We define a broad class of local Lagrangian intersections which we call quasi-minimally degenerate (QMD) before developing techniques for studying their local Floer homology. In some cases, one may think of such intersections as modeled on minimally degenerate functions as defined by Kirwan. One major result of this paper is: if [Formula: see text] are two Lagrangian submanifolds whose intersection decomposes into QMD sets, there is a spectral sequence converging to their Floer homology [Formula: see text] whose [Formula: see text] page is obtained from local data given by the QMD pieces. The [Formula: see text] terms are the singular homologies of submanifolds with boundary that come from perturbations of the QMD sets. We then give some applications of these techniques toward studying affine varieties, reproducing some prior results.
Abstract. Consider the linear general eigenvalue problem Ay = λBy , where A and B are both invertible and Hermitian N × N matrices. In this paper we construct a set of meromorphic functions, the Krein eigenvalues, whose zeros correspond to the real eigenvalues of the general eigenvalue problem. The Krein eigenvalues are generated by the Krein matrix, which is constructed through projections on the positive and negative eigenspaces of B. The number of Krein eigenvalues depends on the number of negative eigenvalues for B. These constructions not only allow for us to determine solutions to the general eigenvalue problem, but also to determine the Krein signature for each real eigenvalue. Furthermore, by applying our formulation to the simplest case of the general eigenvalue problem (where B has one negative eigenvalue), we are able to show an interlacing theorem between the eigenvalues for the general problem and the eigenvalues of A.
In this note, we determine the structure of the associative algebra generated by the differential operators
μ
¯
,
∂
¯
,
∂
\overline{\mu },\overline{\partial },\partial
, and
μ
\mu
that act on complex-valued differential forms of almost complex manifolds. This is done by showing that it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential
[
d
,
−
]
\left[d,-]
, as well as its cohomology with respect to all its inner differentials.
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