, this is also a slightly more general statement of the algebraic translation of an analytic result of Siu. Along the way we write down an injectivity statement for multiplier ideals, and its standard consequences.
We describe the relationship between the notions of $M$-regular sheaf and $GV$-sheaf in the case of abelian varieties. The former is a natural strengthening of the latter, and we provide an algebraic criterion characterizing it among the larger class. Based on this we deduce new basic properties of both $M$-regular and $GV$-sheaves. In the second part we give a number of applications of generation criteria for $M$-regular sheaves to the study of Seshadri constants, Picard bundles, pluricanonical maps on irregular varieties, and semihomogeneous vector bundles. This second part of the paper is based on our earlier preprint math.AG/0306103, with some improved statements and shortened arguments.
We introduce and study the restricted volume of a divisor along a subvariety. Our main result is a description of the irreducible components of the augmented base locus by the vanishing of the restricted volume.
We prove and conjecture results which show that Castelnuovo theory in projective space has a close analogue for abelian varieties. This is related to the geometric Schottky problem: our main result is that a principally polarized abelian variety satisfies a precise version of the Castelnuovo Lemma if and only if it is a Jacobian. This result has a surprising connection to the Trisecant Conjecture. We also give a genus bound for curves in abelian varieties.
We establish a, and conjecture further, relationship between the existence of subvarieties representing minimal cohomology classes on principally polarized abelian varieties, and the generic vanishing of certain sheaf cohomology. The main ingredient is the Generic Vanishing criterion of math.AG/0608127, based on the Fourier-Mukai transform.
This is a survey of M-regularity for coherent sheaves on abelian varieties and its applications, based on lectures given by the second author at the Seattle conference, in August 2005, and at the Luminy GAC workshop in October 2005. Section 3 contains some brief indications of upcoming work.
The purpose of this paper is to define and study systematically some asymptotic invariants associated to base loci of line bundles on smooth projective varieties. We distinguish an open dense subset of the real big cone, called the stable locus, consisting of the set of classes on which the asymptotic base locus is locally constant. The asymptotic invariants define continuous functions on the big cone, whose vanishing characterizes, roughly speaking, the unstable locus. We show that for toric varieties at least, there exists a polyhedral decomposition of the big cone on which these functions are polynomial.
Let X be a smooth complex projective variety of dimension d. It is classical that ample line bundles on X satisfy many beautiful geometric, cohomological, and numerical properties that render their behavior particularly tractable. By contrast, examples due to Cutkosky and others have led to the common impression that the linear series associated to non-ample divisors are in general mired in pathology. However starting with fundamental work of Fujita, Nakayama, and Tsuji, it has recently become apparent that arbitrary effective (or "big") divisors in fact display a surprising number of properties analogous to those of ample line bundles. The key is to study the properties in question from an asymptotic perspective. At the same time, many interesting questions and open problems remain. The purpose of the present expository note is to give an invitation to this circle of ideas. In the hope that this informal overview might serve as a jumping off point for the technical literature in the area, we sketch many examples but provide no proofs. We focus on one particular invariant -- the "volume" of a line bundle -- that measures the rate of growth of the number of sections of powers of the bundle in question.
We carry out a detailed intersection theoretic analysis of the Deligne-Mumford compactification of the divisor on M_{10} consisting of curves sitting on K3 surfaces. This divisor is not of classical Brill-Noether type, and is known to give a counterexample to the Slope Conjecture. The computation relies on the fact that this divisor has four different incarnations as a geometric subvariety of the moduli space of curves, one of them as a higher rank Brill-Noether divisor consisting of curves with an exceptional rank 2 vector bundle. As an application we describe the birational nature of the moduli space of n-pointed curves of genus 10, for all n. We also show that on M_{11} there are effective divisors of minimal slope and having large Iitaka dimension. This seems to contradict the belief that on M_g the classical Brill-Noether divisors are essentially the only ones of slope 6+12/(g+1).
In the present sequel to our previous two papers on regularity on abelian varieties, we give a number of new applications of the theory of $M$-regularity to the study of Seshadri constants, Picard bundles, pluricanonical maps on irregular varieties, and semihomogeneous vector bundles.