We construct a moduli space of stable pairs over a smooth projective variety,
parametrizing morphisms from a fixed coherent sheaf to a varying sheaf
of fixed topological type, subject to a stability condition. This generalizes
the notion used by Pandharipande and Thomas, following Le Potier, where
the fixed sheaf is the structure sheaf of the variety. We then describe the
relevant deformation and obstruction theories. We also show the existence
of the virtual fundamental class in special cases.
Let V be a hypersurface with an isolated singularity at the origin defined
by the holomorphic function f : (Cn , 0) → (C, 0). The Yau algebra L(V ) is defined to
be the Lie algebra of derivations of the moduli algebra A(V ) := On/(f, ∂f , · · · , ∂f ), ∂x1 ∂xn
i.e., L(V ) = Der(A(V ), A(V )) and plays an important role in singularity theory. It is known that L(V ) is a finite dimensional Lie algebra and its dimension λ(V ) is called Yau number. In this article, we generalize the Yau algebra and introduce a new series of k-th Yau algebras Lk(V) which are defined to be the Lie algebras of derivations of the moduli algebras Ak(V) = On/(f,mkJ(f)),k ≥ 0, i.e., Lk(V) = Der(Ak(V),Ak(V)) and where m is the maximal ideal of On. In particular, it is Yau algebra when k = 0. The dimension of Lk(V ) is denoted by λk(V ). These numbers i.e., k-th Yau numbers λk(V ), are new numerical analytic invariants of an isolated singularity. In this paper we studied these new series of Lie algebras Lk(V ) and also compute the Lie algebras L1(V ) for fewnomial isolated singularities. We also formulate a sharp upper estimate conjecture for the λk(V ) of weighted homogeneous isolated hypersurface singularities and we prove this conjecture in case of k = 1 for large class of singularities.
Let V be a hypersurface with an isolated singularity at the origin de-
fined by the holomorphic function f : (Cn, 0) → (C, 0). L(V ) is defined to be the
Lie algebra of derivations of the moduli algebra A(V ) := On/(f, ∂f , · · · , ∂f ), i.e. ∂x1 ∂xn
L(V ) = Der(A(V ), A(V )). The Lie algebra L(V ) is finite dimensional solvable alge- bra and plays an important role in singularity theory. L(V ) is called Yau algebra and λ(V ), the dimension of L(V ), is called Yau number. Fewnomial singularities are those which can be defined by n-nomial in n indeterminates. These singularities were inten- sively studied from a mirror symmetry point of view by many other mathematicians. Here we investigate these singularities from a Yau algebra point of view. In our previous work, we formulated a sharp upper estimate conjecture for the Yau numbers of weighted homogeneous isolated hypersurface singularities and proved that this conjecture holds for binomial isolated hypersurface singularities. In this paper we verify this conjecture for weighted homogeneous fewnomial surface singularities.
Let R = C[x1, x2, · · · , xn]/(f1, · · · , fm) be a positively graded Artinian alge- bra. A long-standing conjecture in algebraic geometry, differential geometry and rational homotopy theory is the non-existence of negative weight derivations on R. Alexsandrov conjectured that there are no negative weight derivations when R is a complete intersec- tion algebra and Yau conjectured there are no negative weight derivations on R when R is the moduli algebra of a weighted homogeneous hypersurface singularity. This prob- lem is also important in rational homotopy theory and differential geometry. In this paper we prove the non-existence of negative weight derivations on R when the degrees of f1, . . . , fm are bounded below by a constant C depending only on the weights of x1, . . . , xn. Moreover this bound C is improved in several special cases.
For every smooth complex projective variety W of dimension d and nonnegative Kodaira dimension, we show
the existence of a universal constant m depending only on d and two natural invariants of the very general fibres of an
Iitaka fibration of W such that the pluricanonical system |mK_W| defines an Iitaka fibration. This is a consequence of a
more general result on polarized adjoint divisors. In order to prove these results we develop a generalized theory of pairs,
singularities, log canonical thresholds, adjunction, etc.