On the Heat Kernel of the Bergmann Metric on Algebraic Varieties

Abstract

Let M be an n-dimensional algebraic variety in a projective space CIP,,+I of dimension n + I with I > O. If we denote the singular set of M by 1: M' then the restriction of the standard Fubini-Study metric of CP,,+1 to M \ 1:M is an incomplete Kahler metric, g, called the Bergmann metric. Let us consider the operators d and tJ defined on C l functions and C l I-forms on M \ 1: M , respectively. We define the domain [g(d) of d to be the set of C l functions f defined on M\1:M such that both f and df are in L2. Similarly, we define the domain .93"(tJ) of tJ to be the set of C l I-forms w such that both wand tJw are in L 2 • We then define the Laplacian ~ with respect to the Bergmann metric by ~ = -tJd with domain 9(£1) given by the set of C 2 functions f such that f E 9(d) and df E 9(tJ). The main purpose of this paper is to show that: Main Result. The Laplacian for jUnctions ~ is essentially selfadjoint on M\1: M . The heat semi-group eAt generated by a has a kernel H(x, y, t) in 9(~) as ajUnction of x or y. Also, H(x, y, t) is symmetric in the variables x and y, and it satisfies the conservation property (H(x, y, t) dy = I lM\r. M for all x E M\1: M . If we denote H(r(x, 51), t) = H(x, 51, t) to be the rotation-ally symmetric heat kernel on the standard Cpll, then for x , y E M \ 1: M and t> 0, we have H(x, y, t) ::; H(rx(Y) ' t). Moreover, equality holds if and only if M is a totally geodesic CP". As an interesting corollary of this upper bound, we derive a universal lower bound for the dll! non-zero eigenvalue for algebraic varieties of degree d.