594 research outputs found
13/2 ways of counting curves
In the past 20 years, compactifications of the families of curves in
algebraic varieties X have been studied via stable maps, Hilbert schemes,
stable pairs, unramified maps, and stable quotients. Each path leads to a
different enumeration of curves. A common thread is the use of a 2-term
deformation/obstruction theory to define a virtual fundamental class. The
richest geometry occurs when X is a nonsingular projective variety of dimension
3.
We survey here the 13/2 principal ways to count curves with special attention
to the 3-fold case. The different theories are linked by a web of conjectural
relationships which we highlight. Our goal is to provide a guide for graduate
students looking for an elementary route into the subject.Comment: Typo fixed, In "Moduli spaces", LMS Lecture Note Series, 411 (2014),
282-333. Cambridge University Pres
Algebraic cobordism of bundles on varieties
The double point relation defines a natural theory of algebraic cobordism for
bundles on varieties. We construct a simple basis (over the rationals) of the
corresponding cobordism groups over Spec(C) for all dimensions of varieties and
ranks of bundles. The basis consists of split bundles over products of
projective spaces. Moreover, we prove the full theory for bundles on varieties
is an extension of scalars of standard algebraic cobordism.Comment: 24 pages, reference adde
Almost closed 1-forms
We construct an algebraic almost closed 1-form with zero scheme not
expressible (even locally) as the critical locus of a holomorphic function on a
nonsingular variety. The result answers a question of Behrend-Fantechi. We
correct here an error in our paper [MPT] where an incorrect construction with
the same claimed properties was proposed.Comment: 16 page
Curves on K3 surfaces and modular forms
We study the virtual geometry of the moduli spaces of curves and sheaves on
K3 surfaces in primitive classes. Equivalences relating the reduced
Gromov-Witten invariants of K3 surfaces to characteristic numbers of stable
pairs moduli spaces are proven. As a consequence, we prove the Katz-Klemm-Vafa
conjecture evaluating integrals (in all genera) in terms of
explicit modular forms. Indeed, all K3 invariants in primitive classes are
shown to be governed by modular forms.
The method of proof is by degeneration to elliptically fibered rational
surfaces. New formulas relating reduced virtual classes on K3 surfaces to
standard virtual classes after degeneration are needed for both maps and
sheaves. We also prove a Gromov-Witten/Pairs correspondence for toric 3-folds.
Our approach uses a result of Kiem and Li to produce reduced classes. In
Appendix A, we answer a number of questions about the relationship between the
Kiem-Li approach, traditional virtual cycles, and symmetric obstruction
theories.
The interplay between the boundary geometry of the moduli spaces of curves,
K3 surfaces, and modular forms is explored in Appendix B by A. Pixton.Comment: An incorrect example in Appendix A, pointed out to us by Dominic
Joyce, has been replaced by a reference to a new paper arXiv:1204.3958
containing a corrected exampl
The 3-fold vertex via stable pairs
The theory of stable pairs in the derived category yields an enumerative
geometry of curves in 3-folds. We evaluate the equivariant vertex for stable
pairs on toric 3-folds in terms of weighted box counting. In the toric
Calabi-Yau case, the result simplifies to a new form of pure box counting. The
conjectural equivalence with the DT vertex predicts remarkable identities. The
equivariant vertex governs primary insertions in the theory of stable pairs for
toric varieties. We consider also the descendent vertex and conjecture the
complete rationality of the descendent theory for stable pairs.Comment: Typos fixed. 40 pages, 8 figure
ALMOST CLOSED 1-FORMS
We construct an algebraic almost closed 1-form with zero scheme not expressible (even locally) as the critical locus of a holomorphic function on a non-singular variety. The result answers a question of Behrend-Fantechi. We correct here an error in our paper (D. Maulik, R Pandharipande and R. P. Thomas, Curves on K3 surfaces and modular forms, J. Topol. 3 (2010) 937-996. arXiv:1001.2719v3), where an incorrect construction with the same claimed properties was propose
Stable pairs and BPS invariants
We define the BPS invariants of Gopakumar-Vafa in the case of irreducible
curve classes on Calabi-Yau 3-folds. The main tools are the theory of stable
pairs in the derived category and Behrend's constructible function approach to
the virtual class. We prove that for irreducible classes the stable pairs
generating function satisfies the strong BPS rationality conjectures.
We define the contribution of each curve to the BPS invariants. A curve
only contributes to the BPS invariants in genera lying between the geometric
genus and arithmetic genus of . Complete formulae are derived for
nonsingular and nodal curves.
A discussion of primitive classes on K3 surfaces from the point of view of
stable pairs is given in the Appendix via calculations of Kawai-Yoshioka. A
proof of the Yau-Zaslow formula for rational curve counts is obtained. A
connection is made to the Katz-Klemm-Vafa formula for BPS counts in all genera.Comment: Fixed typo pointed out by Filippo Vivian
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